Regression analysis with most recent data

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timer Asked: Nov 16th, 2016

Question description

Choose any one of the examples (not the homework problems) in the text from chapters 8, 9, 10, or 11 involving a regression and do the same regression analysis with data updated to include the most recent data you can find. Are the conclusions drawn in the book still valid? Why or why not?

QUANTITATIVE INVESTMENT ANALYSIS Second Edition Richard A. DeFusco, CFA Dennis W. McLeavey, CFA Jerald E. Pinto, CFA David E. Runkle, CFA John Wiley & Sons, Inc. QUANTITATIVE INVESTMENT ANALYSIS Second Edition Richard A. DeFusco, CFA Dennis W. McLeavey, CFA Jerald E. Pinto, CFA David E. Runkle, CFA John Wiley & Sons, Inc. c 2004, 2007 by CFA Institute. All rights reserved. Copyright ⃝ Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our Web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Quantitative investment analysis / Richard A. DeFusco . . . [et al.].— 2nd ed. p. cm.—(The CFA Institute investment series) Includes bibliographical references. ISBN-13 978-0-470-05220-4 (cloth) ISBN-10 0-470-05220-1 (cloth) 1. Investment analysis—Mathematical models. I. DeFusco, Richard Armand. HG4529.Q35 2006 332.601’5195—dc22 2006052578 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 To Margo, Rachel, and Rebekah R.A.D. To Jan, Christine, and Andy D.W.M. In memory of Irwin T. Vanderhoof, CFA J.E.P. To Patricia, Anne, and Sarah D.E.R. CONTENTS Foreword xiii Acknowledgments xvii Introduction CHAPTER 1 The Time Value of Money 1 Introduction 2 Interest Rates: Interpretation 3 The Future Value of a Single Cash Flow 3.1 The Frequency of Compounding 3.2 Continuous Compounding 3.3 Stated and Effective Rates 4 The Future Value of a Series of Cash Flows 4.1 Equal Cash Flows—Ordinary Annuity 4.2 Unequal Cash Flows 5 The Present Value of a Single Cash Flow 5.1 Finding the Present Value of a Single Cash Flow 5.2 The Frequency of Compounding 6 The Present Value of a Series of Cash Flows 6.1 The Present Value of a Series of Equal Cash Flows 6.2 The Present Value of an Infinite Series of Equal Cash Flows—Perpetuity 6.3 Present Values Indexed at Times Other Than t = 0 6.4 The Present Value of a Series of Unequal Cash Flows 7 Solving for Rates, Number of Periods, or Size of Annuity Payments 7.1 Solving for Interest Rates and Growth Rates 7.2 Solving for the Number of Periods 7.3 Solving for the Size of Annuity Payments 7.4 Review of Present and Future Value Equivalence 7.5 The Cash Flow Additivity Principle xix 1 1 1 3 8 10 12 13 13 15 15 15 17 19 19 23 24 26 27 27 30 30 35 36 vii viii CHAPTER 2 Discounted Cash Flow Applications 1 Introduction 2 Net Present Value and Internal Rate of Return 2.1 Net Present Value and the Net Present Value Rule 2.2 The Internal Rate of Return and the Internal Rate of Return Rule 2.3 Problems with the IRR Rule 3 Portfolio Return Measurement 3.1 Money-Weighted Rate of Return 3.2 Time-Weighted Rate of Return 4 Money Market Yields CHAPTER 3 Statistical Concepts and Market Returns 1 Introduction 2 Some Fundamental Concepts 2.1 The Nature of Statistics 2.2 Populations and Samples 2.3 Measurement Scales 3 Summarizing Data Using Frequency Distributions 4 The Graphic Presentation of Data 4.1 The Histogram 4.2 The Frequency Polygon and the Cumulative Frequency Distribution 5 Measures of Central Tendency 5.1 The Arithmetic Mean 5.2 The Median 5.3 The Mode 5.4 Other Concepts of Mean 6 Other Measures of Location: Quantiles 6.1 Quartiles, Quintiles, Deciles, and Percentiles 6.2 Quantiles in Investment Practice 7 Measures of Dispersion 7.1 The Range 7.2 The Mean Absolute Deviation 7.3 Population Variance and Population Standard Deviation 7.4 Sample Variance and Sample Standard Deviation 7.5 Semivariance, Semideviation, and Related Concepts 7.6 Chebyshev’s Inequality 7.7 Coefficient of Variation 7.8 The Sharpe Ratio 8 Symmetry and Skewness in Return Distributions 9 Kurtosis in Return Distributions 10 Using Geometric and Arithmetic Means Contents 39 39 39 40 42 45 47 47 49 54 61 61 61 62 62 63 65 72 73 74 76 77 81 84 85 94 94 98 100 100 101 103 106 110 111 113 115 118 123 127 Contents CHAPTER 4 Probability Concepts 1 Introduction 2 Probability, Expected Value, and Variance 3 Portfolio Expected Return and Variance of Return 4 Topics in Probability 4.1 Bayes’ Formula 4.2 Principles of Counting CHAPTER 5 Common Probability Distributions 1 Introduction 2 Discrete Random Variables 2.1 The Discrete Uniform Distribution 2.2 The Binomial Distribution 3 Continuous Random Variables 3.1 Continuous Uniform Distribution 3.2 The Normal Distribution 3.3 Applications of the Normal Distribution 3.4 The Lognormal Distribution 4 Monte Carlo Simulation CHAPTER 6 Sampling and Estimation 1 Introduction 2 Sampling 2.1 Simple Random Sampling 2.2 Stratified Random Sampling 2.3 Time-Series and Cross-Sectional Data 3 Distribution of the Sample Mean 3.1 The Central Limit Theorem 4 Point and Interval Estimates of the Population Mean 4.1 Point Estimators 4.2 Confidence Intervals for the Population Mean 4.3 Selection of Sample Size 5 More on Sampling 5.1 Data-Mining Bias 5.2 Sample Selection Bias 5.3 Look-Ahead Bias 5.4 Time-Period Bias CHAPTER 7 Hypothesis Testing 1 Introduction 2 Hypothesis Testing ix 129 129 129 152 161 161 166 171 171 171 173 175 185 186 189 197 200 206 215 215 215 216 217 219 221 222 225 225 227 233 235 236 238 240 240 243 243 244 x Contents 3 Hypothesis Tests Concerning the Mean 3.1 Tests Concerning a Single Mean 3.2 Tests Concerning Differences between Means 3.3 Tests Concerning Mean Differences 4 Hypothesis Tests Concerning Variance 4.1 Tests Concerning a Single Variance 4.2 Tests Concerning the Equality (Inequality) of Two Variances 5 Other Issues: Nonparametric Inference 5.1 Tests Concerning Correlation: The Spearman Rank Correlation Coefficient 5.2 Nonparametric Inference: Summary CHAPTER 8 Correlation and Regression 1 Introduction 2 Correlation Analysis 2.1 Scatter Plots 2.2 Correlation Analysis 2.3 Calculating and Interpreting the Correlation Coefficient 2.4 Limitations of Correlation Analysis 2.5 Uses of Correlation Analysis 2.6 Testing the Significance of the Correlation Coefficient 3 Linear Regression 3.1 Linear Regression with One Independent Variable 3.2 Assumptions of the Linear Regression Model 3.3 The Standard Error of Estimate 3.4 The Coefficient of Determination 3.5 Hypothesis Testing 3.6 Analysis of Variance in a Regression with One Independent Variable 3.7 Prediction Intervals 3.8 Limitations of Regression Analysis CHAPTER 9 Multiple Regression and Issues in Regression Analysis 1 Introduction 2 Multiple Linear Regression 2.1 Assumptions of the Multiple Linear Regression Model 2.2 Predicting the Dependent Variable in a Multiple Regression Model 2.3 Testing Whether All Population Regression Coefficients Equal Zero 2.4 Adjusted R 2 3 Using Dummy Variables in Regressions 4 Violations of Regression Assumptions 4.1 Heteroskedasticity 4.2 Serial Correlation 4.3 Multicollinearity 253 254 261 265 269 269 271 275 276 279 281 281 281 281 282 283 287 289 297 300 300 303 306 309 310 318 321 324 325 325 325 331 336 338 340 341 345 345 351 356 Contents 4.4 Heteroskedasticity, Serial Correlation, Multicollinearity: Summarizing the Issues 5 Model Specification and Errors in Specification 5.1 Principles of Model Specification 5.2 Misspecified Functional Form 5.3 Time-Series Misspecification (Independent Variables Correlated with Errors) 5.4 Other Types of Time-Series Misspecification 6 Models with Qualitative Dependent Variables CHAPTER 10 Time-Series Analysis 1 Introduction 2 Challenges of Working with Time Series 3 Trend Models 3.1 Linear Trend Models 3.2 Log-Linear Trend Models 3.3 Trend Models and Testing for Correlated Errors 4 Autoregressive (AR) Time-Series Models 4.1 Covariance-Stationary Series 4.2 Detecting Serially Correlated Errors in an Autoregressive Model 4.3 Mean Reversion 4.4 Multiperiod Forecasts and the Chain Rule of Forecasting 4.5 Comparing Forecast Model Performance 4.6 Instability of Regression Coefficients 5 Random Walks and Unit Roots 5.1 Random Walks 5.2 The Unit Root Test of Nonstationarity 6 Moving-Average Time-Series Models 6.1 Smoothing Past Values with an n-Period Moving Average 6.2 Moving-Average Time-Series Models for Forecasting 7 Seasonality in Time-Series Models 8 Autoregressive Moving-Average Models 9 Autoregressive Conditional Heteroskedasticity Models 10 Regressions with More than One Time Series 11 Other Issues in Time Series 12 Suggested Steps in Time-Series Forecasting CHAPTER 11 Portfolio Concepts 1 Introduction 2 Mean–Variance Analysis 2.1 The Minimum-Variance Frontier and Related Concepts 2.2 Extension to the Three-Asset Case 2.3 Determining the Minimum-Variance Frontier for Many Assets 2.4 Diversification and Portfolio Size xi 359 359 359 360 368 372 372 375 375 375 377 377 380 385 386 386 387 391 391 394 397 399 400 403 407 407 409 412 416 417 420 424 425 429 429 429 430 439 442 445 xii Contents 2.5 Portfolio Choice with a Risk-Free Asset 2.6 The Capital Asset Pricing Model 2.7 Mean–Variance Portfolio Choice Rules: An Introduction 3 Practical Issues in Mean–Variance Analysis 3.1 Estimating Inputs for Mean–Variance Optimization 3.2 Instability in the Minimum-Variance Frontier 4 Multifactor Models 4.1 Factors and Types of Multifactor Models 4.2 The Structure of Macroeconomic Factor Models 4.3 Arbitrage Pricing Theory and the Factor Model 4.4 The Structure of Fundamental Factor Models 4.5 Multifactor Models in Current Practice 4.6 Applications 4.7 Concluding Remarks 449 458 460 464 464 470 473 474 475 478 484 485 493 509 Appendices 511 References 521 Glossary 527 About the CFA Program 541 About the Authors 543 Index 545 FOREWORD HOW QUANTITATIVE INVESTMENT ANALYSIS CAN IMPROVE PORTFOLIO DECISION MAKING I am a Quant. By my own self-admission, I use quantitative investment techniques in the management of investment portfolios. However, when I tell people that I am a Quant, they often respond: ‘‘But Mark, aren’t you a lawyer?’’ Well, yes, but . . . The fact is that Quants come from all walks of life. Whether we are called Quants, Quant Jocks, Gear Heads, Computer Monkeys, or any of the other monikers that are attached to investors who like to scribble equations on a piece of paper, we all share a common denominator—the use of quantitative analysis to make better investment decisions. You don’t have to be a rocket scientist with a Ph.D. in an esoteric mathematical field to be a Quant (although there are, I suspect, several former rocket scientists who have found working in the financial markets to be both fun and profitable). Anyone can become a Quant—even a lawyer. But let’s take a step back. Why should any investor want to use quantitative tools in the management of investment portfolios? There are three reasons why Quants are so popular. First, the financial markets are very complicated places. There are many interwoven variables that can affect the price of securities in an investment portfolio. For example, the stock price of a public company can be affected by macroeconomic factors such as the level of interest rates, current account deficits, government spending, and economic cycles. These factors may affect the cost of capital at which a corporation finances its new projects, or influence the spending patterns of the company’s customers, or provide economic impetus through government spending programs. In addition to macro variables, the value of a company’s stock can be affected by factors that are peculiar to the company itself. Factors such as cash flow, working capital, bookto-market value, earnings growth rates, dividend policy, and debt-to-equity ratios affect the individual value of each public company. These are considered to be the fundamental factors that have an impact on the specific company as opposed to the broader stock market. Then we come to the financial market variables that affect a company’s valuation. Its ‘‘beta’’ or measure of systematic risk will impact the expected return for the company and, in turn, its stock price. The famous Capital Asset Pricing Model that measures a stock’s beta is really just a linear regression equation of the type described in Chapter 8. Last, there are behavioral variables that can affect security values. Such behavior as herding, overconfidence, overreaction to earnings announcements, and momentum trading can all impact the price of a company’s stock. These behavioral variables can have a lasting impact on a stock price (remember the technology bubble of 1998–2001 when tech stocks were going to take over the world?) as well as generate a significant amount of ‘‘noise’’ around a security’s true value. xiii xiv Foreword Considering all of these variables together at one time to determine the true value of a security can be an overwhelming task without some framework in which to analyze their impact. It is simply not possible for the human mind alone (at least, not mine) to be able to weigh the impact of individual company specific factors such as price-to-earnings ratios, macroeconomic variables such as government spending programs, investor behavioral patterns such as momentum trading, and other potentially influential variables in a rigorous fashion within the human brain. This is where Quantitative Investment Analysis can help. Factor modeling techniques such as those described in Chapter 11 can be used to supplement the intuition of the human mind to produce a quantitative framework that digests the large number of plausible variables that can impact the price of a security. Further, given the many variables that can affect a security’s value, it is not possible to consider each variable in isolation. The economic factors that cause a security’s price to go up or down are interwoven in a complex web such that the variables must be considered together to determine their collective impact on the price of a security. This is where the value of Chapters 8 and 9 are most useful. These two chapters provide the basic knowledge for building regression equations to study the impact of economic factors on security prices. The regression techniques provided in Chapters 8 and 9 can be used to filter out which variables have a significant impact on the price of a security, and which variables just provide ‘‘noise.’’ In addition, Chapter 9 introduces the reader to ‘‘dummy variables.’’ Despite their name, you don’t have to be a dummy like me to use them. Dummy variables are a neat way to study different states of the world and their impact on security prices. They are often referred to as ‘‘binary’’ variables because they divide the world into two states for observation, for example, financial markets up versus financial markets down; Republicans in control of the White House versus Democrats in control of the White House; Chicago Cubs win (almost never) versus Chicago Cubs lose; and so on. This last variable—the record of the Chicago Cubs—I can attest has no impact on security valuations, although, as a long-standing and suffering Cub fan, it does have an impact on my morale. As another example, consider a recent research paper where I studied the behavior of private equity managers in the way they price their private equity portfolios depending on whether the public stock markets were doing well versus when the public stock markets were doing poorly. To conduct this analysis, I ran a regression equation using dummy variables to divide the world into two states: up public stock markets versus down public stock markets. By using dummy variables in this manner, I was able to observe different behavioral patterns of private equity managers in how they marked up or down their private equity portfolios depending on the performance of the public stock markets. The second reason Quantitative Investment Analysis will add value to the reader is that it provides the basic tools to consider a breadth of economic factors and securities. It is not only the fact that there are many interwoven economic variables that impact the value of a security, the sheer number of securities in the market place can be daunting. Therefore, most investors only look at a subset of the investable securities in the market. Consider the U.S. stock market. Generally, this market is divided into three categories based on company size: large-cap, mid-cap, and small-cap stocks. This division is less so because there might be ‘‘size’’ effects in valuation, but rather, because of the pragmatic limitation that asset managers simply cannot analyze stocks beyond a certain number. So traditional fundamental investors select different parts of the U.S. stock market in which to conduct their security analysis. However, the division of the stock market into size categories effectively establishes barriers for investment managers. There is no reason, for example, why a portfolio Foreword xv manager with insight into how earnings surprises affect stock prices cannot invest across the whole range of stock market capitalization. This is where Chapters 6 and 7 can be useful. The quantitative skills of sampling, estimation, and hypothesis testing can be used to analyze large baskets of data. This allows portfolio managers to invest across a broad universe of stocks, breaking down traditional barriers such as cap-size restrictions. When viewed in this light, quantitative analysis does not displace the fundamental stock picking skills of traditional asset managers. Rather, quantitative analysis extends the portfolio manager’s insight with respect to company, macro, and market variables to a broader array of investment opportunities. This also has implications for the statistical tools and probability concepts provided in Chapters 3 and 4. The larger the data set to be analyzed the greater the reliability of the parameter estimation derived from that data set. Breadth of economic analysis will improve not only the statistical reliability of the quantitative analysis, but will also increase the predictability of the relationships between economic factors and stock price movement. The statistical tools provided in this book allow the portfolio manager to realize the full potential of his or her skill across a larger universe of securities than may have previously been achieved. Another example might help. Every year the California Public Employees’ Retirement System (CalPERS), my former employer, publishes a list of the most poorly governed companies in the United States. This list has now been published for 16 years and has been very successful. Early on in the process, the selection was conducted on a subset of the U.S. stock market. However, this process has evolved to consider every U.S. stock held in CalPERS’s portfolio regardless of stock market capitalization range. This requires the analysis of up to 1,800 stocks every year based on both economic factors and governance variables. The sheer number of securities in this data sample could not be analyzed without the application of quantitative screening tools to expand the governance universe for CalPERS. Last, Quantitative Investment Analysis can provide a certain amount of discipline to the investment process. We are all human, and as humans, we are subject to making mistakes. If I were to recount all of the investment mistakes that I have made over my career, this Foreword would exceed the length of the chapters in this book. Just as a brief example, one of my ‘‘better calls’’ was Starbucks Coffee. Early on when Starbucks was just getting started, I visited one of their shops to see what the buzz was all about. At that time a Latte Grande was selling for about $1.50. I recall that I thought this was an outrageous price and I can remember distinctly saying: ‘‘Oh, this is a dumb idea, this will never catch on!’’ Ah yes . . . So back to quantitative techniques—how can they help? In this instance, they could have helped me remove my human biases and to think more analytically about Starbucks’ prospects. If I had taken the time to conduct an empirical review using the quantitative tools provided in this text, I would have seen the fundamental value underlying that buck-fifty Latte. The fact is that we are all subject to behavioral biases such as overconfidence, momentum, and overreaction. Not only can these be analyzed as discussed above, they can be revealed and discounted when we make our investment decisions. Perhaps the single biggest behavioral hurdle to overcome for investors is the inability to sell a security when its value declines. All too often we become almost emotionally attached to the securities in our portfolio such that we find it hard to sell a security that begins to decline in price. Yet, this is precisely, where Quantitative Investment Analysis can help because it is dispassionate. Quantitative tools and modeling techniques can take the emotion and cognitive biases out of the portfolio decision-making process. As portfolio managers, our goal is to be objective, critical, and demanding. Unfortunately, sometimes our embedded habits and opinions can get in the way. However, quantitative models are unemotional and they can root xvi Foreword out our cognitive biases in a way that we simply cannot do ourselves by looking in the mirror (in fact, when I look in the mirror I see someone who is six feet and four inches tall and incredibly good looking but then my wife Mary reminds me that I am only six feet and one inch tall and she had better offers). All in all, the investor will appreciate the methods, models, and techniques provided in this text. This book serves as an excellent introduction to those investors who are just beginning to use quantitative tools in their portfolio management process as well as an excellent reference guide for those already converted. Quantitative investing is not difficult to grasp—even a lawyer can do it. Mark J. P. Anson CEO, Hermes Pensions Management CEO, British Telecomm Pension Scheme mark@hermes.co.uk ACKNOWLEDGMENTS W e would like to thank the many individuals who played important roles in producing this book. Robert R. Johnson, CFA, Managing Director of the CFA and CIPM Programs Division, saw the need for specialized curriculum materials and initiated this project. We appreciate his support for the timely revision of this textbook. Senior executives in the CFA Program Division have generously given their advice and time in the writing of both editions of this book. Philip J. Young, CFA, provided continuous assistance in writing the book’s learning outcome statements and participated in final manuscript reviews. Jan R. Squires, CFA, contributed an orientation stressing motivation and testability. Mary K. Erickson, CFA, made contributions to the accuracy of the text. John D. Stowe, CFA, supplied suggestions for revising several chapters. The Executive Advisory Board of the Candidate Curriculum Committee provided invaluable input: James Bronson, CFA, Chair; Peter Mackey, CFA, Immediate Past Chair; and members, Alan Meder, CFA, Victoria Rati, CFA, and Matt Scanlan, CFA, as well as the Candidate Curriculum Committee Working Body. The manuscript reviewers for this edition were Philip Fanara, Jr., CFA; Jane Farris, CFA; David M. Jessop; Lisa M. Joublanc, CFA; Asjeet S. Lamba, CFA; Mario Lavallee, CFA; William L. Randolph, CFA; Eric N. Remole; Vijay Singal, CFA; Zoe L. Van Schyndel, CFA; Charlotte Weems, CFA; and Lavone F. Whitmer, CFA. We thank them for their excellent work. We also appreciate the many comments received from those who used the first edition. Jacques R. Gagne, CFA, Gregory M. Noronha, CFA, and Sanjiv Sabherwal provided highly detailed proofreading of the individual chapters. We thank each for their dedicated and painstaking work. We are also indebted to Dr. Sabherwal for his expert assistance in running regressions, revising in-chapter examples, and creating some of the end-of-chapter problems/solutions. Fiona D. Russell provided incisive copyediting that substantially contributed to the book’s accuracy and readability. Wanda A. Lauziere of the CFA Program Division, the project manager for the revision, expertly guided the manuscript from planning through production and made many other contributions to all aspects of the revision. Finally, we thank Ibbotson Associates of Chicago for generously providing us with EnCorr AnalyzerTM . xvii INTRODUCTION CFA Institute is pleased to provide you with this Investment Series covering major areas in the field of investments. These texts are thoroughly grounded in the highly regarded CFA Program Candidate Body of Knowledge (CBOK®) that draws upon hundreds of practicing investment professionals and serves as the anchor for the three levels of the CFA Examinations. In the year this series is being launched, more than 120,000 aspiring investment professionals will each devote over 250 hours of study to master this material as well as other elements of the Candidate Body of Knowledge in order to obtain the coveted CFA charter. We provide these materials for the same reason we have been chartering investment professionals for over 40 years: to improve the competency and ethical character of those serving the capital markets. PARENTAGE One of the valuable attributes of this series derives from its parentage. In the 1940s, a handful of societies had risen to form communities that revolved around common interests and work in what we now think of as the investment industry. Understand that the idea of purchasing common stock as an investment—as opposed to casino speculation—was only a couple of decades old at most. We were only 10 years past the creation of the U.S. Securities and Exchange Commission and laws that attempted to level the playing field after robber baron and stock market panic episodes. In January 1945, in what is today CFA Institute Financial Analysts Journal, a fundamentally driven professor and practitioner from Columbia University and Graham-Newman Corporation wrote an article making the case that people who research and manage portfolios should have some sort of credential to demonstrate competence and ethical behavior. This person was none other than Benjamin Graham, the father of security analysis and future mentor to a well-known modern investor, Warren Buffett. The idea of creating a credential took a mere 16 years to drive to execution but by 1963, 284 brave souls, all over the age of 45, took an exam and launched the CFA credential. What many do not fully understand was that this effort had at its root a desire to create a profession where its practitioners were professionals who provided investing services to individuals in need. In so doing, a fairer and more productive capital market would result. A profession—whether it be medicine, law, or other—has certain hallmark characteristics. These characteristics are part of what attracts serious individuals to devote the energy of their life’s work to the investment endeavor. First, and tightly connected to this Series, there must be a body of knowledge. Second, there needs to be some entry requirements such as those required to achieve the CFA credential. Third, there must be a commitment to continuing education. Fourth, a profession must serve a purpose beyond one’s direct selfish interest. In this case, by properly conducting one’s affairs and putting client interests first, the investment xix xx Introduction professional can work as a fair-minded cog in the wheel of the incredibly productive global capital markets. This encourages the citizenry to part with their hard-earned savings to be redeployed in fair and productive pursuit. As C. Stewart Sheppard, founding executive director of the Institute of Chartered Financial Analysts said, ‘‘Society demands more from a profession and its members than it does from a professional craftsman in trade, arts, or business. In return for status, prestige, and autonomy, a profession extends a public warranty that it has established and maintains conditions of entry, standards of fair practice, disciplinary procedures, and continuing education for its particular constituency. Much is expected from members of a profession, but over time, more is given.’’ ‘‘The Standards for Educational and Psychological Testing,’’ put forth by the American Psychological Association, the American Educational Research Association, and the National Council on Measurement in Education, state that the validity of professional credentialing examinations should be demonstrated primarily by verifying that the content of the examination accurately represents professional practice. In addition, a practice analysis study, which confirms the knowledge and skills required for the competent professional, should be the basis for establishing content validity. For more than 40 years, hundreds upon hundreds of practitioners and academics have served on CFA Institute curriculum committees sifting through and winnowing all the many investment concepts and ideas to create a body of knowledge and the CFA curriculum. One of the hallmarks of curriculum development at CFA Institute is its extensive use of practitioners in all phases of the process. CFA Institute has followed a formal practice analysis process since 1995. The effort involves special practice analysis forums held, most recently, at 20 locations around the world. Results of the forums were put forth to 70,000 CFA charterholders for verification and confirmation of the body of knowledge so derived. What this means for the reader is that the concepts contained in these texts were driven by practicing professionals in the field who understand the responsibilities and knowledge that practitioners in the industry need to be successful. We are pleased to put this extensive effort to work for the benefit of the readers of the Investment Series. BENEFITS This series will prove useful both to the new student of capital markets, who is seriously contemplating entry into the extremely competitive field of investment management, and to the more seasoned professional who is looking for a user-friendly way to keep one’s knowledge current. All chapters include extensive references for those who would like to dig deeper into a given concept. The workbooks provide a summary of each chapter’s key points to help organize your thoughts, as well as sample questions and answers to test yourself on your progress. For the new student, the essential concepts that any investment professional needs to master are presented in a time-tested fashion. This material, in addition to university study and reading the financial press, will help you better understand the investment field. I believe that the general public seriously underestimates the disciplined processes needed for the best investment firms and individuals to prosper. These texts lay the basic groundwork for many of the processes that successful firms use. Without this base level of understanding and an appreciation for how the capital markets work to properly price securities, you may not find Introduction xxi competitive success. Furthermore, the concepts herein give a genuine sense of the kind of work that is to be found day to day managing portfolios, doing research, or related endeavors. The investment profession, despite its relatively lucrative compensation, is not for everyone. It takes a special kind of individual to fundamentally understand and absorb the teachings from this body of work and then convert that into application in the practitioner world. In fact, most individuals who enter the field do not survive in the longer run. The aspiring professional should think long and hard about whether this is the field for him or herself. There is no better way to make such a critical decision than to be prepared by reading and evaluating the gospel of the profession. The more experienced professional understands that the nature of the capital markets requires a commitment to continuous learning. Markets evolve as quickly as smart minds can find new ways to create an exposure, to attract capital, or to manage risk. A number of the concepts in these pages were not present a decade or two ago when many of us were starting out in the business. Hedge funds, derivatives, alternative investment concepts, and behavioral finance are examples of new applications and concepts that have altered the capital markets in recent years. As markets invent and reinvent themselves, a best-in-class foundation investment series is of great value. Those of us who have been at this business for a while know that we must continuously hone our skills and knowledge if we are to compete with the young talent that constantly emerges. In fact, as we talk to major employers about their training needs, we are often told that one of the biggest challenges they face is how to help the experienced professional, laboring under heavy time pressure, keep up with the state of the art and the more recently educated associates. This series can be part of that answer. CONVENTIONAL WISDOM It doesn’t take long for the astute investment professional to realize two common characteristics of markets. First, prices are set by conventional wisdom, or a function of the many variables in the market. Truth in markets is, at its essence, what the market believes it is and how it assesses pricing credits or debits on those beliefs. Second, as conventional wisdom is a product of the evolution of general theory and learning, by definition conventional wisdom is often wrong or at the least subject to material change. When I first entered this industry in the mid-1970s, conventional wisdom held that the concepts examined in these texts were a bit too academic to be heavily employed in the competitive marketplace. Many of those considered to be the best investment firms at the time were led by men who had an eclectic style, an intuitive sense of markets, and a great track record. In the rough-and-tumble world of the practitioner, some of these concepts were considered to be of no use. Could conventional wisdom have been more wrong? If so, I’m not sure when. During the years of my tenure in the profession, the practitioner investment management firms that evolved successfully were full of determined, intelligent, intellectually curious investment professionals who endeavored to apply these concepts in a serious and disciplined manner. Today, the best firms are run by those who carefully form investment hypotheses and test them rigorously in the marketplace, whether it be in a quant strategy, in comparative shopping for stocks within an industry, or in many hedge fund strategies. Their goal is to create investment processes that can be replicated with some statistical reliability. I believe xxii Introduction those who embraced the so-called academic side of the learning equation have been much more successful as real-world investment managers. THE TEXTS Approximately 35 percent of the Candidate Body of Knowledge is represented in the initial four texts of the series. Additional texts on corporate finance and international financial statement analysis are in development, and more topics may be forthcoming. One of the most prominent texts over the years in the investment management industry has been Maginn and Tuttle’s Managing Investment Portfolios: A Dynamic Process. The third edition updates key concepts from the 1990 second edition. Some of the more experienced members of our community, like myself, own the prior two editions and will add this to our library. Not only does this tome take the concepts from the other readings and put them in a portfolio context, it also updates the concepts of alternative investments, performance presentation standards, portfolio execution and, very importantly, managing individual investor portfolios. To direct attention, long focused on institutional portfolios, toward the individual will make this edition an important improvement over the past. Quantitative Investment Analysis focuses on some key tools that are needed for today’s professional investor. In addition to classic time value of money, discounted cash flow applications, and probability material, there are two aspects that can be of value over traditional thinking. First are the chapters dealing with correlation and regression that ultimately figure into the formation of hypotheses for purposes of testing. This gets to a critical skill that many professionals are challenged by: the ability to sift out the wheat from the chaff. For most investment researchers and managers, their analysis is not solely the result of newly created data and tests that they perform. Rather, they synthesize and analyze primary research done by others. Without a rigorous manner by which to understand quality research, not only can you not understand good research, you really have no basis by which to evaluate less rigorous research. What is often put forth in the applied world as good quantitative research lacks rigor and validity. Second, the last chapter on portfolio concepts moves the reader beyond the traditional capital asset pricing model (CAPM) type of tools and into the more practical world of multifactor models and to arbitrage pricing theory. Many have felt that there has been a CAPM bias to the work put forth in the past, and this chapter helps move beyond that point. Equity Asset Valuation is a particularly cogent and important read for anyone involved in estimating the value of securities and understanding security pricing. A well-informed professional would know that the common forms of equity valuation—dividend discount modeling, free cash flow modeling, price/earnings models, and residual income models (often known by trade names)—can all be reconciled to one another under certain assumptions. With a deep understanding of the underlying assumptions, the professional investor can better understand what other investors assume when calculating their valuation estimates. In my prior life as the head of an equity investment team, this knowledge would give us an edge over other investors. Fixed Income Analysis has been at the frontier of new concepts in recent years, greatly expanding horizons over the past. This text is probably the one with the most new material for the seasoned professional who is not a fixed-income specialist. The application of option and derivative technology to the once staid province of fixed income has helped contribute to an Introduction xxiii explosion of thought in this area. And not only does that challenge the professional to stay up to speed with credit derivatives, swaptions, collateralized mortgage securities, mortgage backs, and others, but it also puts a strain on the world’s central banks to provide oversight and the risk of a correlated event. Armed with a thorough grasp of the new exposures, the professional investor is much better able to anticipate and understand the challenges our central bankers and markets face. I hope you find this new series helpful in your efforts to grow your investment knowledge, whether you are a relatively new entrant or a grizzled veteran ethically bound to keep up to date in the ever-changing market environment. CFA Institute, as a long-term committed participant of the investment profession and a not-for-profit association, is pleased to give you this opportunity. Jeff Diermeier, CFA President and Chief Executive Officer CFA Institute September 2006 QUANTITATIVE INVESTMENT ANALYSIS CHAPTER 1 THE TIME VALUE OF MONEY 1. INTRODUCTION As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows. To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date. The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts. The chapter is organized as follows: Section 2 introduces some terminology used throughout the chapter and supplies some economic intuition for the variables we will discuss. Section 3 tackles the problem of determining the worth at a future point in time of an amount invested today. Section 4 addresses the future worth of a series of cash flows. These two sections provide the tools for calculating the equivalent value at a future date of a single cash flow or series of cash flows. Sections 5 and 6 discuss the equivalent value today of a single future cash flow and a series of future cash flows, respectively. In Section 7, we explore how to determine other quantities of interest in time value of money problems. 2. INTEREST RATES: INTERPRETATION In this chapter, we will continually refer to interest rates. In some cases, we assume a particular value for the interest rate; in other cases, the interest rate will be the unknown quantity we seek to determine. Before turning to the mechanics of time value of money problems, we must illustrate the underlying economic concepts. In this section, we briefly explain the meaning and interpretation of interest rates. Time value of money concerns equivalence relationships between cash flows occurring on different dates. The idea of equivalence relationships is relatively simple. Consider the following exchange: You pay $10,000 today and in return receive $9,500 today. Would you 1 2 Quantitative Investment Analysis accept this arrangement? Not likely. But what if you received the $9,500 today and paid the $10,000 one year from now? Can these amounts be considered equivalent? Possibly, because a payment of $10,000 a year from now would probably be worth less to you than a payment of $10,000 today. It would be fair, therefore, to discount the $10,000 received in one year; that is, to cut its value based on how much time passes before the money is paid. An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. If $9,500 today and $10,000 in one year are equivalent in value, then $10,000 − $9,500 = $500 is the required compensation for receiving $10,000 in one year rather than now. The interest rate—the required compensation stated as a rate of return—is $500/$9,500 = 0.0526 or 5.26 percent. Interest rates can be thought of in three ways. First, they can be considered required rates of return—that is, the minimum rate of return an investor must receive in order to accept the investment. Second, interest rates can be considered discount rates. In the example above, 5.26 percent is that rate at which we discounted the $10,000 future amount to find its value today. Thus, we use the terms ‘‘interest rate’’ and ‘‘discount rate’’ almost interchangeably. Third, interest rates can be considered opportunity costs. An opportunity cost is the value that investors forgo by choosing a particular course of action. In the example, if the party who supplied $9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent on the money. So we can view 5.26 percent as the opportunity cost of current consumption. Economics tells us that interest rates are set in the marketplace by the forces of supply and demand, where investors are suppliers of funds and borrowers are demanders of funds. Taking the perspective of investors in analyzing market-determined interest rates, we can view an interest rate r as being composed of a real risk-free interest rate plus a set of four premiums that are required returns or compensation for bearing distinct types of risk: r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium The real risk-free interest rate is the single-period interest rate for a completely risk-free security if no inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption. • The inflation premium compensates investors for expected inflation and reflects the average inflation rate expected over the maturity of the debt. Inflation reduces the purchasing power of a unit of currency—the amount of goods and services one can buy with it. The sum of the real risk-free interest rate and the inflation premium is the nominal risk-free interest rate.1 Many countries have governmental short-term debt whose interest rate can be considered to represent the nominal risk-free interest rate in that country. The interest rate on a 90-day U.S. Treasury bill (T-bill), for example, represents the nominal risk-free interest rate over that time horizon.2 U.S. T-bills can be bought and sold in large quantities with minimal transaction costs and are backed by the full faith and credit of the U.S. government. • 1 Technically, 1 plus the nominal rate equals the product of 1 plus the real rate and 1 plus the inflation rate. As a quick approximation, however, the nominal rate is equal to the real rate plus an inflation premium. In this discussion we focus on approximate additive relationships to highlight the underlying concepts. 2 Other developed countries issue securities similar to U.S. Treasury bills. The French government issues BTFs or negotiable fixed-rate discount Treasury bills (Bons du Trésor à taux fixe et à intérêts précomptés) with maturities of 3, 6, and 12 months. The Japanese government issues a short-term Treasury bill with maturities of 6 and 12 months. The German government issues at discount both Treasury financing 3 Chapter 1 The Time Value of Money The default risk premium compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount. • The liquidity premium compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly. U.S. T-bills, for example, do not bear a liquidity premium because large amounts can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position. • The maturity premium compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general (holding all else equal). The difference between the interest rate on longer-maturity, liquid Treasury debt and that on short-term Treasury debt reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well). • Using this insight into the economic meaning of interest rates, we now turn to a discussion of solving time value of money problems, starting with the future value of a single cash flow. 3. THE FUTURE VALUE OF A SINGLE CASH FLOW In this section, we introduce time value associated with a single cash flow or lump-sum investment. We describe the relationship between an initial investment or present value (PV), which earns a rate of return (the interest rate per period) denoted as r, and its future value (FV), which will be received N years or periods from today. The following example illustrates this concept. Suppose you invest $100 (PV = $100) in an interest-bearing bank account paying 5 percent annually. At the end of the first year, you will have the $100 plus the interest earned, 0.05× $100 = $5, for a total of $105. To formalize this one-period example, we define the following terms: PV = present value of the investment FVN = future value of the investment N periods from today r = rate of interest per period For N = 1, the expression for the future value of amount PV is FV1 = PV(1 + r) (1-1) For this example, we calculate the future value one year from today as FV1 = $100(1.05) = $105. Now suppose you decide to invest the initial $100 for two years with interest earned and credited to your account annually (annual compounding). At the end of the first year (the paper (Finanzierungsschätze des Bundes or, for short, Schätze) and Treasury discount paper (Bubills) with maturities up to 24 months. In the United Kingdom, the British government issues gilt-edged Treasury bills with maturities ranging from 1 to 364 days. The Canadian government bond market is closely related to the U.S. market; Canadian Treasury bills have maturities of 3, 6, and 12 months. 4 Quantitative Investment Analysis beginning of the second year), your account will have $105, which you will leave in the bank for another year. Thus, with a beginning amount of $105 (PV = $105), the amount at the end of the second year will be $105(1.05) = $110.25. Note that the $5.25 interest earned during the second year is 5 percent of the amount invested at the beginning of Year 2. Another way to understand this example is to note that the amount invested at the beginning of Year 2 is composed of the original $100 that you invested plus the $5 interest earned during the first year. During the second year, the original principal again earns interest, as does the interest that was earned during Year 1. You can see how the original investment grows: Original investment Interest for the first year ($100 × 0.05) Interest for the second year based on original investment ($100 × 0.05) Interest for the second year based on interest earned in the first year (0.05× $5.00 interest on interest) Total $100.00 5.00 5.00 0.25 $110.25 The $5 interest that you earned each period on the $100 original investment is known as simple interest (the interest rate times the principal). Principal is the amount of funds originally invested. During the two-year period, you earn $10 of simple interest. The extra $0.25 that you have at the end of Year 2 is the interest you earned on the Year 1 interest of $5 that you reinvested. The interest earned on interest provides the first glimpse of the phenomenon known as compounding. Although the interest earned on the initial investment is important, for a given interest rate it is fixed in size from period to period. The compounded interest earned on reinvested interest is a far more powerful force because, for a given interest rate, it grows in size each period. The importance of compounding increases with the magnitude of the interest rate. For example, $100 invested today would be worth about $13,150 after 100 years if compounded annually at 5 percent, but worth more than $20 million if compounded annually over the same time period at a rate of 13 percent. To verify the $20 million figure, we need a general formula to handle compounding for any number of periods. The following general formula relates the present value of an initial investment to its future value after N periods: FVN = PV(1 + r)N (1-2) where r is the stated interest rate per period and N is the number of compounding periods. In the bank example, FV2 = $100(1 + 0.05)2 = $110.25. In the 13 percent investment example, FV100 = $100(1.13)100 = $20,316,287.42. The most important point to remember about using the future value equation is that the stated interest rate, r, and the number of compounding periods, N , must be compatible. Both variables must be defined in the same time units. For example, if N is stated in months, then r should be the one-month interest rate, unannualized. A time line helps us to keep track of the compatibility of time units and the interest rate per time period. In the time line, we use the time index t to represent a point in time a stated number of periods from today. Thus the present value is the amount available for investment today, indexed as t = 0. We can now refer to a time N periods from today as t = N . The time line in Figure 1-1 shows this relationship. 5 Chapter 1 The Time Value of Money 0 1 2 3 ... N − 1 N FVN = PV(1 + r)N PV FIGURE 1-1 The Relationship Between an Initial Investment, PV, and Its Future Value, FV In Figure 1-1, we have positioned the initial investment, PV, at t = 0. Using Equation 1-2, we move the present value, PV, forward to t = N by the factor (1 + r)N . This factor is called a future value factor. We denote the future value on the time line as FV and position it at t = N . Suppose the future value is to be received exactly 10 periods from today’s date (N = 10). The present value, PV, and the future value, FV, are separated in time through the factor (1 + r).10 The fact that the present value and the future value are separated in time has important consequences: • • • We can add amounts of money only if they are indexed at the same point in time. For a given interest rate, the future value increases with the number of periods. For a given number of periods, the future value increases with the interest rate. To better understand these concepts, consider three examples that illustrate how to apply the future value formula. EXAMPLE 1-1 The Future Value of a Lump Sum with Interim Cash Reinvested at the Same Rate You are the lucky winner of your state’s lottery of $5 million after taxes. You invest your winnings in a five-year certificate of deposit (CD) at a local financial institution. The CD promises to pay 7 percent per year compounded annually. This institution also lets you reinvest the interest at that rate for the duration of the CD. How much will you have at the end of five years if your money remains invested at 7 percent for five years with no withdrawals? Solution: To solve this problem, compute the future value of the $5 million investment using the following values in Equation 1-2: PV = $5,000,000 r = 7% = 0.07 N =5 FVN = PV(1 + r)N = $5,000,000(1.07)5 6 Quantitative Investment Analysis = $5, 000, 000(1.402552) = $7, 012, 758.65 At the end of five years, you will have $7,012,758.65 if your money remains invested at 7 percent with no withdrawals. In this and most examples in this chapter, note that the factors are reported at six decimal places but the calculations may actually reflect greater precision. For example, the reported 1.402552 has been rounded up from 1.40255173 (the calculation is actually carried out with more than eight decimal places of precision by the calculator or spreadsheet). Our final result reflects the higher number of decimal places carried by the calculator or spreadsheet.3 EXAMPLE 1-2 The Future Value of a Lump Sum with No Interim Cash An institution offers you the following terms for a contract: For an investment of ¥2,500,000, the institution promises to pay you a lump sum six years from now at an 8 percent annual interest rate. What future amount can you expect? Solution: Use the following data in Equation 1-2 to find the future value: PV = ¥2,500,000 r = 8% = 0.08 N =6 FVN = PV(1 + r)N = ¥2,500,000(1.08)6 = ¥2,500,000(1.586874) = ¥3,967,186 You can expect to receive ¥3,967,186 six years from now. Our third example is a more complicated future value problem that illustrates the importance of keeping track of actual calendar time. 3 We could also solve time value of money problems using tables of interest rate factors. Solutions using tabled values of interest rate factors are generally less accurate than solutions obtained using calculators or spreadsheets, so practitioners prefer calculators or spreadsheets. 7 Chapter 1 The Time Value of Money EXAMPLE 1-3 The Future Value of a Future Lump Sum A pension fund manager estimates that his corporate sponsor will make a $10 million contribution five years from now. The rate of return on plan assets has been estimated at 9 percent per year. The pension fund manager wants to calculate the future value of this contribution 15 years from now, which is the date at which the funds will be distributed to retirees. What is that future value? Solution: By positioning the initial investment, PV, at t = 5, we can calculate the future value of the contribution using the following data in Equation 1-2: PV = $10,000,000 r = 9% = 0.09 N = 10 FVN = PV(1 + r)N = $10,000,000(1.09)10 = $10,000,000(2.367364) = $23,673,636.75 This problem looks much like the previous two, but it differs in one important respect: its timing. From the standpoint of today (t = 0), the future amount of $23,673,636.75 is 15 years into the future. Although the future value is 10 years from its present value, the present value of $10 million will not be received for another five years. 0 1 2 3 4 5 $10,000,000 ... 15 $23,673,636.75 FIGURE 1-2 The Future Value of a Lump Sum, Initial Investment; Not at t = 0 As Figure 1-2 shows, we have followed the convention of indexing today as t = 0 and indexing subsequent times by adding 1 for each period. The additional contribution of $10 million is to be received in five years, so it is indexed as t = 5 and appears as such in the figure. The future value of the investment in 10 years is then indexed at t = 15; that is, 10 years following the receipt of the $10 million contribution at t = 5. Time lines like this one can be extremely useful when dealing with more-complicated problems, especially those involving more than one cash flow. 8 Quantitative Investment Analysis In a later section of this chapter, we will discuss how to calculate the value today of the $10 million to be received five years from now. For the moment, we can use Equation 1-2. Suppose the pension fund manager in Example 1-3 above were to receive $6,499,313.86 today from the corporate sponsor. How much will that sum be worth at the end of five years? How much will it be worth at the end of 15 years? PV = $6, 499, 313.86 r = 9% = 0.09 N =5 FVN = PV(1 + r)N = $6,499,313.86(1.09)5 = $6,499,313.86(1.538624) = $10, 000, 000 at the five-year mark and PV = $6,499,313.86 r = 9% = 0.09 N = 15 FVN = PV(1 + r)N = $6,499,313.86(1.09)15 = $6,499,313.86(3.642482) = $23,673,636.74 at the 15-year mark These results show that today’s present value of about $6.5 million becomes $10 million after five years and $23.67 million after 15 years. 3.1. The Frequency of Compounding In this section, we examine investments paying interest more than once a year. For instance, many banks offer a monthly interest rate that compounds 12 times a year. In such an arrangement, they pay interest on interest every month. Rather than quote the periodic monthly interest rate, financial institutions often quote an annual interest rate that we refer to as the stated annual interest rate or quoted interest rate. We denote the stated annual interest rate by rs . For instance, your bank might state that a particular CD pays 8 percent compounded monthly. The stated annual interest rate equals the monthly interest rate multiplied by 12. In this example, the monthly interest rate is 0.08/12 = 0.0067 or 0.67 percent.4 This rate is strictly a quoting convention because (1 + 0.0067)12 = 1.083, not 1.08; the term (1 + rs ) is not meant to be a future value factor when compounding is more frequent than annual. 4 To avoid rounding errors when using a financial calculator, divide 8 by 12 and then press the %i key, rather than simply entering 0.67 for %i, so we have (1 + 0.08/12)12 = 1.083000. Chapter 1 The Time Value of Money 9 With more than one compounding period per year, the future value formula can be expressed as ! rs "mN (1-3) FVN = PV 1 + m where rs is the stated annual interest rate, m is the number of compounding periods per year, and N now stands for the number of years. Note the compatibility here between the interest rate used, rs /m, and the number of compounding periods, mN. The periodic rate, rs /m, is the stated annual interest rate divided by the number of compounding periods per year. The number of compounding periods, mN, is the number of compounding periods in one year multiplied by the number of years. The periodic rate, rs /m, and the number of compounding periods, mN, must be compatible. EXAMPLE 1-4 The Future Value of a Lump Sum with Quarterly Compounding Continuing with the CD example, suppose your bank offers you a CD with a two-year maturity, a stated annual interest rate of 8 percent compounded quarterly, and a feature allowing reinvestment of the interest at the same interest rate. You decide to invest $10,000. What will the CD be worth at maturity? Solution: Compute the future value with Equation 1-3 as follows: PV = $10, 000 rs = 8% = 0.08 m=4 rs /m = 0.08/4 = 0.02 N =2 mN = 4(2) = 8 interest periods ! rs "mN FVN = PV 1 + m = $10,000(1.02)8 = $10,000(1.171659) = $11,716.59 At maturity, the CD will be worth $11,716.59. The future value formula in Equation 1-3 does not differ from the one in Equation 1-2. Simply keep in mind that the interest rate to use is the rate per period and the exponent is the number of interest, or compounding, periods. 10 Quantitative Investment Analysis EXAMPLE 1-5 The Future Value of a Lump Sum with Monthly Compounding An Australian bank offers to pay you 6 percent compounded monthly. You decide to invest A$1 million for one year. What is the future value of your investment if interest payments are reinvested at 6 percent? Solution: Use Equation 1-3 to find the future value of the one-year investment as follows: PV = A$1,000,000 rs = 6% = 0.06 m = 12 rs /m = 0.06/12 = 0.0050 N =1 mN = 12(1) = 12 interest periods ! rs "mN FVN = PV 1 + m = A$1,000,000(1.005)12 = A$1,000,000(1.061678) = A$1,061,677.81 If you had been paid 6 percent with annual compounding, the future amount would be only A$1,000,000(1.06) = A$1,060,000 instead of A$1,061,677.81 with monthly compounding. 3.2. Continuous Compounding The preceding discussion on compounding periods illustrates discrete compounding, which credits interest after a discrete amount of time has elapsed. If the number of compounding periods per year becomes infinite, then interest is said to compound continuously. If we want to use the future value formula with continuous compounding, we need to find the limiting value of the future value factor for m → ∞ (infinitely many compounding periods per year) in Equation 1-3. The expression for the future value of a sum in N years with continuous compounding is FVN = PVers N (1-4) The term ers N is the transcendental number e ≈ 2.7182818 raised to the power rs N . Most financial calculators have the function e x . 11 Chapter 1 The Time Value of Money EXAMPLE 1-6 The Future Value of a Lump Sum with Continuous Compounding Suppose a $10,000 investment will earn 8 percent compounded continuously for two years. We can compute the future value with Equation 1-4 as follows: PV = $10,000 rs = 8% = 0.08 N =2 FVN = PVers N = $10,000e0.08(2) = $10,000(1.173511) = $11,735.11 With the same interest rate but using continuous compounding, the $10,000 investment will grow to $11,735.11 in two years, compared with $11,716.59 using quarterly compounding as shown in Example 1-4. Table 1-1 shows how a stated annual interest rate of 8 percent generates different ending dollar amounts with annual, semiannual, quarterly, monthly, daily, and continuous compounding for an initial investment of $1 (carried out to six decimal places). As Table 1-1 shows, all six cases have the same stated annual interest rate of 8 percent; they have different ending dollar amounts, however, because of differences in the frequency of compounding. With annual compounding, the ending amount is $1.08. More frequent compounding results in larger ending amounts. The ending dollar amount with continuous compounding is the maximum amount that can be earned with a stated annual rate of 8 percent. Table 1-1 also shows that a $1 investment earning 8.16 percent compounded annually grows to the same future value at the end of one year as a $1 investment earning 8 percent compounded semiannually. This result leads us to a distinction between the stated annual TABLE 1-1 The Effect of Compounding Frequency on Future Value Frequency Annual Semiannual Quarterly Monthly Daily Continuous rs /m 8%/1 = 8% 8%/2 = 4% 8%/4 = 2% 8%/12 = 0.6667% 8%/365 = 0.0219% mN 1×1=1 2×1=2 4×1=4 12 × 1 = 12 365 × 1 = 365 Future Value of $1 $1.00(1.08) = $1.08 $1.00(1.04)2 = $1.081600 $1.00(1.02)4 = $1.082432 $1.00(1.006667)12 = $1.083000 $1.00(1.000219)365 = $1.083278 $1.00e0.08(1) = $1.083287 12 Quantitative Investment Analysis interest rate and the effective annual rate (EAR).5 For an 8 percent stated annual interest rate with semiannual compounding, the EAR is 8.16 percent. 3.3. Stated and Effective Rates The stated annual interest rate does not give a future value directly, so we need a formula for the EAR. With an annual interest rate of 8 percent compounded semiannually, we receive a periodic rate of 4 percent. During the course of a year, an investment of $1 would grow to $1(1.04)2 = $1.0816, as illustrated in Table 1-1. The interest earned on the $1 investment is $0.0816 and represents an effective annual rate of interest of 8.16 percent. The effective annual rate is calculated as follows: EAR = (1 + Periodic interest rate)m − 1 (1-5) The periodic interest rate is the stated annual interest rate divided by m, where m is the number of compounding periods in one year. Using the example in Table 1-1, we can solve for EAR as follows: (1.04)2 − 1 = 8.16 percent. The concept of EAR extends to continuous compounding. Suppose we have a rate of 8 percent compounded continuously. We can find the EAR in the same way as above by finding the appropriate future value factor. In this case, a $1 investment would grow to $1e0.08(1.0) = $1.0833. The interest earned for one year represents an effective annual rate of 8.33 percent and is larger than the 8.16 percent EAR with semiannual compounding because interest is compounded more frequently. With continuous compounding, we can solve for the effective annual rate as follows: EAR = ers − 1 (1-6) We can reverse the formulas for EAR with discrete and continuous compounding to find a periodic rate that corresponds to a particular effective annual rate. Suppose we want to find the appropriate periodic rate for a given effective annual rate of 8.16 percent with semiannual compounding. We can use Equation 1-5 to find the periodic rate: 0.0816 = (1 + Periodic rate)2 − 1 1.0816 = (1 + Periodic rate)2 (1.0816)1/2 − 1 = Periodic rate 1.04 − 1 = Periodic rate 4% = Periodic rate 5 Among the terms used for the effective annual return on interest-bearing bank deposits are annual percentage yield (APY) in the United States and equivalent annual rate (EAR) in the United Kingdom. By contrast, the annual percentage rate (APR) measures the cost of borrowing expressed as a yearly rate. In the United States, the APR is calculated as a periodic rate times the number of payment periods per year and, as a result, some writers use APR as a general synonym for the stated annual interest rate. Nevertheless, APR is a term with legal connotations; its calculation follows regulatory standards that vary internationally. Therefore, ‘‘stated annual interest rate’’ is the preferred general term for an annual interest rate that does not account for compounding within the year. 13 Chapter 1 The Time Value of Money To calculate the continuously compounded rate (the stated annual interest rate with continuous compounding) corresponding to an effective annual rate of 8.33 percent, we find the interest rate that satisfies Equation 1-6: 0.0833 = ers − 1 1.0833 = ers To solve this equation, we take the natural logarithm of both sides. (Recall that the natural log of ers is lners = rs .) Therefore, ln 1.0833 = rs , resulting in rs = 8 percent. We see that a stated annual rate of 8 percent with continuous compounding is equivalent to an EAR of 8.33 percent. 4. THE FUTURE VALUE OF A SERIES OF CASH FLOWS In this section, we consider series of cash flows, both even and uneven. We begin with a list of terms commonly used when valuing cash flows that are distributed over many time periods. An annuity is a finite set of level sequential cash flows. An ordinary annuity has a first cash flow that occurs one period from now (indexed at t = 1). • An annuity due has a first cash flow that occurs immediately (indexed at t = 0). • A perpetuity is a perpetual annuity, or a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now. • • 4.1. Equal Cash Flows—Ordinary Annuity Consider an ordinary annuity paying 5 percent annually. Suppose we have five separate deposits of $1,000 occurring at equally spaced intervals of one year, with the first payment occurring at t = 1. Our goal is to find the future value of this ordinary annuity after the last deposit at t = 5. The increment in the time counter is one year, so the last payment occurs five years from now. As the time line in Figure 1-3 shows, we find the future value of each $1,000 deposit as of t = 5 with Equation 1-2, FVN = PV(1 + r)N . The arrows in Figure 1-3 extend from the payment date to t = 5. For instance, the first $1,000 deposit made at t = 1 will compound over four periods. Using Equation 1-2, we find that the future value of the first deposit at t = 5 is $1,000(1.05)4 = $1,215.51. We calculate the future value of all other payments in a similar fashion. (Note that we are finding the future value at t = 5, so the last payment does not earn any interest.) With all values now at t = 5, we can add the future values to arrive at the future value of the annuity. This amount is $5,525.63. We can arrive at a general annuity formula if we define the annuity amount as A, the number of time periods as N , and the interest rate per period as r. We can then define the future value as FVN = A[(1 + r)N −1 + (1 + r)N −2 + (1 + r)N −3 + · · · + (1 + r)1 + (1 + r)0 ] which simplifies to FVN = A # (1 + r)N − 1 r $ (1-7) 14 Quantitative Investment Analysis 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 Sum at t = 5 $1,000(1.05)4 $1,000(1.05)3 $1,000(1.05)2 $1,000(1.05)1 = = = = $1,215.506250 $1,157.625000 $1,102.500000 $1,050.000000 $1,000(1.05)0 = $1,000.000000 $5,525.63 FIGURE 1-3 The Future Value of a Five-Year Ordinary Annuity The term in brackets is the future value annuity factor. This factor gives the future value of an ordinary annuity of $1 per period. Multiplying the future value annuity factor by the annuity amount gives the future value of an ordinary annuity. For the ordinary annuity in Figure 1-3, we find the future value annuity factor from Equation 1-7 as # $ (1.05)5 − 1 = 5.525631 0.05 With an annuity amount A = $1, 000, the future value of the annuity is $1,000(5.525631) = $5,525.63, an amount that agrees with our earlier work. The next example illustrates how to find the future value of an ordinary annuity using the formula in Equation 1-7. EXAMPLE 1-7 The Future Value of an Annuity Suppose your company’s defined contribution retirement plan allows you to invest up to ¤20,000 per year. You plan to invest ¤20,000 per year in a stock index fund for the next 30 years. Historically, this fund has earned 9 percent per year on average. Assuming that you actually earn 9 percent a year, how much money will you have available for retirement after making the last payment? Solution: Use Equation 1-7 to find the future amount: A = ¤20,000 r = 9% = 0.09 N = 30 (1 + r)N − 1 (1.09)30 − 1 = = 136.307539 r 0.09 FVN = ¤20,000(136.307539) FV annuity factor = = ¤2,726,150.77 Assuming the fund continues to earn an average of 9 percent per year, you will have ¤2,726,150.77 available at retirement. 15 Chapter 1 The Time Value of Money TABLE 1-2 A Series of Unequal Cash Flows and Their Future Values at 5 Percent Time t t t t t =1 =2 =3 =4 =5 Cash Flow $1,000 $2,000 $4,000 $5,000 $6,000 Future Value at Year 5 $1,000(1.05)4 = $1,215.51 $2,000(1.05)3 = $2,315.25 $4,000(1.05)2 = $4,410.00 $5,000(1.05)1 = $5,250.00 $6,000(1.05)0 = $6,000.00 Sum = $19,190.76 4.2. Unequal Cash Flows In many cases, cash flow streams are unequal, precluding the simple use of the future value annuity factor. For instance, an individual investor might have a savings plan that involves unequal cash payments depending on the month of the year or lower savings during a planned vacation. One can always find the future value of a series of unequal cash flows by compounding the cash flows one at a time. Suppose you have the five cash flows described in Table 1-2, indexed relative to the present (t = 0). All of the payments shown in Table 1-2 are different. Therefore, the most direct approach to finding the future value at t = 5 is to compute the future value of each payment as of t = 5 and then sum the individual future values. The total future value at Year 5 equals $19,190.76, as shown in the third column. Later in this chapter, you will learn shortcuts to take when the cash flows are close to even; these shortcuts will allow you to combine annuity and single-period calculations. 5. THE PRESENT VALUE OF A SINGLE CASH FLOW 5.1. Finding the Present Value of a Single Cash Flow Just as the future value factor links today’s present value with tomorrow’s future value, the present value factor allows us to discount future value to present value. For example, with a 5 percent interest rate generating a future payoff of $105 in one year, what current amount invested at 5 percent for one year will grow to $105? The answer is $100; therefore, $100 is the present value of $105 to be received in one year at a discount rate of 5 percent. Given a future cash flow that is to be received in N periods and an interest rate per period of r, we can use the formula for future value to solve directly for the present value as follows: FVN = PV(1 + r)N # $ 1 PV = FVN (1 + r)N (1-8) PV = FVN (1 + r)−N We see from Equation 1-8 that the present value factor, (1 + r)−N , is the reciprocal of the future value factor, (1 + r)N . 16 Quantitative Investment Analysis EXAMPLE 1-8 The Present Value of a Lump Sum An insurance company has issued a Guaranteed Investment Contract (GIC) that promises to pay $100,000 in six years with an 8 percent return rate. What amount of money must the insurer invest today at 8 percent for six years to make the promised payment? Solution: We can use Equation 1-8 to find the present value using the following data: FVN = $100,000 r = 8% = 0.08 N =6 PV = FVN (1 + r)−N $ # 1 = $100, 000 (1.08)6 = $100, 000(0.6301696) = $63, 016.96 We can say that $63,016.96 today, with an interest rate of 8 percent, is equivalent to $100,000 to be received in six years. Discounting the $100,000 makes a future $100,000 equivalent to $63,016.96 when allowance is made for the time value of money. As the time line in Figure 1-4 shows, the $100,000 has been discounted six full periods. 0 1 2 3 4 5 6 PV = $63,016.96 $100,000 = FV FIGURE 1-4 The Present Value of a Lump Sum to Be Received at Time t = 6 EXAMPLE 1-9 The Projected Present Value of a More Distant Future Lump Sum Suppose you own a liquid financial asset that will pay you $100,000 in 10 years from today. Your daughter plans to attend college four years from today, and you want to know what the asset’s present value will be at that time. Given an 8 percent discount rate, what will the asset be worth four years from today? 17 Chapter 1 The Time Value of Money Solution: The value of the asset is the present value of the asset’s promised payment. At t = 4, the cash payment will be received six years later. With this information, you can solve for the value four years from today using Equation 1-8: FVN = $100,000 r = 8% = 0.08 N =6 PV = FVN (1 + r)−N $ # 1 = $100,000 (1.08)6 = $100,000(0.6301696) = $63,016.96 0 ... $46,319.35 4 ... 10 $63,016.96 $100,000 FIGURE 1-5 The Relationship between Present Value and Future Value The time line in Figure 1-5 shows the future payment of $100,000 that is to be received at t = 10. The time line also shows the values at t = 4 and at t = 0. Relative to the payment at t = 10, the amount at t = 4 is a projected present value, while the amount at t = 0 is the present value (as of today). Present value problems require an evaluation of the present value factor, (1 + r)−N . Present values relate to the discount rate and the number of periods in the following ways: For a given discount rate, the farther in the future the amount to be received, the smaller that amount’s present value. • Holding time constant, the larger the discount rate, the smaller the present value of a future amount. • 5.2. The Frequency of Compounding Recall that interest may be paid semiannually, quarterly, monthly, or even daily. To handle interest payments made more than once a year, we can modify the present value formula (Equation 1-8) as follows: Recall that rs is the quoted interest rate and equals the periodic 18 Quantitative Investment Analysis interest rate multiplied by the number of compounding periods in each year. In general, with more than one compounding period in a year, we can express the formula for present value as ! rs "−mN PV = FVN 1 + (1-9) m where m = number of compounding periods per year rs = quoted annual interest rate N = number of years The formula in Equation 1-9 is quite similar to that in Equation 1-8. As we have already noted, present value and future value factors are reciprocals. Changing the frequency of compounding does not alter this result. The only difference is the use of the periodic interest rate and the corresponding number of compounding periods. The following example illustrates Equation 1-9. EXAMPLE 1-10 The Present Value of a Lump Sum with Monthly Compounding The manager of a Canadian pension fund knows that the fund must make a lump-sum payment of C$5 million 10 years from now. She wants to invest an amount today in a GIC so that it will grow to the required amount. The current interest rate on GICs is 6 percent a year, compounded monthly. How much should she invest today in the GIC? Solution: Use Equation 1-9 to find the required present value: FVN = C$5,000,000 rs = 6% = 0.06 m = 12 rs /m = 0.06/12 = 0.005 N = 10 mN = 12(10) = 120 ! rs "−mN PV = FVN 1 + m = C$5,000,000(1.005)−120 = C$5,000,000(0.549633) = C$2,748,163.67 In applying Equation 1-9, we use the periodic rate (in this case, the monthly rate) and the appropriate number of periods with monthly compounding (in this case, 10 years of monthly compounding, or 120 periods). 19 Chapter 1 The Time Value of Money 6. THE PRESENT VALUE OF A SERIES OF CASH FLOWS Many applications in investment management involve assets that offer a series of cash flows over time. The cash flows may be highly uneven, relatively even, or equal. They may occur over relatively short periods of time, longer periods of time, or even stretch on indefinitely. In this section, we discuss how to find the present value of a series of cash flows. 6.1. The Present Value of a Series of Equal Cash Flows We begin with an ordinary annuity. Recall that an ordinary annuity has equal annuity payments, with the first payment starting one period into the future. In total, the annuity makes N payments, with the first payment at t = 1 and the last at t = N . We can express the present value of an ordinary annuity as the sum of the present values of each individual annuity payment, as follows: PV = A A A A A + + + ··· + + (1 + r) (1 + r)2 (1 + r)3 (1 + r)N −1 (1 + r)N (1-10) where A = the annuity amount r = the interest rate per period corresponding to the frequency of annuity payments (for example, annual, quarterly, or monthly) N = the number of annuity payments Because the annuity payment (A) is a constant in this equation, it can be factored out as a common term. Thus the sum of the interest factors has a shortcut expression: ⎡ ⎤ 1 1 − ⎢ (1 + r)N ⎥ ⎥ PV = A ⎢ ⎣ ⎦ r (1-11) In much the same way that we computed the future value of an ordinary annuity, we find the present value by multiplying the annuity amount by a present value annuity factor (the term in brackets in Equation 1-11). EXAMPLE 1-11 The Present Value of an Ordinary Annuity Suppose you are considering purchasing a financial asset that promises to pay ¤1,000 per year for five years, with the first payment one year from now. The required rate of return is 12 percent per year. How much should you pay for this asset? Solution: To find the value of the financial asset, use the formula for the present value of an ordinary annuity given in Equation 1-11 with the following data: 20 Quantitative Investment Analysis A = ¤1,000 r = 12% = 0.12 N =5 ⎡ ⎤ 1 1 − ⎢ (1 + r)N ⎥ ⎥ PV = A ⎢ ⎣ ⎦ r ⎡ ⎤ 1 1 − ⎢ (1.12)5 ⎥ ⎥ = ¤1,000 ⎢ ⎣ ⎦ 0.12 = ¤1,000(3.604776) = ¤3,604.78 The series of cash flows of ¤1,000 per year for five years is currently worth ¤3,604.78 when discounted at 12 percent. Keeping track of the actual calendar time brings us to a specific type of annuity with level payments: the annuity due. An annuity due has its first payment occurring today (t = 0). In total, the annuity due will make N payments. Figure 1-6 presents the time line for an annuity due that makes four payments of $100. As Figure 1-6 shows, we can view the four-period annuity due as the sum of two parts: a $100 lump sum today and an ordinary annuity of $100 per period for three periods. At a 12 percent discount rate, the four $100 cash flows in this annuity due example will be worth $340.18.6 Expressing the value of the future series of cash flows in today’s dollars gives us a convenient way of comparing annuities. The next example illustrates this approach. 0 $100 1 $100 2 $100 3 $100 FIGURE 1-6 An Annuity Due of $100 per Period 6 There is an alternative way to calculate the present value of an annuity due. Compared to an ordinary annuity, the payments in an annuity due are each discounted one less period. Therefore, we can modify Equation 1-11 to handle annuities due by multiplying the right-hand side of the equation by (1 + r): PV(Annuity due) = A{[1 − (1 + r)−N ]/r}(1 + r). 21 Chapter 1 The Time Value of Money EXAMPLE 1-12 The Present Value of an Immediate Cash Flow Plus an Ordinary Annuity You are retiring today and must choose to take your retirement benefits either as a lump sum or as an annuity. Your company’s benefits officer presents you with two alternatives: an immediate lump sum of $2 million or an annuity with 20 payments of $200,000 a year with the first payment starting today. The interest rate at your bank is 7 percent per year compounded annually. Which option has the greater present value? (Ignore any tax differences between the two options.) Solution: To compare the two options, find the present value of each at time t = 0 and choose the one with the larger value. The first option’s present value is $2 million, already expressed in today’s dollars. The second option is an annuity due. Because the first payment occurs at t = 0, you can separate the annuity benefits into two pieces: an immediate $200,000 to be paid today (t = 0) and an ordinary annuity of $200,000 per year for 19 years. To value this option, you need to find the present value of the ordinary annuity using Equation 1-11 and then add $200,000 to it. A = $200,000 N = 19 r = 7% = 0.07 ⎡ 1 ⎢ 1 − (1 + r)N PV = A ⎢ ⎣ r ⎡ ⎤ ⎥ ⎥ ⎦ ⎤ 1 1 − ⎢ (1.07)19 ⎥ ⎥ = $200,000 ⎢ ⎣ ⎦ 0.07 = $200,000(10.335595) = $2,067,119.05 The 19 payments of $200,000 have a present value of $2,067,119.05. Adding the initial payment of $200,000 to $2,067,119.05, we find that the total value of the annuity option is $2,267,119.05. The present value of the annuity is greater than the lump sum alternative of $2 million. We now look at another example reiterating the equivalence of present and future values. 22 Quantitative Investment Analysis EXAMPLE 1-13 The Projected Present Value of an Ordinary Annuity A German pension fund manager anticipates that benefits of ¤1 million per year must be paid to retirees. Retirements will not occur until 10 years from now at time t = 10. Once benefits begin to be paid, they will extend until t = 39 for a total of 30 payments. What is the present value of the pension liability if the appropriate annual discount rate for plan liabilities is 5 percent compounded annually? Solution: This problem involves an annuity with the first payment at t = 10. From the perspective of t = 9, we have an ordinary annuity with 30 payments. We can compute the present value of this annuity with Equation 1-11 and then look at it on a time line. A = ¤1,000,000 r = 5% = 0.05 N = 30 ⎡ ⎤ 1 ⎢ 1 − (1 + r)N ⎥ ⎥ PV = A ⎢ ⎣ ⎦ r ⎡ ⎤ 1 1 − ⎢ (1.05)30 ⎥ ⎥ = ¤1,000,000 ⎢ ⎣ ⎦ 0.05 = ¤1,000,000(15.372451) = ¤15,372,451.03 0................................ 9 10 11 12 . . . . . . . . . . . . . . . . . . . . . 39 ¤ 1 ¤ 1 ¤ 1. . . . . . . . . . . . . . . . . . . . . ¤1 FIGURE 1-7 The Present Value of an Ordinary Annuity with First Payment at Time t = 10 (in millions) On the time line, we have shown the pension payments of ¤1 million extending from t = 10 to t = 39. The bracket and arrow indicate the process of finding the present value of the annuity, discounted back to t = 9. The present value of the pension 23 Chapter 1 The Time Value of Money benefits as of t = 9 is ¤15,372,451.03. The problem is to find the present value today (at t = 0). Now we can rely on the equivalence of present value and future value. As Figure 1-7 shows, we can view the amount at t = 9 as a future value from the vantage point of t = 0. We compute the present value of the amount at t = 9 as follows: FVN = ¤15,372,451.03(the present value at t = 9) N =9 r = 5% = 0.05 PV = FVN (1 + r)−N = ¤15,372,451.03(1.05)−9 = ¤15,372,451.03(0.644609) = ¤9,909,219.00 The present value of the pension liability is ¤9,909,219.00. Example 1-13 illustrates three procedures emphasized in this chapter: finding the present or future value of any cash flow series; recognizing the equivalence of present value and appropriately discounted future value; and • keeping track of the actual calendar time in a problem involving the time value of money. • • 6.2. The Present Value of an Infinite Series of Equal Cash Flows—Perpetuity Consider the case of an ordinary annuity that extends indefinitely. Such an ordinary annuity is called a perpetuity (a perpetual annuity). To derive a formula for the present value of a perpetuity, we can modify Equation 1-10 to account for an infinite series of cash flows: PV = A ∞ # + t=1 1 (1 + r)t $ (1-12) As long as interest rates are positive, the sum of present value factors converges and PV = A r (1-13) To see this, look back at Equation 1-11, the expression for the present value of an ordinary annuity. As N (the number of periods in the annuity) goes to infinity, the term 1/(1 + r)N goes to 0 and Equation 1-11 simplifies to Equation 1-13. This equation will reappear when we value dividends from stocks because stocks have no predefined life span. (A stock paying constant dividends is similar to a perpetuity.) With the first payment a year from now, a 24 Quantitative Investment Analysis perpetuity of $10 per year with a 20 percent required rate of return has a present value of $10/0.2 = $50. Equation 1-13 is valid only for a perpetuity with level payments. In our development above, the first payment occurred at t = 1; therefore, we compute the present value as of t = 0. Other assets also come close to satisfying the assumptions of a perpetuity. Certain government bonds and preferred stocks are typical examples of financial assets that make level payments for an indefinite period of time. EXAMPLE 1-14 The Present Value of a Perpetuity The British government once issued a type of security called a consol bond, which promised to pay a level cash flow indefinitely. If a consol bond paid £100 per year in perpetuity, what would it be worth today if the required rate of return were 5 percent? Solution: To answer this question, we can use Equation 1-13 with the following data: A = £100 r = 5% = 0.05 PV = A/r = £100/0.05 = £2,000 The bond would be worth £2,000. 6.3. Present Values Indexed at Times Other Than t = 0 In practice with investments, analysts frequently need to find present values indexed at times other than t = 0. Subscripting the present value and evaluating a perpetuity beginning with $100 payments in Year 2, we find PV1 = $100/0.05 = $2,000 at a 5 percent discount rate. Further, we can calculate today’s PV as PV0 = $2,000/1.05 = $1,904.76. Consider a similar situation in which cash flows of $6 per year begin at the end of the 4th year and continue at the end of each year thereafter, with the last cash flow at the end of the 10th year. From the perspective of the end of the third year, we are facing a typical seven-year ordinary annuity. We can find the present value of the annuity from the perspective of the end of the third year and then discount that present value back to the present. At an interest rate of 5 percent, the cash flows of $6 per year starting at the end of the fourth year will be worth $34.72 at the end of the third year (t = 3) and $29.99 today (t = 0). The next example illustrates the important concept that an annuity or perpetuity beginning sometime in the future can be expressed in present value terms one period prior to the first payment. That present value can then be discounted back to today’s present value. 25 Chapter 1 The Time Value of Money EXAMPLE 1-15 The Present Value of a Projected Perpetuity Consider a level perpetuity of £100 per year with its first payment beginning at t = 5. What is its present value today (at t = 0), given a 5 percent discount rate? Solution: First, we find the present value of the perpetuity at t = 4 and then discount that amount back to t = 0. (Recall that a perpetuity or an ordinary annuity has its first payment one period away, explaining the t = 4 index for our present value calculation.) i. Find the present value of the perpetuity at t = 4: A = £100 r = 5% = 0.05 PV = A/r = £100/0.05 = £2,000 ii. Find the present value of the future amount at t = 4. From the perspective of t = 0, the present value of £2,000 can be considered a future value. Now we need to find the present value of a lump sum: FVN = £2,000(the present value at t = 4) r = 5% = 0.05 N =4 PV = FVN (1 + r)−N = £2,000(1.05)−4 = £2,000(0.822702) = £1,645.40 Today’s present value of the perpetuity is £1,645.40. As discussed earlier, an annuity is a series of payments of a fixed amount for a specified number of periods. Suppose we own a perpetuity. At the same time, we issue a perpetuity obligating us to make payments; these payments are the same size as those of the perpetuity we own. However, the first payment of the perpetuity we issue is at t = 5; payments then continue on forever. The payments on this second perpetuity exactly offset the payments received from the perpetuity we own at t = 5 and all subsequent dates. We are left with level nonzero net cash flows at t = 1, 2, 3, and 4. This outcome exactly fits the definition of an annuity with four payments. Thus we can construct an annuity as the difference between two perpetuities with equal, level payments but differing starting dates. The next example illustrates this result. 26 Quantitative Investment Analysis EXAMPLE 1-16 The Present Value of an Ordinary Annuity as the Present Value of a Current Minus Projected Perpetuity Given a 5 percent discount rate, find the present value of a four-year ordinary annuity of £100 per year starting in Year 1 as the difference between the following two level perpetuities: Perpetuity 1 £100 per year starting in Year 1 (first payment at t = 1) Perpetuity 2 £100 per year starting in Year 5 (first payment at t = 5) Solution: If we subtract Perpetuity 2 from Perpetuity 1, we are left with an ordinary annuity of £100 per period for four years (payments at t = 1, 2, 3, 4). Subtracting the present value of Perpetuity 2 from that of Perpetuity 1, we arrive at the present value of the four-year ordinary annuity: i. ii. iii. iv. PV0 (Perpetuity 1) = £100/0.05 = £2,000 PV4 (Perpetuity 2) = £100/0.05 = £2,000 PV0 (Perpetuity 2) = £2,000/(1.05)4 = £1,645.40 PV0 (Annuity) = PV0 (Perpetuity 1) − PV0 (Perpetuity 2) = £2,000 − £1,645.40 = £354.60 The four-year ordinary annuity’s present value is equal to £2,000 − £1,645.40 = £354.60. 6.4. The Present Value of a Series of Unequal Cash Flows When we have unequal cash flows, we must first find the present value of each individual cash flow and then sum the respective present values. For a series with many cash flows, we usually use a spreadsheet. Table 1-3 lists a series of cash flows with the time periods in the first column, cash flows in the second column, and each cash flow’s present value in the third column. The last row of Table 1-3 shows the sum of the five present values. We could calculate the future value of these cash flows by computing them one at a time using the single-payment future value formula. We already know the present value of TABLE 1-3 A Series of Unequal Cash Flows and Their Present Values at 5 Percent Time Period Cash Flow 1 2 3 4 5 $1,000 $2,000 $4,000 $5,000 $6,000 Present Value at Year 0 $1,000(1.05)−1 = $952.38 $2,000(1.05)−2 = $1,814.06 $4,000(1.05)−3 = $3,455.35 $5,000(1.05)−4 = $4,113.51 $6,000(1.05)−5 = $4,701.16 Sum = $15,036.46 27 Chapter 1 The Time Value of Money this series, however, so we can easily apply time-value equivalence. The future value of the series of cash flows from Table 1-2, $19,190.76, is equal to the single $15,036.46 amount compounded forward to t = 5: PV = $15,036.46 N =5 r = 5% = 0.05 FVN = PV(1 + r)N = $15,036.46(1.05)5 = $15,036.46(1.276282) = $19,190.76 7. SOLVING FOR RATES, NUMBER OF PERIODS, OR SIZE OF ANNUITY PAYMENTS In the previous examples, certain pieces of information have been made available. For instance, all problems have given the rate of interest, r, the number of time periods, N , the annuity amount, A, and either the present value, PV, or future value, FV. In real-world applications, however, although the present and future values may be given, you may have to solve for either the interest rate, the number of periods, or the annuity amount. In the subsections that follow, we show these types of problems. 7.1. Solving for Interest Rates and Growth Rates Suppose a bank deposit of ¤100 is known to generate a payoff of ¤111 in one year. With this information, we can infer the interest rate that separates the present value of ¤100 from the future value of ¤111 by using Equation 1-2, FVN = PV(1 + r)N , with N = 1. With PV, FV, and N known, we can solve for r directly: 1 + r = FV/PV 1 + r = ¤111/¤100 = 1.11 r = 0.11, or 11% The interest rate that equates ¤100 at t = 0 to ¤111 at t = 1 is 11 percent. Thus we can state that ¤100 grows to ¤111 with a growth rate of 11 percent. As this example shows, an interest rate can also be considered a growth rate. The particular application will usually dictate whether we use the term ‘‘interest rate’’ or ‘‘growth rate.’’ Solving Equation 1-2 for r and replacing the interest rate r with the growth rate g produces the following expression for determining growth rates: g = (FVN /PV)1/N − 1 Following are two examples that use the concept of a growth rate. (1-14) 28 Quantitative Investment Analysis EXAMPLE 1-17 Calculating a Growth Rate (1) For 1998, Limited Brands, Inc., recorded net sales of $8,436 million. For 2002, Limited Brands recorded net sales of $8,445 million, only slightly higher than in 1998. Over the four-year period from the end of 1998 to the end of 2002, what was the rate of growth of Limited Brands’ net sales? Solution: To solve this problem, we can use Equation 1-14, g = (FVN /PV)1/N − 1. We denote net sales in 1998 as PV and net sales in 2002 as FV4 . We can then solve for the growth rate as follows: g= , 4 $8,445/$8,436 − 1 √ 4 = 1.001067 − 1 = 1.000267 − 1 = 0.000267 The calculated growth rate of approximately 0.03 percent a year, barely more than zero, confirms the initial impression that Limited Brands’ net sales were essentially flat during the 1998–2002 period. EXAMPLE 1-18 Calculating a Growth Rate (2) In Example 1-17, we found that Limited Brands’ compound growth rate of net sales was close to zero for 1998 to 2002. As a retailer, Limited Brands’ sales depend both on the number of stores (or selling square feet or meters) and sales per store (or sales per average selling square foot or meter). In fact, Limited Brands decreased its number of stores during the 1998–2002 period. In 1998, Limited Brands operated 5,382 stores, whereas in 2002 it operated 4,036 stores. In this case, we can speak of a positive compound rate of decrease or a negative compound growth rate. What was the growth rate in number of stores operated? Solution: Using Equation 1-14, we find g= , 4 4,036/5,382 − 1 √ 4 = 0.749907 − 1 = 0.930576 − 1 = −0.069424 The rate of growth in stores operated was approximately −6.9 percent during the 1998–2002 period. Note that we can also refer to −6.9 percent as the compound 29 Chapter 1 The Time Value of Money annual growth rate because it is the single number that compounds the number of stores in 1998 forward to the number of stores in 2002. Table 1-4 lists the number of stores operated by Limited Brands from 1998 to 2002. TABLE 1-4 Number of Limited Brands Stores, 1998–2002 Year Number of Stores 1998 1999 2000 2001 2002 5,382 5,023 5,129 4,614 4,036 (1 + g)t 5,023/5,382 = 0.933296 5,129/5,023 = 1.021103 4,614/5,129 = 0.899591 4,036/4,614 = 0.874729 t 0 1 2 3 4 Source: www.limited.com. Table 1-4 also shows 1 plus the one-year growth rate in number of stores. We can compute the 1 plus four-year cumulative growth in number of stores from 1998 to 2002 as the product of quantities (1 + one-year growth rate). We arrive at the same result as when we divide the ending number of stores, 4,036, by the beginning number of stores, 5,382: .... 4,036 5,023 5,129 4,614 4,036 = 5,382 5,382 5,023 5,129 4,614 = (1 + g1 )(1 + g2 )(1 + g3 )(1 + g4 ) 0.749907 = (0.933296)(1.021103)(0.899591)(0.874729) The right-hand side of the equation is the product of 1 plus the one-year growth rate in number of stores operated for each year. Recall that, using Equation 1-14, we took the fourth root of 4,036/5,382 = 0.749907. In effect, we were solving for the single value of g which, when compounded over four periods, gives the correct product of 1 plus the one-year growth rates.7 In conclusion, we do not need to compute intermediate growth rates as in Table 1-4 to solve for a compound growth rate g. Sometimes, however, the intermediate growth rates are interesting or informative. For example, during one year (2000), Limited Brands increased its number of stores. We can also analyze the variability in growth rates when we conduct an analysis as in Table 1-4. How did Limited Brands maintain approximately the same revenues during the period although it operated increasingly fewer stores? Elsewhere in Limited Brands’ disclosures, the company noted that its sales per average selling square foot increased during the period. 7 The compound growth rate that we calculate here is an example of a geometric mean, specifically the geometric mean of the growth rates. We define the geometric mean in the chapter on statistical concepts. 30 Quantitative Investment Analysis The compound growth rate is an excellent summary measure of growth over multiple time periods. In our Limited Brands example, the compound growth rate of −6.9 percent is the single growth rate that, when added to 1, compounded over four years, and multiplied by the 1998 number of stores operated, yields the 2002 number of stores operated. 7.2. Solving for the Number of Periods In this section, we demonstrate how to solve for the number of periods given present value, future value, and interest or growth rates. EXAMPLE 1-19 The Number of Annual Compounding Periods Needed for an Investment to Reach a Specific Value You are interested in determining how long it will take an investment of ¤10,000,000 to double in value. The current interest rate is 7 percent compounded annually. How many years will it take ¤10,000,000 to double to ¤20,000,000? Solution: Use Equation 1-2, FVN = PV(1 + r)N , to solve for the number of periods, N , as follows: (1 + r)N = FVN /PV = 2 N ln(1 + r) = ln(2) N = ln(2)/ln(1 + r) = ln(2)/ln(1.07) = 10.24 With an interest rate of 7 percent, it will take approximately 10 years for the initial ¤10,000,000 investment to grow to ¤20,000,000. Solving for N in the expression (1.07)N = 2.0 requires taking the natural logarithm of both sides and using the rule that ln(x N ) = N ln(x). Generally, we find that N = [ln(FV/PV)]/ln(1 + r). Here, N = ln(¤20,000,000/¤10,000,000)/ln(1.07) = ln(2)/ln(1.07) = 10.24.8 7.3. Solving for the Size of Annuity Payments In this section, we discuss how to solve for annuity payments. Mortgages, auto loans, and retirement savings plans are classic examples of applications of annuity formulas. 8 To quickly approximate the number of periods, practitioners sometimes use an ad hoc rule called the Rule of 72: Divide 72 by the stated interest rate to get the approximate number of years it would take to double an investment at the interest rate. Here, the approximation gives 72/7 = 10.3 years. The Rule of 72 is loosely based on the observation that it takes 12 years to double an amount at a 6 percent interest rate, giving 6 × 12 = 72. At a 3 percent rate, one would guess it would take twice as many years, 3 × 24 = 72. 31 Chapter 1 The Time Value of Money EXAMPLE 1-20 The Annuity Payments Needed to Reach a Future Value with Monthly Compounding You are planning to purchase a $120,000 house by making a down payment of $20,000 and borrowing the remainder with a 30-year fixed-rate mortgage with monthly payments. The first payment is due at t = 1. Current mortgage interest rates are quoted at 8 percent with monthly compounding. What will your monthly mortgage payments be? Solution: The bank will determine the mortgage payments such that at the stated periodic interest rate, the present value of the payments will be equal to the amount borrowed⎡(in this case, $100,000). With this fact in mind, we can use Equation 1-11, ⎤ 1 ⎢ 1 − (1 + r)N ⎥ ⎥, to solve for the annuity amount, A, as the present value PV = A ⎢ ⎣ ⎦ r divided by the present value annuity factor: PV = $100,000 rs = 8% = 0.08 m = 12 rs /m = 0.08/12 = 0.006667 N = 30 mN = 12 × 30 = 360 Present value annuity factor = 1− 1 1 1− [1 + (rs /m)]mN (1.006667)360 = rs /m 0.006667 = 136.283494 A = PV/Present value annuity factor = $100,000/136.283494 = $733.76 The amount borrowed, $100,000, is equivalent to 360 monthly payments of $733.76 with a stated interest rate of 8 percent. The mortgage problem is a relatively straightforward application of finding a level annuity payment. Next, we turn to a retirement-planning problem. This problem illustrates the complexity of the situation in which an individual wants to retire with a specified retirement income. Over the course of a life cycle, the individual may be able to save only a small amount during 32 Quantitative Investment Analysis the early years but then may have the financial resources to save more during later years. Savings plans often involve uneven cash flows, a topic we will examine in the last part of this chapter. When dealing with uneven cash flows, we take maximum advantage of the principle that dollar amounts indexed at the same point in time are additive—the cash flow additivity principle. EXAMPLE 1-21 The Projected Annuity Amount Needed to Fund a Future Annuity Inflow Jill Grant is 22 years old (at t = 0) and is planning for her retirement at age 63 (at t = 41). She plans to save $2,000 per year for the next 15 years (t = 1 to t = 15). She wants to have retirement income of $100,000 per year for 20 years, with the first retirement payment starting at t = 41. How much must Grant save each year from t = 16 to t = 40 in order to achieve her retirement goal? Assume she plans to invest in a diversified stock-and-bond mutual fund that will earn 8 percent per year on average. Solution: To help solve this problem, we set up the information on a time line. As Figure 1-8 shows, Grant will save $2,000 (an outflow) each year for Years 1 to 15. Starting in Year 41, Grant will start to draw retirement income of $100,000 per year for 20 years. In the time line, the annual savings is recorded in parentheses ($2) to show that it is an outflow. The problem is to find the savings, recorded as X , from Year 16 to Year 40. 0 1 ($2) 2 ... 15 ($2) ... ($2) 16 (X) 17 ... 40 (X) ... (X) 41 $100 42 ... 60 $100 ... $100 FIGURE 1-8 Solving for Missing Annuity Payments (in thousands) Solving this problem involves satisfying the following relationship: the present value of savings (outflows) equals the present value of retirement income (inflows). We could bring all the dollar amounts to t = 40 or to t = 15 and solve for X . Let us evaluate all dollar amounts at t = 15 (we encourage the reader to repeat the problem by bringing all cash flows to t = 40). As of t = 15, the first payment of X will be one period away (at t = 16). Thus we can value the stream of X s using the formula for the present value of an ordinary annuity. This problem involves three series of level cash flows. The basic idea is that the present value of the retirement income must equal the present value of Grant’s savings. Our strategy requires the following steps: 1. Find the future value of the savings of $2,000 per year and index it at t = 15. This value tells us how much Grant will have saved. 33 Chapter 1 The Time Value of Money 2. Find the present value of the retirement income at t = 15. This value tells us how much Grant needs to meet her retirement goals (as of t = 15). Two substeps are necessary. First, calculate the present value of the annuity of $100,000 per year at t = 40. Use the formula for the present value of an annuity. (Note that the present value is indexed at t = 40 because the first payment is at t = 41.) Next, discount the present value back to t = 15 (a total of 25 periods). 3. Now compute the difference between the amount Grant has saved (Step 1) and the amount she needs to meet her retirement goals (Step 2). Her savings from t = 16 to t = 40 must have a present value equal to the difference between the future value of her savings and the present value of her retirement income. Our goal is to determine the amount Grant should save in each of the 25 years from t = 16 to t = 40. We start by bringing the $2,000 savings to t = 15, as follows: A = $2,000 r = 8% = 0.08 N = 15 # $ (1 + r)N − 1 FV = A r $ # (1.08)15 − 1 = $2,000 0.08 = $2,000(27.152114) = $54,304.23 At t = 15, Grant’s initial savings will have grown to $54,304.23. Now we need to know the value of Grant’s retirement income at t = 15. As stated earlier, computing the retirement present value requires two substeps. First, find the present value at t = 40 with the formula in Equation 1-11; second, discount this present value back to t = 15. Now we can find the retirement income present value at t = 40: A = $100,000 r = 8% = 0.08 N = 20 ⎡ ⎤ 1 1 − ⎢ (1 + r)N ⎥ ⎥ PV = A ⎢ ⎣ ⎦ r ⎡ ⎤ 1 ⎢ 1 − (1.08)20 ⎥ ⎥ = $100,000 ⎢ ⎣ ⎦ 0.08 34 Quantitative Investment Analysis = $100,000(9.818147) = $981,814.74 The present value amount is as of t = 40, so we must now discount it back as a lump sum to t = 15: FVN = $981,814.74 N = 25 r = 8% = 0.08 PV = FVN (1 + r)−N = $981,814.74(1.08)−25 = $981,814.74(0.146018) = $143,362.53 Now recall that Grant will have saved $54,304.23 by t = 15. Therefore, in present value terms, the annuity from t = 16 to t = 40 must equal the difference between the amount already saved ($54,304.23) and the amount required for retirement ($143,362.53). This amount is equal to $143,362.53 − $54,304.23 = $89,058.30. Therefore, we must now find the annuity payment, A, from t = 16 to t = 40 that has a present value of $89,058.30. We find the annuity payment as follows: PV = $89,058.30 r = 8% = 0.08 N = 25 ⎡ 1 ⎢ 1 − (1 + r)N Present value annuity factor = ⎢ ⎣ r ⎡ ⎤ ⎥ ⎥ ⎦ ⎤ 1 1 − ⎢ (1.08)25 ⎥ ⎥ =⎢ ⎣ ⎦ 0.08 = 10.674776 A = PV/Present value annuity factor = $89,058.30/10.674776 = $8,342.87 35 Chapter 1 The Time Value of Money Grant will need to increase her savings to $8,342.87 per year from t = 16 to t = 40 to meet her retirement goal of having a fund equal to $981,814.74 after making her last payment at t = 40. 7.4. Review of Present and Future Value Equivalence As we have demonstrated, finding present and future values involves moving amounts of money to different points on a time line. These operations are possible because present value and future value are equivalent measures separated in time. Table 1-5 illustrates this equivalence; it lists the timing of five cash flows, their present values at t = 0, and their future values at t = 5. To interpret Table 1-5, start with the third column, which shows the present values. Note that each $1,000 cash payment is discounted back the appropriate number of periods to find the present value at t = 0. The present value of $4,329.48 is exactly equivalent to the series of cash flows. This information illustrates an important point: A lump sum can actually generate an annuity. If we place a lump sum in an account that earns the stated interest rate for all periods, we can generate an annuity that is equivalent to the lump sum. Amortized loans, such as mortgages and car loans, are examples of this principle. To see how a lump sum can fund an annuity, assume that we place $4,329.48 in the bank today at 5 percent interest. We can calculate the size of the annuity payments by using Equation 1-11. Solving for A, we find PV 1 − [1/(1 + r)N ] r $4,329.48 = 1 − [1/(1.05)5 ] 0.05 = $1,000 A= Table 1-6 shows how the initial investment of $4,329.48 can actually generate five $1,000 withdrawals over the next five years. To interpret Table 1-6, start with an initial present value of $4,329.48 at t = 0. From t = 0 to t = 1, the initial investment earns 5 percent interest, generating a future value TABLE 1-5 The Equivalence of Present and Future Values Time 1 2 3 4 5 Cash Flow $1,000 $1,000 $1,000 $1,000 $1,000 Present Value at t = 0 $1,000(1.05)−1 = $952.38 $1,000(1.05) = $907.03 $1,000(1.05)−3 = $863.84 $1,000(1.05)−4 = $822.70 $1,000(1.05)−5 = $783.53 Sum: $4,329.48 −2 Future Value at t = 5 $1,000(1.05)4 = $1,215.51 $1,000(1.05)3 = $1,157.63 $1,000(1.05)2 = $1,102.50 $1,000(1.05)1 = $1,050.00 $1,000(1.05)0 = $1,000.00 Sum: $5,525.64 36 Quantitative Investment Analysis TABLE 1-6 How an Initial Present Value Funds an Annuity Time Period Amount Available at the Beginning of the Time Period 1 2 3 4 5 $4,329.48 $3,545.95 $2,723.25 $1,859.41 $952.38 Ending Amount Before Withdrawal $4,329.48(1.05) = $4,545.95 $3,545.95(1.05) = $3,723.25 $2,723.25(1.05) = $2,859.41 $1,859.41(1.05) = $1,952.38 $952.38(1.05) = $1,000 Withdrawal Amount Available After Withdrawal $1,000 $1,000 $1,000 $1,000 $1,000 $3,545.95 $2,723.25 $1,859.41 $952.38 $0 of $4,329.48(1.05) = $4,545.95. We then withdraw $1,000 from our account, leaving $4,545.95 − $1,000 = $3,545.95 (the figure reported in the last column for time period 1). In the next period, we earn one year’s worth of interest and then make a $1,000 withdrawal. After the fourth withdrawal, we have $952.38, which earns 5 percent. This amount then grows to $1,000 during the year, just enough for us to make the last withdrawal. Thus the initial present value, when invested at 5 percent for five years, generates the $1,000 five-year ordinary annuity. The present value of the initial investment is exactly equivalent to the annuity. Now we can look at how future value relates to annuities. In Table 1-5, we reported that the future value of the annuity was $5,525.64. We arrived at this figure by compounding the first $1,000 payment forward four periods, the second $1,000 forward three periods, and so on. We then added the five future amounts at t = 5. The annuity is equivalent to $5,525.64 at t = 5 and $4,329.48 at t = 0. These two dollar measures are thus equivalent. We can verify the equivalence by finding the present value of $5,525.64, which is $5,525.64 × (1.05)−5 = $4,329.48. We found this result above when we showed that a lump sum can generate an annuity. To summarize what we have learned so far: A lump sum can be seen as equivalent to an annuity, and an annuity can be seen as equivalent to its future value. Thus present values, future values, and a series of cash flows can all be considered equivalent as long as they are indexed at the same point in time. 7.5. The Cash Flow Additivity Principle The cash flow additivity principle—the idea that amounts of money indexed at the same point in time are additive—is one of the most important concepts in time value of money mathematics. We have already mentioned and used this principle; this section provides a reference example for it. Consider the two series of cash flows shown on the time line in Figure 1-9. The series are denoted A and B. If we assume that the annual interest rate is 2 percent, we can find the future value of each series of cash flows as follows. Series A’s future value is $100(1.02) + $100 = $202. Series B’s future value is $200(1.02) + $200 = $404. The future value of (A + B) is $202 + $404 = $606 by the method we have used up to this point. The alternative way to find the future value is to add the cash flows of each series, A and B (call it A + B), and then find the future value of the combined cash flow, as shown in Figure 1-9. 37 Chapter 1 The Time Value of Money t=0 t=1 t=2 $100 $100 t=1 t=2 $200 $200 t=1 t=2 $300 $300 A t=0 B t=0 A+B FIGURE 1-9 The Additivity of Two Series of Cash Flows The third time line in Figure 1-9 shows the combined series of cash flows. Series A has a cash flow of $100 at t = 1, and Series B has a cash flow of $200 at t = 1. The combined series thus has a cash flow of $300 at t = 1. We can similarly calculate the cash flow of the combined series at t = 2. The future value of the combined series (A + B) is $300(1.02) + $300 = $606—the same result we found when we added the future values of each series. The additivity and equivalence principles also appear in another common situation. Suppose cash flows are $4 at the end of the first year and $24 (actually separate payments of $4 and $20) at the end of the second year. Rather than finding present values of the first year’s $4 and the second year’s $24, we can treat this situation as a $4 annuity for two years and a second-year $20 lump sum. If the discount rate were 6 percent, the $4 annuity would have a present value of $7.33 and the $20 lump sum a present value of $17.80, for a total of $25.13. CHAPTER 2 DISCOUNTED CASH FLOW APPLICATIONS 1. INTRODUCTION As investment analysts, much of our work includes evaluating transactions involving present and future cash flows. In the chapter on the time value of money (TVM), we presented the mathematics needed to solve those problems and illustrated the techniques for the major problem types. In this chapter we turn to applications. Analysts must master the numerous applications of TVM or discounted cash flow analysis in equity, fixed income, and derivatives analysis as they study each of those topics individually. In this chapter, we present a selection of important TVM applications: net present value and internal rate of return as tools for evaluating cash flow streams, portfolio return measurement, and the calculation of money market yields. Important in themselves, these applications also introduce concepts that reappear in many other investment contexts. The chapter is organized as follows. Section 2 introduces two key TVM concepts, net present value and internal rate of return. Building on these concepts, Section 3 discusses a key topic in investment management, portfolio return measurement. Investment managers often face the task of investing funds for the short term; to understand the choices available, they need to understand the calculation of money market yields. The chapter thus concludes with a discussion of that topic in Section 4. 2. NET PRESENT VALUE AND INTERNAL RATE OF RETURN In applying discounted cash flow analysis in all fields of finance, we repeatedly encounter two concepts, net present value and internal rate of return. In the following sections we present these keystone concepts. We could explore the concepts of net present value and internal rate of return in many contexts, because their scope of application covers all areas of finance. Capital budgeting, however, can serve as a representative starting point. Capital budgeting is important not only in corporate finance but also in security analysis, because both equity and fixed income analysts must be able to assess how well managers are investing the assets of their companies. There are three chief areas of financial decision-making in most businesses. Capital budgeting is the allocation of funds to relatively long-range projects or investments. From the perspective of capital budgeting, a company is a portfolio of projects and investments. Capital structure is 39 40 Quantitative Investment Analysis the choice of long-term financing for the investments the company wants to make. Working capital management is the management of the company’s short-term assets (such as inventory) and short-term liabilities (such as money owed to suppliers). 2.1. Net Present Value and the Net Present Value Rule Net present value (NPV) describes a way to characterize the value of an investment, and the net present value rule is a method for choosing among alternative investments. The net present value of an investment is the present value of its cash inflows minus the present value of its cash outflows. The word ‘‘net’’ in net present value refers to subtracting the present value of the investment’s outflows (costs) from the present value of its inflows (benefits) to arrive at the net benefit. The steps in computing NPV and applying the NPV rule are as follows: 1. Identify all cash flows associated with the investment—all inflows and outflows.1 2. Determine the appropriate discount rate or opportunity cost, r, for the investment project.2 3. Using that discount rate, find the present value of each cash flow. (Inflows have a positive sign and increase NPV; outflows have a negative sign and decrease NPV.) 4. Sum all present values. The sum of the present values of all cash flows (inflows and outflows) is the investment’s net present value. 5. Apply the NPV rule: If the investment’s NPV is positive, an investor should undertake it; if the NPV is negative, the investor should not undertake it. If an investor has two candidates for investment but can only invest in one (i.e., mutually exclusive projects), the investor should choose the candidate with the higher positive NPV. What is the meaning of the NPV rule? In calculating the NPV of an investment proposal, we use an estimate of the opportunity cost of capital as the discount rate. The opportunity cost of capital is the alternative return that investors forgo in undertaking the investment. When NPV is positive, the investment adds value because it more than covers the opportunity cost of the capital needed to undertake it. So a company undertaking a positive NPV investment increases shareholders’ wealth. An individual investor making a positive NPV investment increases personal wealth, but a negative NPV investment decreases wealth. When working problems using the NPV rule, it will be helpful to refer to the following formula: NPV = 1 N ! t=0 CFt (1 + r)t (2-1) In developing cash flow estimates, we observe two principles. First, we include only the incremental cash flows resulting from undertaking the project; we do not include sunk costs (costs that have been committed prior to the project). Second, we account for tax effects by using after-tax cash flows. For a full discussion of these and other issues in capital budgeting, see Brealey and Myers (2003). 2 The weighted-average cost of capital (WACC) is often used to discount cash flows. This value is a weighted average of the after-tax required rates of return on the company’s common stock, preferred stock, and long-term debt, where the weights are the fraction of each source of financing in the company’s target capital structure. For a full discussion of the issues surrounding the cost of capital, see Brealey and Myers (2003). Chapter 2 Discounted Cash Flow Applications 41 where CFt = the expected net cash flow at time t N = the investment’s projected life r = the discount rate or opportunity cost of capital As always, we state the inputs on a compatible time basis: If cash flows are annual, N is the project’s life in years and r is an annual rate. For instance, suppose you are reviewing a proposal that requires an initial outlay of $2 million (CF0 = −$2 million). You expect that the proposed investment will generate net positive cash flows of CF1 = $0.50 million at the end of Year 1, CF2 = $0.75 million at the end of Year 2, and CF3 = $1.35 million at the end of Year 3. Using 10 percent as a discount rate, you calculate the NPV as follows: NPV = −$2 + $0.50/(1.10) + $0.75/(1.10)2 + $1.35/(1.10)3 = −$2 + $0.454545 + $0.619835 + $1.014275 = $0.088655 million Because the NPV of $88,655 is positive, you accept the proposal under the NPV rule. Consider an example in which a research and development program is evaluated using the NPV rule. EXAMPLE 2-1 Evaluating a Research and Development Program Using the NPV Rule As an analyst covering the RAD Corporation, you are evaluating its research and development (R&D) program for the current year. Management has announced that it intends to invest $1 million in R&D. Incremental net cash flows are forecasted to be $150,000 per year in perpetuity. RAD Corporation’s opportunity cost of capital is 10 percent. 1. State whether RAD’s R&D program will benefit shareholders, as judged by the NPV rule. 2. Evaluate whether your answer to Part 1 changes if RAD Corporation’s opportunity cost of capital is 15 percent rather than 10 percent. Solution to 1: The constant net cash flows of $150,000, which we can denote as CF, form a perpetuity. The present value of the perpetuity is CF/r, so we calculate the project’s NPV as NPV = CF0 + CF/r = −$1,000,000 + $150,000/0.10 = $500,000 With an opportunity cost of 10 percent, the present value of the program’s cash inflows is $1.5 million. The program’s cost is an immediate outflow of $1 million; therefore, its net present value is $500,000. As NPV is positive, you conclude that RAD Corporation’s R&D program will benefit shareholders. 42 Quantitative Investment Analysis Solution to 2: With an opportunity cost of capital of 15 percent, you compute the NPV as you did above, but this time you use a 15 percent discount rate: NPV = −$1,000,000 + $150,000/0.15 = $0 With a higher opportunity cost of capital, the present value of the inflows is smaller and the program’s NPV is smaller: At 15 percent, the NPV exactly equals $0. At NPV = 0, the program generates just enough cash flow to compensate shareholders for the opportunity cost of making the investment. When a company undertakes a zero-NPV project, the company becomes larger but shareholders’ wealth does not increase. 2.2. The Internal Rate of Return and the Internal Rate of Return Rule Financial managers often want a single number that represents the rate of return generated by an investment. The rate of return computation most often used in investment applications (including capital budgeting) is the internal rate of return (IRR). The internal rate of return rule is a second method for choosing among investment proposals. The internal rate of return is the discount rate that makes net present value equal to zero. It equates the present value of the investment’s costs (outflows) to the present value of the investment’s benefits (inflows). The rate is ‘‘internal’’ because it depends only on the cash flows of the investment; no external data are needed. As a result, we can apply the IRR concept to any investment that can be represented as a series of cash flows. In the study of bonds, we encounter IRR under the name of yield to maturity. Later in this chapter, we will explore IRR as the money-weighted rate of return for portfolios. Before we continue, however, we must add a note of caution about interpreting IRR: Even if our cash flow projections are correct, we will realize a compound rate of return that is equal to IRR over the life of the investment only if we can reinvest all interim cash flows at exactly the IRR. Suppose the IRR for a project is 15 percent but we consistently reinvest the cash generated by the project at a lower rate. In this case, we will realize a return that is less than 15 percent. (This principle can work in our favor if we can reinvest at rates above 15 percent.) To return to the definition of IRR, in mathematical terms we said the following: NPV = CF0 + CF1 CF2 CFN + + ··· + =0 (1 + IRR)1 (1 + IRR)2 (1 + IRR)N (2-2) Again, the IRR in Equation 2-2 must be compatible with the timing of the cash flows. If the cash flows are quarterly, we have a quarterly IRR in Equation 2-2. We can then state the IRR on an annual basis. For some simple projects, the cash flow at t = 0, CF0 , captures the single capital outlay or initial investment; cash flows after t = 0 are the positive returns to the investment. In such cases, we can say CF0 = −Investment (the negative sign indicates an outflow). Thus we can rearrange Equation 2-2 in a form that is helpful in those cases: Investment = CF1 CF2 CFN + + ··· + (1 + IRR)1 (1 + IRR)2 (1 + IRR)N Chapter 2 Discounted Cash Flow Applications 43 For most real-life problems, financial analysts use software, spreadsheets, or financial calculators to solve this equation for IRR, so you should familiarize yourself with such tools.3 The investment decision rule using IRR, the IRR rule, states the following: ‘‘Accept projects or investments for which the IRR is greater than the opportunity cost of capital.’’ The IRR rule uses the opportunity cost of capital as a hurdle rate, or rate that a project’s IRR must exceed for the project to be accepted. Note that if the opportunity cost of capital is equal to the IRR, then the NPV is equal to 0. If the project’s opportunity cost is less than the IRR, the NPV is greater than 0 (using a discount rate less than the IRR will make the NPV positive). With these comments in mind, we work through two examples that involve the internal rate of return. EXAMPLE 2-2 Evaluating a Research and Development Program Using the IRR Rule In the previous RAD Corporation example, the initial outlay is $1 million and the program’s cash flows are $150,000 in perpetuity. Now you are interested in determining the program’s internal rate of return. Address the following: 1. Write the equation for determining the internal rate of return of this R&D program. 2. Calculate the IRR. Solution to 1: Finding the IRR is equivalent to finding the discount rate that makes the NPV equal to 0. Because the program’s cash flows are a perpetuity, you can set up the NPV equation as NPV = −Investment + CF/IRR = 0 NPV = −$1,000,000 + $150,000/IRR = 0 or as Investment = CF/IRR $1,000,000 = $150,000/IRR Solution to 2: We can solve for IRR as IRR = $150,000/$1,000,000 = 0.15 or 15 percent. The solution of 15 percent accords with the definition of IRR. In Example 2-1, you found that a discount rate of 15 percent made the program’s NPV equal to 0. By definition, therefore, the program’s IRR must be 15 percent. If the opportunity cost of capital is also 15 percent, the R&D program just covers its opportunity costs and neither 3 In some real-world capital budgeting problems, the initial investment (which has a minus sign) may be followed by subsequent cash inflows (which have plus signs) and outflows (which have minus signs). In these instances, the project can have more than one IRR. The possibility of multiple solutions is a theoretical limitation of IRR. 44 Quantitative Investment Analysis increases nor decreases shareholder wealth. If it is less than 15 percent, the IRR rule indicates that management should invest in the program because it more than covers its opportunity cost. If the opportunity cost is greater than 15 percent, the IRR rule tells management to reject the R&D program. For a given opportunity cost, the IRR rule and the NPV rule lead to the same decision in this example. EXAMPLE 2-3 The IRR and NPV Rules Side by Side The Japanese company Kageyama Ltd. is considering whether or not to open a new factory to manufacture capacitors used in cell phones. The factory will require an investment of ¥1,000 million. The factory is expected to generate level cash flows of ¥294.8 million per year in each of the next five years. According to information in its financial reports, Kageyama’s opportunity cost of capital for this type of project is 11 percent. 1. Determine whether the project will benefit Kageyama’s shareholders using the NPV rule. 2. Determine whether the project will benefit Kageyama’s shareholders using the IRR rule. Solution to 1: The cash flows can be grouped into an initial outflow of ¥1,000 million and an ordinary annuity of five inflows of ¥294.8 million. The expression for the present value of an annuity is A[1 − (1 + r)−N ]/r, where A is the level annuity payment. Therefore, with amounts shown in millions of Japanese yen, NPV = −1,000 + 294.8[1 − (1.11)−5 ]/0.11 = −1,000 + 1,089.55 = 89.55 Because the project’s NPV is positive ¥89.55 million, it should benefit Kageyama’s shareholders. Solution to 2: The IRR of the project is the solution to NPV = −1,000 + 294.8[1 − (1 + IRR)−5 ]/IRR = 0 This project’s positive NPV tells us that the internal rate of return must be greater than 11 percent. Using a calculator, we find that IRR is 0.145012 or 14.50 percent. Table 2-1 gives the keystrokes on most financial calculators. Because the IRR of 14.50 percent is greater than the opportunity cost of the project, the project should benefit Kageyama’s shareholders. Whether it uses the IRR rule or the NPV rule, Kageyama makes the same decision: Build the factory. 45 Chapter 2 Discounted Cash Flow Applications TABLE 2-1 Computing IRR Notation Used on Most Calculators N %i compute PV PMT FV Numerical Value for This Problem 5 X −1,000 294.8 n/a (=0) In the previous example, value creation is evident: For a single ¥1,000 million payment, Kageyama creates a project worth ¥1,089.55 million, a value increase of ¥89.55 million. Another perspective on value creation comes from converting the initial investment into a capital charge against the annual operating cash flows that the project generates. Recall that the project generates an annual operating cash flow of ¥294,800,000. If we subtract a capital charge of ¥270,570,310 (the amount of a five-year annuity having a present value of ¥1,000 million at 11 percent), we find ¥294,800,000 − ¥270,570,310 = ¥24,229,690. The amount of ¥24,229,690 represents the profit in each of the next five years after taking into account opportunity costs. The present value of a five-year annuity of ¥24,229,690 at an 11 percent cost of capital is exactly what we calculated as the project’s NPV: ¥89.55 million. Therefore, we can also calculate NPV by converting the initial investment to an annual capital charge against cash flow. 2.3. Problems with the IRR Rule The IRR and NPV rules give the same accept or reject decision when projects are independent—that is, when the decision to invest in one project does not affect the decision to undertake another. When a company cannot finance all the projects it would like to undertake—that is, when projects are mutually exclusive—it must rank the projects from most profitable to least. However, rankings according to IRR and NPV may not be the same. The IRR and NPV rules rank projects differently when the size or scale of the projects differs (measuring size by the investment needed to undertake the project), or • the timing of the projects’ cash flows differs. • When the IRR and NPV rules conflict in ranking projects, we should take directions from the NPV rule. Why that preference? The NPV of an investment represents the expected addition to shareholder wealth from an investment, and we take the maximization of shareholder wealth to be a basic financial objective of a company. To illustrate the preference for the NPV rule, consider first the case of projects that differ in size. Suppose that a company has only ¤30,000 available to invest.4 The company has available two one-period investment projects described as A and B in Table 2-2. 4 Or suppose the two projects require the same physical or other resources, so that only one can be undertaken. 46 Quantitative Investment Analysis TABLE 2-2 IRR and NPV for Mutually Exclusive Projects of Different Size Project A B Investment at t = 0 Cash Flow at t = 1 −¤10,000 −¤30,000 ¤15,000 ¤42,000 IRR NPV at 8% 50% 40% ¤3,888.89 ¤8,888.89 Project A requires an immediate investment of ¤10,000. This project will make a single cash payment of $15,000 at t = 1. Because the IRR is the discount rate that equates the present value of the future cash flow with the cost of the investment, the IRR equals 50 percent. If we assume that the opportunity cost of capital is 8 percent, then the NPV of Project A is ¤3,888.89. We compute the IRR and NPV of Project B as 40 percent and ¤8,888.89, respectively. The IRR and NPV rules indicate that we should undertake both projects, but to do so we would need ¤40,000—more money than is available. So we need to rank the projects. How do the projects rank according to IRR and NPV? The IRR rule ranks Project A, with the higher IRR, first. The NPV rule, however, ranks Project B, with the higher NPV, first—a conflict with the IRR rule’s ranking. Choosing Project A because it has the higher IRR would not lead to the largest increase in shareholders’ wealth. Investing in Project A effectively leaves ¤20,000 (¤30,000 minus A’s cost) uninvested. Project A increases wealth by almost ¤4,000, but Project B increases wealth by almost ¤9,000. The difference between the two projects’ scale creates the inconsistency in the ranking between the two rules. IRR and NPV can also rank projects of the same scale differently when the timing of cash flows differs. We can illustrate this principle with Projects A and D, presented in Table 2-3. TABLE 2-3 IRR and NPV for Mutually Exclusive Projects with Different Timing of Cash Flows Project A D CF0 − ¤10,000 − ¤10,000 CF1 CF2 CF3 IRR NPV at 8% ¤15,000 0 0 ¤21,220 0 50.0% 28.5% ¤3,888.89 ¤6,845.12 0 The terms CF0 , CF1 , CF2 , and CF3 represent the cash flows at time periods 0, 1, 2, and 3. The IRR for Project A is the same as it was in the previous example. The IRR for Project D is found as follows: −10,000 + 21,220 =0 (1 + IRR)3 The IRR for Project D is 28.5 percent, compared with 50 percent for Project A. IRRs and IRR rankings are not affected by any external interest rate or discount rate because a project’s cash flows alone determine the internal rate of return. The IRR calculation furthermore assumes reinvestment at the IRR, so we generally cannot interpret them as achievable rates of return. For Project D, for example, to achieve a 28.5 percent return we would need to earn 28.5 percent on ¤10,000 for the first year, earn 28.5 percent on ¤10,000(1.285) = ¤12,850 the second year, Chapter 2 Discounted Cash Flow Applications 47 and earn 28.5 percent on ¤10,000(1.285)2 = ¤16,512.25 the third year.5 A reinvestment rate such as 50 percent or 28.5 percent may be quite unrealistic. By contrast, the calculation of NPV uses an external market-determined discount rate, and reinvestment is assumed to take place at that discount rate. NPV rankings can depend on the external discount rate chosen. Here, Project D has a larger but more distant cash inflow (¤21,220 versus ¤15,000). As a result, Project D has a higher NPV than Project A at lower discount rates.6 The NPV rule’s assumption about reinvestment rates is more realistic and more economically relevant because it incorporates the market-determined opportunity cost of capital as a discount rate. As a consequence, the NPV is the expected addition to shareholder wealth from an investment. In summary, when dealing with mutually exclusive projects, choose among them using the NPV rule when the IRR rule and NPV rule conflict.7 3. PORTFOLIO RETURN MEASUREMENT Suppose you are an investor and you want to assess the success of your investments. You face two related but distinct tasks. The first is performance measurement, which involves calculating returns in a logical and consistent manner. Accurate performance measurement provides the basis for your second task, performance appraisal.8 Performance measurement is thus of great importance for all investors and investment managers because it is the foundation for all further analysis. In our discussion of portfolio return measurement, we will use the fundamental concept of holding period return (HPR), the return that an investor earns over a specified holding period. For an investment that makes one cash payment at the end of the holding period, HPR = (P1 − P0 + D1 )/P0 , where P0 is the initial investment, P1 is the price received at the end of the holding period, and D1 is the cash paid by the investment at the end of the holding period. Particularly when we measure performance over many periods, or when the portfolio is subject to additions and withdrawals, portfolio performance measurement is a challenging task. Two of the measurement tools available are the money-weighted rate of return measure and the time-weighted rate of return measure. The first measure we discuss, the money-weighted rate of return, implements a concept we have already covered in the context of capital budgeting: internal rate of return. 3.1. Money-Weighted Rate of Return The first performance measurement concept that we will discuss is an internal rate of return calculation. In investment management applications, the internal rate of return is called the 5 The ending amount ¤10,000(1.285)3 = ¤21,218 differs from the ¤21,220 amount listed in Table 2-3 because we rounded IRR. 6 There is a crossover discount rate above which Project A has a higher NPV than Project D. This crossover rate is 18.94 percent. 7 Technically, different reinvestment rate assumptions account for this conflict between the IRR and NPV rules. The IRR rule assumes that the company can earn the IRR on all reinvested cash flows, but the NPV rule assumes that cash flows are reinvested at the company’s opportunity cost of capital. The NPV assumption is far more realistic. For further details on this and other topics in capital budgeting, see Brealey and Myers (2003). 8 The term ‘‘performance evaluation’’ has been used as a synonym for performance appraisal. In later chapters we will discuss one performance appraisal tool, the Sharpe ratio. 48 Quantitative Investment Analysis TABLE 2-4 Cash Flows Time 0 1 Outlay $200 to purchase the first share $225 to purchase the second share Proceeds 1 2 2 $5 dividend received from first share (and not reinvested) $10 dividend ($5 per share × 2 shares) received $470 received from selling two shares at $235 per share money-weighted rate of return because it accounts for the timing and amount of all dollar flows into and out of the portfolio.9 To illustrate the money-weighted return, consider an investment that covers a two-year horizon. At time t = 0, an investor buys one share at $200. At time t = 1, he purchases an additional share at $225. At the end of Year 2, t = 2, he sells both shares for $235 each. During both years, the stock pays a per-share dividend of $5. The t = 1 dividend is not reinvested. Table 2-4 shows the total cash inflows and outflows. The money-weighted return on this portfolio is its internal rate of return for the two-year period. The portfolio’s internal rate of return is the rate, r, for which the present value of the cash inflows minus the present value of the cash outflows equals 0, or PV(outflows) = PV(inflows) $200 + $225 $5 $480 = + (1 + r) (1 + r) (1 + r)2 The left-hand side of this equation details the outflows: $200 at time t = 0 and $225 at time t = 1. The $225 outflow is discounted back one period because it occurs at t = 1. The right-hand side of the equation shows the present value of the inflows: $5 at time t = 1 (discounted back one period) and $480 (the $10 dividend plus the $470 sale proceeds) at time t = 2 (discounted back two periods). To solve for the money-weighted return, we use either a financial calculator that allows us to enter cash flows or a spreadsheet with an IRR function.10 The first step is to group net cash flows by time. For this example, we have −$200 for the t = 0 net cash flow, −$220 = −$225 + $5 for the t = 1 net cash flow, and $480 for the t = 2 net cash flow. After entering these cash flows, we use the spreadsheet’s or calculator’s IRR function to find that the money-weighted rate of return is 9.39 percent.11 Now we take a closer look at what has happened to the portfolio during each of the two years. In the first year, the portfolio generated a one-period holding period return of 9 In the United States, the money-weighted return is frequently called the dollar-weighted return. We follow a standard presentation of the money-weighted return as an IRR concept. 10 In this particular case we could solve for r by solving the quadratic equation 480x 2 − 220x − 200 = 0 with x = 1/(1 + r), using standard results from algebra. In general, however, we rely on a calculator or spreadsheet software to compute a money-weighted rate of return. 11 Note that the calculator or spreadsheet will give the IRR as a periodic rate. If the periods are not annual, we annualize the periodic rate. Chapter 2 Discounted Cash Flow Applications 49 ($5 + $225 − $200)/$200 = 15 percent. At the beginning of the second year, the amount invested is $450, calculated as $225 (per share price of stock) × 2 shares, because the $5 dividend was spent rather than reinvested. At the end of the second year, the proceeds from the liquidation of the portfolio are $470 (as detailed in Table 2-4) plus $10 in dividends (as also detailed in Table 2-4). So in the second year the portfolio produced a holding period return of ($10 + $470 − $450)/$450 = 6.67 percent. The mean holding period return was (15% + 6.67%)/2 = 10.84 percent. The money-weighted rate of return, which we calculated as 9.39 percent, puts a greater weight on the second year’s relatively poor performance (6.67 percent) than the first year’s relatively good performance (15 percent), as more money was invested in the second year than in the first. That is the sense in which returns in this method of calculating performance are ‘‘money weighted.’’ As a tool for evaluating investment managers, the money-weighted rate of return has a serious drawback. Generally, the investment manager’s clients determine when money is given to the investment manager and how much money is given. As we have seen, those decisions may significantly influence the investment manager’s money-weighted rate of return. A general principle of evaluation, however, is that persons or entities should be judged only on the basis of their own actions, or actions under their control. An evaluation tool should isolate the effects of the investment manager’s actions. The next section presents a tool that is effective in that respect. 3.2. Time-Weighted Rate of Return An investment measure that is not sensitive to the additions and withdrawals of funds is the time-weighted rate of return. In the investment management industry, the time-weighted rate of return is the preferred performance measure. The time-weighted rate of return measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period. In contrast to the money-weighted rate of return, the time-weighted rate of return is not affected by cash withdrawals or additions to the portfolio. The term ‘‘time-weighted’’ refers to the fact that returns are averaged over time. To compute an exact time-weighted rate of return on a portfolio, take the following three steps: 1. Price the portfolio immediately prior to any significant addition or withdrawal of funds. Break the overall evaluation period into subperiods based on the dates of cash inflows and outflows. 2. Calculate the holding period return on the portfolio for each subperiod. 3. Link or compound holding period returns to obtain an annual rate of return for the year (the time-weighted rate of return for the year). If the investment is for more than one year, take the geometric mean of the annual returns to obtain the time-weighted rate of return over that measurement period. Let us return to our money-weighted example and calculate the time-weighted rate of return for that investor’s portfolio. In that example, we computed the holding period returns on the portfolio, Step 2 in the procedure for finding time-weighted rate of return. Given that the portfolio earned returns of 15 percent during the first year and 6.67 percent during the second year, what is the portfolio’s time-weighted rate of return over an evaluation period of two years? 50 Quantitative Investment Analysis We find this time-weighted return by taking the geometric mean of the two holding period returns, Step 3 in the procedure above. The calculation of the geometric mean exactly mirrors the calculation of a compound growth rate. Here, we take the product of 1 plus the holding period return for each period to find the terminal value at t = 2 of $1 invested at t = 0. We then take the square root of this product and subtract 1 to get the geometric mean. We interpret the result as the annual compound growth rate of $1 invested in the portfolio at t = 0. Thus we have (1 + Time-weighted return)2 = (1.15)(1.0667) " Time-weighted return = (1.15)(1.0667) − 1 = 10.76% The time-weighted return on the portfolio was 10.76 percent, compared with the moneyweighted return of 9.39 percent, which gave larger weight to the second year’s return. We can see why investment managers find time-weighted returns more meaningful. If a client gives an investment manager more funds to invest at an unfavorable time, the manager’s money-weighted rate of return will tend to be depressed. If a client adds funds at a favorable time, the money-weighted return will tend to be elevated. The time-weighted rate of return removes these effects. In defining the steps to calculate an exact time-weighted rate of return, we said that the portfolio should be valued immediately prior to any significant addition or withdrawal of funds. With the amount of cash flow activity in many portfolios, this task can be costly. We can often obtain a reasonable approximation of the time-weighted rate of return by valuing the portfolio at frequent, regular intervals, particularly if additions and withdrawals are unrelated to market movements. The more frequent the valuation, the more accurate the approximation. Daily valuation is commonplace. Suppose that a portfolio is valued daily over the course of a year. To compute the time-weighted return for the year, we first compute each day’s holding period return: rt = MVEt − MVBt MVBt where MVBt equals the market value at the beginning of day t and MVEt equals the market value at the end of day t. We compute 365 such daily returns, denoted r1 , r2 , . . . , r365 . We obtain the annual return for the year by linking the daily holding period returns in the following way: (1 + r1 ) × (1 + r2 ) × · · · × (1 + r365 ) − 1. If withdrawals and additions to the portfolio happen only at day’s end, this annual return is a precise timeweighted rate of return for the year. Otherwise, it is an approximate time-weighted return for the year. If we have a number of years of data, we can calculate a time-weighted return for each year individually, as above. If ri is the time-weighted return for year i, we calculate an annualized time-weighted return as the geometric mean of N annual returns, as follows: r TW = [(1 + r1 ) × (1 + r2 ) × · · · × (1 + rN )]1/N − 1 Example 2-4 illustrates the calculation of the time-weighted rate of return. 51 Chapter 2 Discounted Cash Flow Applications EXAMPLE 2-4 Time-Weighted Rate of Return Strubeck Corporation sponsors a pension plan for its employees. It manages part of the equity portfolio in-house and delegates management of the balance to Super Trust Company. As chief investment officer of Strubeck, you want to review the performance of the in-house and Super Trust portfolios over the last four quarters. You have arranged for outflows and inflows to the portfolios to be made at the very beginning of the quarter. Table 2-5 summarizes the inflows and outflows as well as the two portfolios’ valuations. In the table, the ending value is the portfolio’s value just prior to the cash inflow or outflow at the beginning of the quarter. The amount invested is the amount each portfolio manager is responsible for investing. TABLE 2-5 Cash Flows for the In-House Strubeck Account and the Super Trust Account Quarter In-House Account Beginning value Beginning of period inflow (outflow) Amount invested Ending value Super Trust Account Beginning value Beginning of period inflow (outflow) Amount invested Ending value 1 2 3 4 $4,000,000 $6,000,000 $5,775,000 $6,720,000 $1,000,000 $5,000,000 $6,000,000 ($500,000) $5,500,000 $5,775,000 $225,000 $6,000,000 $6,720,000 ($600,000) $6,120,000 $5,508,000 $10,000,000 $13,200,000 $12,240,000 $5,659,200 $2,000,000 $12,000,000 $13,200,000 ($1,200,000) $12,000,000 $12,240,000 ($7,000,000) $5,240,000 $5,659,200 ($400,000) $5,259,200 $5,469,568 Based on the information given, address the following: 1. Calculate the time-weighted rate of return for the in-house account. 2. Calculate the time-weighted rate of return for the Super Trust account. Solution to 1: To calculate the time-weighted rate of return for the in-house account, we compute the quarterly holding period returns for the account and link them into an annual return. The in-house account’s time-weighted rate of return is 27 percent, calculated as follows: 1Q HPR : r1 = ($6,000,000 − $5,000,000)/$5,000,000 = 0.20 2Q HPR : r2 = ($5,775,000 − $5,500,000)/$5,500,000 = 0.05 3Q HPR : r3 = ($6,720,000 − $6,000,000)/$6,000,000 = 0.12 52 Quantitative Investment Analysis 4Q HPR : r4 = ($5,508,000 − $6,120,000)/$6,120,000 = −0.10 (1 + r1 )(1 + r2 )(1 + r3 )(1 + r4 ) − 1 = (1.20)(1.05)(1.12)(0.90) − 1 = 0.27 or 27% Solution to 2: The account managed by Super Trust has a time-weighted rate of return of 26 percent, calculated as follows: 1Q HPR : r1 = ($13,200,000 − $12,000,000)/$12,000,000 = 0.10 2Q HPR : r2 = ($12,240,000 − $12,000,000)/$12,000,000 = 0.02 3Q HPR : r3 = ($5,659,200 − $5,240,000)/$5,240,000 = 0.08 4Q HPR : r4 = ($5,469,568 − $5,259,200)/$5,259,200 = 0.04 (1 + r1 )(1 + r2 )(1 + r3 )(1 + r4 ) − 1 = (1.10)(1.02)(1.08)(1.04) − 1 = 0.26 or 26% The in-house portfolio’s time-weighted rate of return is higher than the Super Trust portfolio’s by 100 basis points. Having worked through this exercise, we are ready to look at a more detailed case. EXAMPLE 2-5 Time-Weighted and Money-Weighted Rates of Return Side by Side Your task is to compute the investment performance of the Walbright Fund during 2003. The facts are as follows: • • • • • • On 1 January 2003, the Walbright Fund had a market value of $100 million. During the period 1 January 2003 to 30 April 2003, the stocks in the fund showed a capital gain of $10 million. On 1 May 2003, the stocks in the fund paid a total dividend of $2 million. All dividends were reinvested in additional shares. Because the fund’s performance had been exceptional, institutions invested an additional $20 million in Walbright on 1 May 2003, raising assets under management to $132 million ($100 + $10 + $2 + $20). On 31 December 2003, Walbright received total dividends of $2.64 million. The fund’s market value on 31 December 2003, not including the $2.64 million in dividends, was $140 million. The fund made no other interim cash payments during 2003. Based on the information given, address the following: 1. Compute the Walbright Fund’s time-weighted rate of return. Chapter 2 Discounted Cash Flow Applications 2. Compute the Walbright Fund’s money-weighted rate of return. 3. Interpret the differences between the time-weighted and money-weighted rates of return. Solution to 1: Because interim cash flows were made on 1 May 2003, we must compute two interim total returns and then link them to obtain an annual return. Table 2-6 lists the relevant market values on 1 January, 1 May, and 31 December as well as the associated interim four-month (1 January to 1 May) and eight-month (1 May to 31 December) holding period returns. TABLE 2-6 Cash Flows for the Walbright Fund 1 January 2003 1 May 2003 31 December 2003 Beginning portfolio value = $100 million Dividends received before additional investment = $2 million Ending portfolio value = $110 million $2 + $10 = 12% Holding period return = $100 New investment = $20 million Beginning market value for last 2/3 of year = $132 million Dividends received = $2.64 million Ending portfolio value = $140 million $2.64 + $140 − $132 = 8.06% Holding period return = $132 Now we must geometrically link the four- and eight-month returns to compute an annual return. We compute the time-weighted return as follows: Time-weighted return = 1.12 × 1.0806 − 1 = 0.2103 In this instance, we compute a time-weighted rate of return of 21.03 percent for one year. The four-month and eight-month intervals combine to equal one year. (Taking the square root of the product 1.12 × 1.0806 would be appropriate only if 1.12 and 1.0806 each applied to one full year.) Solution to 2: To calculate the money-weighted return, we find the discount rate that sets the present value of the outflows (purchases) equal to the present value of the inflows (dividends and future payoff). The initial market value of the fund and all additions to it are treated as cash outflows. (Think of them as expenditures.) Withdrawals, receipts, and the ending market value of the fund are counted as inflows. (The ending market value is the amount investors receive on liquidating the fund.) Because interim cash flows have occurred at four-month intervals, we must solve for the four-month internal rate of return. Table 2-6 details the cash flows and their timing. The present value equation (in millions) is as follows: PV(outflows) = PV(inflows) $100 + $2 $20 $2 $2.64 $140 + = + + 1 1 1 3 (1 + r) (1 + r) (1 + r) (1 + r) (1 + r)3 53 54 Quantitative Investment Analysis The left-hand side of the equation shows the investments in the fund or outflows: a $100 million initial investment followed by the $2 million dividend reinvested and an additional $20 million of new investment (both occurring at the end of the first four-month interval, which makes the exponent in the denominator 1). The right-hand side of the equation shows the payoffs or inflows: the $2 million dividend at the first four-month interval followed by the $2.64 million dividend and the terminal market value of $140 million (both occurring at the end of the third four-month interval, which makes the exponent in the denominator 3). The second four-month interval has no cash flow. We can bring all the terms to the right of the equal sign, arranging them in order of time. After simplification, 0 = −$100 − $20 $142.64 + 1 (1 + r) (1 + r)3 Using a spreadsheet or IRR-enabled calculator, we use −100, −20, 0, and $142.64 for the t = 0, t = 1, t = 2, and t = 3 net cash flows, respectively.12 Using either tool, we get a four-month IRR of 6.28 percent. The quick way to annualize this is to multiply by 3. A more accurate way is (1.0628)3 − 1 = 0.20 or 20 percent. Solution to 3: In this example, the time-weighted return (21.03 percent) is greater than the money-weighted return (20 percent). The Walbright Fund’s performance was relatively poorer during the eight-month period, when the fund owned more shares, than it was overall. This fact is reflected in a lower money-weighted rate of return compared with time-weighted rate of return, as the money-weighted return is sensitive to the timing and amount of withdrawals and additions to the portfolio. The accurate measurement of portfolio returns is important to the process of evaluating portfolio managers. In addition to considering returns, however, analysts must also weigh risk. When we worked through Example 2-4, we stopped short of suggesting that inhouse management was superior to Super Trust because it earned a higher time-weighted rate of return. With risk in focus, we can talk of risk-adjusted performance and make comparisons—but only cautiously. In later chapters, we will discuss the Sharpe ratio, an important risk-adjusted performance measure that we might apply to an investment manager’s time-weighted rate of return. For now, we have illustrated the major tools for measuring the return on a portfolio. 4. MONEY MARKET YIELDS In our discussion of internal rate of return and net present value, we referred to the opportunity cost of capital as a market-determined rate. In this section, we begin a discussion of discounted cash flow analysis in actual markets by considering short-term debt markets. To understand the various ways returns are presented in debt markets, we must discuss some of the conventions for quoting yields on money-market instruments. The money market 12 By convention, we denote outflows with a negative sign, and we need 0 as a placeholder for t = 2. 55 Chapter 2 Discounted Cash Flow Applications is the market for short-term debt instruments (one-year maturity or less). Some instruments require the issuer to repay the lender the amount borrowed plus interest. Others are pure discount instruments that pay interest as the difference between the amount borrowed and the amount paid back. In the U.S. money market, the classic example of a pure discount instrument is the U.S. Treasury bill (T-bill) issued by the federal government. The face value of a T-bill is the amount the U.S. government promises to pay back to a T-bill investor. In buying a T-bill, investors pay the face amount less the discount, and receive the face amount at maturity. The discount is the reduction from the face amount that gives the price for the T-bill. This discount becomes the interest that accumulates, because the investor receives the face amount at maturity. Thus, investors earn a dollar return equal to the discount if they hold the instrument to maturity. T-bills are by far the most important class of money-market instruments in the United States. Other types of money-market instruments include commercial paper and bankers’ acceptances, which are discount instruments, and negotiable certificates of deposit, which are interest-bearing instruments. The market for each of these instruments has its own convention for quoting prices or yields. The remainder of this section examines the quoting conventions for T-bills and other money-market instruments. In most instances, the quoted yields must be adjusted for use in other present value problems. Pure discount instruments such as T-bills are quoted differently from U.S. government bonds. T-bills are quoted on a bank discount basis, rather than on a price basis. The bank discount basis is a quoting convention that annualizes, based on a 360-day year, the discount as a percentage of face value. Yield on a bank discount basis is computed as follows: rBD = D 360 F t (2-3) where rBD = the annualized yield on a bank discount basis D = the dollar discount, which is equal to the difference between the face value of the bill, F , and its purchase price, P0 F = the face value of the T-bill t = the actual number of days remaining to maturity 360 = bank convention of the number of days in a year The bank discount yield (often called simply the discount yield) takes the dollar discount from par, D, and expresses it as a fraction of the face value (not the price) of the T-bill. This fraction is then multiplied by the number of periods of length t in one year (that is, 360/t), where the year is assumed to have 360 days. Annualizing in this fashion assumes simple interest (no compounding). Consider the following example. EXAMPLE 2-6 The Bank Discount Yield Suppose a T-bill with a face value (or par value) of $100,000 and 150 days until maturity is selling for $98,000. What is its bank discount yield? 56 Quantitative Investment Analysis Solution: For this example, the dollar discount, D, is $2,000. The yield on a bank discount basis is 4.8 percent, as computed with Equation 2-3: rBD = $2,000 360 = 4.8% $100,000 150 The bank discount formula takes the T-bill’s dollar discount from face or par as a fraction of face value, 2 percent, and then annualizes by the factor 360/150 = 2.4. The price of discount instruments such as T-bills is quoted using discount yields, so we typically translate discount yield into price. Suppose we know the bank discount yield of 4.8 percent but do not know the price. We solve for the dollar discount, D, as follows: D = rBD F t 360 With rBD = 4.8 percent, the dollar discount is D = 0.048 × $100,000 × 150/360 = $2,000. Once we have computed the dollar discount, the purchase price for the T-bill is its face value minus the dollar discount, F − D = $100,000 − $2,000 = $98,000. Yield on a bank discount basis is not a meaningful measure of investors’ return, for three reasons. First, the yield is based on the face value of the bond, not on its purchase price. Returns from investments should be evaluated relative to the amount that is invested. Second, the yield is annualized based on a 360-day year rather than a 365-day year. Third, the bank discount yield annualizes with simple interest, which ignores the opportunity to earn interest on interest (compound interest). We can extend Example 2-6 to discuss three often-used alternative yield measures. The first is the holding period return over the remaining life of the instrument (150 days in the case of the T-bill in Example 2-6). It determines the return that an investor will earn by holding the instrument to maturity; as used here, this measure refers to an unannualized rate of return (or periodic rate of return). In fixed income markets, this holding period return is also called a holding period yield (HPY).13 For an instrument that makes one cash payment during its life, HPY is HPY = P 1 − P0 + D1 P0 (2-4) where P0 = the initial purchase price of the instrument P1 = the price received for the instrument at its maturity D1 = the cash distribution paid by the instrument at its maturity (i.e., interest) 13 Bond-market participants often use the term ‘‘yield’’ when referring to total returns (returns incorporating both price change and income), as in yield to maturity. In other cases, yield refers to returns from income alone (as in current yield, which is annual interest divided by price). As used in this book and by many writers, holding period yield is a bond market synonym for holding period return, total return, and horizon return. Chapter 2 Discounted Cash Flow Applications 57 When we use this expression to calculate the holding period yield for an interest-bearing instrument (for example, coupon-bearing bonds), we need to observe an important detail: The purchase and sale prices must include any accrued interest added to the trade price because the bond was traded between interest payment dates. Accrued interest is the coupon interest that the seller earns from the last coupon date but does not receive as a coupon, because the next coupon date occurs after the date of sale.14 For pure discount securities, all of the return is derived by redeeming the bill for more than its purchase price. Because the T-bill is a pure discount instrument, it makes no interest payment and thus D1 = 0. Therefore, the holding period yield is the dollar discount divided by the purchase price, HPY = D/P0 , where D = P1 − P0 . The holding period yield is the amount that is annualized in the other measures. For the T-bill in Example 2-6, the investment of $98,000 will pay $100,000 in 150 days. The holding period yield on this investment using Equation 2-4 is ($100,000 − $98,000)/$98,000 = $2,000/$98,000 = 2.0408 percent. For this example, the periodic return of 2.0408 percent is associated with a 150-day period. If we were to use the T-bill rate of return as the opportunity cost of investing, we would use a discount rate of 2.0408 percent for the 150-day T-bill to find the present value of any other cash flow to be received in 150 days. As long as the other cash flow has risk characteristics similar to those of the T-bill, this approach is appropriate. If the other cash flow were riskier than the T-bill, then we could use the T-bill’s yield as a base rate, to which we would add a risk premium. The formula for the holding period yield is the same regardless of the currency of denomination. The second measure of yield is the effective annual yield (EAY). The EAY takes the quantity 1 plus the holding period yield and compounds it forward to one year, then subtracts 1 to recover an annualized return that accounts for the effect of interest-on-interest.15 EAY = (1 + HPY)365/t − 1 (2-5) In our example, we can solve for EAY as follows: EAY = (1.020408)365/150 − 1 = 1.050388 − 1 = 5.0388% This example illustrates a general rule: The bank discount yield is less than the effective annual yield. The third alternative measure of yield is the money market yield (also known as the CD equivalent yield). This convention makes the quoted yield on a T-bill comparable to yield quotations on interest-bearing money-market instruments that pay interest on a 360-day basis. In general, the money market yield is equal to the annualized holding period yield; assuming a 360-day year, rMM = (HPY)(360/t). Compared to the bank discount yield, the money market yield is computed on the purchase price, so rMM = (rBD )(F /P0 ). This equation 14 The price with accrued interest is called the full price. Trade prices are quoted ‘‘clean’’ (without accrued interest), but accrued interest, if any, is added to the purchase price. For more on accrued interest, see Fabozzi (2004). 15 Effective annual yield was called the effective annual rate (Equation 1-5) in the chapter on the time value of money. 58 Quantitative Investment Analysis TABLE 2-7 Three Commonly Used Yield Measures Holding Period Yield (HPY) HPY = P1 − P0 + D1 P0 Effective Annual Yield (EAY) EAY = (1 + HPY)365/t − 1 Money Market Yield (CD Equivalent Yield) r MM = 360r BD 360 − (t)(r BD ) shows that the money market yield is larger than the bank discount yield. In practice, the following expression is more useful because it does not require knowing the T-bill price: rMM = 360rBD 360 − (t)(rBD ) (2-6) For the T-bill example, the money market yield is rMM = (360)(0.048)/[360 − (150)(0.048)] = 4.898 percent.16 Table 2-7 summarizes the three yield measures we have discussed. The next example will help you consolidate your knowledge of these yield measures. EXAMPLE 2-7 Using the Appropriate Discount Rate You need to find the present value of a cash flow of $1,000 that is to be received in 150 days. You decide to look at a T-bill maturing in 150 days to determine the relevant interest rate for calculating the present value. You have found a variety of yields for the 150-day bill. Table 2-8 presents this information. TABLE 2-8 Short-Term Money Market Yields Holding period yield Bank discount yield Money market yield Effective annual yield 2.0408% 4.8% 4.898% 5.0388% Which yield or yields are appropriate for finding the present value of the $1,000 to be received in 150 days? Solution: The holding period yield is appropriate, and we can also use the money market yield and effective annual yield after converting them to a holding period yield. 16 Some national markets use the money market yield formula, rather than the bank discount yield formula, to quote the yields on discount instruments such as T-bills. In Canada, the convention is to quote Treasury bill yields using the money market formula assuming a 365-day year. Yields for German Treasury discount paper with a maturity less than one year and French BTFs (T-bills) are computed with the money market formula assuming a 360-day year. Chapter 2 Discounted Cash Flow Applications • 59 Holding period yield (2.0408 percent). This yield is exactly what we want. Because it applies to a 150-day period, we can use it in a straightforward fashion to find the present value of the $1,000 to be received in 150 days. (Recall the principle that discount rates must be compatible with the time period.) The present value is PV = $1,000 = $980.00 1.020408 Now we can see why the other yield measures are inappropriate or not as easily applied. Bank discount yield (4.8 percent). We should not use this yield measure to determine the present value of the cash flow. As mentioned earlier, the bank discount yield is based on the face value of the bill and not on its price. • Money market yield (4.898 percent). To use the money market yield, we need to convert it to the 150-day holding period yield by dividing it by (360/150). After obtaining the holding period yield 0.04898/(360/150) = 0.020408, we use it to discount the $1,000 as above. • Effective annual yield (5.0388 percent). This yield has also been annualized, so it must be adjusted to be compatible with the timing of the cash flow. We can obtain the holding period yield from the EAY as follows: • (1.050388)150/365 − 1 = 0.020408 Recall that when we found the effective annual yield, the exponent was 365/150, or the number of 150-day periods in a 365-day year. To shrink the effective annual yield to a 150-day yield, we use the reciprocal of the exponent that we used to annualize. In Example 2-7, we converted two short-term measures of annual yield to a holding period yield for a 150-day period. That is one type of conversion. We frequently also need to convert periodic rates to annual rates. The issue can arise both in money markets and in longer-term debt markets. As an example, many bonds (long-term debt instruments) pay interest semiannually. Bond investors compute IRRs for bonds, known as yields to maturity (YTM). If the semiannual yield to maturity is 4 percent, how do we annualize it? An exact approach, taking account of compounding, would be to compute (1.04)2 − 1 = 0.0816 or 8.16 percent. This is what we have been calling an effective annual yield. An approach used in U.S. bond markets, however, is to double the semiannual YTM: 4% × 2 = 8%. The yield to maturity calculated this way, ignoring compounding, has been called a bond-equivalent yield. Annualizing a semiannual yield by doubling is putting the yield on a bond-equivalent basis. In practice, the result, 8 percent, would be referred to simply as the bond’s yield to maturity. In money markets, if we annualized a six-month-period yield by doubling it, in order to make the result comparable to bonds’ YTMs, we would also say that the result was a bond-equivalent yield. CHAPTER 3 STATISTICAL CONCEPTS AND MARKET RETURNS 1. INTRODUCTION Statistical methods provide a powerful set of tools for analyzing data and drawing conclusions from them. Whether we are analyzing asset returns, earnings growth rates, commodity prices, or any other financial data, statistical tools help us quantify and communicate the data’s important features. This chapter presents the basics of describing and analyzing data, the branch of statistics known as descriptive statistics. The chapter supplies a set of useful concepts and tools, illustrated in a variety of investment contexts. One theme of our presentation, reflected in the chapter’s title, is the demonstration of the statistical methods that allow us to summarize return distributions.1 We explore four properties of return distributions: • • • • where the returns are centered (central tendency), how far returns are dispersed from their center (dispersion), whether the distribution of returns is symmetrically shaped or lopsided (skewness), and whether extreme outcomes are likely (kurtosis). These same concepts are generally applicable to the distributions of other types of data, too. The chapter is organized as follows. After defining some basic concepts in Section 2, in Sections 3 and 4 we discuss the presentation of data: Section 3 describes the organization of data in a table format, and Section 4 describes the graphic presentation of data. We then turn to the quantitative description of how data are distributed: Section 5 focuses on measures that quantify where data are centered, or measures of central tendency. Section 6 presents other measures that describe the location of data. Section 7 presents measures that quantify the degree to which data are dispersed. Sections 8 and 9 describe additional measures that provide a more accurate picture of data. Section 10 discusses investment applications of concepts introduced in Section 5. 2. SOME FUNDAMENTAL CONCEPTS Before starting the study of statistics with this chapter, it may be helpful to examine a picture of the overall field. In the following, we briefly describe the scope of statistics and its branches 1 Ibbotson Associates (www.ibbotson.com) generously provided much of the data used in this chapter. We also draw on Dimson, Marsh, and Staunton’s (2002) history and study of world markets, as well as other sources. 61 62 Quantitative Investment Analysis of study. We explain the concepts of population and sample. Data come in a variety of types, affecting the ways they can be measured and the appropriate statistical methods for analyzing them. We conclude by discussing the basic types of data measurement. 2.1. The Nature of Statistics The term statistics can have two broad meanings, one referring to data and the other to method. A company’s average earnings per share (EPS) for the last 20 quarters or its average returns for the past 10 years are statistics. We may also analyze historical EPS to forecast future EPS or use the company’s past returns to infer its risk. The totality of methods we employ to collect and analyze data is also called statistics. Statistical methods include descriptive statistics and statistical inference (inferential statistics). Descriptive statistics is the study of how data can be summarized effectively to describe the important aspects of large data sets. By consolidating a mass of numerical details, descriptive statistics turns data into information. Statistical inference involves making forecasts, estimates, or judgments about a larger group from the smaller group actually observed. The foundation for statistical inference is probability theory, and both statistical inference and probability theory will be discussed in later chapters. Our focus in this chapter is solely on descriptive statistics. 2.2. Populations and Samples Throughout the study of statistics we make a critical distinction between a population and a sample. In this section, we explain these two terms as well as the related terms ‘‘parameter’’ and ‘‘sample statistic.’’2 • Definition of Population. A population is defined as all members of a specified group. Any descriptive measure of a population characteristic is called a parameter. Although a population can have many parameters, investment analysts are usually concerned with only a few, such as the mean value, the range of investment returns, and the variance. Even if it is possible to observe all the members of a population, it is often too expensive in terms of time or money to attempt to do so. For example, if the population is all telecommunications customers worldwide and an analyst is interested in their purchasing plans, she will find it too costly to observe the entire population. The analyst can address this situation by taking a sample of the population. • Definition of Sample. A sample is a subset of a population. In taking a sample, the analyst hopes it is characteristic of the population. The field of statistics known as sampling deals with taking samples in appropriate ways to achieve the objective of representing the population well. A later chapter addresses the details of sampling. Earlier, we mentioned statistics in the sense of referring to data. Just as a parameter is a descriptive measure of a population characteristic, a sample statistic (statistic, for short) is a descriptive measure of a sample characteristic. 2 This chapter introduces many statistical concepts and formulas. To make it easy to locate them, we have set off some of the more important ones with bullet points. Chapter 3 Statistical Concepts and Market Returns • 63 Definition of Sample Statistic. A sample statistic (or statistic) is a quantity computed from or used to describe a sample. We devote much of this chapter to explaining and illustrating the use of statistics in this sense. The concept is critical also in statistical inference, which addresses such problems as estimating an unknown population parameter using a sample statistic. 2.3. Measurement Scales To choose the appropriate statistical methods for summarizing and analyzing data, we need to distinguish among different measurement scales or levels of measurement. All data measurements are taken on one of four major scales: nominal, ordinal, interval, or ratio. Nominal scales represent the weakest level of measurement: They categorize data but do not rank them. If we assigned integers to mutual funds that follow different investment strategies, the number 1 might refer to a small-cap value fund, the number 2 to a large-cap value fund, and so on for each possible style. This nominal scale categorizes the funds according to their style but does not rank them. Ordinal scales reflect a stronger level of measurement. Ordinal scales sort data into categories that are ordered with respect to some characteristic. For example, the Morningstar and Standard & Poor’s star ratings for mutual funds represent an ordinal scale in which one star represents a group of funds judged to have had relatively the worst performance, with two, three, four, and five stars representing groups with increasingly better performance, as evaluated by those services. An ordinal scale may also involve numbers to identify categories. For example, in ranking balanced mutual funds based on their five-year cumulative return, we might assign the number 1 to the top 10 percent of funds, and so on, so that the number 10 represents the bottom 10 percent of funds. The ordinal scale is stronger than the nominal scale because it reveals that a fund ranked 1 performed better than a fund ranked 2. The scale tells us nothing, however, about the difference in performance between funds ranked 1 and 2 compared with the difference in performance between funds ranked 3 and 4, or 9 and 10. Interval scales provide not only ranking but also assurance that the differences between scale values are equal. As a result, scale values can be added and subtracted meaningfully. The Celsius and Fahrenheit scales are interval measurement scales. The difference in temperature between 10◦ C and 11◦ C is the same amount as the difference between 40◦ C and 41◦ C. We can state accurately that 12◦ C = 9◦ C + 3◦ C, for example. Nevertheless, the zero point of an interval scale does not reflect complete absence of what is being measured; it is not a true zero point or natural zero. Zero degrees Celsius corresponds to the freezing point of water, not the absence of temperature. As a consequence of the absence of a true zero point, we cannot meaningfully form ratios on interval scales. As an example, 50◦ C, although five times as large a number as 10◦ C, does not represent five times as much temperature. Also, questionnaire scales are often treated as interval scales. If an investor is asked to rank his risk aversion on a scale from 1 (extremely risk-averse) to 7 (extremely risk-loving), the difference between a response of 1 and a response of 2 is sometimes assumed to represent the same difference in risk aversion as the difference between a response of 6 and a response of 7. When that assumption can be justified, the data are measured on interval scales. Ratio scales represent the strongest level of measurement. They have all the characteristics of interval measurement scales as well as a true zero point as the origin. With ratio scales, we 64 Quantitative Investment Analysis can meaningfully compute ratios as well as meaningfully add and subtract amounts within the scale. As a result, we can apply the widest range of statistical tools to data measured on a ratio scale. Rates of return are measured on a ratio scale, as is money. If we have twice as much money, then we have twice the purchasing power. Note that the scale has a natural zero—zero means no money. EXAMPLE 3-1 Identifying Scales of Measurement State the scale of measurement for each of the following: 1. 2. 3. 4. Credit ratings for bond issues.3 Cash dividends per share. Hedge fund classification types.4 Bond maturity in years. Solution to 1: Credit ratings are measured on an ordinal scale. A rating places a bond issue in a category, and the categories are ordered with respect to the expected probability of default. But the difference in the expected probability of default between AA− and A+, for example, is not necessarily equal to that between BB− and B+. In other words, letter credit ratings are not measured on an interval scale. Solution to 2: Cash dividends per share are measured on a ratio scale. For this variable, 0 represents the complete absence of dividends; it is a true zero point. Solution to 3: Hedge fund classification types are measured on a nominal scale. Each type groups together hedge funds with similar investment strategies. In contrast to credit ratings for bonds, however, hedge fund classification schemes do not involve a ranking. Thus such classification schemes are not measured on an ordinal scale. Solution to 4: Bond maturity is measured on a ratio scale. Now that we have addressed the important preliminaries, we can discuss summarizing and describing data. 3 Credit ratings for a bond issue gauge the bond issuer’s ability to meet the promised principal and interest payments on the bond. For example, one rating agency, Standard & Poor’s, assigns bond issues to one of the following ratings, given in descending order of credit quality (increasing probability of default): AAA, AA+, AA, AA−, A+, A, A−, BBB+, BBB, BBB−, BB+, BB, BB−, B, CCC+, CCC−, CC, C, CI, D. For more information on credit risk and credit ratings, see Fabozzi (2004a). 4 ‘‘Hedge fund’’ refers to investment vehicles with legal structures that result in less regulatory oversight than other pooled investment vehicles such as mutual funds. Hedge fund classification types group hedge funds by the kind of investment strategy they pursue. 65 Chapter 3 Statistical Concepts and Market Returns 3. SUMMARIZING DATA USING FREQUENCY DISTRIBUTIONS In this section, we discuss one of the simplest ways to summarize data—the frequency distribution. • Definition of Frequency Distribution. A frequency distribution is a tabular display of data summarized into a relatively small number of intervals. Frequency distributions help in the analysis of large amounts of statistical data, and they work with all types of measurement scales. Rates of return are the fundamental units that analysts and portfolio managers use for making investment decisions, and we can use frequency distributions to summarize rates of return. When we analyze rates of return, our starting point is the holding period return (also called the total return). • Holding Period Return Formula. The holding period return for time period t, Rt , is Rt = Pt − Pt−1 + Dt Pt−1 (3-1) where Pt = price per share at the end of time period t Pt−1 = price per share at the end of time period t − 1, the time period immediately preceding time period t Dt = cash distributions received during time period t Thus the holding period return for time period t is the capital gain (or loss) plus distributions divided by the beginning-period price. (For common stocks, the distribution is a dividend; for bonds, the distribution is a coupon payment.) Equation 3-1 can be used to define the holding period return on any asset for a day, week, month, or year simply by changing the interpretation of the time interval between successive values of the time index, t. The holding period return, as defined in Equation 3-1, has two important characteristics. First, it has an element of time attached to it. For example, if a monthly time interval is used between successive observations for price, then the rate of return is a monthly figure. Second, rate of return has no currency unit attached to it. For instance, suppose that prices are denominated in euros. The numerator and denominator of Equation 3-1 would be expressed in euros, and the resulting ratio would not have any units because the units in the numerator and denominator would cancel one another. This result holds regardless of the currency in which prices are denominated.5 With these concerns noted, we now turn to the frequency distribution of the holding period returns on the S&P 500 Index.6 First, we examine annual rates of return; then we 5 Note, however, that if price and cash distributions in the expression for holding period return were not in one’s home currency, one would generally convert those variables to one’s home currency before calculating the holding period return. Because of exchange rate fluctuations during the holding period, holding period returns on an asset computed in different currencies would generally differ. 6 We use the total return series on the S&P 500 from January 1926 to December 2002 provided by Ibbotson Associates. 66 Quantitative Investment Analysis look at monthly rates of return. The annual rates of return on the S&P 500 calculated with Equation 3-1 span the period January 1926 to December 2002, for a total of 77 annual observations. Monthly return data cover the period January 1926 to December 2002, for a total of 924 monthly observations. We can state a basic procedure for constructing a frequency distribution as follows: • Construction of a Frequency Distribution. 1. 2. 3. 4. 5. Sort the data in ascending order. Calculate the range of the data, defined as Range = Maximum value − Minimum value. Decide on the number of intervals in the frequency distribution, k. Determine interval width as Range/k. Determine the intervals by successively adding the interval width to the minimum value, to determine the ending points of intervals, stopping after reaching an interval that includes the maximum value. 6. Count the number of observations falling in each interval. 7. Construct a table of the intervals listed from smallest to largest that shows the number of observations falling in each interval. In Step 4, when rounding the interval width, round up rather than down, to ensure that the final interval includes the maximum value of the data. As the above procedure makes clear, a frequency distribution groups data into a set of intervals.7 An interval is a set of values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers all the values represented in the data. The actual number of observations in a given interval is called the absolute frequency, or simply the frequency. The frequency distribution is the list of intervals together with the corresponding measures of frequency. To illustrate the basic procedure, suppose we have 12 observations sorted in ascending order: −4.57, −4.04, −1.64, 0.28, 1.34, 2.35, 2.38, 4.28, 4.42, 4.68, 7.16, and 11.43. The minimum observation is −4.57 and the maximum observation is +11.43, so the range is +11.43 − (−4.57) = 16. If we set k = 4, the interval width is 16/4 = 4. Table 3-1 shows the repeated addition of the interval width of 4 to determine the endpoints for the intervals (Step 5). Thus the intervals are [−4.57 to −0.57), [−0.57 to 3.43), [3.43 to 7.43), and [7.43 to 11.43].8 Table 3-2 summarizes Steps 5 through 7. TABLE 3-1 Endpoints of Intervals −4.57 + 4.00 = −0.57 −0.57 + 4.00 = 3.43 3.43 + 4.00 = 7.43 7.43 + 4.00 = 11.43 7 Intervals are also sometimes called classes, ranges, or bins. The notation [−4.57 to −0.57) means −4.57 ≤ observation < −0.57. In this context, a square bracket indicates that the endpoint is included in the interval. 8 67 Chapter 3 Statistical Concepts and Market Returns TABLE 3-2 Frequency Distribution Interval A B C D Absolute Frequency −4.57 ≤ observation < −0.57 −0.57 ≤ observation < 3.43 3.43 ≤ observation < 7.43 7.43 ≤ observation ≤ 11.43 3 4 4 1 Note that the intervals do not overlap, so each observation can be placed uniquely into one interval. In practice, we may want to refine the above basic procedure. For example, we may want the intervals to begin and end with whole numbers for ease of interpretation. We also need to explain the choice of the number of intervals, k. We turn to these issues in discussing the construction of frequency distributions for the S&P 500. We first consider the case of constructing a frequency distribution for the annual returns on the S&P 500 over the period 1926 to 2002. During that period, the return on the S&P 500 had a minimum value of −43.34 percent (in 1931) and a maximum value of +53.99 percent (in 1933). Thus the range of the data was +54% − (−43%) = 97%, approximately. The question now is the number k of intervals into which we should group observations. Although some guidelines for setting k have been suggested in statistical literature, the setting of a useful value for k often involves inspecting the data and exercising judgment. How much detail should we include? If we use too few intervals, we will summarize too much and lose pertinent characteristics. If we use too many intervals, we may not summarize enough. We can establish an appropriate value for k by evaluating the usefulness of the resulting interval width. A large number of empty intervals may indicate that we are trying to organize the data to present too much detail. Starting with a relatively small interval width, we can see whether or not the intervals are mostly empty and whether or not the value of k associated with that interval width is too large. If intervals are mostly empty or k is very large, we can consider increasingly larger intervals (smaller values of k) until we have a frequency distribution that effectively summarizes the distribution. For the annual S&P 500 series, return intervals of 1 percent width would result in 97 intervals and many of them would be empty because we have only 77 annual observations. We need to keep in mind that the purpose of a frequency distribution is to summarize the data. Suppose that for ease of interpretation we want to use an interval width stated in whole rather than fractional percents. A 2 percent interval width would have many fewer empty intervals than a 1 percent interval width and effectively summarize the data. A 2 percent interval width would be associated with 97/2 = 48.5 intervals, which we can round up to 49 intervals. That number of intervals will cover 2% × 49 = 98%. We can confirm that if we start the smallest 2 percent interval at the whole number −44.0 percent, the final interval ends at −44.0% + 98% = 54% and includes the maximum return in the sample, 53.99 percent. In so constructing the frequency distribution, we will also have intervals that end and begin at a value of 0 percent, allowing us to count the negative and positive returns in the data. Without too much work, we have found an effective way to summarize the data. We will use return intervals of 2 percent, beginning with −44% ≤ Rt < −42% (given as ‘‘−44% to −42%’’ in the table) and ending with 52% ≤ Rt ≤ 54%. Table 3-3 shows the frequency distribution for the annual total returns on the S&P 500. 68 3 2 0 7 1 1 1 0 1 0 4 1 1 0 0 0 0 1 3.90% 2.60% 0.00% 9.09% 1.30% 1.30% 1.30% 0.00% 1.30% 0.00% 5.19% 1.30% 1.30% 0.00% 0.00% 0.00% 0.00% 1.30% 0.00% 0.00% 1.30% 0.00% 1.30% 0.00% 23 25 25 17 18 19 20 5 6 6 10 4 5 5 5 2 2 3 1 1 2 2 1 1 Cumulative Frequency 29.87% 32.47% 32.47% 22.08% 23.38% 24.68% 25.97% 6.49% 7.79% 7.79% 12.99% 5.19% 6.49% 6.49% 6.49% 2.60% 2.60% 3.90% 1.30% 1.30% 2.60% 2.60% 1.30% 1.30% Relative Frequency Cumulative 4 4 1 3 1 1 2 6 3 5 2 1 1 5 4 0 4 0 0 2 0 1 0 0 2 8.0% to 10.0% 10.0% to 12.0% 12.0% to 14.0% 14.0% to 16.0% 16.0% to 18.0% 18.0% to 20.0% 20.0% to 22.0% 22.0% to 24.0% 24.0% to 26.0% 26.0% to 28.0% 28.0% to 30.0% 30.0% to 32.0% 32.0% to 34.0% 34.0% to 36.0% 36.0% to 38.0% 38.0% to 40.0% 40.0% to 42.0% 42.0% to 44.0% 44.0% to 46.0% 46.0% to 48.0% 48.0% to 50.0% 50.0% to 52.0% 52.0% to 54.0% Frequency 4.0% to 16.0% 6.0% to 8.0% Return Interval Note: The lower class limit is the weak inequality (≤) and the upper class limit is the strong inequality (<). Source: Frequency distribution generated with Ibbotson Associates EnCorr Analyzer. −2.0% to 0.0% 0.0% to 2.0% 2.0% to 4.0% −10.0% to −8.0% −8.0% to −6.0% −6.0% to −4.0% −4.0% to −2.0% −18.0% to −16.0% −16.0% to −14.0% −14.0% to −12.0% −12.0% to −10.0% −26.0% to −24.0% −24.0% to −22.0% −22.0% to −20.0% −20.0% to −18.0% −32.0% to −30.0% −30.0% to −28.0% −28.0% to −26.0% 0 0 1 0 1 0 −44.0% to −42.0% −42.0% to −40.0% −40.0% to −38.0% −38.0% to −36.0% −36.0% to −34.0% −34.0% to −32.0% Frequency Return Interval Relative Frequency TABLE 3-3 Frequency Distribution for the Annual Total Return on the S&P 500, 1926–2002 1.30% 0.00% 0.00% 2.60% 0.00% 0.00% 2.60% 0.00% 6.49% 5.19% 0.00% 5.19% 6.49% 2.60% 1.30% 1.30% 2.60% 7.79% 3.90% 1.30% 3.90% 1.30% 1.30% 5.19% 5.19% Relative Frequency 75 75 75 77 72 72 74 74 64 68 68 72 55 57 58 59 41 47 50 34 37 38 39 29 33 Cumulative Frequency 97.40% 97.40% 97.40% 100.00% 93.51% 93.51% 96.10% 96.10% 83.12% 88.31% 88.31% 93.51% 71.43% 74.03% 75.32% 76.62% 53.25% 61.04% 64.94% 44.16% 48.05% 49.35% 50.65% 37.66% 42.86% Relative Frequency Cumulative Chapter 3 Statistical Concepts and Market Returns 69 Table 3-3 includes three other useful ways to present data, which we can compute once we have established the frequency distribution: the relative frequency, the cumulative frequency (also called the cumulative absolute frequency), and the cumulative relative frequency. • Definition of Relative Frequency. The relative frequency is the absolute frequency of each interval divided by the total number of observations. The cumulative relative frequency cumulates (adds up) the relative frequencies as we move from the first to the last interval. It tells us the fraction of observations that are less than the upper limit of each interval. Examining the frequency distribution given in Table 3-3, we see that the first return interval, −44 percent to −42 percent, has one observation; its relative frequency is 1/77 or 1.30 percent. The cumulative frequency for this interval is 1 because only one observation is less than −42 percent. The cumulative relative frequency is thus 1/77 or 1.30 percent. The next return interval has zero observations; therefore, its cumulative frequency is 0 plus 1 and its cumulative relative frequency is 1.30 percent (the cumulative relative frequency from the previous interval). We can find the other cumulative frequencies by adding the (absolute) frequency to the previous cumulative frequency. The cumulative frequency, then, tells us the number of observations that are less than the upper limit of each return interval. As Table 3-3 shows, return intervals have frequencies from 0 to 7 in this sample. The interval encompassing returns between −10 percent and −8 percent (−10% ≤ Rt < −8%) has the most observations, seven. Next most frequent are returns between 18 percent and 20 percent (18% ≤ Rt < 20%), with six observations. From the cumulative frequency column, we see that the number of negative returns is 23. The number of positive returns must then be equal to 77 − 23, or 54. We can express the number of positive and negative outcomes as a percentage of the total to get a sense of the risk inherent in investing in the stock market. During the 77-year period, the S&P 500 had negative annual returns 29.9 percent of the time (that is, 23/77). This result appears in the fifth column of Table 3-3, which reports the cumulative relative frequency. The frequency distribution gives us a sense of not only where most of the observations lie but also whether the distribution is evenly distributed, lopsided, or peaked. In the case of the S&P 500, we can see that more than half of the outcomes are positive and most of those annual returns are larger than 10 percent. (Only 11 of the 54 positive annual returns—about 20 percent—were between 0 and 10 percent.) Table 3-3 permits us to make an important further point about the choice of the number of intervals related to equity returns in particular. From the frequency distribution in Table 3-3, we can see that only five outcomes fall between −44 percent and −16 percent and between 38 percent and 54 percent. Stock return data are frequently characterized by a few very large or small outcomes. We could have collapsed the return intervals in the tails of the frequency distribution by choosing a smaller value of k, but then we would have lost the information about how extremely poorly or well the stock market had performed. A risk manager may need to know the worst possible outcomes and thus may want to have detailed information on the tails (the extreme values). A frequency distribution with a relatively large value of k is useful for that. A portfolio manager or analyst may be equally interested in detailed information on the tails; however, if the manager or analyst wants a picture only of where most of the observations lie, he might prefer to use an interval width of 4 percent (25 intervals beginning at −44 percent), for example. The frequency distribution for monthly returns on the S&P 500 looks quite different from that for annual returns. The monthly return series from January 1926 to December 70 Quantitative Investment Analysis TABLE 3-4 Frequency Distribution for the Monthly Total Return on the S&P 500, January 1926 to December 2002 Return Interval Absolute Frequency Relative Frequency Cumulative Absolute Frequency Cumulative Relative Frequency −30.0% to −28.0% −28.0% to −26.0% −26.0% to −24.0% −24.0% to −22.0% −22.0% to −20.0% −20.0% to −18.0% −18.0% to −16.0% −16.0% to −14.0% −14.0% to −12.0% −12.0% to −10.0% −10.0% to −8.0% −8.0% to −6.0% −6.0% to −4.0% −4.0% to −2.0% −2.0% to 0.0% 0.0% to 2.0% 2.0% to 4.0% 4.0% to 6.0% 6.0% to 8.0% 8.0% to 10.0% 10.0% to 12.0% 12.0% to 14.0% 14.0% to 16.0% 16.0% to 18.0% 18.0% to 20.0% 20.0% to 22.0% 22.0% to 24.0% 24.0% to 26.0% 26.0% to 28.0% 28.0% to 30.0% 30.0% to 32.0% 32.0% to 34.0% 34.0% to 36.0% 36.0% to 38.0% 38.0% to 40.0% 40.0% to 42.0% 42.0% to 44.0% 1 0 1 1 2 2 2 3 5 6 20 30 54 90 138 182 153 126 58 21 14 6 2 3 0 0 0 1 0 0 0 0 0 0 2 0 1 0.11% 0.00% 0.11% 0.11% 0.22% 0.22% 0.22% 0.32% 0.54% 0.65% 2.16% 3.25% 5.84% 9.74% 14.94% 19.70% 16.56% 13.64% 6.28% 2.27% 1.52% 0.65% 0.22% 0.32% 0.00% 0.00% 0.00% 0.11% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.22% 0.00% 0.11% 1 1 2 3 5 7 9 12 17 23 43 73 127 217 355 537 690 816 874 895 909 915 917 920 920 920 920 921 921 921 921 921 921 921 923 923 924 0.11% 0.11% 0.22% 0.32% 0.54% 0.76% 0.97% 1.30% 1.84% 2.49% 4.65% 7.90% 13.74% 23.48% 38.42% 58.12% 74.68% 88.31% 94.59% 96.86% 98.38% 99.03% 99.24% 99.57% 99.57% 99.57% 99.57% 99.68% 99.68% 99.68% 99.68% 99.68% 99.68% 99.68% 99.89% 99.89% 100.00% Note: The lower class limit is the weak inequality (≤) and the upper class limit is the strong inequality (<). The relative frequency is the absolute frequency or cumulative frequency divided by the total number of observations. Source: Frequency distribution generated with Ibbotson Associates EnCorr Analyzer. Chapter 3 Statistical Concepts and Market Returns 71 2002 has 924 observations. Returns range from a minimum of approximately −30 percent to a maximum of approximately +43 percent. With such a large quantity of monthly data we must summarize to get a sense of the distribution, and so we group the data into 37 equally spaced return intervals of 2 percent. The gains from summarizing in this way are substantial. Table 3-4 presents the resulting frequency distribution. The absolute frequencies appear in the second column, followed by the relative frequencies. The relative frequencies are rounded to two decimal places. The cumulative absolute and cumulative relative frequencies appear in the fourth and fifth columns, respectively. The advantage of a frequency distribution is evident in Table 3-4, which tells us that the vast majority of observations (599/924 = 65 percent) lie in the four intervals spanning −2 percent to +6 percent. Altogether, we have 355 negative returns and 569 positive returns. Almost 62 percent of the monthly outcomes are positive. Looking at the cumulative relative frequency in the last column, we see that the interval −2 percent to 0 percent shows a cumulative frequency of 38.42 percent, for an upper return limit of 0 percent. This means that 38.42 percent of the observations lie below the level of 0 percent. We can also see that not many observations are greater than +12 percent or less than −12 percent. Note that the frequency distributions of annual and monthly returns are not directly comparable. On average, we should expect the returns measured at shorter intervals (for example, months) to be smaller than returns measured over longer periods (for example, years). Next, we construct a frequency distribution of average inflation-adjusted returns over 1900–2000 for 16 major equity markets. EXAMPLE 3-2 Constructing a Frequency Distribution How have equities rewarded investors in different countries in the long run? To answer this question, we could examine the average annual returns directly.9 The worth of a nominal level of return depends on changes in the purchasing power of money, however, and internationally there have been a variety of experiences with price inflation. It is preferable, therefore, to compare the average real or inflation-adjusted returns earned by investors in different countries. Dimson, Marsh, and Staunton (2002) presented authoritative evidence on asset returns in 16 countries for the 101 years 1900–2000. Table 3-5 excerpts their findings for average inflation-adjusted returns. Table 3-6 summarizes the data in Table 3-5 into six intervals spanning 4 percent to 10 percent. As Table 3-6 shows, there is substantial variation internationally of average real equity returns. Three-fourths of the observations fall in one of three intervals: 6.0 to 7.0 percent, 7.0 to 8.0 percent, or 9.0 to 10.0 percent. Most average real equity returns are between 6.0 percent and 10 percent; the cumulative relative frequency of returns less than 6.0 percent was only 12.50 percent. 9 The average or arithmetic mean of a set of values equals the sum of the values divided by the number of values summed. To find the arithmetic mean of 101 annual returns, for example, we sum the 101 annual returns and then divide the total by 101. Among the most familiar of statistical concepts, the arithmetic mean is explained in more detail later in the chapter. 72 Quantitative Investment Analysis TABLE 3-5 Real (Inflation-Adjusted) Equity Returns: Sixteen Major Equity Markets, 1900–2000 Country Arithmetic Mean Australia Belgium Canada Denmark France Germany Ireland Italy Japan Netherlands South Africa Spain Sweden Switzerland United Kingdom United States 9.0% 4.8% 7.7% 6.2% 6.3% 8.8% 7.0% 6.8% 9.3% 7.7% 9.1% 5.8% 9.9% 6.9% 7.6% 8.7% Source: Dimson, Marsh, and Staunton (2002), Table 4-3. Swiss equities date from 1911. TABLE 3-6 Frequency Distribution of Average Real Equity Returns Return Interval 4.0% to 5.0% 5.0% to 6.0% 6.0% to 7.0% 7.0% to 8.0% 8.0% to 9.0% 9.0% to 10% Absolute Frequency Relative Frequency Cumulative Absolute Frequency 1 1 4 4 2 4 6.25% 6.25% 25.00% 25.00% 12.50% 25.00% 1 2 6 10 12 16 Cumulative Relative Frequency 6.25% 12.50% 37.50% 62.50% 75.00% 100.00% Note: Relative frequencies are rounded to sum to 100%. 4. THE GRAPHIC PRESENTATION OF DATA A graphical display of data allows us to visualize important characteristics quickly. For example, we may see that the distribution is symmetrically shaped, and this finding may influence which probability distribution we use to describe the data. In this section, we discuss the histogram, the frequency polygon, and the cumulative frequency distribution as methods for displaying data graphically. We construct all of these graphic presentations with the information contained in the frequency distribution of the S&P 500 shown in either Table 3-3 or Table 3-4. Chapter 3 Statistical Concepts and Market Returns 73 4.1. The Histogram A histogram is the graphical equivalent of a frequency distribution. • Definition of Histogram. A histogram is a bar chart of data that have been grouped into a frequency distribution. The advantage of the visual display is that we can see quickly where most of the observations lie. To see how a histogram is constructed, look at the return interval 18% ≤ Rt < 20% in Table 3-3. This interval has an absolute frequency of 6. Therefore, we erect a bar or rectangle with a height of 6 over that return interval on the horizontal axis. Continuing with this process for all other return intervals yields a histogram. Figure 3-1 presents the histogram of the annual total return series on the S&P 500 from 1926 to 2002. In the histogram in Figure 3-1, the height of each bar represents the absolute frequency for each return interval. The return interval −10% ≤ Rt < −8% has a frequency of 7 and is represented by the tallest bar in the histogram. Because there are no gaps between the interval limits, there are no gaps between the bars of the histogram. Many of the return intervals have zero frequency; therefore, they have no height in the histogram. Frequency 7 6 5 4 3 2 1 2.0% !34.0 % !3 !2 0.0% 8.0% -! !2 26.0 4.0% % -! !2 22.0 0.0% % -! !1 18.0 6.0% % -! !1 14.0 2.0% % -! 10.0 !8 .0% % ! !4 6 . 0 .0% % -! 0.0% 2.0% -2 4.0% .0% 8.0% 6.0% - 10 12.0 .0% %14.0 16.0 % %18.0 20.0 % %22.0 24.0 % %26.0 28.0 % %30.0 32.0 % %34.0 36.0 % %38.0 40.0 % %42.0 44.0 % %46.0 48.0 % %50.0 52.0 % %54.0 % % 38.0 -! !3 !3 6.0% 0.0% !4 !4 4.0% -! 42.0 % 0 Return Intervals FIGURE 3-1 Histogram of S&P 500 Annual Total Returns: 1926 to 2002 Source: Ibbotson EnCorr Analyzer. 74 Quantitative Investment Analysis Frequency 200 180 160 140 120 100 80 60 40 20 0.0% -! 16.0 !1 % 4.0% -! 12.0 !1 % 0.0% -! 8 . 0% !6 .0% -! 4.0% !2 .0% - 0. 0% 2.0% - 4. 0% 6.0% - 8. 0% 10.0 %12.0 % 14.0 %16.0 % 18.0 %20.0 % 22.0 %24.0 % 26.0 %28.0 % 30.0 %32.0 % 34.0 %36.0 % 38.0 %40.0 % 42.0 %44.0 % !2 %- 8.0% !1 !2 2.0 -! 6.0% !2 !3 0.0% -! 28.0 % 24.0 % 0 Return Intervals FIGURE 3-2 Histogram of S&P 500 Monthly Total Returns: January 1926 to December 2002 Figure 3-2 presents the histogram for the distribution of monthly returns on the S&P 500. Somewhat more symmetrically shaped than the histogram of annual returns shown in Figure 3-1, this histogram also appears more bell-shaped than the distribution of annual returns. 4.2. The Frequency Polygon and the Cumulative Frequency Distribution Two other graphical tools for displaying data are the frequency polygon and the cumulative frequency distribution. To construct a frequency polygon, we plot the midpoint of each interval on the x-axis and the absolute frequency for that interval on the y-axis; we then connect neighboring points with a straight line. Figure 3-3 shows the frequency polygon for the 924 monthly returns for the S&P 500 from January 1926 to December 2002. In Figure 3-3, we have replaced the bars in the histogram with points connected with straight lines. For example, the return interval 0 percent to 2 percent has an absolute frequency of 182. In the frequency polygon, we plot the return-interval midpoint of 1 percent and a frequency of 182. We plot all other points in a similar way.10 This form of visual display adds a degree of continuity to the representation of the distribution. 10 Even though the upper limit on the interval is not a return falling in the interval, we still average it with the lower limit to determine the midpoint. 75 Chapter 3 Statistical Concepts and Market Returns Frequency 200 180 160 140 120 100 80 60 40 20 43% 39% 35% 31% 27% 23% 19% 15% 11% 7% 3% !1% !5% !9% !13% !17% !21% !25% !29% 0 Return Interval Midpoints FIGURE 3-3 Frequency Polygon of S&P 500 Monthly Total Returns: January 1926 to December 2002 Source: Ibbotson Associates. Another form of line graph is the cumulative frequency distribution. Such a graph can plot either the cumulative absolute or cumulative relative frequency against the upper interval limit. The cumulative frequency distribution allows us to see how many or what percent of the observations lie below a certain value. To construct the cumulative frequency distribution, we graph the returns in the fourth or fifth column of Table 3-4 against the upper limit of each return interval. Figure 3-4 presents a graph of the cumulative absolute distribution for the monthly returns on the S&P 500. Notice that the cumulative distribution tends to flatten out when returns are extremely negative or extremely positive. The steep slope in the middle of Figure 3-4 reflects the fact that most of the observations lie in the neighborhood of −2 percent to 6 percent. We can further examine the relationship between the relative frequency and the cumulative relative frequency by looking at the two return intervals reproduced in Table 3-7. The first return interval (0 percent to 2 percent) has a cumulative relative frequency of 58.12 percent. The next return interval (2 percent to 4 percent) has a cumulative relative frequency of 74.68 percent. The change in the cumulative relative frequency as we move from one interval to the next is the next interval’s relative frequency. For instance, as we go from the first return interval (0 percent to 2 percent) to the next return interval (2 percent to 4 percent), the change in the cumulative relative frequency is 74.68% − 58.12% = 16.56%. (Values in the table have been rounded to two decimal places.) The fact that the slope is steep indicates that these frequencies are large. As you can see in the graph of the cumulative distribution, the slope of the curve changes as we move from the first return interval to the last. A fairly small slope for the cumulative distribution for the first few return intervals tells us that these return intervals do not contain many observations. You can go back to the frequency distribution in Table 3-4 and verify that the cumulative absolute frequency is only 23 observations (the 76 Quantitative Investment Analysis Cumulative Frequency 1000 900 800 700 600 500 400 300 200 100 44% 40% 36% 32% 28% 24% 20% 16% 12% 8% 4% 0% !4% !8% !12% !16% !20% !24% !28% 0 Return Interval Upper Limits FIGURE 3-4 Cumulative Absolute Frequency Distribution of S&P 500 Monthly Total Returns: January 1926 to December 2002 Source: Ibbotson Associates. TABLE 3-7 Selected Class Frequencies for the S&P 500 Monthly Returns Return Interval Absolute Frequency Relative Frequency Cumulative Absolute Frequency Cumulative Relative Frequency 0.0% to 2.0% 2.0% to 4.0% 182 153 19.70% 16.56% 537 690 58.12% 74.68% cumulative relative frequency is 2.49 percent) up to the 10th return interval (−12 percent to −10 percent). In essence, the slope of the cumulative absolute distribution at any particular interval is proportional to the number of observations in that interval. 5. MEASURES OF CENTRAL TENDENCY So far, we have discussed methods we can use to organize and present data so that they are more understandable. The frequency distribution of an asset class’s return series, for example, reveals the nature of the risks that investors may encounter in a particular asset class. As an illustration, the histogram for the annual returns on the S&P 500 clearly shows that large positive and negative annual returns are common. Although frequency distributions and histograms provide a convenient way to summarize a series of observations, these methods are just a first step toward describing the data. In this section we discuss the use of quantitative measures that explain characteristics of data. Our focus is on measures of central tendency and other measures of location or location parameters. A measure of central tendency specifies 77 Chapter 3 Statistical Concepts and Market Returns where the data are centered. Measures of central tendency are probably more widely used than any other statistical measure because they can be computed and applied easily. Measures of location include not only measures of central tendency but other measures that illustrate the location or distribution of data. In the following subsections we explain the common measures of central tendency—the arithmetic mean, the median, the mode, the weighted mean, and the geometric mean. We also explain other useful measures of location, including quartiles, quintiles, deciles, and percentiles. 5.1. The Arithmetic Mean Analysts and portfolio managers often want one number that describes a representative possible outcome of an investment decision. The arithmetic mean is by far the most frequently used measure of the middle or center of data. • Definition of Arithmetic Mean. The arithmetic mean is the sum of the observations divided by the number of observations. We can compute the arithmetic mean for both populations and samples, known as the population mean and the sample mean, respectively. 5.1.1. The Population Mean The population mean is the arithmetic mean computed for a population. If we can define a population adequately, then we can calculate the population mean as the arithmetic mean of all the observations or values in the population. For example, analysts examining the fiscal 2002 year-over-year growth in same-store sales of major U.S. wholesale clubs might define the population of interest to include only three companies: BJ’s Wholesale Club (NYSE: BJ), Costco Wholesale Corporation (Nasdaq: COST), and Sam’s Club, part of Wal-Mart Stores (NYSE: WMT).11 As another example, if a portfolio manager’s investment universe (the set of securities she must choose from) is the Nikkei–Dow Jones Average, the relevant population is the 225 shares on the First Section of the Tokyo Stock Exchange that compose the Nikkei. • Population Mean Formula. The population mean, µ, is the arithmetic mean value of a population. For a finite population, the population mean is µ= N ! i=1 N Xi (3-2) where N is the number of observations in the entire population and Xi is the ith observation. The population mean is an example of a parameter. The population mean is unique; that is, a given population has only one mean. To illustrate the calculation, we can take the case of the population mean of current price-to-earnings ratio (P/E) of stocks of U.S. companies running major wholesale clubs as of the beginning of September 2003. As of that date, the current P/Es for BJ, COST, and WMT were 16.73, 22.02, and 29.30, respectively, according to First Call/Thomson Financial. Thus the population mean current P/E on that date was µ = (16.73 + 22.02 + 29.30)/3 = 68.05/3 = 22.68. 11 A wholesale club implements a store format dedicated mostly to bulk sales in warehouse-sized stores to customers who pay membership dues. As of the early 2000s, those three wholesale clubs dominated the segment in the United States. 78 Quantitative Investment Analysis 5.1.2. The Sample Mean The sample mean is the arithmetic mean computed for a sample. Many times we cannot observe every member of a set; instead, we observe a subset or sample of the population. The concept of the mean can be applied to the observations in a sample with a slight change in notation. • Sample Mean Formula. The sample mean or average, X (read ‘‘X-bar’’), is the arithmetic mean value of a sample: n ! Xi X = i=1 n (3-3) where n is the number of observations in the sample. Equation 3-3 tells us to sum the values of the observations (Xi ) and divide the sum by the number of observations. For example, if the sample of P/E multiples contains the values 35, 30, 22, 18, 15, and 12, the sample mean P/E is 132/6 = 22. The sample mean is also called the arithmetic average.12 As we discussed earlier, the sample mean is a statistic (that is, a descriptive measure of a sample). Means can be computed for individual units or over time. For instance, the sample might be the 2003 return on equity (ROE) for the 300 companies in the Financial Times Stock Exchange (FTSE) Eurotop 300, an index of Europe’s 300 largest companies. In this case, we calculate mean ROE in 2003 as an average across 300 individual units. When we examine the characteristics of some units at a specific point in time (such as ROE for the FTSE Eurotop 300), we are examining cross-sectional data. The mean of these observations is called a cross-sectional mean. On the other hand, if our sample consists of the historical monthly returns on the FTSE Eurotop 300 for the past five years, then we have time-series data. The mean of these observations is called a time-series mean. We will examine specialized statistical methods related to the behavior of time series in the chapter on times-series analysis. Next, we show an example of finding the sample mean return for equities in 16 European countries for 2002. In this case, the mean is cross-sectional because we are averaging individual country returns. EXAMPLE 3-3 Calculating a Cross-Sectional Mean The MSCI EAFE (Europe, Australasia, and Far East) Index is a free float-adjusted market capitalization index designed to measure developed-market equity performance excluding the United States and Canada.13 As of the end of 2002, the EAFE consisted of 21 developed market country indexes, including indexes for 16 European markets, 2 12 Statisticians prefer the term ‘‘mean’’ to ‘‘average.’’ Some writers refer to all measures of central tendency (including the median and mode) as averages. The term ‘‘mean’’ avoids any possibility of confusion. 13 The term ‘‘free float-adjusted’’ means that the weights of companies in the index reflect the value of the shares actually available for investment. 79 Chapter 3 Statistical Concepts and Market Returns Australasian markets (Australia and New Zealand), and 3 Far Eastern markets (Hong Kong, Japan, and Singapore). Suppose we are interested in the local currency performance of the 16 European markets in the EAFE in 2002, a severe bear market year. We want to find the sample mean total return for 2002 across these 16 markets. The return series reported in Table 3-8 are in local currency (that is, returns are for investors living in the country). Because this return is not stated in any single investor’s home currency, it is not a return any single investor would earn. Rather, it is an average of returns in 16 local currencies. TABLE 3-8 Total Returns for European Equity Markets, 2002 Market Total Return in Local Currency Austria Belgium Denmark Finland France Germany Greece Ireland Italy Netherlands Norway Portugal Spain Sweden Switzerland United Kingdom −2.97% −29.71% −29.67% −41.65% −33.99% −44.05% −39.06% −38.97% −23.64% −34.27% −29.73% −28.29% −29.47% −43.07% −25.84% −25.66% Source: www.mscidata.com. Using the data in Table 3-8, calculate the sample mean return for the 16 equity markets in 2002. Solution: The calculation applies Equation 3-3 to the returns in Table 3-8: (−2.97 − 29.71 − 29.67 − 41.65 − 33.99 − 44.05 − 39.06 − 38.97 − 23.64 − 34.27 − 29.73 − 28.29 − 29.47 − 43.07 − 25.84 − 25.66)/16 = −500.04/16 = −31.25 percent. In Example 3-3, we can verify that seven markets had returns less than the mean and nine had returns that were greater. We should not expect any of the actual observations to equal the mean, because sample means provide only a summary of the data being analyzed. As an analyst, you will often need to find a few numbers that describe the characteristics of the distribution. The mean is generally the statistic that you will use as a measure of the typical outcome for a distribution. You can then use the mean to compare the performance of 80 1 Quantitative Investment Analysis 2 3 4 5 6 7 9 10 11 12 Fulcrum When the fulcrum is placed at 8, the bar is perfectly balanced. FIGURE 3-5 Center of Gravity Analogy for the Arithmetic Mean two different markets. For example, you might be interested in comparing the stock market performance of investments in Pacific Rim countries with investments in European countries. You can use the mean returns in these markets to compare investment results. 5.1.3. Properties of the Arithmetic Mean The arithmetic mean can be likened to the center of gravity of an object. Figure 3-5 expresses this analogy graphically by plotting nine hypothetical observations on a bar. The nine observations are 2, 4, 4, 6, 10, 10, 12, 12, and 12; the arithmetic mean is 72/9 = 8. The observations are plotted on the bar with various heights based on their frequency (that is, 2 is one unit high, 4 is two units high, and so on). When the bar is placed on a fulcrum, it balances only when the fulcrum is located at the point on the scale that corresponds to the arithmetic mean. As analysts, we often use the mean return as a measure of the typical outcome for an asset. As in the example above, however, some outcomes are above the mean and some are below it. We can calculate the distance between the mean and each outcome and call it a deviation. Mathematically, it is always true that the sum of the deviations around the mean equals 0. We can see this by using the definition of the arithmetic mean shown in Equation 3-3, multiplying n ! both sides of the equation by n: nX = Xi . The sum of the deviations from the mean can thus be calculated as follows: n ! i=1 (Xi − X ) = i=1 n ! i=1 Xi − n ! i=1 X = n ! i=1 Xi − nX = 0 Deviations from the arithmetic mean are important information because they indicate risk. The concept of deviations around the mean forms the foundation for the more complex concepts of variance, skewness, and kurtosis, which we will discuss later in this chapter. An advantage of the arithmetic mean over two other measures of central tendency, the median and mode, is that the mean uses all the information about the size and magnitude of the observations. The mean is also easy to work with mathematically. A property and potential drawback of the arithmetic mean is its sensitivity to extreme values. Because all observations are used to compute the mean, the arithmetic mean can be pulled sharply upward or downward by extremely large or small observations, respectively. For example, suppose we compute the arithmetic mean of the following seven numbers: 1, 2, 3, 4, 5, 6, and 1,000. The mean is 1,021/7 = 145.86 or approximately 146. Because the magnitude of the mean, 146, is so much larger than that of the bulk of the observations (the first six), we might question how well it represents the location of the data. In practice, although an extreme value or outlier in a financial dataset may only represent a rare value in Chapter 3 Statistical Concepts and Market Returns 81 the population, it may also reflect an error in recording the value of an observation, or an observation generated from a different population from that producing the other observations in the sample. In the latter two cases in particular, the arithmetic mean could be misleading. Perhaps the most common approach in such cases is to report the median in place of or in addition to the mean.14 We discuss the median next. 5.2. The Median A second important measure of central tendency is the median. • Definition of Median. The median is the value of the middle item of a set of items that has been sorted into ascending or descending order. In an odd-numbered sample of n items, the median occupies the (n + 1)/2 position. In an even-numbered sample, we define the median as the mean of the values of items occupying the n/2 and (n + 2)/2 positions (the two middle items).15 Earlier we gave the current P/Es of three wholesale clubs as 16.73, 22.02, and 29.30. With an odd number of observations (n = 3), the median occupies the (n + 1)/2 = 4/2 = 2nd position. The median P/E was 22.02. The P/E value of 22.02 is the ‘‘middlemost’’ observation: One lies above it, and one lies below it. Whether we use the calculation for an even- or oddnumbered sample, an equal number of observations lie above and below the median. A distribution has only one median. A potential advantage of the median is that, unlike the mean, extreme values do not affect it. The median, however, does not use all the information about the size and magnitude of the observations; it focuses only on the relative position of the ranked observations. Calculating the median is also more complex; to do so, we need to order the observations from smallest to largest, determine whether the sample size is even or odd, and, on that basis, apply one of two calculations. Mathematicians express this disadvantage by saying that the median is less mathematically tractable than the mean. To demonstrate finding the median, we use the data from Example 3-3, reproduced in Table 3-9 in ascending order, of the 2002 total return for European equities. Because this sample has 16 observations, the median is the mean of the values in the sorted array that occupy the 16/2 = 8th and 18/2 = 9th positions. Norway’s return occupies the eighth position with a return of −29.73 percent, and Belgium’s return occupies the ninth 14 Other approaches to handling extreme values involve variations of the arithmetic mean. The trimmed mean is computed by excluding a stated small percentage of the lowest and highest values and then computing an arithmetic mean of the remaining values. For example, a 5 percent trimmed mean discards the lowest 2.5 percent and the largest 2.5 percent of values and computes the mean of the remaining 95 percent of values. A trimmed mean is used in sports competitions when judges’ lowest and highest scores are discarded in computing a contestant’s score. A Winsorized mean assigns a stated percent of the lowest values equal to one specified low value, and a stated percent of the highest values equal to one specified high value, then computes a mean from the restated data. For example, a 95 percent Winsorized mean sets the bottom 2.5 percent of values equal to the 2.5th percentile value and the upper 2.5 percent of values equal to the 97.5th percentile value. (Percentile values are defined later.) 15 The notation M is occasionally used for the median. Just as for the mean, we may distinguish between d a population median and a sample median. With the understanding that a population median divides a population in half while a sample median divides a sample in half, we follow general usage in using the term ‘‘median’’ without qualification, for the sake of brevity. 82 Quantitative Investment Analysis TABLE 3-9 Total Returns for European Equity Markets, 2002 (in ascending order) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Market Germany Sweden Finland Greece Ireland Netherlands France Norway Belgium Denmark Spain Portugal Switzerland United Kingdom Italy Austria Total Return in Local Currency −44.05% −43.07% −41.65% −39.06% −38.97% −34.27% −33.99% −29.73% −29.71% −29.67% −29.47% −28.29% −25.84% −25.66% −23.64% −2.97% Source: www.mscidata.com. position with a return of −29.71 percent. The median, as the mean of these two returns, is (−29.73 − 29.71)/2 = −29.72 percent. Note that the median is not influenced by extremely large or small outcomes. Had Germany’s total return been a much lower value or Austria’s total return a much larger value, the median would not have changed. Using a context that arises often in practice, Example 3-4 shows how to use the mean and median in a sample with extreme values. EXAMPLE 3-4 Median and Arithmetic Mean: The Case of the Price–Earnings Ratio Suppose a client asks you for a valuation analysis on the seven-stock U.S. common stock portfolio given in Table 3-10. The stocks are equally weighted in the portfolio. One valuation measure that you use is P/E, the ratio of share price to earnings per share (EPS). Many variations exist for the denominator in the P/E, but you are examining P/E defined as current price divided by the current mean of all analysts’ EPS estimates for the company for the current fiscal year (‘‘Consensus Current EPS’’ in the table).16 The values in Table 3-10 are as of 11 September 2003. For comparison purposes, the consensus current P/E on the S&P 500 was 23.63 at that time. Using the data in Table 3-10, address the following: 16 For more information on price multiples, see Stowe, Robinson, Pinto, and McLeavey (2002). 83 Chapter 3 Statistical Concepts and Market Returns 1. Calculate the arithmetic mean P/E. 2. Calculate the median P/E. 3. Evaluate the mean and median P/Es as measures of central tendency for the above portfolio. TABLE 3-10 P/Es for a Client Portfolio Stock Exponent Inc. (Nasdaq: EXPO) Express Scripts (Nasdaq: ESRX) General Dynamics (NYSE: GD) Limited Brands (NYSE: LTD) Merant plc (Nasdaq: MRNT) Microsoft Corporation (Nasdaq: MSFT) O’Reilly Automotive, Inc. (Nasdaq: ORLY) Consensus Current EPS Consensus Current P/E 1.23 3.19 4.95 1.06 0.03 1.11 1.84 13.68 19.07 17.56 15.60 443.33 25.61 21.01 Source: First Call/Thomson Financial. Solution to 1: The mean P/E is (13.68 + 19.07 + 17.56 + 15.60 + 443.33 + 25.61 + 21.01)/7 = 555.86/7 = 79.41. Solution to 2: The P/Es listed in ascending order are: 13.68 15.60 17.56 19.07 21.01 25.61 443.33 The sample has an odd number of observations with n = 7, so the median occupies the (n + 1)/2 = 8/2 = 4th position in the sorted list. Therefore, the median P/E is 19.07. Solution to 3: Merant’s P/E of approximately 443 tremendously influences the value of the portfolio’s arithmetic mean P/E. The mean P/E of 79 is much larger than the P/E of six of the seven stocks in the portfolio. The mean P/E also misleadingly suggests an orientation to stocks with high P/Es. The mean P/E of the stocks excluding Merant, or excluding the largest- and smallest-P/E stocks (Merant and Exponent), is below the S&P 500’s P/E of 23.63. The median P/E of 19.07 appears to better represent the central tendency of the P/Es. It frequently happens that when a company’s EPS is close to zero—at a low point in the business cycle, for example—its P/E is extremely high. The high P/E in those circumstances reflects an anticipated future recovery of earnings. Extreme P/E values need to be investigated and handled with care. For reasons related to this example, analysts often use the median of price multiples to characterize the valuation of industry groups. 84 Quantitative Investment Analysis 5.3. The Mode The third important measure of central tendency is the mode. • Definition of Mode. The mode is the most frequently occurring value in a distribution.17 A distribution can have more than one mode or even no mode. When a distribution has one most frequently occurring value, the distribution is said to be unimodal. If a distribution has two most frequently occurring values, then it has two modes, and we say it is bimodal. If the distribution has three most frequently occurring values, then it is trimodal. When all the values in a data set are different, the distribution has no mode because no value occurs more frequently than any other value. Stock return data and other data from continuous distributions may not have a modal outcome. When such data are grouped into intervals, however, we often find an interval (possibly more than one) with the highest frequency: the modal interval (or intervals). For example, the frequency distribution for the monthly returns on the S&P 500 has a modal interval of 0 percent to 2 percent, as shown in Figure 3-2; this return interval has 182 observations out of a total of 924. The modal interval always has the highest bar in the histogram. The mode is the only measure of central tendency that can be used with nominal data. When we categorize mutual funds into different styles and assign a number to each style, the mode of these categorized data is the most frequent mutual fund style. EXAMPLE 3-5 Calculating a Mode Table 3-11 gives the credit ratings on senior unsecured debt as of September 2002 of nine U.S. department stores rated by Moody’s Investors Service. In descending order of credit quality (increasing expected probability of default), Moody’s ratings are Aaa, Aa1, Aa2, Aa3, A1, A2, A3, Baa1, Baa2, Baa3, Ba1, Ba2, Ba3, B1, B2, B3, Caa, Ca, and C.18 Using the data in Table 3-11, address the following concerning the senior unsecured debt of U.S. department stores: 1. State the modal credit rating. 2. State the median credit rating. Solution to 1: The group of companies represents seven distinct credit ratings, ranging from A2 to B1. To make our task easy, we first organize the ratings into a frequency distribution. All credit ratings have a frequency of 1 except for Baa1, which has a frequency of 3. Therefore, the modal credit rating of U.S. department stores as of the date of the 17 The notation Mo is occasionally used for the mode. Just as for the mean and the median, we may distinguish between a population mode and a sample mode. With the understanding that a population mode is the value with the greatest probability of occurrence, while a sample mode is the most frequently occurring value in the sample, we follow general usage in using the term ‘‘mode’’ without qualification, for the sake of brevity. 18 For more information on credit risk and credit ratings, see Fabozzi (2004a). 85 Chapter 3 Statistical Concepts and Market Returns Moody’s report was Baa1. Moody’s considers bonds rated Baa1 to be medium-grade obligations—they are neither highly protected nor poorly secured. TABLE 3-11 Senior Unsecured Debt Ratings: U.S. Department Stores, September 2002 Company Credit Rating Dillards, Inc. Federated Department Stores, Inc. Kohl’s Corporation May’s Department Stores Company Neiman Marcus Group, Inc. Nordstom, Inc. Penney, JC, Company, Inc. Saks Incorporated Sears, Roebuck and Co. Ba3 Baa1 A3 A2 Baa2 Baa1 Ba2 B1 Baa1 Source: Moody’s Investors Service. Solution to 2: For the group, n = 9, an odd number. The group’s median occupies the (n + 1)/2 = 10/2 = 5th position. We see from Table 3-12 that Baa1 occupies the fifth position. Therefore, the median credit rating as of September 2002 was Baa1. TABLE 3-12 Senior Unsecured Debt Ratings: U.S. Department Stores, Distribution of Credit Ratings Credit Rating A2 A3 Baa1 Baa2 Ba2 Ba3 B1 Frequency 1 1 3 1 1 1 1 5.4. Other Concepts of Mean Earlier we explained the arithmetic mean, which is a fundamental concept for describing the central tendency of data. Other concepts of mean are very important in investments, however. In the following, we discuss such concepts. 5.4.1. The Weighted Mean The concept of weighted mean arises repeatedly in portfolio analysis. In the arithmetic mean, all observations are equally weighted by the factor 1/n (or 1/N ). In working with portfolios, we need the more general concept of weighted mean to allow different weights on different observations. To illustrate the weighted mean concept, an investment manager with $100 million to invest might allocate $70 million to equities and $30 million to bonds. The portfolio has a weight of 0.70 on stocks and 0.30 on bonds. How do we calculate the return on this portfolio? 86 Quantitative Investment Analysis TABLE 3-13 Total Returns for Canadian Equities and Bonds, 1998–2002 Year Equities Bonds 1998 1999 2000 2001 2002 −1.6% 31.7% 7.4% −12.6% −12.4% 9.1% −1.1% 10.3% 8.0% 8.7% Source: www.fidelity.ca and www.money.msn.ca. The portfolio’s return clearly involves an averaging of the returns on the stock and bond investments. The mean that we compute, however, must reflect the fact that stocks have a 70 percent weight in the portfolio and bonds have a 30 percent weight. The way to reflect this weighting is to multiply the return on the stock investment by 0.70 and the return on the bond investment by 0.30, then sum the two results. This sum is an example of a weighted mean. It would be incorrect to take an arithmetic mean of the return on the stock and bond investments, equally weighting the returns on the two asset classes. Consider a portfolio invested in Canadian stocks and bonds in which the stock component is indexed on the S&P/TSX Composite Index and the bond component is indexed on the RBC Capital Markets Canadian Bond Market Index. These indexes represent the broad Canadian equity and bond markets, respectively. The portfolio manager allocates 60 percent of the portfolio to Canadian stocks and 40 percent to Canadian bonds. Table 3-13 presents total returns for these indexes for 1998 to 2002. • Weighted Mean Formula. The weighted mean X w (read ‘‘X -bar sub-w’’) for a set of observations X1 , X2 , . . . , Xn with corresponding weights of w1 , w2 , . . . , wn is computed as Xw = n ! wi Xi (3-4) i=1 where the sum of the weights equals 1; that is, ! i wi = 1. In the context of portfolios, a positive weight represents an asset held long and a negative weight represents an asset held short.19 The return on the portfolio under consideration is the weighted average of the return on Canadian stocks and Canadian bonds (the weight on stocks is 0.60; that on bonds is 0.40). Apart from expenses, if the portfolio tracks the indexes perfectly, we find, using Equation 3-4, that 19 The formula for the weighted mean can be compared to the formula for the arithmetic mean. For a set of observations X1 , X2 , . . . , Xn , let the weights w1 , w2 , . . . , wn all equal 1/n. Under this assumption, the n ! Xi . This is the formula for the arithmetic mean. Therefore, formula for the weighted mean is (1/n) i=1 the arithmetic mean is a special case of the weighted mean in which all the weights are equal. Chapter 3 Statistical Concepts and Market Returns 87 Portfolio return for 1998 = wstock Rstock + wbonds Rbonds = 0.60(−1.6%) + 0.40(9.1%) = 2.7% It should be clear that the correct mean to compute in this example is the weighted mean and not the arithmetic mean. If we had computed the arithmetic mean for 1998, we would have calculated a return equal to 1/2(−1.6%) + 1/2(9.1%) = (−1.6% + 9.1%)/2 = 3.8%. Given that the portfolio manager invested 60 percent in stocks and 40 percent in bonds, the arithmetic mean would underweight the investment in stocks and overweight the investment in bonds, resulting in a number for portfolio return that is too high by 1.1 percentage points (3.8% − 2.7%). Now suppose that the portfolio manager maintains constant weights of 60 percent in stocks and 40 percent in bonds for all five years. This method is called a constant-proportions strategy. Because value is price multiplied by quantity, price fluctuation causes portfolio weights to change. As a result, the constant-proportions strategy requires rebalancing to restore the weights in stocks and bonds to their target levels. Assuming that the portfolio manager is able to accomplish the necessary rebalancing, we can compute the portfolio returns in 1999, 2000, 2001, and 2002 with Equation 3-4 as follows: Portfolio return for 1999 = 0.60(31.7) + 0.40(−1.1) = 18.6% Portfolio return for 2000 = 0.60(7.4) + 0.40(10.3) = 8.6% Portfolio return for 2001 = 0.60(−12.6) + 0.40(8.0) = −4.4% Portfolio return for 2002 = 0.60(−12.4) + 0.40(8.7) = −4.0% We can now find the time-series mean of the returns for 1998 through 2002 using Equation 3-3 for the arithmetic mean. The time-series mean total return for the portfolio is (2.7 + 18.6 + 8.6 − 4.4 − 4.0)/5 = 21.5/5 = 4.3 percent. Instead of calculating the portfolio time-series mean return from portfolio annual returns, we can calculate the arithmetic mean bond and stock returns for the five years and then apply the portfolio weights of 0.60 and 0.40, respectively, to those values. The mean stock return is (−1.6 + 31.7 + 7.4 − 12.6 − 12.4)/5 = 12.5/5 = 2.5 percent. The mean bond return is (9.1 − 1.1 + 10.3 + 8.0 + 8.7)/5 = 35.0/5 = 7.0 percent. Therefore, the mean total return for the portfolio is 0.60(2.5) + 0.40(7.0) = 4.3 percent, which agrees with our previous calculation. EXAMPLE 3-6 Portfolio Return as a Weighted Mean Table 3-14 gives information on the estimated average asset allocation of Canadian pension funds as well as four-year asset class returns.20 20 In Table 3-14, equities are represented by the S&P/TSX Composite Index, U.S. equities by the S&P 500, international (non–North American) equities by the MSCI EAFE Index, bonds by the Scotia Capital Markets Universe Bond Index, mortgages by the Scotia Capital Markets Mortgage Index, real estate by the Standard Life Investments pooled real estate fund, and cash and equivalents by 91-day T-bills. 88 Quantitative Investment Analysis TABLE 3-14 Asset Allocation for Average Canadian Pension Fund as of 31 March 2003 Asset Class Asset Allocation (Weight) Asset Class Return (%) 34.6 10.8 6.4 34.0 1.3 4.5 8.4 0.6 −9.3 −10.5 6.0 9.0 10.2 4.2 Equities U.S. equities International equities Bonds Mortgages Real estate Cash and equivalents Source: Standard Life Investments, Inc. Using the information in Table 3-14, calculate the mean return earned by the average Canadian pension fund over the four years ending 31 March 2003. Solution: Converting the percent asset allocation to decimal form, we find the mean return as a weighted average of the asset class returns. We have Mean portfolio return = 0.346(0.6%) + 0.108(−9.3%) + 0.064(−10.5%) +0.340(6.0%) + 0.013(9.0%) + 0.045(10.2%) +0.084(4.2%) = 0.208% − 1.004% − 0.672% + 2.040% +0.117% + 0.459% + 0.353% = 1.5% The previous examples illustrate the general principle that a portfolio return is a weighted sum. Specifically, a portfolio’s return is the weighted average of the returns on the assets in the portfolio; the weight applied to each asset’s return is the fraction of the portfolio invested in that asset. Market indexes are computed as weighted averages. For market-capitalization indexes such as the CAC-40 in France or the S&P 500 in the United States, each included stock receives a weight corresponding to its outstanding market value divided by the total market value of all stocks in the index. Our illustrations of weighted mean use past data, but they might just as well use forward-looking data. When we take a weighted average of forward-looking data, the weighted mean is called expected value. Suppose we make one forecast for the year-end level of the S&P 500 assuming economic expansion and another forecast for the year-end level of the S&P 500 assuming economic contraction. If we multiply the first forecast by the probability of expansion and the second forecast by the probability of contraction and then add these weighted forecasts, we are calculating the expected value of the S&P 500 at year-end. If we 89 Chapter 3 Statistical Concepts and Market Returns take a weighted average of possible future returns on the S&P 500, we are computing the S&P 500’s expected return. The probabilities must sum to 1, satisfying the condition on the weights in the expression for weighted mean, Equation 3-4. 5.4.2. The Geometric Mean The geometric mean is most frequently used to average rates of change over time or to compute the growth rate of a variable. In investments, we frequently use the geometric mean to average a time series of rates of return on an asset or a portfolio or to compute the growth rate of a financial variable such as earnings or sales. In the chapter on the time value of money, for instance, we computed a sales growth rate (Example 1-17). That growth rate was a geometric mean. Because of the subject’s importance, in a later section we will return to the use of the geometric mean and offer practical perspectives on its use. The geometric mean is defined by the following formula. • Geometric Mean Formula. The geometric mean, G, of a set of observations X1 , X2 , . . . , Xn is " G = n X1 X2 X3 . . . Xn (3-5) with Xi ≥ 0 for i = 1, 2, . . . , n. Equation 3-5 has a solution, and the geometric mean exists, only if the product under the radical sign is non-negative. We impose the restriction that all the observations Xi in Equation 3-5 are greater than or equal to zero. We can solve for the geometric mean using Equation 3-5 directly with any calculator that has an exponentiation key (on most calculators, yx ). We can also solve for the geometric mean using natural logarithms. Equation 3-5 can also be stated as ln G = 1 ln(X1 X2 X3 . . . Xn ) n or as ln G = n ! ln Xi i=1 n When we have computed ln G, then G = eln G (on most calculators, the key for this step is e x ). Risky assets can have negative returns up to −100 percent (if their price falls to zero), so we must take some care in defining the relevant variables to average in computing a geometric mean. We cannot just use the product of the returns for the sample and then take the nth root because the returns for any period could be negative. We must redefine the returns to make them positive. We do this by adding 1.0 to the returns expressed as decimals. The term (1 + Rt ) represents the year-ending value relative to an initial unit of investment at the beginning of the year. As long as we use (1 + Rt ), the observations will never be negative because the biggest negative return is −100 percent. The result is the geometric mean of 1 + Rt ; by then subtracting 1.0 from this result, we obtain the geometric mean of the individual returns Rt . For example, the returns on Canadian stocks as represented by the S&P/TSX Composite Index during the 1998–2002 period were given in Table 3-13 as −0.016, 0.317, 0.074, −0.126, and −0.124, putting the returns into decimal form. Adding 1.0 to those returns produces 0.9840, 1.317, 1.074, 0.874, and 0.876. Using Equation 3-5 90 Quantitative Investment Analysis √ √ we have 5 (0.9840)(1.317)(1.074)(0.874)(0.876) = 5 1.065616 = 1.012792. This number is 1 plus the geometric mean rate of return. Subtracting 1.0 from this result, we have 1.012792 − 1.0 = 0.012792 or approximately 1.3 percent. The geometric mean return for Canadian stocks during the 1998–2002 period was 1.3 percent. An equation that summarizes the calculation of the geometric mean return, RG , is a slightly modified version of Equation 3-5 in which the Xi represent ‘‘1 + return in decimal form.’’ Because geometric mean returns use time series, we use a subscript t indexing time as well. " 1 + RG = T (1 + R1 )(1 + R2 ) . . . (1 + RT ) 1 + RG = #T $ t=1 %1 T (1 + Rt ) which leads to the following formula: • Geometric Mean Return Formula. Given a time series of holding period returns Rt , t = 1, 2, . . . , T , the geometric mean return over the time period spanned by the returns R1 through RT is #T %1 T $ (1 + Rt ) −1 (3-6) RG = t=1 We can use Equation 3-6 to solve for the geometric mean return for any return data series. Geometric mean returns are also referred to as compound returns. If the returns being averaged in Equation 3-6 have a monthly frequency, for example, we may call the geometric mean monthly return the compound monthly return. The next example illustrates the computation of the geometric mean while contrasting the geometric and arithmetic means. EXAMPLE 3-7 Geometric and Arithmetic Mean Returns (1) As a mutual fund analyst, you are examining, as of early 2003, the most recent five years of total returns for two U.S. large-cap value equity mutual funds. Based on the data in Table 3-15, address the following: 1. Calculate the geometric mean return of SLASX. 2. Calculate the arithmetic mean return of SLASX and contrast it to the fund’s geometric mean return. 3. Calculate the geometric mean return of PRFDX. 4. Calculate the arithmetic mean return of PRFDX and contrast it to the fund’s geometric mean return. Solution to 1: Converting the returns on SLASX to decimal form and adding 1.0 to each return produces 1.162, 1.203, 1.093, 0.889, and 0.830. We use Equation 3-6 to find SLASX’s geometric mean return: 91 Chapter 3 Statistical Concepts and Market Returns TABLE 3-15 Total Returns for Two Mutual Funds, 1998–2002 Year Selected American Shares (SLASX) T. Rowe Price Equity Income (PRFDX) 16.2% 20.3% 9.3% −11.1% −17.0% 9.2% 3.8% 13.1% 1.6% −13.0% 1998 1999 2000 2001 2002 Source: American Association of Individual Investors (AAII). " 5 (1.162)(1.203)(1.093)(0.889)(0.830) − 1 √ 5 = 1.127384 − 1 = 1.024270 − 1 = 0.024270 RG = = 2.43% Solution to 2: For SLASX, R = (16.2 + 20.3 + 9.3 − 11.1 − 17.0)/5 = 17.7/5 = 3.54%. The arithmetic mean return for SLASX exceeds the geometric mean return by 3.54 − 2.43 = 1.11% or 111 basis points. Solution to 3: Converting the returns on PRFDX to decimal form and adding 1.0 to each return produces 1.092, 1.038, 1.131, 1.016, and 0.870. We use Equation 3-6 to find PRFDX’s geometric mean return: " 5 (1.092)(1.038)(1.131)(1.016)(0.870) − 1 √ 5 = 1.133171 − 1 = 1.025319 − 1 = 0.025319 RG = = 2.53% Solution to 4: For PRFDX, R = (9.2 + 3.8 + 13.1 + 1.6 − 13.0)/5 = 14.7/5 = 2.94%. The arithmetic mean for PRFDX exceeds the geometric mean return by 2.94 − 2.53 = 0.41% or 41 basis points. The table below summarizes the findings. TABLE 3-16 Mutual Fund Arithmetic and Geometric Mean Returns: Summary of Findings Fund SLASX PRFDX Arithmetic Mean Geometric Mean 3.54% 2.94% 2.43% 2.53% 92 Quantitative Investment Analysis In Example 3-7, for both mutual funds, the geometric mean return was less than the arithmetic mean return. In fact, the geometric mean is always less than or equal to the arithmetic mean.21 The only time that the two means will be equal is when there is no variability in the observations—that is, when all the observations in the series are the same.22 In Example 3-7, there was variability in the funds’ returns; thus for both funds, the geometric mean was strictly less than the arithmetic mean. In general, the difference between the arithmetic and geometric means increases with the variability in the period-by-period observations.23 This relationship is also illustrated by Example 3-7. Even casual inspection reveals that the returns of SLASX are more variable than those of PRFDX, and consequently, the spread between the arithmetic and geometric mean returns is larger for SLASX (111 basis points) than for PRFDX (41 basis points).24 The arithmetic and geometric mean also rank the two funds differently. Although SLASX has the higher arithmetic mean return, PRFDX has the higher geometric mean return. How should the analyst interpret this result? The geometric mean return represents the growth rate or compound rate of return on an investment. One dollar invested in SLASX at the beginning of 1998 would have grown to (1.162)(1.203)(1.093)(0.889)(0.830) = $1.127, which is equal to 1 plus the geometric mean return compounded over five periods: (1.0243)5 = $1.127, confirming that the geometric mean is the compound rate of return. For PRFDX, one dollar would have grown to a larger amount, (1.092)(1.038)(1.131)(1.016)(0.870) = $1.133, equal to (1.0253)5 . With its focus on the profitability of an investment over a multiperiod horizon, the geometric mean is of key interest to investors. The arithmetic mean return, focusing on average single-period performance, is also of interest. Both arithmetic and geometric means have a role to play in investment management, and both are often reported for return series. Example 3-8 highlights these points in a simple context. EXAMPLE 3-8 Geometric and Arithmetic Mean Returns (2) A hypothetical investment in a single stock initially costs ¤100. One year later, the stock is trading at ¤200. At the end of the second year, the stock price falls back to the original purchase price of ¤100. No dividends are paid during the two-year period. Calculate the arithmetic and geometric mean annual returns. Solution: First, we need to find the Year 1 and Year 2 annual returns with Equation 3-1. Return in Year 1 = 200/100 − 1 = 100% 21 This statement can be proved using Jensen’s inequality that the average value of a function is less than or equal to the function evaluated at the mean if the function is concave from below—the case for ln(X ). 22 For instance, suppose the return for each of the three years is 10 percent. The arithmetic mean is 10 percent. To find the geometric mean, we first express the returns as (1 + Rt ) and then find the geometric mean: [(1.10)(1.10)(1.10)]1/3 − 1.0 = 10 percent. The two means are the same. 23 We will soon introduce standard deviation as a measure of variability. Holding the arithmetic mean return constant, the geometric mean return decreases for an increase in standard deviation. 24 We will introduce formal measures of variability later. But note, for example, the 20.4 percentage point swing in returns between 2000 and 2001 for SLASX versus the 11.5 percentage point for PRFDX. Chapter 3 Statistical Concepts and Market Returns 93 Return in Year 2 = 100/200 − 1 = −50% The arithmetic mean of the annual returns is (100% − 50%)/2 = 25%. Before we find the geometric mean, we must convert the percentage rates of return to (1 + Rt ). After this adjustment, the geometric mean from Equation 3-6 is √ 2.0 × 0.50 − 1 = 0%. The geometric mean return of 0 percent accurately reflects that the ending value of the investment in Year 2 equals the starting value in Year 1. The compound rate of return on the investment is 0 percent. The arithmetic mean return reflects the average of the one-year returns. 5.4.3. The Harmonic Mean The arithmetic mean, the weighted mean, and the geometric mean are the most frequently used concepts of mean in investments. A fourth concept, the harmonic mean, X H , is appropriate in a limited number of applications.25 • Harmonic Mean Formula. The harmonic mean of a set of observations X1 , X2 , . . . , Xn is n ! X H = n/ (1/Xi ) (3-7) i=1 with Xi > 0 for i = 1, 2, . . . , n. The harmonic mean is the value obtained by summing the reciprocals of the observations—terms of the form 1/Xi —then averaging that sum by dividing it by the number of observations n, and, finally, taking the reciprocal of the average. The harmonic mean may be viewed as a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. The harmonic mean is a relatively specialized concept of the mean that is appropriate when averaging ratios (‘‘amount per unit’’) when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units. The concept is best explained through an illustration. A well-known application arises in the investment strategy known as cost averaging, which involves the periodic investment of a fixed amount of money. In this application, the ratios we are averaging are prices per share at purchases dates, and we are applying those prices to a constant amount of money to yield a variable number of shares. Suppose an investor purchases ¤1,000 of a security each month for n = 2 months. The share prices are ¤10 and ¤15 at the two purchase dates. What is the average price paid for the security? In this example, in the first month we purchase ¤1,000/¤10 = 100 shares, and in the second month we purchase ¤1,000/¤15 = 66.67, or 166.67 shares in total. Dividing the total euro amount invested, ¤2,000, by the total number of shares purchased, 166.67, gives an average price paid of ¤2,000/166.67 = ¤12. The average price paid is in fact the harmonic mean of the asset’s prices at the purchase dates. Using Equation 3-7, the harmonic mean 25 The terminology ‘‘harmonic’’ arises from its use relative to a type of series involving reciprocals known as a harmonic series. 94 Quantitative Investment Analysis price is 2/[(1/10) + (1/15)] = ¤12. The value ¤12 is less than the arithmetic mean purchase price (¤10 + ¤15)/2 = ¤12.5. However, we could find the correct value of ¤12 using the weighted mean formula, where the weights on the purchase prices equal the shares purchased at a given price as a proportion of the total shares purchased. In our example, the calculation would be (100/166.67)¤10.00 + (66.67/166.67)¤15.00 = ¤12. If we had invested varying amounts of money at each date, we could not use the harmonic mean formula. We could, however, still use the weighted mean formula in a manner similar to that just described. A mathematical fact concerning the harmonic, geometric, and arithmetic means is that unless all the observations in a dataset have the same value, the harmonic mean is less than the geometric mean, which in turn is less than the arithmetic mean. In the illustration given, the harmonic mean price was indeed less than the arithmetic mean price. 6. OTHER MEASURES OF LOCATION: QUANTILES Having discussed measures of central tendency, we now examine an approach to describing the location of data that involves identifying values at or below which specified proportions of the data lie. For example, establishing that 25, 50, and 75 percent of the annual returns on a portfolio are at or below the values −0.05, 0.16, and 0.25, respectively, provides concise information about the distribution of portfolio returns. Statisticians use the word quantile (or fractile) as the most general term for a value at or below which a stated fraction of the data lies. In the following, we describe the most commonly used quantiles—quartiles, quintiles, deciles, and percentiles—and their application in investments. 6.1. Quartiles, Quintiles, Deciles, and Percentiles We know that the median divides a distribution in half. We can define other dividing lines that split the distribution into smaller sizes. Quartiles divide the distribution into quarters, quintiles into fifths, deciles into tenths, and percentiles into hundredths. Given a set of observations, the yth percentile is the value at or below which y percent of observations lie. Percentiles are used frequently, and the other measures can be defined with respect to them. For example, the first quartile (Q 1 ) divides a distribution such that 25 percent of the observations lie at or below it; therefore, the first quartile is also the 25th percentile. The second quartile (Q 2 ) represents the 50th percentile, and the third quartile (Q 3 ) represents the 75th percentile because 75 percent of the observations lie at or below it. When dealing with actual data, we often find that we need to approximate the value of a percentile. For example, if we are interested in the value of the 75th percentile, we may find that no observation divides the sample such that exactly 75 percent of the observations lie at or below that value. The following procedure, however, can help us determine or estimate a percentile. The procedure involves first locating the position of the percentile within the set of observations and then determining (or estimating) the value associated with that position. Let Py be the value at or below which y percent of the distribution lies, or the yth percentile. (For example, P18 is the point at or below which 18 percent of the observations lie; 100 − 18 = 82 percent are greater than P18 .) The formula for the position of a percentile in an array with n entries sorted in ascending order is Ly = (n + 1) y 100 (3-8) Chapter 3 Statistical Concepts and Market Returns 95 where y is the percentage point at which we are dividing the distribution and Ly is the location (L) of the percentile (Py ) in the array sorted in ascending order. The value of Ly may or may not be a whole number. In general, as the sample size increases, the percentile location calculation becomes more accurate; in small samples it may be quite approximate. As an example of the case in which Ly is not a whole number, suppose that we want to determine the third quartile of returns for 2002 (Q 3 or P75 ) for the 16 European equity markets given in Table 3-8. According to Equation 3-8, the position of the third quartile is L75 = (16 + 1)75/100 = 12.75, or between the 12th and 13th items in Table 3-9, which ordered the returns into ascending order. The 12th item in Table 3-9 is the return to equities in Portugal in 2002, −28.29 percent. The 13th item is the return to equities in Switzerland in 2002, −25.84 percent. Reflecting the ‘‘0.75’’ in ‘‘12.75,’’ we would conclude that P75 lies 75 percent of the distance between −28.29 percent and −25.84 percent. To summarize: When the location, Ly , is a whole number, the location corresponds to an actual observation. For example, if Italy had not been included in the sample, then n + 1 would have been 16 and, with L75 = 12, the third quartile would be P75 = X12 , where Xi is defined as the value of the observation in the ith (i = L75 ) position of the data sorted in ascending order (i.e., P75 = −28.29). • When Ly is not a whole number or integer, Ly lies between the two closest integer numbers (one above and one below), and we use linear interpolation between those two places to determine Py . Interpolation means estimating an unknown value on the basis of two known values that surround it (lie above and below it); the term ‘‘linear’’ refers to a straight-line estimate. Returning to the calculation of P75 for the equity returns, we found that Ly = 12.75; the next lower whole number is 12 and the next higher whole number is 13. Using linear interpolation, P75 ≈ X12 + (12.75 − 12)(X13 − X12 ). As above, in the 12th position is the return to equities in Portugal, so X12 = −28.29 percent; X13 = −25.84 percent, the return to equities in Switzerland. Thus our estimate is P75 ≈ X12 + (12.75 − 12)(X13 − X12 ) = −28.29 + 0.75[−25.84 − (−28.29)] = −28.29 + 0.75(2.45) = −28.29 + 1.84 = −26.45 percent. In words, −28.29 and −25.84 bracket P75 from below and above, respectively. Because 12.75 − 12 = 0.75, using linear interpolation we move 75 percent of the distance from −28.29 to −25.84 as our estimate of P75 . We follow this pattern whenever Ly is non-integer: The nearest whole numbers below and above Ly establish the positions of observations that bracket Py and then interpolate between the values of those two observations. • Example 3-9 illustrates the calculation of various quantiles for the dividend yield on the components of a major European equity index. EXAMPLE 3-9 and Quintiles Calculating Percentiles, Quartiles, The DJ EuroSTOXX 50 is an index of Europe’s 50 largest publicly traded companies as measured by market capitalization. Table 3-17 shows the dividend yields on the 50 component stocks in the index as of mid-2003, ranked in ascending order. 96 Quantitative Investment Analysis TABLE 3-17 Dividend Yields on the Components of the DJ EuroSTOXX 50 Dividend Yield No. Company 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 AstraZeneca BP Deutsche Telekom HSBC Holdings Credit Suisse Group L’Oréal SwissRe Roche Holding Munich Re Group General Assicurazioni Vodafone Group Carrefour Nokia Novartis Allianz Koninklije Philips Electronics Siemens Deutsche Bank Telecom Italia AXA Telefonica Nestlé Royal Bank of Scotland Group ABN-AMRO Holding BNP Paribas 17 18 19 20 21 22 23 24 25 0.00% 0.00% 0.00% 0.00% 0.26% 1.09% 1.27% 1.33% 1.36% 1.39% 1.41% 1.51% 1.75% 1.81% 1.92% 2.01% 2.16% 2.27% 2. 27% 2.39% 2.49% 2.55% 2.60% 2.65% 2.65% No. Company 26 27 28 29 30 31 32 33 UBS Tesco Total GlaxoSmithKline BT Group Unilever BASF Santander Central Hispano Banco Bilbao Vizcaya Argentaria Diageo HBOS E.ON Shell Transport and Co. Barclays Royal Dutch Petroleum Co. Fortis Bayer DaimlerChrysler Suez Aviva Eni ING Group Prudential Lloyds TSB AEGON 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Dividend Yield 2.65% 2.95% 3.11% 3.31% 3.34% 3.53% 3.59% 3.66% 3.67% 3.68% 3.78% 3.87% 3.88% 4.06% 4.27% 4.28% 4.45% 4.68% 5.13% 5.15% 5.66% 6.16% 6.43% 7.68% 8.14% Source: http://france.finance.yahoo.com accessed 8 July 2003. Using the data in Table 3-17, address the following: 1. 2. 3. 4. 5. Calculate the 10th and 90th percentiles. Calculate the first, second, and third quartiles. State the value of the median. How many quintiles are there, and to what percentiles do the quintiles correspond? Calculate the value of the first quintile. Solution to 1: In this example, n = 50. Using Equation 3-8, Ly = (n + 1)y/100 for position of the yth percentile, so for the 10th percentile we have L10 = (50 + 1)(10/100) = 5.1 Chapter 3 Statistical Concepts and Market Returns L10 is between the fifth and sixth observations with values X5 = 0.26 and X6 = 1.09. The estimate of the 10th percentile (first decile) for dividend yield is P10 ≈ X5 + (5.1 − 5)(X6 − X5 ) = 0.26 + 0.1(1.09 − 0.26) = 0.26 + 0.1(0.83) = 0.34% For the 90th percentile, L90 = (50 + 1)(90/100) = 45.9 L90 is between the 45th and 46th observations with values X45 = 5.15 and X46 = 5.66, respectively. The estimate of the 90th percentile (ninth decile) is P90 ≈ X45 + (45.9 − 45)(X46 − X45 ) = 5.15 + 0.9(5.66 − 5.15) = 5.15 + 0.9(0.51) = 5.61% Solution to 2: The first, second, and third quartiles correspond to P25 , P50 , and P75 , respectively. L25 = (51)(25/100) = 12.75 L25 is between the 12th and 13th entries with values X12 = 1.51 and X13 = 1.75. P25 = Q1 ≈ X12 + (12.75 − 12)(X13 − X12 ) = 1.51 + 0.75(1.75 − 1.51) = 1.51 + 0.75(0.24) = 1.69% L50 = (51)(50/100) = 25.5 L25 is between the 25th and 26th entries. But these entries share the same value, X25 = X26 = 2.65, so no interpolation is needed. P50 = Q2 = 2.65% L75 = (51)(75/100) = 38.25 L75 is between the 38th and 39th entries with values X38 = 3.88 and X39 = 4.06. P75 = Q3 ≈ X38 + (38.25 − 38)(X39 − X38 ) = 3.88 + 0.25(4.06 − 3.88) = 3.88 + 0.25(0.18) = 3.93% Solution to 3: The median is the 50th percentile, 2.65 percent. This is the same value that we would obtain by taking the mean of the n/2 = 50/2 = 25th item and (n + 2)/2 = 52/2 = 26th item, consistent with the procedure given earlier for the median of an even-numbered sample. 97 98 Quantitative Investment Analysis Solution to 4: There are four quintiles, and they correspond to P20 , P40 , P60 , and P80 . Solution to 5: The first quintile is P20 . L20 = (50 + 1)(20/100) = 10.2 L20 is between the 10th and 11th observations with values X10 = 1.39 and X11 = 1.41. The estimate of the first quintile is P20 ≈ X10 + (10.2 − 10)(X11 − X10 ) = 1.39 + 0.2(1.41 − 1.39) = 1.39 + 0.2(0.02) = 1.394% or 1.39% 6.2. Quantiles in Investment Practice In this section, we discuss the use of quantiles in investments. Quantiles are used in portfolio performance evaluation as well as in investment strategy development and research. Investment analysts use quantiles every day to rank performance—for example, the performance of portfolios. The performance of investment managers is often characterized in terms of the quartile in which they fall relative to the performance of their peer group of managers. The Morningstar mutual fund star rankings, for example, associates the number of stars with percentiles of performance relative to similar-style mutual funds. Another key use of quantiles is in investment research. Analysts refer to a group defined by a particular quantile as that quantile. For example, analysts often refer to the set of companies with returns falling below the 10th percentile cutoff point as the bottom return decile. Dividing data into quantiles based on some characteristic allows analysts to evaluate the impact of that characteristic on a quantity of interest. For instance, empirical finance studies commonly rank companies based on the market value of their equity and then sort them into deciles. The 1st decile contains the portfolio of those companies with the smallest market values, and the 10th decile contains those companies with the largest market value. Ranking companies by decile allows analysts to compare the performance of small companies with large ones. We can illustrate the use of quantiles, in particular quartiles, in investment research using the example of Bauman, Conover, and Miller (1998). That study compared the performance of international growth stocks to value stocks. Typically, value stocks are defined as those for which the market price is relatively low in relation to earnings per share, book value per share, or dividends per share. Growth stocks, on the other hand, have comparatively high prices in relation to those same measures. The Bauman et al. classification criteria were the following valuation measures: price-to-earnings (P/E), price-to-cash flow (P/CF), price-to-book value (P/B), and dividend yield (D/P). They assigned one-fourth of the total sample with the lowest P/E on 30 June of each year from 1986 to 1996 (the value group) to Quartile 1, and the one-fourth with the highest P/E of each year (the growth group) to Quartile 4. The stocks with the second-highest P/E formed Quartile 3, and the stocks with the second-lowest P/E, Quartile 2. The authors repeated this process for each of the four fundamental factors. Treating each 99 Chapter 3 Statistical Concepts and Market Returns TABLE 3-18 Mean Annual Returns of Value and Growth Stocks Based on Selected Characteristics, 1986–1996 Selection Criteria Classification by P/E Median P/E Return Standard deviation Classification by P/CF Median P/CF Return Standard deviation Classification by P/B Median P/B Return Standard deviation Classification by D/P Median D/P Return Standard deviation Total Observations Spread in Return, Q1 to Q4 Q1 (Value) Q2 Q3 Q4 (Growth) 8.7 15.0% 46.5 15.2 13.6% 38.3 24.2 13.5% 42.5 72.5 10.6% 50.4 +4.4% 4.4 15.5% 48.7 8.2 13.7% 41.2 13.3 12.9% 41.9 34.2 11.2% 51.4 +4.3% 0.8 18.1% 69.6 1.4 14.4% 45.9 2.2 12.6% 45.1 4.3 12.4% 57.0 +5.7% 5.6% 14.1% 40.5 3.2% 14.1% 38.7 1.9% 12.5% 38.9 0.6% 9.3% 42.0 +4.8% 28,463 30,240 32,265 25,394 Source: Bauman et al. quartile group as a portfolio composed of equally weighted stocks, they were able to compare the performance of the various value/growth quartiles. Table 1 from their study is reproduced as Table 3-18. Table 3-18 reports each valuation factor’s median, mean return, and standard deviation for each quartile grouping. Moving from Quartile 1 to Quartile 4, P/E, P/CF, and P/B increase, but D/P decreases. Regardless of the selection criteria, international value stocks outperformed international growth stocks during the sample period. Bauman, Conover, and Miller also divided companies into one of four quartiles based on market value of equity. Then they examined the returns to the stocks in the quartiles. Table 7 from their article is reproduced here as Table 3-19. As the table shows, the small-company portfolio had a median market value of $46.6 million and the large-company portfolio had a median value of $2,472.3 million. Large companies were more than 50 times larger than small companies, yet their mean stock returns were less than half those of the small companies (small, 22.0 percent; large, 10.8 percent). Overall, Bauman et al. found two effects. First, international value stocks (as the authors defined them) outperformed international growth stocks. Second, international small stocks outperformed international large stocks. The authors’ next step was to examine how value and growth stocks performed while controlling for size. This step involved constructing 16 different value/growth and size portfolios (4 × 4 = 16) and investigating the interaction between these two fundamental factors. They found that international value stocks outperformed international growth stocks except when market capitalization was very small. For portfolio managers, these findings suggest that value stocks offered investors relatively more favorable returns than did growth stocks in international markets during the specific time period studied. 100 Quantitative Investment Analysis TABLE 3-19 Mean Annual Returns of International Stocks Grouped by Market Capitalization, 1986–1996 Selection Criteria Total Observations Classification by size Median size (millions) Return Standard deviation Q1 (Small) Q2 Q3 Q4 (Large) $46.6 22.0% $209.9 13.6% $583.7 11.1% $2,472.3 10.8% 87.8 45.2 39.5 Spread in Return, Q1 to Q4 32,555 +11.2% 34.0 Source: Bauman et al. 7. MEASURES OF DISPERSION As the well-known researcher Fischer Black has written, ‘‘[t]he key issue in investments is estimating expected return.’’26 Few would disagree with the importance of expected return or mean return in investments: The mean return tells us where returns, and investment results, are centered. To completely understand an investment, however, we also need to know how returns are dispersed around the mean. Dispersion is the variability around the central tendency. If mean return addresses reward, dispersion addresses risk. In this section, we examine the most common measures of dispersion: range, mean absolute deviation, variance, and standard deviation. These are all measures of absolute dispersion. Absolute dispersion is the amount of variability present without comparison to any reference point or benchmark. These measures are used throughout investment practice. The variance or standard deviation of return is often used as a measure of risk pioneered by Nobel laureate Harry Markowitz. William Sharpe, another winner of the Nobel Prize in economics, developed the Sharpe ratio, a measure of risk-adjusted performance. That measure makes use of standard deviation of return. Other measures of dispersion, mean absolute deviation and range, are also useful in analyzing data. 7.1. The Range We encountered range earlier when we discussed the construction of frequency distribution. The simplest of all the measures of dispersion, range can be computed with interval or ratio data. • Definition of Range. The range is the difference between the maximum and minimum values in a dataset: Range = Maximum value − Minimum value 26 Black (1993). (3-9) 101 Chapter 3 Statistical Concepts and Market Returns As an illustration of range, the largest monthly return for the S&P 500 in the period from January 1926 to December 2002 is 42.56 percent (in April 1933) and the smallest is −29.73 percent (in September 1931). The range of returns is thus 72.29 percent [42.56 percent − (−29.73 percent)]. An alternative definition of range reports the maximum and minimum values. This alternative definition provides more information than does the range as defined in Equation 3-9. One advantage of the range is ease of computation. A disadvantage is that the range uses only two pieces of information from the distribution. It cannot tell us how the data are distributed (that is, the shape of the distribution). Because the range is the difference between the maximum and minimum returns, it can reflect extremely large or small outcomes that may not be representative of the distribution.27 7.2. The Mean Absolute Deviation Measures of dispersion can be computed using all the observations in the distribution rather than just the highest and lowest. The question is, how should we measure dispersion? Our previous discussion on properties of the arithmetic mean introduced the notion of distance or deviation from the mean (Xi − X ) as a fundamental piece of information used in statistics. We could compute measures of dispersion as the arithmetic average of the deviations around the mean, but we would encounter a problem: The deviations around the mean always sum to 0. If we computed the mean of the deviations, the result would also equal 0. Therefore, we need to find a way to address the problem of negative deviations canceling out positive deviation. One solution is to examine the absolute deviations around the mean as in the mean absolute deviation. • Mean Absolute Deviation Formula. The mean absolute deviation (MAD) for a sample is n ! & & &Xi − X & MAD = i=1 n (3-10) where X is the sample mean and n is the number of observations in the sample. In calculating MAD, we ignore the signs of the deviations around the mean. For example, if Xi = −11.0 and X = 4.5, the absolute value of the difference is | − 11.0 − 4.5| = | − 15.5| = 15.5. The mean absolute deviation uses all of the observations in the sample and is thus superior to the range as a measure of dispersion. One technical drawback of MAD is that it is difficult to manipulate mathematically compared with the next measure we will introduce, variance.28 Example 3-10 illustrates the use of the range and the mean absolute deviation in evaluating risk. 27 Another distance measure of dispersion that we may encounter, the interquartile range, focuses on the middle rather than the extremes. The interquartile range (IQR) is the difference between the third and first quartiles of a dataset: IQR = Q3 − Q1 . The IQR represents the length of the interval containing the middle 50 percent of the data, with a larger interquartile range indicating greater dispersion, all else equal. 28 In some analytic work such as optimization, the calculus operation of differentiation is important. Variance as a function can be differentiated, but absolute value cannot. 102 Quantitative Investment Analysis EXAMPLE 3-10 The Range and the Mean Absolute Deviation Having calculated mean returns for the two mutual funds in Example 3-7, the analyst is now concerned with evaluating risk. TABLE 3-15 (repeated) Total Returns for Two Mutual Funds, 1998–2002 Year Selected American Shares (SLASX) T. Rowe Price Equity Income (PRFDX) 16.2% 20.3% 9.3% −11.1% −17.0% 9.2% 3.8% 13.1% 1.6% −13.0% 1998 1999 2000 2001 2002 Source: AAII. Based on the data in Table 3-15 repeated above, answer the following: 1. Calculate the range of annual returns for (A) SLASX and (B) PRFDX, and state which mutual fund appears to be riskier based on these ranges. 2. Calculate the mean absolute deviation of returns on (A) SLASX and (B) PRFDX, and state which mutual fund appears to be riskier based on MAD. Solutions to 1: A. For SLASX, the largest return was 20.3 percent and the smallest was −17.0 percent. The range is thus 20.3 − (−17.0) = 37.3%. B. For PFRDX, the range is 13.1 − (−13.0) = 26.1%. With a larger range of returns than PRFDX, SLASX appeared to be the riskier fund during the 1998–2002 period. Solutions to 2: A. The arithmetic mean return for SLASX as calculated in Example 3-7 is 3.54 percent. The MAD of SLASX returns is MAD = |16.2 − 3.54| + |20.3 − 3.54| + |9.3 − 3.54| + | − 11.1 − 3.54| + | − 17.0 − 3.54| 5 = 12.66 + 16.76 + 5.76 + 14.64 + 20.54 5 = 70.36 = 14.1% 5 Chapter 3 Statistical Concepts and Market Returns 103 B. The arithmetic mean return for PRFDX as calculated in Example 3-7 is 2.94 percent. The MAD of PRFDX returns is MAD = |9.2 − 2.94| + |3.8 − 2.94| + |13.1 − 2.94| + |1.6 − 2.94| + | − 13.0 − 2.94| 5 = 6.26 + 0.86 + 10.16 + 1.34 + 15.94 5 = 34.56 = 6.9% 5 SLASX, with a MAD of 14.1 percent, appears to be much riskier than PRFDX, with a MAD of 6.9 percent. 7.3. Population Variance and Population Standard Deviation The mean absolute deviation addressed the issue that the sum of deviations from the mean equals zero by taking the absolute value of the deviations. A second approach to the treatment of deviations is to square them. The variance and standard deviation, which are based on squared deviations, are the two most widely used measures of dispersion. Variance is defined as the average of the squared deviations around the mean. Standard deviation is the positive square root of the variance. The following discussion addresses the calculation and use of variance and standard deviation. 7.3.1. Population Variance If we know every member of a population, we can compute the population variance. Denoted by the symbol σ2 , the population variance is the arithmetic average of the squared deviations around the mean. • Population Variance Formula. The population variance is σ2 = N ! i=1 (Xi − µ)2 N (3-11) where µ is the population mean and N is the size of the population. Given knowledge of the population mean, µ, we can use Equation 3-11 to calculate the sum of the squared differences from the mean, taking account of all N items in the population, and then to find the mean squared difference by dividing the sum by N . Whether a difference from the mean is positive or negative, squaring that difference results in a positive number. Thus variance takes care of the problem of negative deviations from the mean canceling out positive deviations by the operation of squaring those deviations. The P/Es for BJ, COST, and WMT were given earlier as 16.73, 22.02, and 29.30, respectively. We calculated the mean P/E as 22.68. Therefore, the population variance of the P/Es is (1/3)[(16.73 − 22.68)2 + (22.02 − 22.68)2 + (29.30 − 22.68)2 ] = (1/3)(−5.952 + −0.662 + 6.622 ) = (1/3)(35.4025 + 0.4356 + 43.8244) = (1/3)(79.6625) = 26.5542. 104 Quantitative Investment Analysis 7.3.2. Population Standard Deviation Because the variance is measured in squared units, we need a way to return to the original units. We can solve this problem by using standard deviation, the square root of the variance. Standard deviation is more easily interpreted than the variance because standard deviation is expressed in the same unit of measurement as the observations. • Population Standard Deviation Formula. The population standard deviation, defined as the positive square root of the population variance, is σ= ' ( N (! ( (Xi − µ)2 ( ) i=1 N (3-12) where µ is the population mean and N is the size of the population. Using the example of the P/Es for BJ, COST, and WMT, according √ to Equation 3-12 we would calculate the variance, 26.5542, then take the square root: 26.5542 = 5.1531 or approximately 5.2. Both the population variance and standard deviation are examples of parameters of a distribution. In later chapters, we will introduce the notion of variance and standard deviation as risk measures. In investments, we often do not know the mean of a population of interest, usually because we cannot practically identify or take measurements from each member of the population. We then estimate the population mean with the mean from a sample drawn from the population, and we calculate a sample variance or standard deviation using formulas different from Equations 3-11 and 3-12. We shall discuss these calculations in subsequent sections. However, in investments we sometimes have a defined group that we can consider to be a population. With well-defined populations, we use Equations 3-11 and 3-12, as in the following example. EXAMPLE 3-11 Calculating the Population Standard Deviation Table 3-20 gives the yearly portfolio turnover for the 10 U.S. equity funds that composed the 2002 Forbes Magazine Honor Roll.29 Portfolio turnover, a measure of trading activity, is the lesser of the value of sales or purchases over a year divided by average net assets during the year. The number and identity of the funds on the Forbes Honor Roll changes from year to year. 29 Forbes magazine annually selects U.S. equity mutual funds meeting certain criteria for its Honor Roll. The criteria relate to capital preservation (performance in bear markets), continuity of management (the fund must have a manager with at least six years’ tenure), diversification, accessibility (disqualifying funds that are closed to new investors), and after-tax long-term performance. 105 Chapter 3 Statistical Concepts and Market Returns TABLE 3-20 Portfolio Turnover: 2002 Forbes Honor Roll Mutual Funds Fund Yearly Portfolio Turnover FPA Capital Fund (FPPTX) Mairs & Power Growth Fund (MPGFX) Muhlenkamp Fund (MUHLX) Longleaf Partners Fund (LLPFX) Heartland Value Fund (HRTVX) Scudder–Dreman High Return Equity-A (KDHAX) Clipper Fund (CFIMX) Weitz Value Fund (WVALX) Third Avenue Value Fund (TAVFX) Dodge & Cox Stock Fund (DODGX) 23% 8% 11% 18% 56% 29% 23% 13% 16% 10% Source: Forbes (2003). Based on the data in Table 3-20, address the following: 1. Calculate the population mean portfolio turnover for the period used by Forbes for the ten 2002 Honor Roll funds. 2. Calculate the population variance and population standard deviation of portfolio turnover. 3. Explain the use of the population formulas in this example. Solution to 1: µ = (23 + 8 + 11 + 18 + 56 + 29 + 23 + 13 + 16 + 10)/10 = 207/ 10 = 20.7 percent N ! (Xi − µ)2 i=1 Solution to 2: Having established that µ = 20.7, we can calculate σ2 = N by first calculating the numerator in the expression and then dividing by N = 10. The numerator (the sum of the squared differences from the mean) is (23 − 20.7)2 + (8 − 20.7)2 + (11 − 20.7)2 + (18 − 20.7)2 + (56 − 20.7)2 + (29 − 20.7)2 + (23 − 20.7)2 + (13 − 20.7)2 + (16 − 20.7)2 + (10 − 20.7)2 = 1,784.1 Thus σ2 = 1,784.1/10 = 178.41. √ To calculate standard deviation, σ = 178.41 = 13.357 percent. (The unit of variance is percent squared, so the unit of standard deviation is percent.) Solution to 3: If the population is clearly defined to be the Forbes Honor Roll funds in one specific year (2002), and if portfolio turnover is understood to refer to the specific 106 Quantitative Investment Analysis one-year period reported upon by Forbes, the application of the population formulas to variance and standard deviation is appropriate. The results of 178.41 and 13.357 are, respectively, the cross-sectional variance and standard deviation in yearly portfolio turnover for the 2002 Forbes Honor Roll Funds.30 7.4. Sample Variance and Sample Standard Deviation 7.4.1. Sample Variance In many instances in investment management, a subset or sample of the population is all that we can observe. When we deal with samples, the summary measures are called statistics. The statistic that measures the dispersion in a sample is called the sample variance. • Sample Variance Formula. The sample variance is s2 = n ! i=1 (Xi − X )2 (3-13) n−1 where X is the sample mean and n is the number of observations in the sample. Equation 3-13 tells us to take the following steps to compute the sample variance: i. Calculate the sample mean, X . ii. Calculate each observation’s squared deviation from the sample mean, (Xi − X )2 . n ! iii. Sum the squared deviations from the mean: (Xi − X )2 . i=1 iv. Divide the sum of squared deviations from the mean by n − 1: n ! i=1 (Xi − X )2 /(n − 1). We will illustrate the calculation of the sample variance and the sample standard deviation in Example 3-12. We use the notation s2 for the sample variance to distinguish it from population variance, 2 σ . The formula for sample variance is nearly the same as that for population variance except for the use of the sample mean, X , in place of the population mean, µ, and a different divisor. In the case of the population variance, we divide by the size of the population, N . For the sample variance, however, we divide by the sample size minus 1, or n − 1. By using n − 1 (rather than n) as the divisor, we improve the statistical properties of the sample variance. In statistical terms, the sample variance defined in Equation 3-13 is an unbiased estimator of the population variance.31 The quantity n − 1 is also known as the number of degrees of freedom in estimating the population variance. To estimate the population variance with s2 , we must first calculate the mean. Once we have computed the sample mean, there are only n − 1 independent deviations from it. 30 In fact, we could not properly use the Honor Roll funds to estimate the population variance of portfolio turnover (for example) of any other differently defined population, because the Honor Roll funds are not a random sample from any larger population of U.S. equity mutual funds. 31 We discuss this concept further in the chapter on sampling. 107 Chapter 3 Statistical Concepts and Market Returns 7.4.2. Sample Standard Deviation Just as we computed a population standard deviation, we can compute a sample standard deviation by taking the positive square root of the sample variance. • Sample Standard Deviation Formula. The sample standard deviation, s, is s= ' ( n (! ( (Xi − X )2 ( ) i=1 (3-14) n−1 where X is the sample mean and n is the number of observations in the sample. To calculate the sample standard deviation, we first compute the sample variance using the steps given. We then take the square root of the sample variance. Example 3-12 illustrates the calculation of the sample variance and standard deviation for the two mutual funds introduced earlier. EXAMPLE 3-12 Calculating Sample Variance and Sample Standard Deviation After calculating the geometric and arithmetic mean returns of two mutual funds in Example 3-7, we calculated two measures of dispersions for those funds, the range and mean absolute deviation of returns, in Example 3-10. We now calculate the sample variance and sample standard deviation of returns for those same two funds. TABLE 3-15 (repeated) Total Returns for Two Mutual Funds, 1998–2002 Year Selected American Shares (SLASX) T. Rowe Price Equity Income (PRFDX) 16.2% 20.3% 9.3% −11.1% −17.0% 9.2% 3.8% 13.1% 1.6% −13.0% 1998 1999 2000 2001 2002 Source: AAII. Based on the data in Table 3-15 repeated above, answer the following: 1. Calculate the sample variance of return for (A) SLASX and (B) PRFDX. 2. Calculate sample standard deviation of return for (A) SLASX and (B) PRFDX. 3. Contrast the dispersion of returns as measured by standard deviation of return and mean absolute deviation of return for each of the two funds. 108 Quantitative Investment Analysis Solution to 1: To calculate the sample variance, we use Equation 3-13. (Deviation answers are all given in percent squared.) A. SLASX i. The sample mean is R = (16.2 + 20.3 + 9.3 − 11.1 − 17.0)/5 = 17.7/5 = 3.54%. ii. The squared deviations from the mean are (16.2 − 3.54)2 = (12.66)2 = 160.2756 (20.3 − 3.54)2 = (16.76)2 = 280.8976 (9.3 − 3.54)2 = (5.76)2 = 33.1776 (−11.1 − 3.54)2 = (−14.64)2 = 214.3296 (−17.0 − 3.54)2 = (−20.54)2 = 421.8916 iii. The sum of the squared deviations from the mean is 160.2756 + 280.8976 + 33.1776 + 214.3296 + 421.8916 = 1,110.5720. iv. Divide the sum of the squared deviations from the mean by n − 1: 1,110.5720/(5 − 1) = 1,110.5720/4 = 277.6430. B. PRFDX i. The sample mean is R = (9.2 + 3.8 + 13.1 + 1.6 − 13.0)/5 = 14.7/5 = 2.94%. ii. The squared deviations from the mean are (9.2 − 2.94)2 = (6.26)2 = 39.1876 (3.8 − 2.94)2 = (0.86)2 = 0.7396 (13.1 − 2.94)2 = (10.16)2 = 103.2256 (1.6 − 2.94)2 = (−1.34)2 = 1.7956 (−13.0 − 2.94)2 = (−15.94)2 = 254.0836 iii. The sum of the squared deviations from the mean is 39.1876 + 0.7396 + 103.2256 + 1.7956 + 254.0836 = 399.032. iv. Divide the sum of the squared deviations from the mean by n − 1: 399.032/4 = 99.758. Solution to 2: To find the standard deviation, we take the positive square root of variance. √ A. For SLASX, σ = √277.6430 = 16.66% or 16.7 percent. B. For PRFDX, σ = 99.758 = 9.99% or 10.0 percent. Solution to 3: Table 3-21 summarizes the results from Part 2 for standard deviation and incorporates the results for MAD from Example 3-10. Note that the mean absolute deviation is less than the standard deviation. The mean absolute deviation will always be less than or equal to the standard deviation because the standard deviation gives more weight to large deviations than to small ones (remember, the deviations are squared). 109 Chapter 3 Statistical Concepts and Market Returns TABLE 3-21 Two Mutual Funds: Comparison of Standard Deviation and Mean Absolute Deviation Fund Standard Deviation Mean Absolute Deviation 16.7% 10.0% 14.1% 6.9% SLASX PRFDX Because the standard deviation is a measure of dispersion about the arithmetic mean, we usually present the arithmetic mean and standard deviation together when summarizing data. When we are dealing with data that represent a time series of percent changes, presenting the geometric mean—representing the compound rate of growth—is also very helpful. Table 3-22 presents the historical geometric and arithmetic mean returns, along with the historical standard deviation of returns, for various equity return series. We present these statistics for nominal (rather than inflation-adjusted) returns so we can observe the original magnitudes of the returns. TABLE 3-22 Equity Market Returns: Means and Standard Deviations Return Series Geometric Mean Arithmetic Mean Standard Deviation 12.20% 0.97% 20.49 5.65 I. Ibbotson Associates Series: 1926–2002 S&P 500 (Annual) S&P 500 (Monthly) 10.20% 0.81% II. Dimson et al. (2002) Series (Annual): 1900–2000 Australia Belgium Canada Denmark France Germany Ireland Italy Japan Netherlands South Africa Spain Sweden Switzerland United Kingdom United States 11.9% 8.2% 9.7% 8.9% 12.1% 9.7% 9.5% 12.0% 12.5% 9.0% 12.0% 10.0% 11.6% 7.6% 10.1% 10.1% 13.3% 10.5% 11.0% 10.7% 14.5% 15.2% 11.5% 16.1% 15.9% 11.0% 14.2% 12.1% 13.9% 9.3% 11.9% 12.0% Source: Ibbotson EnCorr Analyzer; Dimson et al. 18.2% 24.1% 16.6% 21.7% 24.6% 36.4% 22.8% 34.2% 29.5% 22.7% 23.7% 22.8% 23.5% 19.7% 21.8% 19.9% 110 Quantitative Investment Analysis 7.5. Semivariance, Semideviation, and Related Concepts An asset’s variance or standard deviation of returns is often interpreted as a measure of the asset’s risk. Variance and standard deviation of returns take account of returns above and below the mean, but investors are concerned only with downside risk, for example, returns below the mean. As a result, analysts have developed semivariance, semideviation, and related dispersion measures that focus on downside risk. Semivariance is defined as the average squared deviation below the mean. Semideviation (sometimes called semistandard deviation) is the positive square root of semivariance. To compute the sample semivariance, for example, we take the following steps: i. Calculate the sample mean. ii. Identify the observations that are smaller than the mean (discarding observations equal to and greater than the mean); suppose there are n∗ observations smaller than the mean. iii. Compute the sum of the squared negative deviations from the mean (using the n∗ observations that are smaller than the mean). iv. Divide the sum of the squared negative deviations from Step iii by n∗ − 1. A formula for semivariance is ! (Xi − X )2 /(n∗ − 1) for all Xi 1. Table 3-23 illustrates the proportion of the observations that must lie within a certain number of standard deviations around the sample mean. When k = 1.25, for example, the inequality states that the minimum proportion of the observations that lie within ±1.25s is 1 − 1/(1.25)2 = 1 − 0.64 = 0.36 or 36 percent. The most frequently cited facts that result from Chebyshev’s inequality are that a two-standard-deviation interval around the mean must contain at least 75 percent of the observations and a three-standard-deviation interval around the mean must contain at least 89 percent of the observations, no matter how the data are distributed. The importance of Chebyshev’s inequality stems from its generality. The inequality holds for samples and populations and for discrete and continuous data regardless of the shape of the distribution. As we shall see in the chapter on sampling, we can make much more precise interval statements if we can assume that the sample is drawn from a population that follows a specific distribution called the normal distribution. Frequently, however, we cannot confidently assume that distribution. TABLE 3-23 Proportions from Chebyshev’s Inequality k 1.25 1.50 2.00 2.50 3.00 4.00 Interval Around the Sample Mean X X X X X X ± 1.25s ± 1.50s ± 2s ± 2.50s ± 3s ± 4s Proportion 36% 56% 75% 84% 89% 94% Note: Standard deviation is denoted as s. 32 For negatively skewed returns, semivariance is greater than one-half variance; for positively skewed returns, semivariance is less than one-half variance. See Estrada (2003). We discuss skewness later in this chapter. 33 As discussed in the chapter on probability concepts and the chapter on portfolio concepts, we can find a portfolio’s variance as a straightforward function of the variances and correlations of the component securities. There is no similar procedure for semivariance and target semivariance. We also cannot take the derivative of semivariance or target semivariance. 112 Quantitative Investment Analysis The next example illustrates the use of Chebyshev’s inequality. EXAMPLE 3-13 Applying Chebyshev’s Inequality According to Table 3-22, the arithmetic mean monthly return and standard deviation of monthly returns on the S&P 500 were 0.97 percent and 5.65 percent, respectively, during the 1926–2002 period, totaling 924 monthly observations. Using this information, address the following: 1. Calculate the endpoints of the interval that must contain at least 75 percent of monthly returns according to Chebyshev’s inequality. 2. What are the minimum and maximum number of observations that must lie in the interval computed in Part 1, according to Chebyshev’s inequality? Solution to 1: According to Chebyshev’s inequality, at least 75 percent of the observations must lie within two standard deviations of the mean, X ± 2s. For the monthly S&P 500 return series, we have 0.97% ± 2(5.65%) = 0.97% ± 11.30%. Thus the lower endpoint of the interval that must contain at least 75 percent of the observations is 0.97% − 11.30% = −10.33%, and the upper endpoint is 0.97% + 11.30% = 12.27%. Solution to 2: For a sample size of 924, at least 0.75(924) = 693 observations must lie in the interval from −10.33% to 12.27% that we computed in Part 1. Chebyshev’s inequality gives the minimum percentage of observations that must fall within a given interval around the mean, but it does not give the maximum percentage. Table 34, which gave the frequency distribution of monthly returns on the S&P 500, is excerpted below. The data in the excerpted table are consistent with the prediction of Chebyshev’s inequality. The set of intervals running from −10.0% to 12.0% is just slightly narrower than the two-standard-deviation interval −10.33% to 12.27%. A total of 886 observations (approximately 96 percent of observations) fall in the range from −10.0% to 12.0%. TABLE 3-4 (Excerpt) Frequency Distribution for the Monthly Total Return on the S&P 500, January 1926 to December 2002 Return Interval −10.0% to −8.0% −8.0% to −6.0% −6.0% to −4.0% −4.0% to −2.0% −2.0% to 0.0% 0.0% to 2.0% 2.0% to 4.0% 4.0% to 6.0% 6.0% to 8.0% 8.0% to 10.0% 10.0% to 12.0% Absolute Frequency 20 30 54 90 138 182 153 126 58 21 14 886 Chapter 3 Statistical Concepts and Market Returns 113 7.7. Coefficient of Variation We noted earlier that standard deviation is more easily interpreted than variance because standard deviation uses the same units of measurement as the observations. We may sometimes find it difficult to interpret what standard deviation means in terms of the relative degree of variability of different sets of data, however, either because the datasets have markedly different means or because the datasets have different units of measurement. In this section we explain a measure of relative dispersion, the coefficient of variation that can be useful in such situations. Relative dispersion is the amount of dispersion relative to a reference value or benchmark. We can illustrate the problem of interpreting the standard deviation of datasets with markedly different means using two hypothetical samples of companies. The first sample, composed of small companies, includes companies with 2003 sales of ¤50 million, ¤75 million, ¤65 million, and ¤90 million. The second sample, composed of large companies, includes companies with 2003 sales of ¤800 million, ¤825 million, ¤815 million, and ¤840 million. We can verify using Equation 3-14 that the standard deviation of sales in both samples is ¤16.8 million.34 In the first sample, the largest observation, ¤90 million, is 80 percent larger than the smallest observation, ¤50 million. In the second sample, the largest observation is only 5 percent larger than the smallest observation. Informally, a standard deviation of ¤16.8 million represents a high degree of variability relative to the first sample, which reflects mean 2003 sales of ¤70 million, but a small degree of variability relative to the second sample, which reflects mean 2003 sales of ¤820 million. The coefficient of variation is helpful in situations such as that just described. • Coefficient of Variation Formula. The coefficient of variation, CV, is the ratio of the standard deviation of a set of observations to their mean value:35 CV = s/X (3-15) where s is the sample standard deviation and X is the sample mean. When the observations are returns, for example, the coefficient of variation measures the amount of risk (standard deviation) per unit of mean return. Expressing the magnitude of variation among observations relative to their average size, the coefficient of variation permits direct comparisons of dispersion across different datasets. Reflecting the correction for scale, the coefficient of variation is a scale-free measure (that is, it has no units of measurement). We can illustrate the application of the coefficient of variation using our earlier example of two samples of companies. The coefficient of variation for the first sample is (¤16.8 million)/(¤70 million) = 0.24; the coefficient of variation for the second sample is (¤16.8 million)/(¤820 million) = 0.02. This confirms our intuition that the first sample had much greater variability in sales than the second sample. Note that 0.24 and 0.02 are pure numbers in the sense that they are free of units of measurement (because we divided the standard deviation by the mean, which is measured in the same units as the standard deviation). If we need to compare the dispersion among data sets stated in different units of measurement, the coefficient of variation can be useful because it is free from units of measurement. Example 3-14 illustrates the calculation of the coefficient of variation. second sample was created by adding ¤750 million to each observation in the first sample. Standard deviation (and variance) has the property of remaining unchanged if we add a constant amount to each observation. 35 The reader will also encounter CV defined as 100(s/X ), which states CV as a percentage. 34 The 114 Quantitative Investment Analysis EXAMPLE 3-14 Calculating the Coefficient of Variation Table 3-24 summarizes annual mean returns and standard deviations for several major U.S. asset classes, using an option in Ibbotson EnCorr Analyzer to convert monthly return statistics to annual ones. TABLE 3-24 Arithmetic Mean Annual Return and Standard Deviation of Returns, U.S. Asset Classes, 1926–2002 Asset Class S&P 500 U.S. small stock U.S. long-term corporate U.S. long-term government U.S. 30-day T-bill Arithmetic Mean Return Standard Deviation of Return 12.3% 16.9% 6.1% 5.8% 3.8% 21.9% 35.1% 7.2% 8.2% 0.9% Source: Ibbotson EnCorr Analyzer. Using the information in Table 3-24, address the following: 1. Calculate the coefficient of variation for each asset class given. 2. Rank the asset classes from most risky to least risky using CV as a measure of relative dispersion. 3. Determine whether there is more difference between the absolute or the relative riskiness of the S&P 500 and U.S. small stocks. Use the standard deviation as a measure of absolute risk and CV as a measure of relative risk. Solution to 1: S&P 500: CV = 21.9%/12.3% = 1.780 U.S. small stock: CV = 35.1%/16.9% = 2.077 U.S. long-term corporate: CV = 7.2%/6.1% = 1.180 U.S. long-term government: CV = 8.2%/5.8% = 1.414 U.S. 30-day T-bill: CV = 0.9%/3.8% = 0.237 Solution to 2: Based on CV, the ranking is U.S. small stocks (most risky), S&P 500, U.S. long-term governments, U.S. long-term corporates, and U.S. 30-day T-bills (least risky). Solution to 3: As measured both by standard deviation and CV, U.S. small stocks were riskier than the S&P 500. However, the CVs reveal less difference between small stock and S&P 500 return variability than that suggested by the standard deviations alone. The standard deviation of small stock returns was (35.1 − 21.9)/21.9 = 0.603 or about 60 percent larger than S&P 500 returns, compared with a difference in the CV of (2.077 − 1.780)/1.780 = 0.167 or 17 percent. Chapter 3 Statistical Concepts and Market Returns 115 7.8. The Sharpe Ratio Although CV was designed as a measure of relative dispersion, its inverse reveals something about return per unit of risk because the standard deviation of returns is commonly used as a measure of investment risk. For example, a portfolio with a mean monthly return of 1.19 percent and a standard deviation of 4.42 percent has an inverse CV of 1.19%/4.42% = 0.27. This result indicates that each unit of standard deviation represents a 0.27 percent return. A more precise return–risk measure recognizes the existence of a risk-free return, a return for virtually zero standard deviation. With a risk-free asset, an investor can choose a risky portfolio, p, and then combine that portfolio with the risk-free asset to achieve any desired level of absolute risk as measured by standard deviation of return, sp . Consider a graph with mean return on the vertical axis and standard deviation of return on the horizontal axis. Any combination of portfolio p and the risk-free asset lies on a ray (line) with slope equal to the quantity (Mean return − Risk-free return) divided by sp . The ray giving investors choices offering the most reward (return in excess of the risk-free rate) per unit of risk is the one with the highest slope. The ratio of excess return to standard deviation of return for a portfolio p—the slope of the ray passing through p—is a single-number measure of a portfolio’s performance known as the Sharpe ratio, after its developer, William F. Sharpe. • Sharpe Ratio Formula. The Sharpe ratio for a portfolio p, based on historical returns, is defined as Rp − RF (3-16) Sh = sp where R p is the mean return to the portfolio, R F is the mean return to a risk-free asset, and sp is the standard deviation of return on the portfolio.36 The numerator of the Sharpe measure is the portfolio’s mean return minus the mean return on the risk-free asset over the sample period. The R p − R F term measures the extra reward that investors receive for the added risk taken. We call this difference the mean excess return on portfolio p. Thus the Sharpe ratio measures the reward, in terms of mean excess return, per unit of risk, as measured by standard deviation of return. Those risk-averse investors who make decisions only in terms of mean return and standard deviation of return prefer portfolios with larger Sharpe ratios to those with smaller Sharpe ratios. 36 The equation presents the ex post or historical Sharpe ratio. We can also think of the Sharpe ratio for a portfolio going forward based on our expectations for mean return, the risk-free return, and the standard deviation of return; this would be the ex ante Sharpe ratio. One may also encounter an alternative calculation for the Sharpe ratio in which the denominator is the standard deviation of the series (Portfolio return − Risk-free return) rather than the standard deviation of portfolio return; in practice, the two standard deviation calculations generally yield very similar results. For more information on the Sharpe ratio (which has also been called the Sharpe measure, the reward-to-variability ratio, and the excess return to variability measure), see Elton, Gruber, Brown, and Goetzmann (2003) and Sharpe (1994). 116 Quantitative Investment Analysis To illustrate the calculation of the Sharpe ratio, consider the performance of the S&P 500 and U.S. small stocks during the 1926–2002 period, as given previously in Table 3-24. Using the mean U.S. T-bill return to represent the risk-free rate, we find 12.3 − 3.8 = 0.39 21.9 16.9 − 3.8 = 0.37 U.S. small stocks: Sh = 35.1 S&P 500: Sh = Although U.S. small stocks earned higher mean returns, they performed slightly less well than the S&P 500, as measured by the Sharpe ratio. The Sharpe ratio is a mainstay of performance evaluation. We must issue two cautions concerning its use, one related to interpreting negative Sharpe ratios and the other to conceptual limitations. Finance theory tells us that in the long run, investors should be compensated with additional mean return above the risk-free rate for bearing additional risk, at least if the risky portfolio is well diversified. If investors are so compensated, the numerator of the Sharpe ratio will be positive. Nevertheless, we often find that portfolios exhibit negative Sharpe ratios when the ratio is calculated over periods in which bear markets for equities dominate. This raises a caution when dealing with negative Sharpe ratios. With positive Sharpe ratios, a portfolio’s Sharpe ratio decreases if we increase risk, all else equal. That result is intuitive for a risk-adjusted performance measure. With negative Sharpe ratios, however, increasing risk results in a numerically larger Sharpe ratio (for example, doubling risk may increase the Sharpe ratio from −1 to −0.5). Therefore, in a comparison of portfolios with negative Sharpe ratios, we cannot generally interpret the larger Sharpe ratio (the one closer to zero) to mean better risk-adjusted performance.37 Practically, to make an interpretable comparison in such cases using the Sharpe ratio, we may need to increase the evaluation period such that one or more of the Sharpe ratios becomes positive; we might also consider using a different performance evaluation metric. The conceptual limitation of the Sharpe ratio is that it considers only one aspect of risk, standard deviation of return. Standard deviation is most appropriate as a risk measure for portfolio strategies with approximately symmetric return distributions. Strategies with option elements have asymmetric returns. Relatedly, an investment strategy may produce frequent small gains but have the potential for infrequent but extremely large losses.38 Such a strategy is sometimes described as picking up coins in front of a bulldozer; for example, some hedge fund strategies tend to produce that return pattern. Calculated over a period in which the strategy is working (a large loss has not occurred), this type of strategy would have a high Sharpe ratio. In this case, the Sharpe ratio would give an overly optimistic picture of risk-adjusted performance because standard deviation would incompletely measure the risk assumed.39 Therefore, before applying the Sharpe ratio to evaluate a manager, we should judge whether standard deviation adequately describes the risk of the manager’s investment strategy. 37 If the standard deviations are equal, however, the portfolio with the negative Sharpe ratio closer to zero is superior. 38 This statement describes a return distribution with negative skewness. We discuss skewness later in this chapter. 39 For more information, see Amin and Kat (2003). 117 Chapter 3 Statistical Concepts and Market Returns Example 3-15 illustrates the calculation of the Sharpe ratio in a portfolio performance evaluation context. EXAMPLE 3-15 Calculating the Sharpe Ratio In earlier examples, we computed the various statistics for two mutual funds, Selected American Shares (SLASX) and T. Rowe Price Equity Income (PRFDX), for a five-year period ending in December 2002. Table 3-25 summarizes selected statistics for these two mutual funds for a longer period, the 10-year period ending in 2002. TABLE 3-25 Mutual Fund Mean Return and Standard Deviation of Return, 1993–2002 Fund SLASX PRFDX Arithmetic Mean Standard Deviation of Return 12.58% 11.64% 19.44% 13.65% Source: AAII. The U.S. 30-day T-bill rate is frequently used as a proxy for the risk-free rate. Table 3-26 gives the annual return on T-bills for the 1993–2002 period. TABLE 3-26 Annualized U.S. 30-Day T-Bill Rates of Return, 1993–2002 Year Return 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2.90% 3.90% 5.60% 5.21% 5.26% 4.86% 4.68% 5.89% 3.83% 1.65% Source: Ibbotson Associates. Using the information in Tables 3-25 and 3-26, address the following: 1. Calculate the Sharpe ratios for SLASX and PRFDX during the 1993–2002 period. 118 Quantitative Investment Analysis 2. State which fund had superior risk-adjusted performance during this period, as measured by the Sharpe ratio. Solution to 1: We already have in hand the means of the portfolio return and standard deviations of returns. The mean annual risk-free rate of return from 1993 to 2002, using U.S. T-bills as a proxy, is (2.90 + 3.90 + 5.60 + 5.21 + 5.26 + 4.86 + 4.68 + 5.89 + 3.83 + 1.65)/10 = 43.78/10 = 4.38 percent. 12.58 − 4.38 = 0.42 19.44 11.64 − 4.38 = 0.53 = 13.65 SLASX: Sh,SLASX = PRFDX: Sh,PRFDX Solution to 2: PRFDX had a higher positive Sharpe ratio than SLASX during the period. As measured by the Sharpe ratio, PRFDX’s performance was superior. 8. SYMMETRY AND SKEWNESS IN RETURN DISTRIBUTIONS Mean and variance may not adequately describe an investment’s distribution of returns. In calculations of variance, for example, the deviations around the mean are squared, so we do not know whether large deviations are likely to be positive or negative. We need to go beyond measures of central tendency and dispersion to reveal other important characteristics of the distribution. One important characteristic of interest to analysts is the degree of symmetry in return distributions. If a return distribution is symmetrical about its mean, then each side of the distribution is a mirror image of the other. Thus equal loss and gain intervals exhibit the same frequencies. Losses from −5 percent to −3 percent, for example, occur with about the same frequency as gains from 3 percent to 5 percent. One of the most important distributions is the normal distribution, depicted in Figure 36. This symmetrical, bell-shaped distribution plays a central role in the mean–variance model of portfolio selection; it is also used extensively in financial risk management. The normal distribution has the following characteristics: Its mean and median are equal. It is completely described by two parameters—its mean and variance. • Roughly 68 percent of its observations lie between plus and minus one standard deviation from the mean; 95 percent lie between plus and minus two standard deviations; and 99 percent lie between plus and minus three standard deviations. • • A distribution that is not symmetrical is called skewed. A return distribution with positive skew has frequent small losses and a few extreme gains. A return distribution with negative skew has frequent small gains and a few extreme losses. Figure 3-7 shows positively and negatively skewed distributions. The positively skewed distribution shown has a long tail on its 119 Chapter 3 Statistical Concepts and Market Returns Probability = 68.3% !5 !4 !3 !2 !1 0 EV 1 2 3 4 5 4 5 4 5 Standard Deviations Probability = 95.5% !5 !4 !3 !2 !1 0 EV 1 2 3 Standard Deviations Probability = 99.7% !5 !4 !3 !2 !1 0 EV 1 2 3 Standard Deviations FIGURE 3-6 Properties of a Normal Distribution (EV = Expected Value) Source: Reprinted with permission from Fixed Income Readings for the Chartered Financial Analyst® Program. Copyright 2000, Frank J. Fabozzi Associates, New Hope, PA. 120 Distribution Skewed to the Right (Positively Skewed) Quantitative Investment Analysis Distribution Skewed to the Left (Negatively Skewed) FIGURE 3-7 Properties of a Skewed Distribution Source: Reprinted with permission from Fixed Income Readings for the Chartered Financial Analyst®Program. Copyright 2000, Frank J. Fabozzi Associates, New Hope, PA. right side; the negatively skewed distribution has a long tail on its left side. For the positively skewed unimodal distribution, the mode is less than the median, which is less than the mean. For the negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode.40 Investors should be attracted by a positive skew because the mean return falls above the median. Relative to the mean return, positive skew amounts to a limited, though frequent, downside compared with a somewhat unlimited, but less frequent, upside. Skewness is the name given to a statistical measure of skew. (The word ‘‘skewness’’ is also sometimes used interchangeably for ‘‘skew.’’) Like variance, skewness is computed using each observation’s deviation from its mean. Skewness (sometimes referred to as relative skewness) is computed as the average cubed deviation from the mean standardized by dividing by the standard deviation cubed to make the measure free of scale.41 A symmetric distribution has skewness of 0, a positively skewed distribution has positive skewness, and a negatively skewed distribution has negative skewness, as given by this measure. We can illustrate the principle behind the measure by focusing on the numerator. Cubing, unlike squaring, preserves the sign of the deviations from the mean. If a distribution is positively skewed with a mean greater than its median, then more than half of the deviations from the mean are negative and less than half are positive. In order for the sum to be positive, the losses must be small and likely, and the gains less likely but more extreme. Therefore, if skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations. A simple example illustrates that a symmetrical distribution has a skewness measure equal to 0. Suppose we have the following data: 1, 2, 3, 4, 5, 6, 7, 8, and 9. The mean outcome is 5, and the deviations are −4, −3, −2, −1, 0, 1, 2, 3, and 4. Cubing the deviations yields −64, −27, −8, −1, 0, 1, 8, 27, and 64, with a sum of 0. The numerator of skewness (and so skewness itself) is thus equal to 0, supporting our claim. Below we give the formula for computing skewness from a sample. 40 As a mnemonic, in this case the mean, median, and mode occur in the same order as they would be listed in a dictionary. 41 We are discussing a moment coefficient of skewness. Some textbooks present the Pearson coefficient of skewness, equal to 3(Mean − Median)/Standard deviation, which has the drawback of involving the calculation of the median. 121 Chapter 3 Statistical Concepts and Market Returns TABLE 3-27 S&P 500 Annual and Monthly Total Returns, 1926–2002: Summary Statistics Return Series S&P 500 (Annual) S&P 500 (Monthly) Number of Periods Arithmetic Mean Standard Deviation Skewness Excess Kurtosis 77 924 12.20% 0.97% 20.49% 5.65% −0.2943 0.3964 −0.2207 9.4645 Source: Ibbotson EnCorr Analyzer. • Sample Skewness Formula. Sample skewness (also called sample relative skewness), SK , is n ! (Xi − X )3 + * n i=1 SK = (3-17) (n − 1)(n − 2) s3 where n is the number of observations in the sample and s is the sample standard deviation.42 The algebraic sign of Equation 3-17 indicates the direction of skew, with a negative SK indicating a negatively skewed distribution and a positive SK indicating a positively skewed distribution. Note that as n becomes large, the expression reduces to the mean cubed deviation, n ! (Xi − X )3 , 1 i=1 SK ≈ . As a frame of reference, for a sample size of 100 or larger taken n s3 from a normal distribution, a skewness coefficient of ±0.5 would be considered unusually large. Table 3-27 shows several summary statistics for the annual and monthly returns on the S&P 500. Earlier we discussed the arithmetic mean return and standard deviation of return, and we shall shortly discuss kurtosis. Table 3-27 reveals that S&P 500 annual returns during this period were negatively skewed while monthly returns were positively skewed, and the magnitude of skewness was greater for the monthly series. We would find for other market series that the shape of the distribution of returns often depends on the holding period examined. Some researchers believe that investors should prefer positive skewness, all else equal—that is, they should prefer portfolios with distributions offering a relatively large frequency of unusually large payoffs.43 Different investment strategies may tend to introduce different types and amounts of skewness into returns. Example 3-16 illustrates the calculation of skewness for a managed portfolio. EXAMPLE 3-16 Calculating Skewness for a Mutual Fund Table 3-28 presents 10 years of annual returns on the T. Rowe Price Equity Income Fund (PRFDX). 42 The term n/[(n − 1)(n − 2)] in Equation 3-17 corrects for a downward bias in small samples. more on the role of skewness in portfolio selection, see Reilly and Brown (2003) and Elton et al. (2003) and the references therein. 43 For 122 Quantitative Investment Analysis TABLE 3-28 Annual Rates of Return: T. Rowe Price Equity Income, 1993–2002 Year Return 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 14.8% 4.5% 33.3% 20.3% 28.8% 9.2% 3.8% 13.1% 1.6% −13.0% Source: AAII. Using the information in Table 3-28, address the following: 1. Calculate the skewness of PRFDX showing two decimal places. 2. Characterize the shape of the distribution of PRFDX returns based on your answer to Part 1. Solution to 1: To calculate skewness, we find the sum of the cubed deviations from the mean, divide by the standard deviation cubed, and then multiply that result by n/[(n − 1)(n − 2)]. Table 3-29 gives the calculations. TABLE 3-29 Calculating Skewness for PRFDX Year Rt Rt − R 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 14.8% 4.5% 33.3% 20.3% 28.8% 9.2% 3.8% 13.1% 1.6% −13.0% 3.16 −7.14 21.66 8.66 17.16 −2.44 −7.84 1.46 −10.04 −24.64 n= R= 10 11.64% s= 13.65% Source: AAII. Sum = s3 = Sum/s3 = n/[(n − 1)(n − 2)] = Skewness = (Rt − R)3 31.554 −363.994 10,161.910 649.462 5,053.030 −14.527 −481.890 3.112 −1,012.048 −14,959.673 −933.064 2,543.302 −0.3669 0.1389 −0.05 Chapter 3 Statistical Concepts and Market Returns 123 Using Equation 3-17, the calculation is: SK = * + −933.064 10 = −0.05 (9)(8) 13.653 In this example, five deviations are negative and five are positive. Two large positive deviations, in 1995 and 1997, are more than offset by a very large negative deviation in 2002 and a moderately large negative deviation in 2001, both bear market years. The result is that skewness is a very small negative number. Solution to 2: Based on this small sample, the distribution of annual returns for the fund appears to be approximately symmetric (or very slightly negatively skewed). The negative and positive deviations from the mean are equally frequent, and large positive deviations approximately offset large negative deviations. 9. KURTOSIS IN RETURN DISTRIBUTIONS In the previous section, we discussed how to determine whether a return distribution deviates from a normal distribution because of skewness. One other way in which a return distribution might differ from a normal distribution is by having more returns clustered closely around the mean (being more peaked) and more returns with large deviations from the mean (having fatter tails). Relative to a normal distribution, such a distribution has a greater percentage of small deviations from the mean return (more small surprises) and a greater percentage of extremely large deviations from the mean return (more big surprises). Most investors would perceive a greater chance of extremely large deviations from the mean as increasing risk. Kurtosis is the statistical measure that tells us when a distribution is more or less peaked than a normal distribution. A distribution that is more peaked than normal is called leptokurtic (lepto from the Greek word for slender); a distribution that is less peaked than normal is called platykurtic (platy from the Greek word for broad); and a distribution identical to the normal distribution in this respect is called mesokurtic (meso from the Greek word for middle). The situation of more-frequent extremely large surprises that we described is one of leptokurtosis.44 Figure 3-8 illustrates a leptokurtic distribution. It is more peaked and has fatter tails than the normal distribution. The calculation for kurtosis involves finding the average of deviations from the mean raised to the fourth power and then standardizing that average by dividing by the standard deviation raised to the fourth power.45 For all normal distributions, kurtosis is equal to 3. Many statistical packages report estimates of excess kurtosis, which is kurtosis minus 3.46 Excess kurtosis thus characterizes kurtosis relative to the normal distribution. A normal or other mesokurtic distribution has excess kurtosis equal to 0. A leptokurtic distribution has 44 Kurtosis has been described as an illness characterized by episodes of extremely rude behavior. measure is free of scale. It is always positive because the deviations are raised to the fourth power. 46 Ibbotson and some software packages, such as Microsoft Excel, label ‘‘excess kurtosis’’ as simply ‘‘kurtosis.’’ This highlights the fact that one should familiarize oneself with the description of statistical quantities in any software packages that one uses. 45 This 124 Quantitative Investment Analysis Fat Tails Distribution Normal Distribution !5 !4 !3 !2 !1 0 1 2 3 4 5 Standard Deviations FIGURE 3-8 Leptokurtic: Fat Tailed Source: Reprinted with permission from Fixed Income Readings for the Chartered Financial Analyst®Program. Copyright 2000, Frank J. Fabozzi Associates, New Hope, PA. excess kurtosis greater than 0, and a platykurtic distribution has excess kurtosis less than 0. A return distribution with positive excess kurtosis—a leptokurtic return distribution—has more frequent extremely large deviations from the mean than a normal distribution. Below is the expression for computing kurtosis from a sample. • Sample Excess Kurtosis Formula. The sample excess kurtosis is ⎛ ⎜ ⎜ n(n + 1) KE = ⎜ ⎜ (n − 1)(n − 2)(n − 3) ⎝ n ! i=1 4 ⎞ (Xi − X ) ⎟ 2 ⎟ ⎟ − 3(n − 1) ⎟ (n − 2)(n − 3) s4 ⎠ (3-18) where n is the sample size and s is the sample standard deviation. In Equation 3-18, sample kurtosis is the first term. Note !that as n becomes large, Equation 3! 4 2 2 (X − X )4 (X − X ) 3n 1 n − 2 = − 3. For a sample of 100 18 approximately equals 3 n n n s4 s4 or larger taken from a normal distribution, a sample excess kurtosis of 1.0 or larger would be considered unusually large. 125 Chapter 3 Statistical Concepts and Market Returns Most equity return series have been found to be leptokurtic. If a return distribution has positive excess kurtosis (leptokurtosis) and we use statistical models that do not account for the fatter tails, we will underestimate the likelihood of very bad or very good outcomes. For example, the return on the S&P 500 for 19 October 1987 was 20 standard deviations away from the mean daily return. Such an outcome is possible with a normal distribution, but its likelihood is almost equal to 0. If daily returns are drawn from a normal distribution, a return four standard deviations or more away from the mean is expected once every 50 years; a return greater than five standard deviations away is expected once every 7,000 years. The return for October 1987 is more likely to have come from a distribution that had fatter tails than from a normal distribution. Looking at Table 3-27 given earlier, the monthly return series for the S&P 500 has very large excess kurtosis, approximately 9.5. It is extremely fat-tailed relative to the normal distribution. By contrast, the annual return series has very slightly negative excess kurtosis (roughly −0.2). The results for excess kurtosis in the table are consistent with research findings that the normal distribution is a better approximation for U.S. equity returns for annual holding periods than for shorter ones (such as monthly).47 The following example illustrates the calculations for sample excess kurtosis for one of the two mutual funds we have been examining. EXAMPLE 3-17 Calculating Sample Excess Kurtosis Having concluded in Example 3-16 that the annual returns on T. Rowe Price Equity Income Fund were approximately symmetrically distributed during the 1993–2002 period, what can we say about the kurtosis of the fund’s return distribution? Table 3-28 (repeated below) recaps the annual returns for the fund. TABLE 3-28 (repeated) Annual Rates of Return: T. Rowe Price Equity Income, 1993–2002 Year Return 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 14.8% 4.5% 33.3% 20.3% 28.8% 9.2% 3.8% 13.1% 1.6% −13.0% Source: AAII. 47 See Campbell, Lo, and MacKinlay (1997) for more details. 126 Quantitative Investment Analysis Using the information from Table 3-28 repeated above, address the following: 1. Calculate the sample excess kurtosis of PRFDX showing two decimal places. 2. Characterize the shape of the distribution of PRFDX returns based on your answer to Part 1 as leptokurtic, mesokurtic, or platykurtic. Solution to 1: To calculate excess kurtosis, we find the sum of the deviations from the mean raised to the fourth power, divide by the standard deviation raised to the fourth power, and then multiply that result by n(n + 1)/[(n − 1)(n − 2)(n − 3)]. This calculation determines kurtosis. Excess kurtosis is kurtosis minus 3(n − 1)2 /[(n − 2)(n − 3)]. Table 3-30 gives the calculations. TABLE 3-30 Calculating Kurtosis for PRFDX Year Rt Rt − R (Rt − R)4 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 14.8% 4.5% 33.3% 20.3% 28.8% 9.2% 3.8% 13.1% 1.6% −13.0% 3.16 −7.14 21.66 8.66 17.16 −2.44 −7.84 1.46 −10.04 −24.64 99.712 2,598.920 220,106.977 5,624.340 86,709.990 35.445 3,778.020 4.544 10,160.963 368,606.351 n= R= 10 11.64% Sum = s4 = Sum/s4 = n(n + 1)/[(n − 1)(n − 2)(n − 3)] = Kurtosis = 3(n − 1)2 /[(n − 2)(n − 3)] = Excess Kurtosis = s= 13.65% Source: AAII. Using Equation 3-18, the calculation is: KE = * + 3(9)2 110 697,725.261 − 4 (9)(8)(7) 13.65 (8)(7) = 4.39 − 4.34 = 0.05 697,725.261 34,716.074 20.098 0.2183 4.39 4.34 0.05 Chapter 3 Statistical Concepts and Market Returns 127 Solution to 2: The distribution of PRFDX’s annual returns appears to be mesokurtic, based on a sample excess kurtosis close to zero. With skewness and excess kurtosis both close to zero, PRFDX’s annual returns appear to have been approximately normally distributed during the period.48 10. USING GEOMETRIC AND ARITHMETIC MEANS With the concepts of descriptive statistics in hand, we will see why the geometric mean is appropriate for making investment statements about past performance. We will also explore why the arithmetic mean is appropriate for making investment statements in a forward-looking context. For reporting historical returns, the geometric mean has considerable appeal because it is the rate of growth or return we would have had to earn each year to match the actual, cumulative investment performance. In our simplified Example 3-8, for instance, we purchased a stock for ¤100 and two years later it was worth ¤100, with an intervening year at ¤200. The geometric mean of 0 percent is clearly the compound rate of growth during the two years. Specifically, the ending amount is the beginning amount times (1 + RG )2 . The geometric mean is an excellent measure of past performance. Example 3-8 illustrated how the arithmetic mean can distort our assessment of historical performance. In that example, the total performance for the two-year period was unambiguously 0 percent. With a 100 percent return for the first year and −50 percent for the second, however, the arithmetic mean was 25 percent. As we noted previously, the arithmetic mean is always greater than or equal to the geometric mean. If we want to estimate the average return over a one-period horizon, we should use the arithmetic mean because the arithmetic mean is the average of one-period returns. If we want to estimate the average returns over more than one period, however, we should use the geometric mean of returns because the geometric mean captures how the total returns are linked over time. As a corollary to using the geometric mean for performance reporting, the use of semilogarithmic rather than arithmetic scales is more appropriate when graphing past performance.49 In the context of reporting performance, a semilogarithmic graph has an arithmetic scale on the horizontal axis for time and a logarithmic scale on the vertical axis for the value of the investment. The vertical axis values are spaced according to the differences between their logarithms. Suppose we want to represent £1, £10, £100, and £1,000 as values of an investment on the vertical axis. Note that each successive value represents a 10-fold increase over the previous value, and each will be equally spaced on the vertical axis because the difference in their logarithms is roughly 2.30; that is, ln 10 − ln 1 = ln 100 − ln 10 = ln 1,000 − ln 100 = 2.30. On a semilogarithmic scale, equal 48 It is useful to know that we can conduct a Jarque-Bera (JB) statistical test of normality based on sample size n, sample skewness, and sample excess kurtosis. We can conclude that a distribution is not normal with no more than a 5 percent chance of being wrong if the quantity JB = n[(SK2 /6) + (KE2 /24)] is 6 or greater for a sample with at least 30 observations. In this mutual fund example, we have only 10 observations and the test described is only correct based on large samples (as a guideline, for n ≥ 30). Gujarati (2003) provides more details on this test. 49 See Campbell (1974) for more information. 128 Quantitative Investment Analysis movements on the vertical axis reflect equal percentage changes, and growth at a constant compound rate plots as a straight line. A plot curving upward reflects increasing growth rates over time. The slopes of a plot at different points may be compared in order to judge relative growth rates. In addition to reporting historical performance, financial analysts need to calculate expected equity risk premiums in a forward-looking context. For this purpose, the arithmetic mean is appropriate. We can illustrate the use of the arithmetic mean in a forward-looking context with an example based on an investment’s future cash flows. In contrasting the geometric and arithmetic means for discounting future cash flows, the essential issue concerns uncertainty. Suppose an investor with $100,000 faces an equal chance of a 100 percent return or a −50 percent return, represented on the tree diagram as a 50/50 chance of a 100 percent return or a −50 percent return per period. With 100 percent √ return in one period and −50 percent return in the other, the geometric mean return is 2(0.5) − 1 = 0. The geometric mean return of 0 percent gives the mode or median of ending wealth after two periods and thus accurately predicts the modal or median ending wealth of $100,000 in this example. Nevertheless, the arithmetic mean return better predicts the arithmetic mean ending wealth. With equal chances of 100 percent or −50 percent returns, consider the four equally likely outcomes of $400,000, $100,000, $100,000, and $25,000 as if they actually occurred. The arithmetic mean ending wealth would be $156,250 = ($400,000 + $100,000 + $100,000 + $25,000)/4. The actual returns would be 300 percent, 0 percent, 0 percent, and −75 percent for a two-period arithmetic mean return of (300 + 0 + 0 − 75)/4 = 56.25 percent. This arithmetic mean return predicts the arithmetic mean ending wealth of $100,000 × 1.5625 = $156,250. Noting that 56.25 percent for two periods is 25 percent per period, we then must discount the expected terminal wealth of $156,250 at the 25 percent arithmetic mean rate to reflect the uncertainty in the cash flows. Uncertainty in cash flows or returns causes the arithmetic mean to be larger than the geometric mean. The more uncertain the returns, the more divergence exists between the arithmetic and geometric means. The geometric mean return approximately equals the arithmetic return minus half the variance of return.50 Zero variance or zero uncertainty in returns would leave the geometric and arithmetic returns approximately equal, but real-world uncertainty presents an arithmetic mean return larger than the geometric. For example, Dimson et al. (2002) reported that from 1900 to 2000, U.S. equities had nominal annual returns with an arithmetic mean of 12 percent and standard deviation of 19.9 percent. They reported the geometric mean as 10.1 percent. We can see the geometric mean is approximately the arithmetic mean minus half of the variance of returns: RG ≈ 0.12 − (1/2)(0.1992 ) = 0.10. 50 See Bodie, Kane, and Marcus (2001). CHAPTER 4 PROBABILITY CONCEPTS 1. INTRODUCTION All investment decisions are made in an environment of risk. The tools that allow us to make decisions with consistency and logic in this setting come under the heading of probability. This chapter presents the essential probability tools needed to frame and address many real-world problems involving risk. We illustrate how these tools apply to such issues as predicting investment manager performance, forecasting financial variables, and pricing bonds so that they fairly compensate bondholders for default risk. Our focus is practical. We explore in detail the concepts that are most important to investment research and practice. One such concept is independence, as it relates to the predictability of returns and financial variables. Another is expectation, as analysts continually look to the future in their analyses and decisions. Analysts and investors must also cope with variability. We present variance, or dispersion around expectation, as a risk concept important in investments. The reader will acquire specific skills in using portfolio expected return and variance. The basic tools of probability, including expected value and variance, are set out in Section 2 of this chapter. Section 3 introduces covariance and correlation (measures of relatedness between random quantities) and the principles for calculating portfolio expected return and variance. Two topics end the chapter: Bayes’ formula and outcome counting. Bayes’ formula is a procedure for updating beliefs based on new information. In several areas, including a widely used option-pricing model, the calculation of probabilities involves defining and counting outcomes. The chapter ends with a discussion of principles and shortcuts for counting. 2. PROBABILITY, EXPECTED VALUE, AND VARIANCE The probability concepts and tools necessary for most of an analyst’s work are relatively few and simple but require thought to apply. This section presents the essentials for working with probability, expectation, and variance, drawing on examples from equity and fixed income analysis. An investor’s concerns center on returns. The return on a risky asset is an example of a random variable, a quantity whose outcomes (possible values) are uncertain. For example, a portfolio may have a return objective of 10 percent a year. The portfolio manager’s focus at the moment may be on the likelihood of earning a return that is less than 10 percent over the next year. Ten percent is a particular value or outcome of the random variable ‘‘portfolio return.’’ Although we may be concerned about a single outcome, frequently our interest may be in a set of outcomes: The concept of ‘‘event’’ covers both. 129 130 • Quantitative Investment Analysis Definition of Event. An event is a specified set of outcomes. We may specify an event to be a single outcome—for example, the portfolio earns a return of 10 percent. (We use italics to highlight statements that define events.) We can capture the portfolio manager’s concerns by defining the event as the portfolio earns a return below 10 percent. This second event, referring as it does to all possible returns greater than or equal to −100 percent (the worst possible return) but less than 10 percent, contains an infinite number of outcomes. To save words, it is common to use a capital letter in italics to represent a defined event. We could define A = the portfolio earns a return of 10 percent and B = the portfolio earns a return below 10 percent. To return to the portfolio manager’s concern, how likely is it that the portfolio will earn a return below 10 percent? The answer to this question is a probability: a number between 0 and 1 that measures the chance that a stated event will occur. If the probability is 0.40 that the portfolio earns a return below 10 percent, there is a 40 percent chance of that event happening. If an event is impossible, it has a probability of 0. If an event is certain to happen, it has a probability of 1. If an event is impossible or a sure thing, it is not random at all. So, 0 and 1 bracket all the possible values of a probability. Probability has two properties, which together constitute its definition. • Definition of Probability. The two defining properties of a probability are as follows: 1. The probability of any event E is a number between 0 and 1: 0 ≤ P(E) ≤ 1. 2. The sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1. P followed by parentheses stands for ‘‘the probability of (the event in parentheses),’’ as in P(E) for ‘‘the probability of event E.’’ We can also think of P as a rule or function that assigns numerical values to events consistent with Properties 1 and 2. In the above definition, the term mutually exclusive means that only one event can occur at a time; exhaustive means that the events cover all possible outcomes. The events A = the portfolio earns a return of 10 percent and B = the portfolio earns a return below 10 percent are mutually exclusive because A and B cannot both occur at the same time. For example, a return of 8.1 percent means that B has occurred and A has not occurred. Although events A and B are mutually exclusive, they are not exhaustive because they do not cover outcomes such as a return of 11 percent. Suppose we define a third event: C = the portfolio earns a return above 10 percent. Clearly, A, B, and C are mutually exclusive and exhaustive events. Each of P(A), P(B), and P(C) is a number between 0 and 1, and P(A) + P(B) + P(C) = 1. The most basic kind of mutually exclusive and exhaustive events is the set of all the distinct possible outcomes of the random variable. If we know both that set and the assignment of probabilities to those outcomes—the probability distribution of the random variable—we have a complete description of the random variable, and we can assign a probability to any event that we might describe.1 The probability of any event is the sum of the probabilities of the distinct outcomes included in the definition of the event. Suppose the event of interest is D = the portfolio earns a return above the risk-free rate, and we know the probability distribution of portfolio returns. Assume the risk-free rate is 4 percent. To calculate P(D), the probability 1 In the chapter on common probability distributions, we describe some of the probability distributions most frequently used in investment applications. Chapter 4 Probability Concepts 131 of D, we would sum the probabilities of the outcomes that satisfy the definition of the event; that is, we would sum the probabilities of portfolio returns greater than 4 percent. Earlier, to illustrate a concept, we assumed a probability of 0.40 for a portfolio earning less than 10 percent, without justifying the particular assumption. We also talked about using a probability distribution of outcomes to calculate the probability of events, without explaining how a probability distribution might be estimated. Making actual financial decisions using inaccurate probabilities might have grave consequences. How, in practice, do we estimate probabilities? This topic is a field of study in itself, but there are three broad approaches to estimating probabilities. In investments, we often estimate the probability of an event as a relative frequency of occurrence based on historical data. This method produces an empirical probability. For example, Amihud and Li (2002) report that of their sample of 16,189 dividend changes for NYSE and Amex stocks during the years 1962 to 2000, 14,911 were increases and 1,278 were decreases. The empirical probability that a dividend change is a dividend decrease for U.S. stocks is thus 1,278/16,189 = 0.08, approximately. We will point out empirical probabilities in several places as they appear in this chapter. Relationships must be stable through time for empirical probabilities to be accurate. We cannot calculate an empirical probability of an event not in the historical record or a reliable empirical probability for a very rare event. There are cases, then, in which we may adjust an empirical probability to account for perceptions of changing relationships. In other cases, we have no empirical probability to use at all. We may also make a personal assessment of probability without reference to any particular data. Each of these three types of probability is a subjective probability, one drawing on personal or subjective judgment. Subjective probabilities are of great importance in investments. Investors, in making buy and sell decisions that determine asset prices, often draw on subjective probabilities. Subjective probabilities appear in various places in this chapter, notably in our discussion of Bayes’ formula. In a more narrow range of well-defined problems, we can sometimes deduce probabilities by reasoning about the problem. The resulting probability is an a priori probability, one based on logical analysis rather than on observation or personal judgment. We will use this type of probability in Example 4-6. The counting methods we discuss later are particularly important in calculating an a priori probability. Because a priori and empirical probabilities generally do not vary from person to person, they are often grouped as objective probabilities. In business and elsewhere, we often encounter probabilities stated in terms of odds—for instance, ‘‘the odds for E’’ or the ‘‘odds against E.’’ For example, as of mid-2003, analysts’ fiscal year 2004 EPS forecasts for one Toronto Stock Exchange—listed company ranged from C$3.98 to C$4.25. Nevertheless, one analyst asserts that the odds for the company beating the highest estimate, C$4.25, are 1 to 7. A second analyst argues that the odds against that happening are 15 to 1. What do those statements imply about the probability of the company’s EPS beating the highest estimate? We interpret probabilities stated in terms of odds as follows: • Probability Stated as Odds. Given a probability P(E), 1. Odds for E = P(E)/[1 − P(E)]. The odds for E are the probability of E divided by 1 minus the probability of E. Given odds for E of ‘‘a to b,’’ the implied probability of E is a/(a + b). In the example, the statement that the odds for the company’s EPS for FY2004 beating C$4.25 are 1 to 7 means that the speaker believes the probability of the event is 1/(1 + 7) = 1/8 = 0.125. 2. Odds against E = [1 − P(E)]/P(E), the reciprocal of odds for E. Given odds against E of ‘‘a to b,’’ the implied probability of E is b/(a + b). 132 Quantitative Investment Analysis The statement that the odds against the company’s EPS for FY2004 beating C$4.25 are 15 to 1 is consistent with a belief that the probability of the event is 1/(1 + 15) = 1/16 = 0.0625. To further explain odds for an event, if P(E) = 1/8, the odds for E are (1/8)/(7/8) = (1/8)(8/7) = 1/7, or ‘‘1 to 7.’’ For each occurrence of E, we expect seven cases of nonoccurrence; out of eight cases in total, therefore, we expect E to happen once, and the probability of E is 1/8. In wagering, it is common to speak in terms of the odds against something, as in Statement 2. For odds of ‘‘15 to 1’’ against E (an implied probability of E of 1/16), a $1 wager on E, if successful, returns $15 in profits plus the $1 staked in the wager. We can calculate the bet’s anticipated profit as follows: Win: Probability = 1/16; Profit = $15 Loss: Probability = 15/16; Profit = −$1 Anticipated profit = (1/16)($15) + (15/16)(−$1) = $0 Weighting each of the wager’s two outcomes by the respective probability of the outcome, if the odds (probabilities) are accurate, the anticipated profit of the bet is $0. EXAMPLE 4-1 Profiting from Inconsistent Probabilities You are examining the common stock of two companies in the same industry in which an important antitrust decision will be announced next week. The first company, SmithCo Corporation, will benefit from a governmental decision that there is no antitrust obstacle related to a merger in which it is involved. You believe that SmithCo’s share price reflects a 0.85 probability of such a decision. A second company, Selbert Corporation, will equally benefit from a ‘‘go ahead’’ ruling. Surprisingly, you believe Selbert stock reflects only a 0.50 probability of a favorable decision. Assuming your analysis is correct, what investment strategy would profit from this pricing discrepancy? Consider the logical possibilities. One is that the probability of 0.50 reflected in Selbert’s share price is accurate. In that case, Selbert is fairly valued but SmithCo is overvalued, as its current share price overestimates the probability of a ‘‘go ahead’’ decision. The second possibility is that the probability of 0.85 is accurate. In that case, SmithCo shares are fairly valued, but Selbert shares, which build in a lower probability of a favorable decision, are undervalued. You diagram the situation as shown in Table 4-1. TABLE 4-1 Worksheet for Investment Problem True Probability of a ‘‘Go Ahead’’ Decision 0.50 SmithCo Selbert Shares Overvalued Shares Fairly Valued 0.85 Shares Fairly Valued Shares Undervalued The 0.50 probability column shows that Selbert shares are a better value than SmithCo shares. Selbert shares are also a better value if a 0.85 probability is accurate. Thus SmithCo shares are overvalued relative to Selbert shares. Chapter 4 Probability Concepts 133 Your investment actions depend on your confidence in your analysis and on any investment constraints you face (such as constraints on selling stock short).2 A conservative strategy would be to buy Selbert shares and reduce or eliminate any current position in SmithCo. The most aggressive strategy is to short SmithCo stock (relatively overvalued) and simultaneously buy the stock of Selbert (relatively undervalued). This strategy is known as a pairs arbitrage trade: a trade in two closely related stocks involving the short sale of one and the purchase of the other. The prices of SmithCo and Selbert shares reflect probabilities that are not consistent. According to one of the most important probability results for investments, the Dutch Book Theorem,3 inconsistent probabilities create profit opportunities. In our example, investors, by their buy and sell decisions to exploit the inconsistent probabilities, should eliminate the profit opportunity and inconsistency. To understand the meaning of a probability in investment contexts, we need to distinguish between two types of probability: unconditional and conditional. Both unconditional and conditional probabilities satisfy the definition of probability stated earlier, but they are calculated or estimated differently and have different interpretations. They provide answers to different questions. The probability in answer to the straightforward question ‘‘What is the probability of this event A?’’ is an unconditional probability, denoted P(A). Unconditional probabilities are also frequently referred to as marginal probabilities.4 Suppose the question is ‘‘What is the probability that the stock earns a return above the risk-free rate (event A)?’’ The answer is an unconditional probability that can be viewed as the ratio of two quantities. The numerator is the sum of the probabilities of stock returns above the risk-free rate. Suppose that sum is 0.70. The denominator is 1, the sum of the probabilities of all possible returns. The answer to the question is P(A) = 0.70. Contrast the question ‘‘What is the probability of A?’’ with the question ‘‘What is the probability of A, given that B has occurred?’’ The probability in answer to this last question is a conditional probability, denoted P(A | B) (read: ‘‘the probability of A given B’’). Suppose we want to know the probability that the stock earns a return above the risk-free rate (event A), given that the stock earns a positive return (event B). With the words ‘‘given that,’’ we are restricting returns to those larger than 0 percent—a new element in contrast to the question that brought forth an unconditional probability. The conditional probability is calculated as the ratio of two quantities. The numerator is the sum of the probabilities of stock 2 Selling short or shorting stock means selling borrowed shares in the hope of repurchasing them later at a lower price. 3 The theorem’s name comes from the terminology of wagering. Suppose someone places a $100 bet on X at odds of 10 to 1 against X , and later he is able to place a $600 bet against X at odds of 1 to 1 against X . Whatever the outcome of X , that person makes a riskless profit (equal to $400 if X occurs or $500 if X does not occur) because the implied probabilities are inconsistent. He is said to have made a Dutch book in X . Ramsey (1931) presented the problem of inconsistent probabilities. See also Lo (1999). 4 In analyses of probabilities presented in tables, unconditional probabilities usually appear at the ends or margins of the table, hence the term marginal probability. Because of possible confusion with the way marginal is used in economics (roughly meaning incremental), we use the term unconditional probability throughout this discussion. 134 Quantitative Investment Analysis returns above the risk-free rate; in this particular case, the numerator is the same as it was in the unconditional case, which we gave as 0.70. The denominator, however, changes from 1 to the sum of the probabilities for all outcomes (returns) above 0 percent. Suppose that number is 0.80, a larger number than 0.70 because returns between 0 and the risk-free rate have some positive probability of occurring. Then P(A | B) = 0.70/0.80 = 0.875. If we observe that the stock earns a positive return, the probability of a return above the risk-free rate is greater than the unconditional probability, which is the probability of the event given no other information. The result is intuitive.5 To review, an unconditional probability is the probability of an event without any restriction; it might even be thought of as a stand-alone probability. A conditional probability, in contrast, is a probability of an event given that another event has occurred. In discussing approaches to calculating probability, we gave one empirical estimate of the probability that a change in dividends is a dividend decrease. That probability was an unconditional probability. Given additional information on company characteristics, could an investor refine that estimate? Investors continually seek an information edge that will help improve their forecasts. In mathematical terms, they are attempting to frame their view of the future using probabilities conditioned on relevant information or events. Investors do not ignore useful information; they adjust their probabilities to reflect it. Thus, the concepts of conditional probability (which we analyze in more detail below), as well as related concepts discussed further on, are extremely important in investment analysis and financial markets. To state an exact definition of conditional probability, we first need to introduce the concept of joint probability. Suppose we ask the question ‘‘What is the probability of both A and B happening?’’ The answer to this question is a joint probability, denoted P(AB) (read: ‘‘the probability of A and B’’). If we think of the probability of A and the probability of B as sets built of the outcomes of one or more random variables, the joint probability of A and B is the sum of the probabilities of the outcomes they have in common. For example, consider two events: the stock earns a return above the risk-free rate (A) and the stock earns a positive return (B). The outcomes of A are contained within (a subset of) the outcomes of B, so P(AB) equals P(A). We can now state a formal definition of conditional probability that provides a formula for calculating it. • Definition of Conditional Probability. The conditional probability of A given that B has occurred is equal to the joint probability of A and B divided by the probability of B (assumed not to equal 0). P(A | B) = P(AB)/P(B), P(B) ̸= 0 (4-1) Sometimes we know the conditional probability P(A | B) and we want to know the joint probability P(AB). We can obtain the joint probability from the following multiplication rule for probabilities, which is Equation 4-1 rearranged. • Multiplication Rule for Probability. The joint probability of A and B can be expressed as P(AB) = P(A | B)P(B) (4-2) 5 In this example, the conditional probability is greater than the unconditional probability. The conditional probability of an event may, however, be greater than, equal to, or less than the unconditional probability, depending on the facts. For instance, the probability that the stock earns a return above the risk-free rate given that the stock earns a negative return is 0. 135 Chapter 4 Probability Concepts Equation 4-2 states that the joint probability of A and B equals the probability of A given B times the probability of B. Because P(AB) = P(BA), the expression P(AB) = P(BA) = P(B | A)P(A) is equivalent to Equation 4-2. EXAMPLE 4-2 Conditional Probabilities and Predictability of Mutual Fund Performance (1) Kahn and Rudd (1995) examined whether historical performance predicts future performance for a sample of mutual funds that included 300 actively managed U.S. domestic equity funds. One approach they used involved calculating each fund’s exposure to a set of style indexes (the term ‘‘style’’ captures the distinctions of growth/value and large-capitalization/mid-capitalization/small-capitalization). After establishing a style benchmark (a comparison portfolio matched to the fund’s style) for each fund, Kahn and Rudd computed the fund’s selection return for two periods. They defined selection return as fund return minus the fund’s style-benchmark return. The first period was October 1990 to March 1992. The top 50 percent of funds by selection return for that period were labeled winners; the bottom 50 percent were labeled losers. Based on selection return in the next period, April 1992 to September 1993, the top 50 percent of funds were tagged as winners and the bottom 50 percent as losers for that period. An excerpt from their results is given in Table 4-2. The winner–winner entry, for example, shows that 79 of the 150 first-period winner funds were also winners in the second period (52.7% = 79/150). Note that the four entries in parentheses in the table can be viewed as conditional probabilities. TABLE 4-2 Equity Selection Returns Period 1: October 1990 to March 1992 Period 2: April 1992 to September 1993 Entries are number of funds (percent of row total in parentheses) Period 1 Winner Period 1 Loser Period 2 Winner Period 2 Loser 79 (52.7%) 71 (47.3%) 71 (47.3%) 79 (52.7%) Source: Kahn and Rudd (1995), Table 3. Based on the data in Table 4-2, answer the following questions: 1. State the four events needed to define the four conditional probabilities. 2. State the four entries of the table as conditional probabilities using the form P(this event | that event) = number. 3. Are the conditional probabilities in Part 2 empirical, a priori, or subjective probabilities? 136 Quantitative Investment Analysis 4. Using information in the table, calculate the probability of the event a fund is a loser in both Period 1 and Period 2. (Note that because 50 percent of funds are categorized as losers in each period, the unconditional probability that a fund is labeled a loser in either period is 0.5.) Solution to 1: The four events needed to define the conditional probabilities are as follows: Fund is a Period 1 winner Fund is a Period 1 loser Fund is a Period 2 loser Fund is a Period 2 winner Solution to 2: From Row 1: P(fund is a Period 2 winner | fund is a Period 1 winner) = 0.527 P(fund is a Period 2 loser | fund is a Period 1 winner) = 0.473 From Row 2: P(fund is a Period 2 winner | fund is a Period 1 loser) = 0.473 P(fund is a Period 2 loser | fund is a Period 1 loser) = 0.527 Solution to 3: These probabilities are calculated from data, so they are empirical probabilities. Solution to 4: The estimated probability is 0.264. With A the event that a fund is a Period 2 loser and B the event that a fund is a Period 1 loser, AB is the event that a fund is a loser in both Period 1 and Period 2. From Table 4-2, P(A | B) = 0.527 and P(B) = 0.50. Thus, using Equation 4-2, we find that P(AB) = P(A | B)P(B) = 0.527(0.50) = 0.2635 or a probability of approximately 0.264. When we have two events, A and B, that we are interested in, we often want to know the probability that either A or B occurs. Here the word ‘‘or’’ is inclusive, meaning that either A or B occurs or that both A and B occur. Put another way, the probability of A or B is the probability that at least one of the two events occurs. Such probabilities are calculated using the addition rule for probabilities. • Addition Rule for Probabilities. Given events A and B, the probability that A or B occurs, or both occur, is equal to the probability that A occurs, plus the probability that B occurs, minus the probability that both A and B occur. Chapter 4 Probability Concepts 137 FIGURE 4-1 Addition Rule for Probabilities P(A or B) = P(A) + P(B) − P(AB) (4-3) If we think of the individual probabilities of A and B as sets built of outcomes of one or more random variables, the first step in calculating the probability of A or B is to sum the probabilities of the outcomes in A to obtain P(A). If A and B share any outcomes, then if we now added P(B) to P(A), we would count twice the probabilities of those shared outcomes. So we add to P(A) the quantity [P(B) − P(AB)], which is the probability of outcomes in B net of the probability of any outcomes already counted when we computed P(A). Figure 4-1 illustrates this process; we avoid double-counting the outcomes in the intersection of A and B by subtracting P(AB). As an example of the calculation, if P(A) = 0.50, P(B) = 0.40, and P(AB) = 0.20, then P(A or B) = 0.50 + 0.40 − 0.20 = 0.70. Only if the two events A and B were mutually exclusive, so that P(AB) = 0, would it be correct to state that P(A or B) = P(A) + P(B). The next example shows how much useful information can be obtained using the few probability rules presented to this point. EXAMPLE 4-3 Probability of a Limit Order Executing You have two buy limit orders outstanding on the same stock. A limit order to buy stock at a stated price is an order to buy at that price or lower. A number of vendors, including an Internet service that you use, supply the estimated probability that a limit order will be filled within a stated time horizon, given the current stock price and the price limit. One buy order (Order 1) was placed at a price limit of $10. The probability that it will execute within one hour is 0.35. The second buy order (Order 2) was placed at a price limit of $9.75; it has a 0.25 probability of executing within the same one-hour time frame. 1. What is the probability that either Order 1 or Order 2 will execute? 2. What is the probability that Order 2 executes, given that Order 1 executes? 138 Quantitative Investment Analysis Solution to 1: The probability is 0.35. The two probabilities that are given are P(Order 1 executes) = 0.35 and P(Order 2 executes) = 0.25. Note that if Order 2 executes, it is certain that Order 1 also executes because the price must pass through $10 to reach $9.75. Thus, P(Order 1 executes | Order 2 executes) = 1 and P(Order 1 executes and Order 2 executes) = P(Order 1 executes | Order 2 executes)P(Order 2 executes) = 1(0.25) = 0.25 To answer the question, we use the addition rule for probabilities: P(Order 1 executes or Order 2 executes) = P(Order 1 executes) + P(Order 2 executes) − P(Order 1 executes and Order 2 executes) = 0.35 + 0.25 − 0.25 = 0.35 Note that the outcomes for which Order 2 executes are a subset of the outcomes for which Order 1 executes. After you count the probability that Order 1 executes, you have counted the probability of the outcomes for which Order 2 also executes. Therefore, the answer to the question is the probability that Order 1 executes, 0.35. Solution to 2: If the first order executes, the probability that the second order executes is 0.714. In the solution to Part 1, you found that P(Order 1 executes and Order 2 executes) = P(Order 1 executes | Order 2 executes)P(Order 2 executes) = 1(0.25) = 0.25. An equivalent way to state this joint probability is useful here: P(Order 1 executes and Order 2 executes) = 0.25 = P(Order 2 executes | Order 1 executes)P(Order 1 executes) Because P(Order 1 executes) = 0.35 was a given, you have one equation with one unknown: 0.25 = P(Order 2 executes | Order 1 executes)(0.35) You conclude that P(Order 2 executes | Order 1 executes) = 0.25/0.35 = 5/7, or about 0.714. You can also use Equation 4-1 to obtain this answer. Of great interest to investment analysts are the concepts of independence and dependence. These concepts bear on such basic investment questions as which financial variables are useful for investment analysis, whether asset returns can be predicted, and whether superior investment managers can be selected based on their past records. Two events are independent if the occurrence of one event does not affect the probability of occurrence of the other event. 139 Chapter 4 Probability Concepts • Definition of Independent Events. Two events A and B are independent if and only if P(A | B) = P(A) or, equivalently, P(B | A) = P(B). When two events are not independent, they are dependent: The probability of occurrence of one is related to the occurrence of the other. If we are trying to forecast one event, information about a dependent event may be useful, but information about an independent event will not be useful. When two events are independent, the multiplication rule for probabilities, Equation 4-2, simplifies because P(A | B) in that equation then equals P(A). • Multiplication Rule for Independent Events. When two events are independent, the joint probability of A and B equals the product of the individual probabilities of A and B. P(AB) = P(A)P(B) (4-4) Therefore, if we are interested in two independent events with probabilities of 0.75 and 0.50, respectively, the probability that both will occur is 0.375 = 0.75(0.50). The multiplication rule for independent events generalizes to more than two events; for example, if A, B, and C are independent events, then P(ABC) = P(A)P(B)P(C). EXAMPLE 4-4 BankCorp’s Earnings per Share (1) As part of your work as a banking industry analyst, you build models for forecasting earnings per share of the banks you cover. Today you are studying BankCorp. The historical record shows that in 55 percent of recent quarters BankCorp’s EPS has increased sequentially, and in 45 percent of quarters EPS has decreased or remained unchanged sequentially.6 At this point in your analysis, you are assuming that changes in sequential EPS are independent. Earnings per share for 2Q:2004 (that is, EPS for the second quarter of 2004) were larger than EPS for 1Q:2004. 1. What is the probability that 3Q:2004 EPS will be larger than 2Q:2004 EPS (a positive change in sequential EPS)? 2. What is the probability that EPS decreases or remains unchanged in the next two quarters? Solution to 1: Under the assumption of independence, the probability that 3Q:2004 EPS will be larger than 2Q:2004 EPS is the unconditional probability of positive change, 0.55. The fact that 2Q:2004 EPS was larger than 1Q:2004 EPS is not useful information, as the next change in EPS is independent of the prior change. Solution to 2: The probability is 0.2025 = 0.45(0.45). 6 Sequential comparisons of quarterly EPS are with the immediate prior quarter. A sequential comparison stands in contrast to a comparison with the same quarter one year ago (another frequent type of comparison). 140 Quantitative Investment Analysis The following example illustrates how difficult it is to satisfy a set of independent criteria even when each criterion by itself is not necessarily stringent. EXAMPLE 4-5 Screening Stocks for Investment You have developed a stock screen—a set of criteria for selecting stocks. Your investment universe (the set of securities from which you make your choices) is the Russell 1000 Index, an index of 1,000 large-capitalization U.S. equities. Your criteria capture different aspects of the selection problem; you believe that the criteria are independent of each other, to a close approximation. Fraction of Russell 1000 Stocks Meeting Criterion Criterion First valuation criterion Second valuation criterion Analyst coverage criterion Profitability criterion for company Financial strength criterion for company 0.50 0.50 0.25 0.55 0.67 How many stocks do you expect to pass your screen? Only 23 stocks out of 1,000 pass through your screen. If you define five events—the stock passes the first valuation criterion, the stock passes the second valuation criterion, the stock passes the analyst coverage criterion, the company passes the profitability criterion, the company passes the financial strength criterion (say events A, B, C, D, and E, respectively)—then the probability that a stock will pass all five criteria, under independence, is P(ABCDE) = P(A)P(B)P(C)P(D)P(E) = (0.50)(0.50)(0.25)(0.55)(0.67) = 0.023031 Although only one of the five criteria is even moderately strict (the strictest lets 25 percent of stocks through), the probability that a stock can pass all five is only 0.023031, or about 2 percent. The size of the list of candidate investments is 0.023031(1, 000) = 23.031, or 23 stocks. An area of intense interest to investment managers and their clients is whether records of past performance are useful in identifying repeat winners and losers. The following example shows how this issue relates to the concept of independence. Chapter 4 Probability Concepts 141 EXAMPLE 4-6 Conditional Probabilities and Predictability of Mutual Fund Performance (2) The purpose of the Kahn and Rudd (1995) study, introduced in Example 4-2, was to address the question of repeat mutual fund winners and losers. If the status of a fund as a winner or a loser in one period is independent of whether it is a winner in the next period, the practical value of performance ranking is questionable. Using the four events defined in Example 4-2 as building blocks, we can define the following events to address the issue of predictability of mutual fund performance: Fund is a Period 1 winner and fund is a Period 2 winner Fund is a Period 1 winner and fund is a Period 2 loser Fund is a Period 1 loser and fund is a Period 2 winner Fund is a Period 1 loser and fund is a Period 2 loser In Part 4 of Example 4-2, you calculated that P(fund is a Period 2 loser and fund is a Period 1 loser) = 0.264 If the ranking in one period is independent of the ranking in the next period, what will you expect P(fund is a Period 2 loser and fund is a Period 1 loser) to be? Interpret the empirical probability 0.264. By the multiplication rule for independent events, P(fund is a Period 2 loser and fund is a Period 1 loser) = P(fund is a Period 2 loser)P(fund is a Period 1 loser). Because 50 percent of funds are categorized as losers in each period, the unconditional probability that a fund is labeled a loser in either period is 0.50. Thus P(fund is a Period 2 loser)P(fund is a Period 1 loser) = 0.50(0.50) = 0.25. If the status of a fund as a loser in one period is independent of whether it is a loser in the prior period, we conclude that P(fund is a Period 2 loser and fund is a Period 1 loser) = 0.25. This probability is a priori because it is obtained from reasoning about the problem. You could also reason that the four events described above define categories and that if funds are randomly assigned to the four categories, there is a 1/4 probability of fund is a Period 1 loser and fund is a Period 2 loser. If the classifications in Period 1 and Period 2 were dependent, then the assignment of funds to categories would not be random. The empirical probability of 0.264 is only slightly above 0.25. Is this apparent slight amount of predictability the result of chance? A test conducted by Kahn and Rudd indicated a 35.6 percent chance of observing the tabled data if the Period 1 and Period 2 rankings were independent. In investments, the question of whether one event (or characteristic) provides information about another event (or characteristic) arises in both time-series settings (through time) and cross-sectional settings (among units at a given point in time). Examples 4-4 and 4-6 illustrated independence in a time-series setting. Example 4-5 illustrated independence in a cross-sectional setting. Independence/dependence relationships are often also explored in both settings using regression analysis, a technique we discuss in a later chapter. 142 Quantitative Investment Analysis In many practical problems, we logically analyze a problem as follows: We formulate scenarios that we think affect the likelihood of an event that interests us. We then estimate the probability of the event, given the scenario. When the scenarios (conditioning events) are mutually exclusive and exhaustive, no possible outcomes are left out. We can then analyze the event using the total probability rule. This rule explains the unconditional probability of the event in terms of probabilities conditional on the scenarios. The total probability rule is stated below for two cases. Part 1 gives the simplest case, in which we have two scenarios. One new notation is introduced: If we have an event or scenario S, the event not-S, called the complement of S, is written S C .7 Note that P(S) + P(S C ) = 1, as either S or not-S must occur. Part 2 states the rule for the general case of n mutually exclusive and exhaustive events or scenarios. • The Total Probability Rule. 1. P(A) = P(AS) + P(AS C ) = P(A | S)P(S) + P(A | S C )P(S C ) (4-5) 2. P(A) = P(AS1 ) + P(AS2 ) + · · · + P(ASn ) = P(A | S1 )P(S1 ) + P(A | S2 )P(S2 ) + · · · + P(A | Sn )P(Sn ) (4-6) where S1 , S2 , . . . , Sn are mutually exclusive and exhaustive scenarios or events. Equation 4-6 states the following: The probability of any event [P(A)] can be expressed as a weighted average of the probabilities of the event, given scenarios [terms such P(A | S1 )]; the weights applied to these conditional probabilities are the respective probabilities of the scenarios [terms such as P(S1 ) multiplying P(A | S1 )], and the scenarios must be mutually exclusive and exhaustive. Among other applications, this rule is needed to understand Bayes’ formula, which we discuss later in the chapter. In the next example, we use the total probability rule to develop a consistent set of views about BankCorp’s earnings per share. EXAMPLE 4-7 BankCorp’s Earnings per Share (2) You are continuing your investigation into whether you can predict the direction of changes in BankCorp’s quarterly EPS. You define four events: Event A = change in sequential EPS is positive next quarter AC = change in sequential EPS is 0 or negative next quarter S = change in sequential EPS is positive in the prior quarter S C = change in sequential EPS is 0 or negative in the prior quarter 7 For Probability 0.55 0.45 0.55 0.45 readers familiar with mathematical treatments of probability, S, a notation usually reserved for a concept called the sample space, is being appropriated to stand for scenario. Chapter 4 Probability Concepts 143 On inspecting the data, you observe some persistence in EPS changes: Increases tend to be followed by increases, and decreases by decreases. The first probability estimate you develop is P(change in sequential EPS is positive next quarter | change in sequential EPS is 0 or negative in the prior quarter) = P(A | S C ) = 0.40. The most recent quarter’s EPS (2Q:2004) is announced, and the change is a positive sequential change (the event S). You are interested in forecasting EPS for 3Q:2004. 1. Write this statement in probability notation: ‘‘the probability that the change in sequential EPS is positive next quarter, given that the change in sequential EPS is positive the prior quarter.’’ 2. Calculate the probability in Part 1. (Calculate the probability that is consistent with your other probabilities or beliefs.) Solution to 1: In probability notation, this statement is written P(A | S). Solution to 2: The probability is 0.673 that the change in sequential EPS is positive for 3Q:2004, given the positive change in sequential EPS for 2Q:2004, as shown below. According to Equation 4-5, P(A) = P(A | S)P(S) + P(A | S C )P(S C ). The values of the probabilities needed to calculate P(A | S) are already known: P(A) = 0.55, P(S) = 0.55, P(S C ) = 0.45, and P(A | S C ) = 0.40. Substituting into Equation 4-5, 0.55 = P(A | S)(0.55) + 0.40(0.45) Solving for the unknown, P(A | S) = [0.55 − 0.40(0.45)]/0.55 = 0.672727, or 0.673. You conclude that P(change in sequential EPS is positive next quarter | change in sequential EPS is positive the prior quarter) = 0.673. Any other probability is not consistent with your other estimated probabilities. Reflecting the persistence in EPS changes, this conditional probability of a positive EPS change, 0.673, is greater than the unconditional probability of an EPS increase, 0.55. In the chapter on statistical concepts and market returns, we discussed the concept of a weighted average or weighted mean. The example highlighted in that chapter was that portfolio return is a weighted average of the returns on the individual assets in the portfolio, where the weight applied to each asset’s return is the fraction of the portfolio invested in that asset. The total probability rule, which is a rule for stating an unconditional probability in terms of conditional probabilities, is also a weighted average. In that formula, probabilities of scenarios are used as weights. Part of the definition of weighted average is that the weights sum to 1. The probabilities of mutually exclusive and exhaustive events do sum to 1 (this is part of the definition of probability). The next weighted average we discuss, the expected value of a random variable, also uses probabilities as weights. The expected value of a random variable is an essential quantitative concept in investments. Investors continually make use of expected values—in estimating the rewards of alternative investments, in forecasting EPS and other corporate financial variables and ratios, and in assessing any other factor that may affect their financial position. The expected value of a random variable is defined as follows: 144 • Quantitative Investment Analysis Definition of Expected Value. The expected value of a random variable is the probabilityweighted average of the possible outcomes of the random variable. For a random variable X , the expected value of X is denoted E(X ). Expected value (for example, expected stock return) looks either to the future, as a forecast, or to the ‘‘true’’ value of the mean (the population mean, discussed in the chapter on statistical concepts and market returns). We should distinguish expected value from the concepts of historical or sample mean. The sample mean also summarizes in a single number a central value. However, the sample mean presents a central value for a particular set of observations as an equally weighted average of those observations. To summarize, the contrast is forecast versus historical, or population versus sample. EXAMPLE 4-8 BankCorp’s Earnings per Share (3) You continue with your analysis of BankCorp’s EPS. In Table 4-3, you have recorded a probability distribution for BankCorp’s EPS for the current fiscal year. TABLE 4-3 Probability Distribution for BankCorp’s EPS Probability EPS 0.15 0.45 0.24 0.16 1.00 $2.60 $2.45 $2.20 $2.00 What is the expected value of BankCorp’s EPS for the current fiscal year? Following the definition of expected value, list each outcome, weight it by its probability, and sum the terms. E(EPS) = 0.15($2.60) + 0.45($2.45) + 0.24($2.20) + 0.16($2.00) = $2.3405 The expected value of EPS is $2.34. An equation that summarizes your calculation in Example 4-8 is E(X ) = P(X1 )X1 + P(X2 )X2 + · · · + P(Xn )Xn = n ! P(Xi )Xi (4-7) i=1 where Xi is one of n possible outcomes of the random variable X .8 8 For simplicity, we model all random variables in this chapter as discrete random variables, which have a countable set of outcomes. For continuous random variables, which are discussed along with discrete random variables in the chapter on common probability distributions, the operation corresponding to summation is integration. 145 Chapter 4 Probability Concepts The expected value is our forecast. Because we are discussing random quantities, we cannot count on an individual forecast being realized (although we hope that, on average, forecasts will be accurate). It is important, as a result, to measure the risk we face. Variance and standard deviation measure the dispersion of outcomes around the expected value or forecast. • Definition of Variance. The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value: σ2 (X ) = E{[X − E(X )]2 } (4-8) The two notations for variance are σ2 (X ) and Var(X ). Variance is a number greater than or equal to 0 because it is the sum of squared terms. If variance is 0, there is no dispersion or risk. The outcome is certain, and the quantity X is not random at all. Variance greater than 0 indicates dispersion of outcomes. Increasing variance indicates increasing dispersion, all else equal. Variance of X is a quantity in the squared units of X . For example, if the random variable is return in percent, variance of return is in units of percent squared. Standard deviation is easier to interpret than variance, as it is in the same units as the random variable. If the random variable is return in percent, standard deviation of return is also in units of percent. • Definition of Standard Deviation. Standard deviation is the positive square root of variance. The best way to become familiar with these concepts is to work examples. EXAMPLE 4-9 BankCorp’s Earnings per Share (4) In Example 4-8, you calculated the expected value of BankCorp’s EPS as $2.34, which is your forecast. Now you want to measure the dispersion around your forecast. Table 4-4 shows your view of the probability distribution of EPS for the current fiscal year. TABLE 4-4 Probability Distribution for BankCorp’s EPS Probability EPS 0.15 0.45 0.24 0.16 1.00 $2.60 $2.45 $2.20 $2.00 146 Quantitative Investment Analysis What are the variance and standard deviation of BankCorp’s EPS for the current fiscal year? The order of calculation is always expected value, then variance, then standard deviation. Expected value has already been calculated. Following the definition of variance above, calculate the deviation of each outcome from the mean or expected value, square each deviation, weight (multiply) each squared deviation by its probability of occurrence, and then sum these terms. σ2 (EPS) = P($2.60)[$2.60 − E(EPS)]2 + P($2.45)[$2.45 − E(EPS)]2 + P($2.20)[$2.20 − E(EPS)]2 + P($2.00)[$2.00 − E(EPS)]2 = 0.15(2.60 − 2.34)2 + 0.45(2.45 − 2.34)2 + 0.24(2.20 − 2.34)2 + 0.16(2.00 − 2.34)2 = 0.01014 + 0.005445 + 0.004704 + $0.018496 = $0.038785 Standard deviation is the positive square root of $0.038785: σ(EPS) = $0.0387851/2 = $0.196939, or approximately $0.20. An equation that summarizes your calculation of variance in Example 4-9 is σ2 (X ) = P(X1 )[X1 − E(X )]2 + P(X2 )[X2 − E(X )]2 + · · · + P(Xn )[Xn − E(X )]2 = n ! i=1 P(Xi )[Xi − E(X )]2 (4-9) where Xi is one of n possible outcomes of the random variable X . In investments, we make use of any relevant information available in making our forecasts. When we refine our expectations or forecasts, we are typically making adjustments based on new information or events; in these cases we are using conditional expected values. The expected value of a random variable X given an event or scenario S is denoted E(X | S). Suppose the random variable X can take on any one of n distinct outcomes X1 , X2 , . . . , Xn (these outcomes form a set of mutually exclusive and exhaustive events). The expected value of X conditional on S is the first outcome, X1 , times the probability of the first outcome given S, P(X1 | S), plus the second outcome, X2 , times the probability of the second outcome given S, P(X2 | S), and so forth. E(X | S) = P(X1 | S)X1 + P(X2 | S)X2 + · · · + P(Xn | S)Xn (4-10) We will illustrate this equation shortly. Parallel to the total probability rule for stating unconditional probabilities in terms of conditional probabilities, there is a principle for stating (unconditional) expected values in terms of conditional expected values. This principle is the total probability rule for expected value. 147 Chapter 4 Probability Concepts • The Total Probability Rule for Expected Value. 1. 2. E(X ) = E(X | S)P(S) + E(X | S C )P(S C ) (4-11) E(X ) = E(X | S1 )P(S1 ) + E(X | S2 )P(S2 ) + · · · + E(X | Sn )P(Sn ) (4-12) where S1 , S2 , . . . , Sn are mutually exclusive and exhaustive scenarios or events. The general case, Equation 4-12, states that the expected value of X equals the expected value of X given Scenario 1, E(X | S1 ), times the probability of Scenario 1, P(S1 ), plus the expected value of X given Scenario 2, E(X | S2 ), times the probability of Scenario 2, P(S2 ), and so forth. To use this principle, we formulate mutually exclusive and exhaustive scenarios that are useful for understanding the outcomes of the random variable. This approach was employed in developing the probability distribution of BankCorp’s EPS in Examples 4-8 and 4-9, as we now discuss. The earnings of BankCorp are interest rate sensitive, benefiting from a declining interest rate environment. Suppose there is a 0.60 probability that BankCorp will operate in a declining interest rate environment in the current fiscal year and a 0.40 probability that it will operate in a stable interest rate environment (assessing the chance of an increasing interest rate environment as negligible). If a declining interest rate environment occurs, the probability that EPS will be $2.60 is estimated at 0.25, and the probability that EPS will be $2.45 is estimated at 0.75. Note that 0.60, the probability of declining interest rate environment, times 0.25, the probability of $2.60 EPS given a declining interest rate environment, equals 0.15, the (unconditional) probability of $2.60 given in the table in Examples 4-8 and 4-9 above. The probabilities are consistent. Also, 0.60(0.75) = 0.45, the probability of $2.45 EPS given in Tables 4-3 and 4-4. The tree diagram in Figure 4-2 shows the rest of the analysis. A declining interest rate environment points us to the node of the tree that branches off into outcomes of $2.60 and $2.45. We can find expected EPS given a declining interest rate environment as follows, using Equation 4-10: E(EPS | declining interest rate environment) = 0.25($2.60) + 0.75($2.45) = $2.4875 FIGURE 4-2 BankCorp’s Forecasted EPS 148 Quantitative Investment Analysis If interest rates are stable, E(EPS | stable interest rate environment) = 0.60($2.20) + 0.40($2.00) = $2.12 Once we have the new piece of information that interest rates are stable, for example, we revise our original expectation of EPS from $2.34 downward to $2.12. Now using the total probability rule for expected value, E(EPS) = E(EPS | declining interest rate environment) P(declining interest rate environment) + E(EPS | stable interest rate environment) P(stable interest rate environment) So E(EPS) = $2.4875(0.60) + $2.12(0.40) = $2.3405 or about $2.34. This amount is identical to the estimate of the expected value of EPS calculated directly from the probability distribution in Example 4-8. Just as our probabilities must be consistent, so must our expected values, unconditional and conditional; otherwise our investment actions may create profit opportunities for other investors at our expense. To review, we first developed the factors or scenarios that influence the outcome of the event of interest. After assigning probabilities to these scenarios, we formed expectations conditioned on the different scenarios. Then we worked backward to formulate an expected value as of today. In the problem just worked, EPS was the event of interest, and the interest rate environment was the factor influencing EPS. We can also calculate the variance of EPS given each scenario: σ2 (EPS | declining interest rate environment) = P($2.60 | declining interest rate environment) × [$2.60|E(EPS | declining interest rate environment)]2 + P($2.45 | declining interest rate environment) × [$2.45 − E(EPS | declining interest rate environment)]2 = 0.25($2.60 − $2.4875)2 + 0.75($2.45 − $2.4875)2 = 0.004219 σ2 (EPS | stable interest rate environment) = P($2.20 | stable interest rate environment) × [$2.20 − E(EPS | stable interest rate environment)]2 + P($2.00 | stable interest rate environment) × [$2.00 − E(EPS | stable interest rate environment)]2 = 0.60($2.20 − $2.12)2 + 0.40($2.00 − $2.12)2 = 0.0096 These are conditional variances, the variance of EPS given a declining interest rate environment and the variance of EPS given a stable interest rate environment. The relationship 149 Chapter 4 Probability Concepts between unconditional variance and conditional variance is a relatively advanced topic.9 The main points are 1) that variance, like expected value, has a conditional counterpart to the unconditional concept and 2) that we can use conditional variance to assess risk given a particular scenario. EXAMPLE 4-10 BankCorp’s Earnings per Share (5) Continuing with BankCorp, you focus now on BankCorp’s cost structure. One model you are researching for BankCorp’s operating costs is Ŷ = a + bX where Ŷ is a forecast of operating costs in millions of dollars and X is the number of branch offices. Ŷ represents the expected value of Y given X , or E(Y | X ). (Ŷ is a notation used in regression analysis, which we discuss in a later chapter.) You interpret the intercept a as fixed costs and b as variable costs. You estimate the equation as Ŷ = 12.5 + 0.65X BankCorp currently has 66 branch offices, and the equation estimates that 12.5 + 0.65(66) = $55.4 million. You have two scenarios for growth, pictured in the tree diagram in Figure 4-3. FIGURE 4-3 BankCorp’s Forecasted Operating Costs 9 The unconditional variance of EPS is the sum of two terms: (1) the expected value (probabilityweighted average) of the conditional variances (parallel to the total probability rules) and (2) the variance of conditional expected values of EPS. The second term arises because the variability in conditional expected value is a source of risk. Term 1 is σ2 (EPS) = P(declining interest rate environment) σ2 (EPS | declining interest rate environment) + P(stable interest rate environment) σ2 (EPS | stable interest rate environment) = 0.60(0.004219) + 0.40(0.0096) = 0.006371. Term 2 is σ2 [E(EPS | interest rate environment)] = 0.60($2.4875 − $2.34)2 + 0.40($2.12 − $2.34)2 = 0.032414. Summing the two terms, unconditional variance equals 0.006371 + 0.032414 = 0.038785. 150 Quantitative Investment Analysis 1. Compute the forecasted operating costs given the different levels of operating costs, using Ŷ = 12.5 + 0.65X . State the probability of each level of the number of branch offices. These are the answers to the questions in the terminal boxes of the tree diagram. 2. Compute the expected value of operating costs under the high growth scenario. Also calculate the expected value of operating costs under the low growth scenario. 3. Answer the question in the initial box of the tree: What are BankCorp’s expected operating costs? Solution to 1: Using Ŷ = 12.5 + 0.65X , from top to bottom, we have Operating Costs Ŷ Ŷ Ŷ Ŷ Probability = 12.5 + 0.65(125) = $93.75 million = 12.5 + 0.65(100) = $77.50 million = 12.5 + 0.65(80) = $64.50 million = 12.5 + 0.65(70) = $58.00 million 0.80(0.50) = 0.40 0.80(0.50) = 0.40 0.20(0.85) = 0.17 0.20(0.15) = 0.03 Sum = 1.00 Solution to 2: Dollar amounts are in millions. E(operating costs | high growth) = 0.50($93.75) + 0.50($77.50) = $85.625 E(operating costs | low growth) = 0.85($64.50) + 0.15($58.00) = $63.525 Solution to 3: Dollar amounts are in millions. E(operating costs) = E(operating costs | high growth)P(high growth) + E(operating costs | low growth)P(low growth) = $85.625(0.80) + $63.525(0.20) = $81.205 BankCorp’s expected operating costs are $81.205 million. We will see conditional probabilities again when we discuss Bayes’ formula. This section has introduced a few problems that can be addressed using probability concepts. The following problem draws on these concepts, as well as on analytical skills. EXAMPLE 4-11 The Default Risk Premium for a One-Period Debt Instrument As the co-manager of a short-term bond portfolio, you are reviewing the pricing of a speculative-grade, one-year-maturity, zero-coupon bond. For this type of bond, the 151 Chapter 4 Probability Concepts return is the difference between the amount paid and the principal value received at maturity. Your goal is to estimate an appropriate default risk premium for this bond. You define the default risk premium as the extra return above the risk-free return that will compensate investors for default risk. If R is the promised return (yield-to maturity) on the debt instrument and RF is the risk-free rate, the default risk premium is R − RF . You assess the probability that the bond defaults as P(the bond defaults) = 0.06. Looking at current money market yields, you find that one-year U.S. Treasury bills (T-bills) are offering a return of 5.8 percent, an estimate of RF . As a first step, you make the simplifying assumption that bondholders will recover nothing in the event of a default. What is the minimum default risk premium you should require for this instrument? The challenge in this type of problem is to find a starting point. In many problems, including this one, an effective first step is to divide up the possible outcomes into mutually exclusive and exhaustive events in an economically logical way. Here, from the viewpoint of a bondholder, the two events that affect returns are the bond defaults and the bond does not default. These two events cover all outcomes. How do these events affect a bondholder’s returns? A second step is to compute the value of the bond for the two events. We have no specifics on bond face value, but we can compute value per $1 or one unit of currency invested. Bond value The Bond Defaults The Bond Does Not Default $0 $(1 + R) The third step is to find the expected value of the bond (per $1 invested). E(bond) = $0 × P(the bond defaults) + $(1 + R)[1 − P(the bond defaults)] So E(bond) = $(1 + R)[1 − P(the bond defaults)]. The expected value of the T-bill per $1 invested is (1 + RF ). In fact, this value is certain because the T-bill is risk free. The next step requires economic reasoning. You want the default premium to be large enough so that you expect to at least break even compared with investing in the T-bill. This outcome will occur if the expected value of the bond equals the expected value of the T-bill per $1 invested. Expected Value of Bond = Expected Value of T-Bill $(1 + R)[1 − P(the bond defaults)] = (1 + RF ) Solving for the promised return on the bond, you find R = {(1 + RF )/[1 − P(the bond defaults)]} − 1. Substituting in the values in the statement of the problem, R = [1.058/(1 − 0.06)] − 1 = 1.12553 − 1 = 0.12553 or about 12.55 percent, and default risk premium is R − RF = 12.55% − 5.8% = 6.75%. You require a default risk premium of at least 675 basis points. You can state the matter as follows: If the bond is priced to yield 12.55 percent, you will earn a 675 basis-point spread and receive the bond principal with 94 percent probability. If the bond defaults, however, you will lose everything. With a premium of 675 basis points, you expect to just break even relative to an investment in T-bills. Because an investment 152 Quantitative Investment Analysis in the zero-coupon bond has variability, if you are risk averse, you will demand that the premium be larger than 675 basis points. This analysis is a starting point. Bondholders usually recover part of their investment after a default. A next step would be to incorporate a recovery rate. In this section, we have treated random variables such as EPS as stand-alone quantities. We have not explored how descriptors such as expected value and variance of EPS may be functions of other random variables. Portfolio return is one random variable that is clearly a function of other random variables, the random returns on the individual securities in the portfolio. To analyze a portfolio’s expected return and variance of return, we must understand these quantities are a function of characteristics of the individual securities’ returns. Looking at the dispersion or variance of portfolio return, we see that the way individual security returns move together or covary is important. To understand the significance of these movements, we need to explore some new concepts, covariance and correlation. The next section, which deals with portfolio expected return and variance of return, introduces these concepts. 3. PORTFOLIO EXPECTED RETURN AND VARIANCE OF RETURN Modern portfolio theory makes frequent use of the idea that investment opportunities can be evaluated using expected return as a measure of reward and variance of return as a measure of risk. The calculation and interpretation of portfolio expected return and variance of return are fundamental skills. In this section, we will develop an understanding of portfolio expected return and variance of return.10 Portfolio return is determined by the returns on the individual holdings. As a result, the calculation of portfolio variance, as a function of the individual asset returns, is more complex than the variance calculations illustrated in the previous section. We work with an example of a portfolio that is 50 percent invested in an S&P 500 Index fund, 25 percent invested in a U.S. long-term corporate bond fund, and 25 percent invested in a fund indexed to the MSCI EAFE Index (representing equity markets in Europe, Australasia, and the Far East). Table 4-5 shows these weights. TABLE 4-5 Portfolio Weights Asset Class S&P 500 U.S. long-term corporate bonds MSCI EAFE Weights 0.50 0.25 0.25 10 Although we outline a number of basic concepts in this section, we do not present mean–variance analysis per se. For a presentation of mean–variance analysis, see the chapter on portfolio concepts, as well as the extended treatments in standard investment textbooks such as Bodie, Kane, and Marcus (2001), Elton, Gruber, Brown, and Goetzmann (2003), Reilly and Brown (2003), and Sharpe, Alexander, and Bailey (1999). 153 Chapter 4 Probability Concepts We first address the calculation of the expected return on the portfolio. In the previous section, we defined the expected value of a random variable as the probability-weighted average of the possible outcomes. Portfolio return, we know, is a weighted average of the returns on the securities in the portfolio. Similarly, the expected return on a portfolio is a weighted average of the expected returns on the securities in the portfolio, using exactly the same weights. When we have estimated the expected returns on the individual securities, we immediately have portfolio expected return. This convenient fact follows from the properties of expected value. • Properties of Expected Value. Let wi be any constant and Ri be a random variable. 1. The expected value of a constant times a random variable equals the constant times the expected value of the random variable. E(wi Ri ) = wi E(Ri ) 2. The expected value of a weighted sum of random variables equals the weighted sum of the expected values, using the same weights. E(w1 R1 + w2 R2 + · · · + wn Rn ) = w1 E(R1 ) + w2 E(R2 ) + · · · + wn E(Rn ) (4-13) Suppose we have a random variable with a given expected value. If we multiply each outcome by 2, for example, the random variable’s expected value is multiplied by 2 as well. That is the meaning of Part 1. The second statement is the rule that directly leads to the expression for portfolio expected return. A portfolio with n securities is defined by its portfolio weights, w1 , w2 , . . . , wn , which sum to 1. So portfolio return, Rp , is Rp = w1 R1 + w2 R2 + · · · + wn Rn . We can state the following principle: • Calculation of Portfolio Expected Return. Given a portfolio with n securities, the expected return on the portfolio is a weighted average of the expected returns on the component securities: E(Rp ) = E(w1 R1 + w2 R2 + · · · + wn Rn ) = w1 E(R1 ) + w2 E(R2 ) + · · · + wn E(Rn ) Suppose we have estimated expected returns on the assets in the portfolio, as given in Table 4-6. We calculate the expected return on the portfolio as 11.75 percent: E(Rp ) = w1 E(R1 ) + w2 E(R2 ) + w3 E(R3 ) = 0.50(13%) + 0.25(6%) + 0.25(15%) = 11.75% TABLE 4-6 Weights and Expected Returns Asset Class S&P 500 U.S. long-term corporate bonds MSCI EAFE Weight Expected Return (%) 0.50 0.25 0.25 13 6 15 154 Quantitative Investment Analysis In the previous section, we studied variance as a measure of dispersion of outcomes around the expected value. Here we are interested in portfolio variance of return as a measure of investment risk. Letting Rp stand for the return on the portfolio, portfolio variance is σ2 (Rp ) = E{[Rp − E(Rp )]2 } according to Equation 4-8. How do we implement this definition? In the chapter on statistical concepts and market returns, we learned how to calculate a historical or sample variance based on a sample of returns. Now we are considering variance in a forward-looking sense. We will use information about the individual assets in the portfolio to obtain portfolio variance of return. To avoid clutter in notation, we write ERp for E(Rp ). We need the concept of covariance. • Definition of Covariance. Given two random variables Ri and Rj , the covariance between Ri and Rj is Cov(Ri , Rj ) = E[(Ri − ERi )(Rj − ERj )] (4-14) Alternative notations are σ(Ri , Rj ) and σij . Equation 4-14 states that the covariance between two random variables is the probabilityweighted average of the cross-products of each random variable’s deviation from its own expected value. We will return to discuss covariance after we establish the need for the concept. Working from the definition of variance, we find σ2 (Rp ) = E[(Rp − ERp )2 ] = E{[w1 R1 + w2 R2 + w3 R3 − E(w1 R1 + w2 R2 + w3 R3 )]2 } = E{[w1 R1 + w2 R2 + w3 R3 − w1 ER1 − w2 ER2 − w3 ER3 ]2 } (using Equation 4−13) = E{[w1 (R1 − ER1 ) + w2 (R2 − ER2 ) + w3 (R3 − ER3 )]2 } (rearranging) = E{[w1 (R1 − ER1 ) + w2 (R2 − ER2 ) + w3 (R3 − ER3 )] ×[w1 (R1 − ER1 ) + w2 (R2 − ER2 ) + w3 (R3 − ER3 )]} (what squaring means) = E[w1 w1 (R1 − ER1 )(R1 − ER1 ) + w1 w2 (R1 − ER1 )(R2 − ER2 ) +w1 w3 (R1 − ER1 )(R3 − ER3 ) + w2 w1 (R2 − ER2 )(R1 − ER1 ) +w2 w2 (R2 − ER2 )(R2 − ER2 ) + w2 w3 (R2 − ER2 )(R3 − ER3 ) +w3 w1 (R3 − ER3 )(R1 − ER1 ) + w3 w2 (R3 − ER3 )(R2 − ER2 ) +w3 w3 (R3 − ER3 )(R3 − ER3 )] = w12 E[(R1 (doing the multiplication) 2 − ER1 ) ] + w1 w2 E[(R1 − ER1 )(R2 − ER2 )] +w1 w3 E[(R1 − ER1 )(R3 − ER3 )] + w2 w1 E[(R2 − ER2 )(R1 − ER1 )] +w22 E[(R2 − ER2 )2 ] + w2 w3 E[(R2 − ER2 )(R3 − ER3 )] +w3 w1 E[(R3 − ER3 )(R1 − ER1 )] + w3 w2 E[(R3 − ER3 )(R2 − ER2 )] 155 Chapter 4 Probability Concepts +w32 E[(R3 − ER3 )2 ] (recalling that the wi terms are constants) = w12 σ2 (R1 ) + w1 w2 Cov(R1 , R2 ) + w1 w3 Cov(R1 , R3 ) +w1 w2 Cov(R1 , R2 ) + w22 σ2 (R2 ) + w2 w3 Cov(R2 , R3 ) +w1 w3 Cov(R1 , R3 ) + w2 w3 Cov(R2 , R3 ) + w32 σ2 (R3 ) (4-15) The last step follows from the definitions of variance and covariance.11 For the italicized covariance terms in Equation 4-15, we used the fact that the order of variables in covariance does not matter: Cov(R2 , R1 ) = Cov(R1 , R2 ), for example. As we will show, the diagonal variance terms σ2 (R1 ), σ2 (R2 ), and σ2 (R3 ) can be expressed as Cov(R1 , R1 ), Cov(R2 , R2 ), and Cov(R3 , R3 ), respectively. Using this fact, the most compact way to state Equation 4-15 is 3 3 ! ! σ2 (RP ) = wi wj Cov(Ri , Rj ). The double summation signs say: ‘‘Set i = 1 and let j run i=1 j=1 from 1 to 3; then set i = 2 and let j run from 1 to 3; next set i = 3 and let j run from 1 to 3; finally, add the nine terms.’’ This expression generalizes for a portfolio of any size n to 2 σ (Rp ) = n n ! ! wi wj Cov(Ri , Rj ) (4-16) i=1 j=1 We see from Equation 4-15 that individual variances of return (the bolded diagonal terms) constitute part, but not all, of portfolio variance. The three variances are actually outnumbered by the six covariance terms off the diagonal. For three assets, the ratio is 1 to 2, or 50 percent. If there are 20 assets, there are 20 variance terms and 20(20) − 20 = 380 off-diagonal covariance terms. The ratio of variance terms to off-diagonal covariance terms is less than 6 to 100, or 6 percent. A first observation, then, is that as the number of holdings increases, covariance12 becomes increasingly important, all else equal. What exactly is the effect of covariance on portfolio variance? The covariance terms capture how the co-movements of returns affect portfolio variance. For example, consider two stocks: One tends to have high returns (relative to its expected return) when the other has low returns (relative to its expected return). The returns on one stock tend to offset the returns on the other stock, lowering the variability or variance of returns on the portfolio. Like variance, the units of covariance are hard to interpret, and we will introduce a more intuitive concept shortly. Meanwhile, from the definition of covariance, we can establish two essential observations about covariance. 1. We can interpret the sign of covariance as follows: Covariance of returns is negative if, when the return on one asset is above its expected value, the return on the other asset tends to be below its expected value (an average inverse relationship between returns). Covariance of returns is 0 if returns on the assets are unrelated. 11 Useful facts about variance and covariance include: 1) The variance of a constant times a random variable equals the constant squared times the variance of the random variable, or σ2 (wR) = w2 σ2 (R); 2) The variance of a constant plus a random variable equals the variance of the random variable, or σ2 (w + R) = σ2 (R) because a constant has zero variance; 3) The covariance between a constant and a random variable is zero. 12 When the meaning of covariance as ‘‘off-diagonal covariance’’ is obvious, as it is here, we omit the qualifying words. Covariance is usually used in this sense. 156 Quantitative Investment Analysis TABLE 4-7 Inputs to Portfolio Expected Return and Variance A. Inputs to Portfolio Expected Return Asset A E(RA ) B E(RB ) C E(RC ) B. Covariance Matrix: The Inputs to Portfolio Variance of Return Asset A B C A B C Cov(R A , R A ) Cov(RB , RA ) Cov(RC , RA ) Cov(RA , RB ) Cov(R B , R B ) Cov(RC , RB ) Cov(RA , RC ) Cov(RB , RC ) Cov(R C , R C ) Covariance of returns is positive when the returns on both assets tend to be on the same side (above or below) their expected values at the same time (an average positive relationship between returns). 2. The covariance of a random variable with itself (own covariance) is its own variance: Cov(R, R) = E{[R − E(R)][R − E(R)]} = E{[R − E(R)]2 } = σ2 (R). A complete list of the covariances constitutes all the statistical data needed to compute portfolio variance of return. Covariances are often presented in a square format called a covariance matrix. Table 4-7 summarizes the inputs for portfolio expected return and variance of return. With three assets, the covariance matrix has 32 = 3 × 3 = 9 entries, but it is customary to treat the diagonal terms, the variances, separately from the off-diagonal terms. These diagonal terms are bolded in Table 4-7. This distinction is natural, as security variance is a single-variable concept. So there are 9 − 3 = 6 covariances, excluding variances. But Cov(RB , RA ) = Cov(RA , RB ), Cov(RC , RA ) = Cov(RA , RC ), and Cov(RC , RB ) = Cov(RB , RC ). The covariance matrix below the diagonal is the mirror image of the covariance matrix above the diagonal. As a result, there are only 6/2 = 3 distinct covariance terms to estimate. In general, for n securities, there are n(n − 1)/2 distinct covariances to estimate and n variances to estimate. Suppose we have the covariance matrix shown in Table 4-8. Taking Equation 4-15 and grouping variance terms together produces the following: σ2 (Rp ) = w12 σ2 (R1 ) + w22 σ2 (R2 ) + w32 σ2 (R3 ) + 2w1 w2 Cov(R1 , R2 ) +2w1 w3 Cov(R1 , R3 ) + 2w2 w3 Cov(R2 , R3 ) (4-17) = (0.50)2 (400) + (0.25)2 (81) + (0.25)2 (441) + 2(0.50)(0.25)(45) +2(0.50)(0.25)(189) + 2(0.25)(0.25)(38) = 100 + 5.0625 + 27.5625 + 11.25 + 47.25 + 4.75 = 195.875 The variance is 195.875. Standard deviation of return is 195.8751/2 = 14 percent. To summarize, the portfolio has an expected annual return of 11.75 percent and a standard deviation of return of 14 percent. Let us look at the first three terms in the calculation above. Their sum, 100 + 5.0625 + 27.5625 = 132.625, is the contribution of the individual variances to portfolio variance. If the returns on the three assets were independent, covariances would be 0 and the standard 157 Chapter 4 Probability Concepts TABLE 4-8 Covariance Matrix S&P 500 U.S. long-term corporate bonds MSCI EAFE S&P 500 U.S. Long-Term Corporate Bonds MSCI EAFE 400 45 189 45 81 38 189 38 441 deviation of portfolio return would be 132.6251/2 = 11.52 percent as compared to 14 percent before. The portfolio would have less risk. Suppose the covariance terms were negative. Then a negative number would be added to 132.625, so portfolio variance and risk would be even smaller. At the same time, we have not changed expected return. For the same expected portfolio return, the portfolio has less risk. This risk reduction is a diversification benefit, meaning a risk-reduction benefit from holding a portfolio of assets. The diversification benefit increases with decreasing covariance. This observation is a key insight of modern portfolio theory. It is even more intuitively stated when we can use the concept of correlation. Then we can say that as long as security returns are not perfectly positively correlated, diversification benefits are possible. Furthermore, the smaller the correlation between security returns, the greater the cost of not diversifying (in terms of risk-reduction benefits forgone), all else equal. • Definition of Correlation. The correlation between two random variables, Ri and Rj , is defined as ρ(Ri , Rj ) = Cov(Ri , Rj )/σ(Ri )σ(Rj ). Alternative notations are Corr(Ri , Rj ) and ρij . Frequently, covariance is substituted out using the relationship Cov(Ri , Rj ) = ρ(Ri , Rj )σ(Ri ) σ(Rj ). The division indicated in the definition makes correlation a pure number (one without a unit of measurement) and places bounds on its largest and smallest possible values. Using the above definition, we can state a correlation matrix from data in the covariance matrix alone. Table 4-9 shows the correlation matrix. For example, the covariance between long-term bonds and MSCI EAFE is 38, from Table 4-8. The standard deviation of long-term bond returns is 811/2 = 9 percent, that of MSCI EAFE returns is 4411/2 = 21 percent, from diagonal terms in Table 4-8. The correlation ρ(Return on long-term bonds, Return on EAFE) is 38/(9%)(21%) = 0.201, rounded to 0.20. The correlation of the S&P 500 with itself equals 1: The calculation is own covariance divided by its standard deviation squared. TABLE 4-9 Correlation Matrix of Returns S&P 500 U.S. long-term corporate bonds MSCI EAFE S&P 500 U.S. Long-Term Corporate Bonds MSCI EAFE 1.00 0.25 0.45 0.25 1.00 0.20 0.45 0.20 1.00 158 • Quantitative Investment Analysis Properties of Correlation. 1. Correlation is a number between −1 and +1 for two random variables, X and Y : −1 ≤ ρ(X , Y ) ≤ +1 2. A correlation of 0 (uncorrelated variables) indicates an absence of any linear (straightline) relationship between the variables.13 Increasingly positive correlation indicates an increasingly strong positive linear relationship (up to 1, which indicates a perfect linear relationship). Increasingly negative correlation indicates an increasingly strong negative (inverse) linear relationship (down to −1, which indicates a perfect inverse linear relationship).14 EXAMPLE 4-12 Portfolio Expected Return and Variance of Return You have a portfolio of two mutual funds, A and B, 75 percent invested in A, as shown in Table 4-10. TABLE 4-10 Mutual Fund Expected Returns, Return Variances, and Covariances Fund A E(RA ) = 20% B E(RB ) = 12% Covariance Matrix Fund A B A 625 120 B 120 196 1. Calculate the expected return of the portfolio. 2. Calculate the correlation matrix for this problem. Carry out the answer to two decimal places. 3. Compute portfolio standard deviation of return. Solution to 1: E(Rp ) = wA E(RA ) + (1 − wA )E(RB ) = 0.75(20%) + 0.25(12%) = 18%. Portfolio weights must sum to 1: wB = 1 − wA . Solution to 2: σ(RA ) = 6251/2 = 25 percent σ(RB ) = 1961/2 = 14 percent. There is one distinct covariance and thus one distinct correlation: ρ(RA , RB ) = Cov(RA , RB )/σ(RA )σ(RB ) = 120/25(14) = 0.342857, or 0.34 13 If the correlation is 0, R1 = a + bR2 + error, with b = 0. the correlation is positive, R1 = a + bR2 + error, with b > 0. If the correlation is negative, b < 0. 14 If 159 Chapter 4 Probability Concepts Table 4-11 shows the correlation matrix. TABLE 4-11 Correlation Matrix A B A B 1.00 0.34 0.34 1.00 Diagonal terms are always equal to 1 in a correlation matrix. Solution to 3: σ2 (Rp ) = wA2 σ2 (RA ) + wB2 σ2 (RB ) + 2wA wB Cov(RA , RB ) = (0.75)2 (625) + (0.25)2 (196) + 2(0.75)(0.25)(120) = 351.5625 + 12.25 + 45 = 408.8125 σ(Rp ) = 408.81251/2 = 20.22 percent How do we estimate return covariance and correlation? Frequently, we make forecasts on the basis of historical covariance or use other methods based on historical return data, such as a market model regression.15 We can also calculate covariance using the joint probability function of the random variables, if that can be estimated. The joint probability function of two random variables X and Y , denoted P(X , Y ), gives the probability of joint occurrences of values of X and Y . For example, P(3, 2), is the probability that X equals 3 and Y equals 2. Suppose that the joint probability function of the returns on BankCorp stock (RA ) and the returns on NewBank stock (RB ) has the simple structure given in Table 4-12. TABLE 4-12 Joint Probability Function of BankCorp and NewBank Returns (Entries are joint probabilities) RA = 25% RA = 12% RA = 10% RB = 20% RB = 16% RB = 10% 0.20 0 0 0 0.50 0 0 0 0.30 The expected return on BankCorp stock is 0.20(25%) + 0.50(12%) + 0.30(10%) = 14%. The expected return on NewBank stock is 0.20(20%) + 0.50(16%) + 0.30(10%) = 15%. The joint probability function above might reflect an analysis based on whether banking industry conditions are good, average, or poor. Table 4-13 presents the calculation of covariance. 15 See any of the textbooks mentioned in Footnote 10. 160 Quantitative Investment Analysis TABLE 4-13 Covariance Calculations Banking Industry Condition Good Average Poor Deviations BankCorp Deviations NewBank Product of Deviations Probability of Condition Probability-Weighted Product 25–14 12–14 10–14 20–15 16–15 10–15 55 −2 20 0.20 0.50 0.30 11 −1 6 Cov(RA , RB ) = 16 Note: Expected return for BankCorp is 14% and for NewBank, 15%. The first and second columns of numbers show, respectively, the deviations of BankCorp and NewBank returns from their mean or expected value. The next column shows the product of the deviations. For example, for good industry conditions, (25 − 14)(20 − 15) = 11(5) = 55. Then 55 is multiplied or weighted by 0.20, the probability that banking industry conditions are good: 55(0.20) = 11. The calculations for average and poor banking conditions follow the same pattern. Summing up these probability-weighted products, we find that Cov(RA , RB ) = 16. A formula for computing the covariance between random variables RA and RB is Cov(RA , RB ) = !! i j P(RA,i , RB,j )(RA,i − ERA )(RB,j − ERB ) (4-18) The formula tells us to sum all possible deviation cross-products weighted by the appropriate joint probability. In the example we just worked, as Table 4-12 shows, only three joint probabilities are nonzero. Therefore, in computing the covariance of returns in this case, we need to consider only three cross-products: Cov(RA , RB ) = P(25, 20)[(25 − 14)(20 − 15)] + P(12, 16)[(12 − 14)(16 − 15)] + P(10, 10)[(10 − 14)(10 − 15)] = 0.20(11)(5) + 0.50(−2)(1) + 0.30(−4)(−5) = 11 − 1 + 6 = 16 One theme of this chapter has been independence. Two random variables are independent when every possible pair of events—one event corresponding to a value of X and another event corresponding to a value of Y —are independent events. When two random variables are independent, their joint probability function simplifies. • Definition of Independence for Random Variables. Two random variables X and Y are independent if and only if P(X , Y ) = P(X )P(Y ). For example, given independence, P(3, 2) = P(3)P(2). We multiply the individual probabilities to get the joint probabilities. Independence is a stronger property than uncorrelatedness because correlation addresses only linear relationships. The following condition holds for independent random variables and, therefore, also holds for uncorrelated random variables. Chapter 4 Probability Concepts • 161 Multiplication Rule for Expected Value of the Product of Uncorrelated Random Variables. The expected value of the product of uncorrelated random variables is the product of their expected values. E(XY ) = E(X )E(Y ) if X and Y are uncorrelated. Many financial variables, such as revenue (price times quantity), are the product of random quantities. When applicable, the above rule simplifies calculating expected value of a product of random variables.16 4. TOPICS IN PROBABILITY In the remainder of the chapter we discuss two topics that can be important in solving investment problems. We start with Bayes’ formula: what probability theory has to say about learning from experience. Then we move to a discussion of shortcuts and principles for counting. 4.1. Bayes’ Formula When we make decisions involving investments, we often start with viewpoints based on our experience and knowledge. These viewpoints may be changed or confirmed by new knowledge and observations. Bayes’ formula is a rational method for adjusting our viewpoints as we confront new information.17 Bayes’ formula and related concepts have been applied in many business and investment decision-making contexts, including the evaluation of mutual fund performance.18 Bayes’ formula makes use of Equation 4-6, the total probability rule. To review, that rule expressed the probability of an event as a weighted average of the probabilities of the event, given a set of scenarios. Bayes’ formula works in reverse; more precisely, it reverses the ‘‘given that’’ information. Bayes’ formula uses the occurrence of the event to infer the probability of the scenario generating it. For that reason, Bayes’ formula is sometimes called an inverse probability. In many applications, including the one illustrating its use in this section, an individual is updating his beliefs concerning the causes that may have produced a new observation. • Bayes’ Formula. Given a set of prior probabilities for an event of interest, if you receive new information, the rule for updating your probability of the event is Updated probability of event given the new information = Probability of the new information given event × Prior probability of event Unconditional probability of the new information 16 Otherwise, the calculation depends on conditional expected value; the calculation can be expressed as E(XY ) = E(X )E(Y | X ). 17 Named after the Reverend Thomas Bayes (1702–61). 18 See Baks, Metrick, and Wachter (2001). 162 Quantitative Investment Analysis In probability notation, this formula can be written concisely as: P(Event | Information) = P(Information | Event) P(Event) P(Information) To illustrate Bayes’ formula, we work through an investment example that can be adapted to any actual problem. Suppose you are an investor in the stock of DriveMed, Inc. Positive earnings surprises relative to consensus EPS estimates often result in positive stock returns, and negative surprises often have the opposite effect. DriveMed is preparing to release last quarter’s EPS result, and you are interested in which of these three events happened: last quarter’s EPS exceeded the consensus EPS estimate, or last quarter’s EPS exactly met the consensus EPS estimate, or last quarter’s EPS fell short of the consensus EPS estimate. This list of the alternatives is mutually exclusive and exhaustive. On the basis of your own research, you write down the following prior probabilities (or priors, for short) concerning these three events: P(EPS exceeded consensus) = 0.45 P(EPS met consensus) = 0.30 • P(EPS fell short of consensus) = 0.25 • • These probabilities are ‘‘prior’’ in the sense that they reflect only what you know now, before the arrival of any new information. The next day, DriveMed announces that it is expanding factory capacity in Singapore and Ireland to meet increased sales demand. You assess this new information. The decision to expand capacity relates not only to current demand but probably also to the prior quarter’s sales demand. You know that sales demand is positively related to EPS. So now it appears more likely that last quarter’s EPS will exceed the consensus. The question you have is, ‘‘In light of the new information, what is the updated probability that the prior quarter’s EPS exceeded the consensus estimate?’’ Bayes’ formula provides a rational method for accomplishing this updating. We can abbreviate the new information as DriveMed expands. The first step in applying Bayes’ formula is to calculate the probability of the new information (here: DriveMed expands), given a list of events or scenarios that may have generated it. The list of events should cover all possibilities, as it does here. Formulating these conditional probabilities is the key step in the updating process. Suppose your view is P(DriveMed expands | EPS exceeded consensus) = 0.75 P(DriveMed expands | EPS met consensus) = 0.20 P(DriveMed expands | EPS fell short of consensus) = 0.05 Conditional probabilities of an observation (here: DriveMed expands) are sometimes referred to as likelihoods. Again, likelihoods are required for updating the probability. Next, you combine these conditional probabilities or likelihoods with your prior proba- Chapter 4 Probability Concepts 163 bilities to get the unconditional probability for DriveMed expanding, P(DriveMed expands), as follows: P(DriveMed expands) = P(DriveMed expands | EPS exceeded consensus) × P(EPS exceeded consensus) + P(DriveMed expands | EPS met consensus) × P(EPS met consensus) + P(DriveMed expands | EPS fell short of consensus) × P(EPS fell short of consensus) = 0.75(0.45) + 0.20(0.30) + 0.05(0.25) = 0.41, or 41% This is Equation 4-6, the total probability rule, in action. Now you can answer your question by applying Bayes’ formula: P(EPS exceeded consensus | DriveMed expands) = P(DriveMed expands | EPS exceeded consensus) P(EPS exceeded consensus) P(DriveMed expands) = (0.75/0.41)(0.45) = 1.829268(0.45) = 0.823171 Prior to DriveMed’s announcement, you thought the probability that DriveMed would beat consensus expectations was 45 percent. On the basis of your interpretation of the announcement, you update that probability to 82.3 percent. This updated probability is called your posterior probability because it reflects or comes after the new information. The Bayes’ calculation takes the prior probability, which was 45 percent, and multiplies it by a ratio—the first term on the right-hand side of the equal sign. The denominator of the ratio is the probability that DriveMed expands, as you view it without considering (conditioning on) anything else. Therefore, this probability is unconditional. The numerator is the probability that DriveMed expands, if last quarter’s EPS actually exceeded the consensus estimate. This last probability is larger than unconditional probability in the denominator, so the ratio (1.83 roughly) is greater than 1. As a result, your updated or posterior probability is larger than your prior probability. Thus, the ratio reflects the impact of the new information on your prior beliefs. EXAMPLE 4-13 Consensus EPS Inferring Whether DriveMed’s EPS Met You are still an investor in DriveMed stock. To review the givens, your prior probabilities are P(EPS exceeded consensus) = 0.45, P(EPS met consensus) = 0.30, and 164 Quantitative Investment Analysis P(EPS fell short of consensus) = 0.25. You also have the following conditional probabilities: P(DriveMed expands | EPS exceeded consensus) = 0.75 P(DriveMed expands | EPS met consensus) = 0.20 P(DriveMed expands | EPS fell short of consensus) = 0.05 Recall that you updated your probability that last quarter’s EPS exceeded the consensus estimate from 45 percent to 82.3 percent after DriveMed announced it would expand. Now you want to update your other priors. 1. Update your prior probability that DriveMed’s EPS met consensus. 2. Update your prior probability that DriveMed’s EPS fell short of consensus. 3. Show that the three updated probabilities sum to 1. (Carry each probability to four decimal places.) 4. Suppose, because of lack of prior beliefs about whether DriveMed would meet consensus, you updated on the basis of prior probabilities that all three possibilities were equally likely: P(EPS exceeded consensus) = P(EPS met consensus) = P(EPS fell short of consensus) = 1/3. What is your estimate of the probability P(EPS exceeded consensus | DriveMed expands)? Solution to 1: The probability is P(EPS met consensus | DriveMed expands) = P(DriveMed expands | EPS met consensus) P(EPS met consensus) P(DriveMed expands) The probability P(DriveMed expands) is found by taking each of the three conditional probabilities in the statement of the problem, such as P(DriveMed expands | EPS exceeded consensus); multiplying each one by the prior probability of the conditioning event, such as P(EPS exceeded consensus); then adding the three products. The calculation is unchanged from the problem in the text above: P(DriveMed expands) = 0.75(0.45) + 0.20(0.30) + 0.05(0.25) = 0.41, or 41 percent. The other probabilities needed, P(DriveMed expands | EPS met consensus) = 0.20 and P(EPS met consensus) = 0.30, are givens. So P(EPS met consensus | DriveMed expands) = [P(DriveMed expands | EPS met consensus)/ P(DriveMed expands)]P(EPS met consensus) = (0.20/0.41)(0.30) = 0.487805(0.30) = 0.146341 After taking account of the announcement on expansion, your updated probability that last quarter’s EPS for DriveMed just met consensus is 14.6 percent compared with your prior probability of 30 percent. Chapter 4 Probability Concepts 165 Solution to 2: P(DriveMed expands) was already calculated as 41 percent. Recall that P(DriveMed expands | EPS fell short of consensus) = 0.05 and P(EPS fell short of consensus) = 0.25 are givens. P(EPS fell short of consensus | DriveMed expands) = [P(DriveMed expands | EPS fell short of consensus)/ P(DriveMed expands)]P(EPS fell short of consensus) = (0.05/0.41)(0.25) = 0.121951(0.25) = 0.030488 As a result of the announcement, you have revised your probability that DriveMed’s EPS fell short of consensus from 25 percent (your prior probability) to 3 percent. Solution to 3: The sum of the three updated probabilities is P(EPS exceeded consensus | DriveMed expands) + P(EPS met consensus | DriveMed expands) + P(EPS fell short of consensus | DriveMed expands) = 0.8232 + 0.1463 + 0.0305 = 1.0000 The three events (EPS exceeded consensus, EPS met consensus, EPS fell short of consensus) are mutually exclusive and exhaustive: One of these events or statements must be true, so the conditional probabilities must sum to 1. Whether we are talking about conditional or unconditional probabilities, whenever we have a complete set of the distinct possible events or outcomes, the probabilities must sum to 1. This calculation serves as a check on your work. Solution to 4: Using the probabilities given in the question, P(DriveMed expands) = P(DriveMed expands | EPS exceeded consensus) P(EPS exceeded consensus) + P(DriveMed expands | EPS met consensus) P(EPS met consensus) + P(DriveMed expands | EPS fell short of consensus) P(EPS fell short of consensus) = 0.75(1/3) + 0.20(1/3) + 0.05(1/3) = 1/3 Not surprisingly, the probability of DriveMed expanding is 1/3 because the decision maker has no prior beliefs or views regarding how well EPS performed relative to the consensus estimate. Now we can use Bayes’ formula to find P(EPS exceeded consensus | DriveMed expands) = [P(DriveMed expands | EPS exceeded consensus)/P(DriveMed expands)]P(EPS exceeded consensus) = [(0.75/(1/3)](1/3) = 0.75 or 75 percent. This probability is identical to your estimate of P(DriveMed expands | EPS exceeded consensus). 166 Quantitative Investment Analysis When the prior probabilities are equal, the probability of information given an event equals the probability of the event given the information. When a decision-maker has equal prior probabilities (called diffuse priors), the probability of an event is determined by the information. 4.2. Principles of Counting The first step in addressing a question often involves determining the different logical possibilities. We may also want to know the number of ways that each of these possibilities can happen. In the back of our mind is often a question about probability. How likely is it that I will observe this particular possibility? Records of success and failure are an example. When we evaluate a market timer’s record, one well-known evaluation method uses counting methods presented in this section.19 An important investment model, the binomial option pricing model, incorporates the combination formula that we will cover shortly. We can also use the methods in this section to calculate what we called a priori probabilities in Section 2. When we can assume that the possible outcomes of a random variable are equally likely, the probability of an event equals the number of possible outcomes favorable for the event divided by the total number of outcomes. In counting, enumeration (counting the outcomes one by one) is of course the most basic resource. What we discuss in this section are shortcuts and principles. Without these shortcuts and principles, counting the total number of outcomes can be very difficult and prone to error. The first and basic principle of counting is the multiplication rule. • Multiplication Rule of Counting. If one task can be done in n1 ways, and a second task, given the first, can be done in n2 ways, and a third task, given the first two tasks, can be done in n3 ways, and so on for k tasks, then the number of ways the k tasks can be done is (n1 )(n2 )(n3 ) . . . (nk ). Suppose we have three steps in an investment decision process. The first step can be done in two ways, the second in four ways, and the third in three ways. Following the multiplication rule, there are (2)(4)(3) = 24 ways in which we can carry out the three steps. Another illustration is the assignment of members of a group to an equal number of positions. For example, suppose you want to assign three security analysts to cover three different industries. In how many ways can the assignments be made? The first analyst may be assigned in three different ways. Then two industries remain. The second analyst can be assigned in two different ways. Then one industry remains. The third and last analyst can be assigned in only one way. The total number of different assignments equals (3)(2)(1) = 6. The compact notation for the multiplication we have just performed is 3! (read: 3 factorial). If we had n analysts, the number of ways we could assign them to n tasks would be n! = n(n − 1)(n − 2)(n − 3) . . . 1 19 Henriksson and Merton (1981). 167 Chapter 4 Probability Concepts or n factorial. (By convention, 0! = 1.) To review, in this application we repeatedly carry out an operation (here, job assignment) until we use up all members of a group (here, three analysts). With n members in the group, the multiplication formula reduces to n factorial.20 The next type of counting problem can be called labeling problems.21 We want to give each object in a group a label, to place it in a category. The following example illustrates this type of problem. A mutual fund guide ranked 18 bond mutual funds by total returns for the year 2000. The guide also assigned each fund one of five risk labels: high risk (four funds), above-average risk (four funds), average risk (three funds), below-average risk (four funds), and low risk (three funds); as 4 + 4 + 3 + 4 + 3 = 18, all the funds are accounted for. How many different ways can we take 18 mutual funds and label 4 of them high risk, 4 above-average risk, 3 average risk, 4 below-average risk, and 3 low risk, so that each fund is labeled? The answer is close to 13 billion. We can label any of 18 funds high risk (the first slot), then any of 17 remaining funds, then any of 16 remaining funds, then any of 15 remaining funds (now we have 4 funds in the high risk group); then we can label any of 14 remaining funds above-average risk, then any of 13 remaining funds, and so forth. There are 18! possible sequences. However, order of assignment within a category does not matter. For example, whether a fund occupies the first or third slot of the four funds labeled high risk, the fund has the same label (high risk). Thus there are 4! ways to assign a given group of four funds to the four high risk slots. Making the same argument for the other categories, in total there are (4!)(4!)(3!)(4!)(3!) equivalent sequences. To eliminate such redundancies from the 18! total, we divide 18! by (4!)(4!)(3!)(4!)(3!). We have 18!/(4!)(4!)(3!)(4!)(3!) = 18!/(24)(24)(6)(24)(6) = 12,864,852,000. This procedure generalizes as follows: • Multinomial Formula (General Formula for Labeling Problems). The number of ways that n objects can be labeled with k different labels, with n1 of the first type, n2 of the second type, and so on, with n1 + n2 + · · · + nk = n, is given by n! n1 !n2 ! . . . nk ! The multinomial formula with two different labels (k = 2) is especially important. This special case is called the combination formula. A combination is a listing in which the order of the listed items does not matter. We state the combination formula in a traditional way, but no new concepts are involved. Using the notation in the formula below, the number of objects with the first label is r = n1 and the number with the second label is n − r = n2 (there are just two categories, so n1 + n2 = n). Here is the formula. • Combination Formula (Binomial Formula). The number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does not matter, is " # n! n = n Cr = r (n − r)!r! 20 The shortest explanation of n factorial is that it is the number of ways to order n objects in a row. In all the problems to which we apply this counting method, we must use up all the members of a group (sampling without replacement). 21 This discussion follows Kemeny, Schleifer, Snell, and Thompson (1972) in terminology and approach. 168 Quantitative Investment Analysis " # n Here n Cr and are shorthand notations for n!/(n − r)!r! (read: n choose r, or n r combination r). If we label the r objects as belongs to the group and the remaining objects as does not belong to the group, whatever the group of interest, the combination formula tells us how many ways we can select a group of size r. We can illustrate this formula with the binomial option pricing model. This model describes the movement of the underlying asset as a series of moves, price up (U) or price down (D). For example, two sequences of five moves containing three up moves, such as UUUDD and UDUUD, result in the same final stock price. At least for an option with a payoff dependent on final stock price, the number but not the order of up moves in a sequence matters. How many sequences of five moves belong to the group with three up moves? The answer is 10, calculated using the combination formula (‘‘5 choose 3’’): 5 C3 = 5!/(5 − 3)!3! = (5)(4)(3)(2)(1)/(2)(1)(3)(2)(1) = 120/12 = 10 ways A useful fact can be illustrated as follows: 5 C3 = 5!/2!3! equals 5 C2 = 5!/3!2!, as 3 + 2 = 5; = 5!/1!4! equals 5 C1 = 5!/4!1!, as 4 + 1 = 5. This symmetrical relationship can save work when we need to calculate many possible combinations. Suppose jurors want to select three companies out of a group of five to receive the first-, second-, and third-place awards for the best annual report. In how many ways can the jurors make the three awards? Order does matter if we want to distinguish among the three awards (the rank within the group of three); clearly the question makes order important. On the other hand, if the question were ‘‘In how many ways can the jurors choose three winners, without regard to place of finish?’’ we would use the combination formula. To address the first question above, we need to count ordered listings such as first place, New Company; second place, Fir Company; third place, Well Company. An ordered listing is known as a permutation, and the formula that counts the number of permutations is known as the permutation formula.22 5 C4 • Permutation Formula. The number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does matter, is n Pr = n! (n − r)! So the jurors have 5 P3 = 5!/(5 − 3)! = (5)(4)(3)(2)(1)/(2)(1) = 120/2 = 60 ways in which they can make their awards. To see why this formula works, note that (5)(4)(3)(2)(1)/(2)(1) reduces to (5)(4)(3), after cancellation of terms. This calculation counts the number of ways to fill three slots choosing from a group of five people, according to the multiplication rule of counting. This number is naturally larger than it would be if order did not matter (compare 60 to the value of 10 for ‘‘5 choose 3’’ that we calculated above). For example, first place, Well Company; second place, Fir Company; third place, New Company contains the same three companies as first place, New Company; second place, Fir Company; third place, Well Company. If we were concerned only with award winners (without regard to 22 A more formal definition states that a permutation is an ordered subset of n distinct objects. Chapter 4 Probability Concepts 169 place of finish), the two listings would count as one combination. But when we are concerned with the order of finish, the listings count as two permutations. Answering the following questions may help you apply the counting methods we have presented in this section. 1. Does the task that I want to measure have a finite number of possible outcomes? If the answer is yes, you may be able to use a tool in this section, and you can go to the second question. If the answer is no, the number of outcomes is infinite, and the tools in this section do not apply. 2. Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use n factorial. If the answer is no, go to the third question. 3. Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes, use the multinomial formula. If the answer is no, go to the fourth question. 4. Do I want to count the number of ways that I can choose r objects from a total of n, when the order in which I list the r objects does not matter (can I give the r objects a label)? If the answer to these questions is yes, the combination formula applies. If the answer is no, go to the fifth question. 5. Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects is important? If the answer is yes, the permutation formula applies. If the answer is no, go to question 6. 6. Can the multiplication rule of counting be used? If it cannot, you may have to count the possibilities one by one, or use more advanced techniques than those presented here.23 23 Feller (1957) contains a very full treatment of counting problems and solution methods. CHAPTER 5 COMMON PROBABILITY DISTRIBUTIONS 1. INTRODUCTION In nearly all investment decisions we work with random variables. The return on a stock and its earnings per share are familiar examples of random variables. To make probability statements about a random variable, we need to understand its probability distribution. A probability distribution specifies the probabilities of the possible outcomes of a random variable. In this chapter, we present important facts about four probability distributions and their investment uses. These four distributions—the uniform, binomial, normal, and lognormal—are used extensively in investment analysis. They are used in such basic valuation models as the Black–Scholes–Merton option pricing model, the binomial option pricing model, and the capital asset pricing model. With the working knowledge of probability distributions provided in this chapter, you will also be better prepared to study and use other quantitative methods such as hypothesis testing, regression analysis, and time-series analysis. After discussing probability distributions, we end the chapter with an introduction to Monte Carlo simulation, a computer-based tool for obtaining information on complex problems. For example, an investment analyst may want to experiment with an investment idea without actually implementing it. Or she may need to price a complex option for which no simple pricing formula exists. In these cases and many others, Monte Carlo simulation is an important resource. To conduct a Monte Carlo simulation, the analyst must identify risk factors associated with the problem and specify probability distributions for them. Hence, Monte Carlo simulation is a tool that requires an understanding of probability distributions. Before we discuss specific probability distributions, we define basic concepts and terms. We then illustrate the operation of these concepts through the simplest distribution, the uniform distribution. That done, we address probability distributions that have more applications in investment work but also greater complexity. 2. DISCRETE RANDOM VARIABLES A random variable is a quantity whose future outcomes are uncertain. The two basic types of random variables are discrete random variables and continuous random variables. A discrete 171 172 Quantitative Investment Analysis random variable can take on at most a countable number of possible values. For example, a discrete random variable X can take on a limited number of outcomes x1 , x2 , . . . , xn (n possible outcomes), or a discrete random variable Y can take on an unlimited number of outcomes y1 , y2 , . . . (without end).1 Because we can count all the possible outcomes of X and Y (even if we go on forever in the case of Y ), both X and Y satisfy the definition of a discrete random variable. By contrast, we cannot count the outcomes of a continuous random variable. We cannot describe the possible outcomes of a continuous random variable Z with a list z1 , z2 , . . . because the outcome (z1 + z2 )/2, not in the list, would always be possible. Rate of return is an example of a continuous random variable. In working with a random variable, we need to understand its possible outcomes. For example, stocks traded on the New York Stock Exchange and Nasdaq are quoted in ticks of $0.01. Quoted stock price is thus a discrete random variable with possible values $0, $0.01, $0.02, . . . But we can also model stock price as a continuous random variable (as a lognormal random variable, to look ahead). In many applications, we have a choice between using a discrete or a continuous distribution. We are usually guided by which distribution is most efficient for the task we face. This opportunity for choice is not surprising, as many discrete distributions can be approximated with a continuous distribution, and vice versa. In most practical cases, a probability distribution is only a mathematical idealization, or approximate model, of the relative frequencies of a random variable’s possible outcomes. EXAMPLE 5-1 The Distribution of Bond Price You are researching a probability model for bond price, and you begin by thinking about the characteristics of bonds that affect price. What are the lowest and the highest possible values for bond price? Why? What are some other characteristics of bonds that may affect the distribution of bond price? The lowest possible value of bond price is 0, when the bond is worthless. Identifying the highest possible value for bond price is more challenging. The promised payments on a coupon bond are the coupons (interest payments) plus the face amount (principal). The price of a bond is the present discounted value of these promised payments. Because investors require a return on their investments, 0 percent is the lower limit on the discount rate that investors would use to discount a bond’s promised payments. At a discount rate of 0 percent, the price of a bond is the sum of the face value and the remaining coupons without any discounting. The discount rate thus places the upper limit on bond price. Suppose, for example, that face value is $1,000 and two $40 coupons remain; the interval $0 to $1,080 captures all possible values of the bond’s price. This upper limit decreases through time as the number of remaining payments decreases. 1 We follow the convention that an uppercase letter represents a random variable and a lowercase letter represents an outcome or specific value of the random variable. Thus X refers to the random variable, and x refers to an outcome of X . We subscript outcomes, as in x1 and x2 , when we need to distinguish among different outcomes in a list of outcomes of a random variable. Chapter 5 Common Probability Distributions 173 Other characteristics of a bond also affect its price distribution. Pull to par value is one such characteristic: As the maturity date approaches, the standard deviation of bond price tends to grow smaller as bond price converges to par value. Embedded options also affect bond price. For example, with bonds that are currently callable, the issuer may retire the bonds at a prespecified premium above par; this option of the issuer cuts off part of the bond’s upside. Modeling bond price distribution is a challenging problem. Every random variable is associated with a probability distribution that describes the variable completely. We can view a probability distribution in two ways. The basic view is the probability function, which specifies the probability that the random variable takes on a specific value: P(X = x) is the probability that a random variable X takes on the value x. (Note that capital X represents the random variable and lowercase x represents a specific value that the random variable may take.) For a discrete random variable, the shorthand notation for the probability function is p(x) = P(X = x). For continuous random variables, the probability function is denoted f (x) and called the probability density function (pdf), or just the density.2 A probability function has two key properties (which we state, without loss of generality, using the notation for a discrete random variable): • • 0 ≤ p(x) ≤ 1, because probability is a number between 0 and 1. The sum of the probabilities p(x) over all values of X equals 1. If we add up the probabilities of all the distinct possible outcomes of a random variable, that sum must equal 1. We are often interested in finding the probability of a range of outcomes rather than a specific outcome. In these cases, we take the second view of a probability distribution, the cumulative distribution function (cdf). The cumulative distribution function, or distribution function for short, gives the probability that a random variable X is less than or equal to a particular value x, P(X ≤ x). For both discrete and continuous random variables, the shorthand notation is F (x) = P(X ≤ x). How does the cumulative distribution function relate to the probability function? The word ‘‘cumulative’’ tells the story. To find F (x), we sum up, or cumulate, values of the probability function for all outcomes less than or equal to x. The function of the cdf is parallel to that of cumulative relative frequency, which we discussed in the chapter on statistical concepts and market returns. Next, we illustrate these concepts with examples and show how we use discrete and continuous distributions. We start with the simplest distribution, the discrete uniform. 2.1. The Discrete Uniform Distribution The simplest of all probability distributions is the discrete uniform distribution. Suppose that the possible outcomes are the integers (whole numbers) 1 to 8, inclusive, and the probability that the random variable takes on any of these possible values is the same for all outcomes (that 2 The technical term for the probability function of a discrete random variable, probability mass function (pmf), is used less frequently. 174 Quantitative Investment Analysis TABLE 5-1 Probability Function and Cumulative Distribution Function for a Discrete Uniform Random Variable X =x Probability Function p(x) = P(X = x) Cumulative Distribution Function F (x) = P(X ≤ x) 1 2 3 4 5 6 7 8 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 is, it is uniform). With eight outcomes, p(x) = 1/8, or 0.125, for all value of X (X = 1, 2, 3, 4, 5, 6, 7, 8); the statement just made is a complete description of this discrete uniform random variable. The distribution has a finite number of specified outcomes, and each outcome is equally likely. Table 5-1 summarizes the two views of this random variable, the probability function and the cumulative distribution function. We can use Table 5-1 to find three probabilities: P(X ≤ 7), P(4 ≤ X ≤ 6), and P(4 < X ≤ 6). The following examples illustrate how to use the cdf to find the probability that a random variable will fall in any interval (for any random variable, not only the uniform). The probability that X is less than or equal to 7, P(X ≤ 7), is the next-to-last entry in the third column, 0.875 or 87.5 percent. • To find P(4 ≤ X ≤ 6), we need to find the sum of three probabilities: p(4), p(5), and p(6). We can find this sum in two ways. We can add p(4), p(5), and p(6) from the second column. Or we can calculate the probability as the difference between two values of the cumulative distribution function: • F (6) = P(X ≤ 6) = p(6) + p(5) + p(4) + p(3) + p(2) + p(1) F (3) = P(X ≤ 3) = p(3) + p(2) + p(1) so P(4 ≤ X ≤ 6) = F (6) − F (3) = p(6) + p(5) + p(4) = 3/8 • So we calculate the second probability as F (6) − F (3) = 3/8. The third probability, P(4 < X ≤ 6), the probability that X is less than or equal to 6 but greater than 4, is p(5) + p(6). We compute it as follows, using the cdf: P(4 < X ≤ 6) = P(X ≤ 6) − P(X ≤ 4) = F (6) − F (4) = p(6) + p(5) = 2/8 So we calculate the third probability as F (6) − F (4) = 2/8. Chapter 5 Common Probability Distributions 175 Suppose we want to check that the discrete uniform probability function satisfies the general properties of a probability function given earlier. The first property is 0 ≤ p(x) ≤ 1. We see that p(x) = 1/8 for all x in the first column of the table. (Note that p(x) equals 0 for numbers x such as −14 or 12.215 that are not in that column.) The first property is satisfied. The second property is that the probabilities sum to 1. The entries in the second column of Table 5-1 do sum to 1. The cdf has two other characteristic properties: • • The cdf lies between 0 and 1 for any x : 0 ≤ F (x) ≤ 1 As we increase x, the cdf either increases or remains constant. Check these statements by looking at the third column in Table 5-1. We now have some experience working with probability functions and cdfs for discrete random variables. Later in this chapter, we will discuss Monte Carlo simulation, a methodology driven by random numbers. As we will see, the uniform distribution has an important technical use: It is the basis for generating random numbers, which in turn produce random observations for all other probability distributions.3 2.2. The Binomial Distribution In many investment contexts, we view a result as either a success or a failure, or as binary (twofold) in some other way. When we make probability statements about a record of successes and failures, or about anything with binary outcomes, we often use the binomial distribution. What is a good model for how a stock price moves through time? Different models are appropriate for different uses. Cox, Ross, and Rubinstein (1979) developed an option pricing model based on binary moves, price up or price down, for the asset underlying the option. Their binomial option pricing model was the first of a class of related option pricing models that have played an important role in the development of the derivatives industry. That fact alone would be sufficient reason for studying the binomial distribution, but the binomial distribution has uses in decision-making as well. The building block of the binomial distribution is the Bernoulli random variable, named after the Swiss probabilist Jakob Bernoulli (1654–1704). Suppose we have a trial (an event that may repeat) that produces one of two outcomes. Such a trial is a Bernoulli trial. If we let Y equal 1 when the outcome is success and Y equal 0 when the outcome is failure, then the probability function of the Bernoulli random variable Y is p(1) = P(Y = 1) = p p(0) = P(Y = 0) = 1 − p where p is the probability that the trial is a success. Our next example is the very first step on the road to understanding the binomial option pricing model. 3 See Hillier and Lieberman (2000). Random numbers initially generated by computers are usually random positive integer numbers that are converted to approximate continuous uniform random numbers between 0 and 1. Then the continuous uniform random numbers are used to produce random observations on other distributions, such as the normal, using various techniques. We will discuss random observation generation further in the section on Monte Carlo simulation. 176 Quantitative Investment Analysis EXAMPLE 5-2 One-Period Stock Price Movement as a Bernoulli Random Variable Suppose we describe stock price movement in the following way. Stock price today is S. Next period stock price can move up or down. The probability of an up move is p, and the probability of a down move is 1 − p. Thus, stock price is a Bernoulli random variable with probability of success (an up move) equal to p. When the stock moves up, ending price is uS, with u equal to 1 plus the rate of return if the stock moves up. For example, if the stock earns 0.01 or 1 percent on an up move, u = 1.01. When the stock moves down, ending price is dS, with d equal to 1 plus the rate of return if the stock moves down. For example, if the stock earns −0.01 or −1 percent on a down move, d = 0.99. Figure 5-1 shows a diagram of this model of stock price dynamics. End-of-Period Stock Price Probability = p Stock Price Moves Up: Stock Price Equals uS Stock Price Today, S Probability = 1 − p Stock Price Moves Down: Stock Price Equals dS FIGURE 5-1 One-Period Stock Price as a Bernoulli Random Variable We will continue with the above example later. In the model of stock price movement in Example 5-2, success and failure at a given trial relate to up moves and down moves, respectively. In the following example, success is a profitable trade and failure is an unprofitable one. EXAMPLE 5-3 A Trading Desk Evaluates Block Brokers (1) You work in equities trading at an institutional money manager that regularly trades with a number of block brokers. Blocks are orders to sell or buy that are too large for the liquidity ordinarily available in dealer networks or stock exchanges. Your firm has known interests in certain kinds of stock. Block brokers call your trading desk when they want to sell blocks of stocks that they think your firm may be interested in buying. You know that these transactions have definite risks. For example, if the broker’s client (the seller of the shares) has unfavorable information on the stock, or if the total amount 177 Chapter 5 Common Probability Distributions he is selling through all channels is not truthfully communicated to you, you may see an immediate loss on the trade. From time to time, your firm audits the performance of block brokers. Your firm calculates the post-trade, market-risk-adjusted dollar returns on stocks purchased from block brokers. On that basis, you classify each trade as unprofitable or profitable. You have summarized the performance of the brokers in a spreadsheet, excerpted in Table 5-2 for November 2003. (The broker names are coded BB001 and BB002.) TABLE 5-2 Block Trading Gains and Losses November 2003 BB001 BB002 Profitable Trades Losing Trades 3 5 9 3 View each trade as a Bernoulli trial. Calculate the percentage of profitable trades with the two block brokers for November 2003. These are estimates of p, the underlying probability of a successful (profitable) trade with each broker. Your firm has logged 3 + 9 = 12 trades (the row total) with block broker BB001. Because 3 of the 12 trades were profitable, the percentage of profitable trades was 3/12 or 25 percent. With broker BB002, the percentage of profitable trades was 5/8 or 62.5 percent. A trade is a Bernoulli trial, and the above calculations provide estimates of the underlying probability of a profitable trade (success) with the two brokers. For broker BB001, your estimate is p̂ = 0.25; for broker BB002, your estimate is p̂ = 0.625.4 In n Bernoulli trials, we can have 0 to n successes. If the outcome of an individual trial is random, the total number of successes in n trials is also random. A binomial random variable X is defined as the number of successes in n Bernoulli trials. A binomial random variable is the sum of Bernoulli random variables Yi , i = 1, 2, . . . , n: X = Y1 + Y 2 + · · · + Y n where Yi is the outcome on the ith trial (1 if a success, 0 if a failure). We know that a Bernoulli random variable is defined by the parameter p. The number of trials, n, is the second parameter of a binomial random variable. The binomial distribution makes these assumptions: • • The probability, p, of success is constant for all trials. The trials are independent. 4 The ‘‘hat’’ over p indicates that it is an estimate of p, the underlying probability of a profitable trade with the broker. 178 Quantitative Investment Analysis The second assumption has great simplifying force. If individual trials were correlated, calculating the probability of a given number of successes in n trials would be much more complicated. Under the above two assumptions, a binomial random variable is completely described by two parameters, n and p. We write X ∼ B(n, p) which we read as ‘‘X has a binomial distribution with parameters n and p.’’ You can see that a Bernoulli random variable is a binomial random variable with n = 1 : Y ∼ B(1, p). Now we can find the general expression for the probability that a binomial random variable shows x successes in n trials. We can think in terms of a model of stock price dynamics that can be generalized to allow any possible stock price movements if the periods are made extremely small. Each period is a Bernoulli trial: With probability p, the stock price moves up; with probability 1 − p, the price moves down. A success is an up move, and x is the number of up moves or successes in n periods (trials). With each period’s moves independent and p constant, the number of up moves in n periods is a binomial random variable. We now develop an expression for P(X = x), the probability function for a binomial random variable. Any sequence of n periods that shows exactly x up moves must show n − x down moves. We have many different ways to order the up moves and down moves to get a total of x up moves, but given independent trials, any sequence with x up moves must occur with probability px (1 − p)n−x . Now we need to multiply this probability by the number of different ways we can get a sequence with x up moves. Using a basic result in counting from the chapter on probability concepts, there are n! (n − x)!x! different sequences in n trials that result in x up moves (or successes) and n − x down moves (or failures). Recall from the chapter on probability concepts that n factorial (n!) is defined as n(n − 1)(n − 2) . . . 1 (and 0! = 1 by convention). For example, 5! = (5)(4)(3)(2)(1) = 120. The combination formula n!/[(n − x)!x!] is denoted by ! n x " (read ‘‘n combination x’’ or ‘‘n choose x’’). For example, over three periods, exactly three different sequences have two up moves: UUD, UDU, and DUU. We confirm this by ! 3 2 " = 3! (3)(2)(1) = =3 (3 − 2)!2! (1)(2)(1) If, hypothetically, each sequence with two up moves had a probability of 0.15, then the total probability of two up moves in three periods would be 3 × 0.15 = 0.45. This example should persuade you that for X distributed B(n, p), the probability of x successes in n trials is given by p(x) = P(X = x) = ! n x " px (1 − p)n−x = n! px (1 − p)n−x (n − x)!x! (5-1) 179 Chapter 5 Common Probability Distributions TABLE 5-3 Binomial Probabilities, p = 0.50 and n = 5 Number of Up Moves, x (1) Number of Possible Ways to Reach x Up Moves (2) 0 1 2 3 4 5 1 5 10 10 5 1 Probability for Each Way (3) 0.500 (1 − 0.50)5 0.501 (1 − 0.50)4 0.502 (1 − 0.50)3 0.503 (1 − 0.50)2 0.504 (1 − 0.50)1 0.505 (1 − 0.50)0 = 0.03125 = 0.03125 = 0.03125 = 0.03125 = 0.03125 = 0.03125 Probability for x, p(x) F (x) = P(X ≤ x) (4) = (2) × (3) (5) 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125 0.03125 0.18750 0.50000 0.81250 0.96875 1.00000 Some distributions are always symmetric, such as the normal, and others are always asymmetric or skewed, such as the lognormal. The binomial distribution is symmetric when the probability of success on a trial is 0.50, but it is asymmetric or skewed otherwise. We illustrate Equation 5-1 (the probability function) and the cdf through the symmetrical case. Consider a random variable distributed B(n = 5, p = 0.50). Table 5-3 contains a complete description of this random variable. The fourth column of Table 5-3 is Column 2, n combination x, times Column 3, px (1 − p)n−x ; Column 4 gives the probability for each value of the number of up moves from the first column. The fifth column, cumulating the entries in the fourth column, is the cumulative distribution function. What would happen if we kept n = 5 but sharply lowered the probability of success on a trial to 10 percent? ‘‘Probability for Each Way’’ for X = 0 (no up moves) would then be about 59 percent: 0.100 (1 − 0.10)5 = 0.59049. Because zero successes could still happen one way (Column 2), p(0) = 59 percent. You may want to check that given p = 0.10, P(X ≤ 2) = 99.14 percent: The probability of two or fewer up moves would be more than 99 percent. The random variable’s probability would be massed on 0, 1, and 2 up moves, and the probability of larger outcomes would be minute. The outcomes of 3 and larger would be the long right tail, and the distribution would be right skewed. On the other hand, if we set p = 0.90, we would have the mirror image of the distribution with p = 0.10. The distribution would be left skewed. With an understanding of the binomial probability function in hand, we can continue with our example of block brokers. EXAMPLE 5-4 A Trading Desk Evaluates Block Brokers (2) You now want to evaluate the performance of the block brokers in Example 5-3. You begin with two questions: 1. If you are paying a fair price on average in your trades with a broker, what should be the probability of a profitable trade? 180 Quantitative Investment Analysis 2. Did each broker meet or miss that expectation on probability? You also realize that the brokers’ performance has to be evaluated in light of the sample’s size, and for that you need to use the binomial probability function (Equation 5-1). You thus address the following (referring to the data in Example 5-3): 3. Under the assumption that the prices of trades were fair, (a) calculate the probability of three or fewer profitable trades with broker BB001. (b) calculate the probability of five or more profitable trades with broker BB002. Solutions to 1 and 2: If the price you trade at is fair, 50 percent of the trades you do with a broker should be profitable.5 The rate of profitable trades with broker BB001 was 25 percent. Therefore, broker BB001 missed your performance expectation. Broker BB002, at 62.5 percent profitable trades, exceeded your expectation. Solution to 3: A. For broker BB001, the number of trades (the trials) was n = 12, and 3 were profitable. You are asked to calculate the probability of three or fewer profitable trades, F (3) = p(3) + p(2) + p(1) + p(0). Suppose the underlying probability of a profitable trade with BB001 is p = 0.50. With n = 12 and p = 0.50, according to Equation 5-1 the probability of three profitable trades is ! " ! " n 12 p(3) = px (1 − p)n−x = (0.503 )(0.509 ) x 3 = 12! 0.5012 = 220(0.000244) = 0.053711 (12 − 3)!3! The probability of exactly 3 profitable trades out of 12 is 5.4 percent if broker BB001 were giving you fair prices. Now you need to calculate the other probabilities: p(2) = [12!/(12 − 2)!2!](0.502 )(0.5010 ) = 66(0.000244) = 0.016113 p(1) = [12!/(12 − 1)!1!](0.501 )(0.5011 ) = 12(0.000244) = 0.00293 p(0) = [12!/(12 − 0)!0!](0.500 )(0.5012 ) = 1(0.000244) = 0.000244 Adding all the probabilities, F (3) = 0.053711 + 0.016113 + 0.00293 + 0.000244 = 0.072998 or 7.3 percent. The probability of doing 3 or fewer profitable trades out of 12 would be 7.3 percent if your trading desk were getting fair prices from broker BB001. B. For broker BB002, you are assessing the probability that the underlying probability of a profitable trade with this broker was 50 percent, despite the good results. The question was framed as the probability of doing five or more profitable trades if the 5 Of course, you need to adjust for the direction of the overall market after the trade (any broker’s record will be helped by a bull market) and perhaps make other risk adjustments. Assume that these adjustments have been made. Chapter 5 Common Probability Distributions 181 underlying probability is 50 percent: 1 − F (4) = p(5) + p(6) + p(7) + p(8). You could calculate F (4) and subtract it from 1, but you can also calculate p(5) + p(6) + p(7) + p(8) directly. You begin by calculating the probability that exactly 5 out of 8 trades would be profitable if BB002 were giving you fair prices: p(5) = ! 8 5 " (0.505 )(0.503 ) = 56(0.003906) = 0.21875 The probability is about 21.9 percent. The other probabilities are p(6) = 28(0.003906) = 0.109375 p(7) = 8(0.003906) = 0.03125 p(8) = 1(0.003906) = 0.003906 So p(5) + p(6) + p(7) + p(8) = 0.21875 + 0.109375 + 0.03125 + 0.003906 = 0.363281 or 36.3 percent.6 A 36.3 percent probability is substantial; the underlying probability of executing a fair trade with BB002 might well have been 0.50 despite your success with BB002 in November 2003. If one of the trades with BB002 had been reclassified from profitable to unprofitable, exactly half the trades would have been profitable. In summary, your trading desk is getting at least fair prices from BB002; you will probably want to accumulate additional evidence before concluding that you are trading at better-than-fair prices. The magnitude of the profits and losses in these trades is another important consideration. If all profitable trades had small profits but all unprofitable trades had large losses, for example, you might lose money on your trades even if the majority of them were profitable. In the next example, the binomial distribution helps in evaluating the performance of an investment manager. EXAMPLE 5-5 Meeting a Tracking Error Objective You work for a pension fund sponsor. You have assigned a new money manager to manage a $500 million portfolio indexed on the MSCI EAFE (Europe, Australasia, and Far East) Index, which is designed to measure developed-market equity performance 6 In this example all calculations were worked through by hand, but binomial probability and cdf functions are also available in computer spreadsheet programs. 182 Quantitative Investment Analysis excluding the United States and Canada. After research, you believe it is reasonable to expect that the manager will keep tracking error within a band of 75 basis points (bps) of the benchmark’s return, on a quarterly basis.7 Tracking error is the total return on the portfolio (gross of fees) minus the total return on the benchmark index—here, the EAFE.8 To quantify this expectation further, you will be satisfied if tracking error is within the 75 bps band 90 percent of the time. The manager meets the objective in six out of eight quarters. Of course, six out of eight quarters is a 75 percent success rate. But how does the manager’s record precisely relate to your expectation of a 90 percent success rate and the sample size, 8 observations? To answer this question, you must find the probability that, given an assumed true or underlying success rate of 90 percent, performance could be as bad as or worse than that delivered. Calculate the probability (by hand or with a spreadsheet). Specifically, you want to find the probability that tracking error is within the 75 bps band in six or fewer quarters out of the eight in the sample. With n = 8 and p = 0.90, this probability is F (6) = p(6) + p(5) + p(4) + p(3) + p(2) + p(1) + p(0). Start with p(6) = (8!/6!2!)(0.906 )(0.102 ) = 28(0.005314) = 0.148803 and work through the other probabilities: p(5) = (8!/5!3!)(0.905 )(0.103 ) = 56(0.00059) = 0.033067 p(4) = (8!/4!4!)(0.904 )(0.104 ) = 70(0.000066) = 0.004593 p(3) = (8!/3!5!)(0.903 )(0.105 ) = 56(0.000007) = 0.000408 p(2) = (8!/2!6!)(0.902 )(0.106 ) = 28(0.000001) = 0.000023 p(1) = (8!/1!7!)(0.901 )(0.107 ) = 8(0.00000009) = 0.00000072 p(0) = (8!/0!8!)(0.900 )(0.108 ) = 1(0.00000001) = 0.00000001 Summing all these probabilities, you conclude that F (6) = 0.148803 + 0.033067 + 0.004593+ 0.000408+ 0.000023 + 0.00000072 + 0.00000001 = 0.186895 or 18.7 percent. There is a moderate 18.7 percent probability that the manager would show the record he did (or a worse record) if he had the skill to meet your expectations 90 percent of the time. You can use other evaluation concepts such as tracking risk, defined as the standard deviation of tracking error, to assess the manager’s performance. The calculation above would be only one input into any conclusions that you reach concerning the manager’s performance. But to answer problems involving success rates, you need to be skilled in using the binomial distribution. 7A basis point is one-hundredth of 1 percent (0.01 percent). practitioners use tracking error to describe what we later call tracking risk, the standard deviation of the differences between the portfolio’s and benchmark’s returns. 8 Some 183 Chapter 5 Common Probability Distributions TABLE 5-4 Mean and Variance of Binomial Random Variables Mean Variance p np p(1 − p) np(1 − p) Bernoulli, B(1, p) Binomial, B(n, p) Two descriptors of a distribution that are often used in investments are the mean and the variance (or the standard deviation, the positive square root of variance).9 Table 5-4 gives the expressions for the mean and variance of binomial random variables. Because a single Bernoulli random variable, Y ∼ B(1, p), takes on the value 1 with probability p and the value 0 with probability 1 − p, its mean or weighted-average outcome is p. Its variance is p(1 − p).10 A general binomial random variable, B(n, p), is the sum of n Bernoulli random variables, and so the mean of a B(n, p) random variable is np. Given that a B(1, p) variable has variance p(1 − p), the variance of a B(n, p) random variable is n times that value, or np(1 − p), assuming that all the trials (Bernoulli random variables) are independent. We can illustrate the calculation for two binomial random variables with differing probabilities as follows: Random Variable B(n = 5, p = 0.50) B(n = 5, p = 0.10) Mean 2.50 = 5(0.50) 0.50 = 5(0.10) Variance 1.25 = 5(0.50)(0.50) 0.45 = 5(0.10)(0.90) For a B(n = 5, p = 0.50) random variable, the expected number of successes is 2.5 with a standard deviation of 1.118 = (1.25)1/2 ; for a B(n = 5, p = 0.10) random variable, the expected number of successes is 0.50 with a standard deviation of 0.67 = (0.45)1/2 . EXAMPLE 5-6 The Expected Number of Defaults in a Bond Portfolio Suppose as a bond analyst you are asked to estimate the number of bond issues expected to default over the next year in an unmanaged high-yield bond portfolio with 25 U.S. issues from distinct issuers. The credit ratings of the bonds in the portfolio are tightly clustered around Moody’s B2/Standard & Poor’s B, meaning that the bonds 9 The mean (or arithmetic mean) is the sum of all values in a distribution or dataset, divided by the number of values summed. The variance is a measure of dispersion about the mean. See the chapters on statistical concepts and market returns for further details on these concepts. 10 We can show that p(1 − p) is the variance of a Bernoulli random variable as follows, noting that a Bernoulli random variable can take on only one of two values, 1 or 0: σ2 (Y ) = E[(Y − EY )2 ] = E[(Y − p)2 ] = (1 − p)2 p + (0 − p)2 (1 − p) = (1 − p)[(1 − p)p + p2 ] = p(1 − p). 184 Quantitative Investment Analysis are speculative with respect to the capacity to pay interest and repay principal. The estimated annual default rate for B2/B rated bonds is 10.7 percent. 1. Over the next year, what is the expected number of defaults in the portfolio, assuming a binomial model for defaults? 2. Estimate the standard deviation of the number of defaults over the coming year. 3. Critique the use of the binomial probability model in this context. Solution to 1: For each bond, we can define a Bernoulli random variable equal to 1 if the bond defaults during the year and zero otherwise. With 25 bonds, the expected number of defaults over the year is np = 25(0.107) = 2.675 or approximately 3. Solution to 2: The variance is np(1 − p) = 25(0.107)(0.893) = 2.388775. The standard deviation is (2.388775)1/2 = 1.55. Thus a two-standard-deviation confidence interval about the expected number of defaults would run from approximately 0 to approximately 6, for example. Solution to 3: An assumption of the binomial model is that the trials are independent. In this context, a trial relates to whether an individual bond issue will default over the next year. Because the issuing companies probably share exposure to common economic factors, the trials may not be independent. Nevertheless, for a quick estimate of the expected number of defaults, the binomial model may be adequate. Earlier, we looked at a simple one-period model for stock price movement. Now we extend the model to describe stock price movement on three consecutive days. Each day is an independent trial. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 − p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. We graph stock price movement in Figure 5-2, where we now associate each of the n = 3 stock price moves with time indexed by t. The shape of the graph suggests why it is called a binomial tree. Each boxed value from which successive moves or outcomes branch in the tree is called a node; in this example, a node is potential value for the stock price at a specified time. We see from the tree that the stock price at t = 3 has four possible values: uuuS, uudS, uddS, and dddS. The probability that the stock price equals any one of these four values is given by the binomial distribution. For example, three sequences of moves result in a final stock price of uudS: These are uud, udu, and duu. These sequences have two up moves out of three moves in total; the combination formula confirms that the number of ways to get two up moves (successes) in three periods (trials) is 3!/(3 − 2)!2! = 3. Next note that each of these sequences, uud, udu, and duu, has probability p2 (1 − p). So P(S3 = uudS) = 3p2 (1 − p), where S3 indicates the stock’s price after three moves. The binomial random variable in this application is the number of up moves. Final stock price distribution is a function of the initial stock price, the number of up moves, and the size of the up moves and down moves. We cannot say that stock price itself is a binomial random variable; rather, it is a function of a binomial random variable, as well as of u and d , and initial price. This richness is actually one key to why this way of modeling stock price is useful: It Chapter 5 Common Probability Distributions 185 FIGURE 5-2 A Binomial Model of Stock Price Movement allows us to choose values of these parameters to approximate various distributions for stock price (using a large number of time periods).11 One distribution that can be approximated is the lognormal, an important continuous distribution model for stock price that we will discuss later. The flexibility extends further. In the tree shown above, the transition probabilities are the same at each node: p for an up move and 1 − p for a down move. That standard formula describes a process in which stock return volatility is constant through time. Option experts, however, sometimes model changing volatility through time using a binomial tree in which the probabilities for up and down moves differ at different nodes. The binomial tree also supplies the possibility of testing a condition or contingency at any node. This flexibility is useful in investment applications such as option pricing. Consider an American call option on a dividend-paying stock. (Recall that an American option can be exercised at any time before expiration, at any node on the tree.) Just before an ex-dividend date, it may be optimal to exercise an American call option on stock to buy the stock and receive the dividend.12 If we model stock price with a binomial tree, we can test, at each node, whether exercising the option is optimal. Also, if we know the value of the call at the four terminal nodes at t = 3 and we have a model for discounting values by one period, we can step backward one period to t = 2 to find the call’s value at the three nodes there. Continuing back recursively, we can find the call’s value today. This type of recursive operation is easily programmed on a computer. As a result, binomial trees can value options even more complex than American calls on stock.13 3. CONTINUOUS RANDOM VARIABLES In the previous section, we considered discrete random variables (i.e., random variables whose set of possible outcomes is countable). In contrast, the possible outcomes of continuous random variables are never countable. If 1.250 is one possible value of a continuous random variable, for example, we cannot name the next higher or lower possible value. Technically, the 11 For example, we can split 20 days into 100 subperiods, taking care to use compatible values for u and d . 12 Cash dividends represent a reduction of a company’s assets. Early exercise may be optimal because the exercise price of options is typically not reduced by the amount of cash dividends, so cash dividends negatively affect the position of an American call option holder. 13 See Chance (2003) for more information on option pricing models. 186 Quantitative Investment Analysis range of possible outcomes of a continuous random variable is the real line (all real numbers between −∞ and +∞) or some subset of the real line. In this section, we focus on the two most important continuous distributions in investment work, the normal and lognormal. As we did with discrete distributions, we introduce the topic through the uniform distribution. 3.1. Continuous Uniform Distribution The continuous uniform distribution is the simplest continuous probability distribution. The uniform distribution has two main uses. As the basis of techniques for generating random numbers, the uniform distribution plays a role in Monte Carlo simulation. As the probability distribution that describes equally likely outcomes, the uniform distribution is an appropriate probability model to represent a particular kind of uncertainty in beliefs in which all outcomes appear equally likely. The pdf for a uniform random variable is ⎧ ⎨ 1 for a < x < b f (x) = b − a ⎩ 0 otherwise For example, with a = 0 and b = 8, f (x) = 1/8 or 0.125. We graph this density in Figure 5-3. The graph of the density function plots as a horizontal line with a value of 0.125. What is the probability that a uniform random variable with limits a = 0 and b = 8 is less than or equal to 3, or F (3) = P(X ≤ 3)? When we were working with the discrete uniform random variable with possible outcomes 1, 2, . . . , 8, we summed individual probabilities: p(1) + p(2) + p(3) = 0.375. In contrast, the probability that a continuous uniform random variable, or any continuous random variable, assumes any given fixed value is 0. To illustrate f(x) 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0 1 2 3 4 FIGURE 5-3 Continuous Uniform Distribution 5 6 7 8 9 x 187 Chapter 5 Common Probability Distributions this point, consider the narrow interval 2.510 to 2.511. Because that interval holds an infinity of possible values, the sum of the probabilities of values in that interval alone would be infinite if each individual value in it had a positive probability. To find the probability F (3), we find the area under the curve graphing the pdf, between 0 to 3 on the x axis. In calculus, this operation is called integrating the probability function f (x) from 0 to 3. This area under the curve is a rectangle with base 3 − 0 = 3 and height 1/8. The area of this rectangle equals base times height: 3(1/8) = 3/8 or 0.375. So F (3) = 3/8 or 0.375. The interval from 0 to 3 is three-eighths of the total length between the limits of 0 and 8, and F (3) is three-eighths of the total probability of 1. The middle line of the expression for the cdf captures this relationship. ⎧ for x ≤ a ⎪ ⎪0 ⎪ ⎪ ⎨ x−a for a < x < b F (x) = b−a ⎪ ⎪ ⎪ ⎪ ⎩ 1 for x ≥ b For our problem, F (x) = 0 for x ≤ 0, F (x) = x/8 for 0 < x < 8, and F (x) = 1 for x ≥ 8. We graph this cdf in Figure 5-4. The mathematical operation that corresponds to finding the area under the curve of a pdf f (x) from a to b is the integral of f (x) from a to b: P(a ≤ X ≤ b) = ' b f (x) d x a (5-2) ( ( where dx is the symbol for summing over small changes dx, and the limits of integration (a and b) can be any real numbers or −∞ and +∞. All probabilities of continuous CDF 1.0 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 FIGURE 5-4 Continuous Uniform Cumulative Distribution 6 7 8 9 x 188 Quantitative Investment Analysis random variables can be computed using Equation 5-2. For the uniform distribution example considered above, F (7) is Equation 5-2 with lower limit a = 0 and upper limit b = 7. The integral corresponding to the cdf of a uniform distribution reduces to the three-line expression given previously. To evaluate Equation 5-2 for nearly all other continuous distributions, including the normal and lognormal, we rely on spreadsheet functions, computer programs, or tables of values to calculate probabilities. Those tools use various numerical methods to evaluate the integral in Equation 5-2. Recall that the probability of a continuous random variable equaling any fixed point is 0. This fact has an important consequence for working with the cumulative distribution function of a continuous random variable: For any continuous random variable X , P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b), because the probabilities at the endpoints a and b are 0. For discrete random variables, these relations of equality are not true, because probability accumulates at points. EXAMPLE 5-7 Is Breached Probability That a Lending Facility Covenant You are evaluating the bonds of a below-investment-grade borrower at a low point in its business cycle. You have many factors to consider, including the terms of the company’s bank lending facilities. The contract creating a bank lending facility such as an unsecured line of credit typically has clauses known as covenants. These covenants place restrictions on what the borrower can do. The company will be in breach of a covenant in the lending facility if the interest coverage ratio, EBITDA/interest, calculated on EBITDA over the four trailing quarters, falls below 2.0. EBITDA is earnings before interest, taxes, depreciation, and amortization.14 Compliance with the covenants will be checked at the end of the current quarter. If the covenant is breached, the bank can demand immediate repayment of all borrowings on the facility. That action would probably trigger a liquidity crisis for the company. With a high degree of confidence, you forecast interest charges of $25 million. Your estimate of EBITDA runs from $40 million on the low end to $60 million on the high end. Address two questions (treating projected interest charges as a constant): 1. If the outcomes for EBITDA are equally likely, what is the probability that EBITDA/interest will fall below 2.0, breaching the covenant? 2. Estimate the mean and standard deviation of EBITDA/interest. For a continuous uniform random variable, the mean is given by µ = (a + b)/2 and the variance is given by σ2 = (b − a)2 /12. Solution to 1: EBITDA/interest is a continuous uniform random variable because all outcomes are equally likely. The ratio can take on values between 1.6 = ($40 million)/($25 million) on the low end and 2.4 = ($60 million/$25 million) on the high end. The range of possible values is 2.4 − 1.6=0.8. What fraction of the possible values 14 For a detailed discussion on the use and misuse of EBITDA, see Moody’s Investors Service Global Credit Research, Putting EBITDA in Perspective (June 2000). Chapter 5 Common Probability Distributions 189 falls below 2.0, the level that triggers default? The distance between 2.0 and 1.6 is 0.40; the value 0.40 is one-half the total length of 0.8, or 0.4/0.8 = 0.50. So the probability that the covenant will be breached is 50 percent. Solution to 2: In Solution 1, we found that the lower limit of EBITDA/interest is 1.6. This lower limit is a. We found that the upper limit is 2.4. This upper limit is b. Using the formula given above, µ = (a + b)/2 = (1.6 + 2.4)/2 = 2.0 The variance of the interest coverage ratio is σ2 = (b − a)2 /12 = (2.4 − 1.6)2 /12 = 0.053333 The standard deviation is the positive square root of the variance, 0.230940 = (0.053333)1/2 . The standard deviation is not particularly useful as a risk measure for a uniform distribution, however. The probability that lies within various standard deviation bands around the mean is sensitive to different specifications of the upper and lower limits (although Chebyshev’s inequality is always satisfied).15 Here, a one-standard-deviation interval around the mean of 2.0 runs from 1.769 to 2.231 and captures 0.462/0.80 = 0.5775 or 57.8 percent of the probability. A two-standarddeviation interval runs from 1.538 to 2.462, which extends past both the lower and upper limits of the random variable. 3.2. The Normal Distribution The normal distribution may be the most extensively used probability distribution in quantitative work. It plays key roles in modern portfolio theory and in a number of risk management technologies. Because it has so many uses, the normal distribution must be thoroughly understood by investment professionals. The role of the normal distribution in statistical inference and regression analysis is vastly extended by a crucial result known as the central limit theorem. The central limit theorem states that the sum (and mean) of a large number of independent random variables is approximately normally distributed.16 The French mathematician Abraham de Moivre (1667–1754) introduced the normal distribution in 1733 in developing a version of the central limit theorem. As Figure 5-5 shows, the normal distribution is symmetrical and bell-shaped. The range of possible outcomes of the normal distribution is the entire real line: all real numbers lying between −∞ and +∞. The tails of the bell curve extend without limit to the left and to the right. The defining characteristics of a normal distribution are as follows: 15 Chebyshev’s 16 The inequality is discussed in the chapter on statistical concepts and market returns. central limit theorem is discussed further in the chapter on sampling. 190 Quantitative Investment Analysis !8 !6 !4 !2 0 2 4 6 8 x FIGURE 5-5 Two Normal Distributions The normal distribution is completely described by two parameters—its mean, µ, and variance, σ2 . We indicate this as X ∼ N (µ, σ2 ) (read ‘‘X follows a normal distribution with mean µ and variance σ2 ’’). We can also define a normal distribution in terms of the mean and the standard deviation, σ (this is often convenient because σ is measured in the same units as X and µ). As a consequence, we can answer any probability question about a normal random variable if we know its mean and variance (or standard deviation). • The normal distribution has a skewness of 0 (it is symmetric). The normal distribution has a kurtosis (measure of peakedness) of 3; its excess kurtosis (kurtosis −3.0) equals 0.17 As a consequence of symmetry, the mean, the median, and the mode are all equal for a normal random variable. • A linear combination of two or more normal random variables is also normally distributed. • These bullet points concern a single variable or univariate normal distribution: the distribution of one normal random variable. A univariate distribution describes a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables. You will encounter the multivariate normal distribution in investment work and reading and should know the following about it. When we have a group of assets, we can model the distribution of returns on each asset individually, or the distribution of returns on the assets as a group. ‘‘As a group’’ means that we take account of all the statistical interrelationships among the return series. One model that has often been used for security returns is the multivariate normal distribution. A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters: 17 If we have a sample of size n from a normal distribution, we may want to know the possible variation in sample skewness and kurtosis. For a normal random variable, the standard deviation of sample skewness is 6/n and the standard deviation of sample kurtosis is 24/n. Chapter 5 Common Probability Distributions • • • 191 the list of the mean returns on the individual securities (n means in total); the list of the securities’ variances of return (n variances in total); and the list of all the distinct pairwise return correlations: n(n − 1)/2 distinct correlations in total.18 The need to specify correlations is a distinguishing feature of the multivariate normal distribution in contrast to the univariate normal distribution. The statement ‘‘assume returns are normally distributed’’ is sometimes used to mean a joint normal distribution. For a portfolio of 30 securities, for example, portfolio return is a weighted average of the returns on the 30 securities. A weighted average is a linear combination. Thus, portfolio return is normally distributed if the individual security returns are (joint) normally distributed. To review, in order to specify the normal distribution for portfolio return, we need the means, the variances, and the distinct pairwise correlations of the component securities. With these concepts in mind, we can return to the normal distribution for one random variable. The curves graphed in Figure 5-5 are the normal density function: " ! 1 −(x − µ)2 f (x) = √ exp for − ∞ < x < +∞ 2σ2 σ 2π (5-3) The two densities graphed in Figure 5-5 correspond to a mean of µ = 0 and standard deviations of σ = 1 and σ = 2. The normal density with µ = 0 and σ = 1 is called the standard normal distribution (or unit normal distribution). Plotting two normal distributions with the same mean and different standard deviations helps us appreciate why standard deviation is a good measure of dispersion for the normal distribution: Observations are much more concentrated around the mean for the normal distribution with σ = 1 than for the normal distribution with σ = 2. Although not literally accurate, the normal distribution can be considered an approximate model for returns. Nearly all the probability of a normal random variable is contained within three standard deviations of the mean. For realistic values of mean return and return standard deviation for many assets, the normal probability of outcomes below −100 percent is very small. Whether the approximation is useful in a given application is an empirical question. For example, the normal distribution is a closer fit for quarterly and yearly holding period returns on a diversified equity portfolio than it is for daily or weekly returns.19 A persistent departure from normality in most equity return series is kurtosis greater than 3, the fat-tails problem. So when we approximate equity return distributions with the normal distribution, we should be aware that the normal distribution tends to underestimate the probability of extreme returns.20 Option returns are skewed. Because the normal is a symmetrical distribution, we should be cautious in using the normal distribution to model the returns on portfolios containing significant positions in options. 18 For example, a distribution with two stocks (a bivariate normal distribution) has two means, two variances, and one correlation: 2(2 − 1)/2. A distribution with 30 stocks has 30 means, 30 variances, and 435 distinct correlations: 30(30 − 1)/2. The return correlation of Dow Chemical with American Express stock is the same as the correlation of American Express with Dow Chemical stock, so these are counted as one distinct correlation. 19 See Fama (1976) and Campbell, Lo, and MacKinlay (1997). 20 Fat tails can be modeled by a mixture of normal random variables or by a Student’s t-distribution with a relatively small number of degrees of freedom. See Kon (1984) and Campbell, Lo, and MacKinlay (1997). We discuss the Student’s t-distribution in the chapter on sampling and estimation. 192 Quantitative Investment Analysis 2.14% −3s 13.59% −2s 34.13% −1s 34.13% m 13.59% 1s 2.14% 2s 3s FIGURE 5-6 Units of Standard Deviation The normal distribution, however, is less suitable as a model for asset prices than as a model for returns. A normal random variable has no lower limit. This characteristic has several implications for investment applications. An asset price can drop only to 0, at which point the asset becomes worthless. As a result, practitioners generally do not use the normal distribution to model the distribution of asset prices. Also note that moving from any level of asset price to 0 translates into a return of −100 percent. Because the normal distribution extends below 0 without limit, it cannot be literally accurate as a model for asset returns. Having established that the normal distribution is the appropriate model for a variable of interest, we can use it to make the following probability statements: Approximately 50 percent of all observations fall in the interval µ ± (2/3)σ. Approximately 68 percent of all observations fall in the interval µ ± σ. Approximately 95 percent of all observations fall in the interval µ ± 2σ. • Approximately 99 percent of all observations fall in the interval µ ± 3σ. • • • One, two, and three standard deviation intervals are illustrated in Figure 5-6. The intervals indicated are easy to remember but are only approximate for the stated probabilities. Moreprecise intervals are µ ± 1.96σ for 95 percent of the observations and µ ± 2.58σ for 99 percent of the observations. In general, we do not observe the population mean or the population standard deviation of a distribution, so we need to estimate them.21 We estimate the population mean, µ, using the sample mean, X (sometimes denoted as µ̂), and estimate the population standard deviation, σ, using the sample standard deviation, s (sometimes denoted as σ̂). Using sample mean and the sample standard deviation to estimate the population mean and population standard deviation, respectively, we can make the following probability statements about a normally distributed random variable X , in which we use the more-precise numbers for standard deviation in stating intervals. 21 A population is all members of a specified group, and the population mean is the arithmetic mean computed for the population. A sample is a subset of a population, and the sample mean is the arithmetic mean computed for the sample. For more information on these concepts, see the chapter on statistical concepts and market returns. Chapter 5 Common Probability Distributions 193 Confidence Intervals for Values of a Normal Random Variable X We expect 90 percent of the values of X to lie within the interval from X − 1.65s to X + 1.65s. We call this interval a 90 percent confidence interval for X . • We expect 95 percent of the values of X to lie within the interval from X − 1.96s to X + 1.96s. We call this interval a 95 percent confidence interval for X . • We expect 99 percent of the values of X to lie within the interval from X − 2.58s to X + 2.58s. We call this interval a 99 percent confidence interval for X . • EXAMPLE 5-8 Portfolio (1) Probabilities for a Common Stock You manage a U.S. core equity portfolio that is sector-neutral to the S&P 500 Index (its industry sector weights approximately match the S&P 500’s). Taking a weighted average of the projected mean returns on the holdings, you forecast a portfolio return of 12 percent. You estimate a standard deviation of annual return of 22 percent, close to the long-run figure for the S&P 500. For the year-ahead return on the portfolio, you are asked to do the following: 1. Calculate and interpret a one-standard-deviation confidence interval for portfolio return, with a normality assumption for returns. 2. Calculate and interpret a 90 percent confidence interval for portfolio return, with a normality assumption for returns. 3. Calculate and interpret a 95 percent confidence interval for portfolio return, with a normality assumption for returns. Solution to 1: A one-standard-deviation confidence interval is X ± s. With X = 12 percent and s = 22 percent, the lower end of a one-standard-deviation interval is −10% = 12% − 22%, and the upper end is 34% = 12% + 22%. The interval thus runs from −10 percent to 34 percent, and you expect approximately 68 percent of portfolio returns to lie within it, under normality. A compact notation for this one-standard-deviation confidence interval is [−10%, 34%]. Solution to 2: A 90 percent confidence interval, with a normality assumption for returns, runs from X − 1.65s to X + 1.65s. So the lower limit is −24.3% = 12% − 1.65(22%), and the upper limit is 48.3% = 12% + 1.65(22%). Compactly, this interval is [−24.3%, 48.3%]. Solution to 3: A 95 percent confidence interval, with a normality assumption for returns, goes from X − 1.96s to X + 1.96s. So the lower limit is −31.12% = 12% − 1.96(22%), and the upper limit is 55.12% = 12% + 1.96(22%). Compactly, this interval is [−31.12%, 55.12%]. The 95 percent and 99 percent confidence intervals are probably the two most frequently used in practice. An approximate 95 percent confidence interval using 2 194 Quantitative Investment Analysis rather than 1.96 standard deviations as the multiplier gives a quick answer and thus is frequently used. The calculation of the lower limit of −31.12 percent in Solution 3 illustrates an earlier point: For many realistic values of mean and standard deviation, the fact that the normal distribution extends to −∞ on the left may not be critical. For a normal distribution, only 2.5 percent of the total probability lies to the left of the mean minus 1.96 standard deviations (and 2.5 percent lies to the right of the mean plus 1.96 standard deviations). Figure 5-7 illustrates these probabilities by showing that 47.5 percent of the total probability lies between the mean and the mean plus (or minus) 1.96 standard deviations. 47.5% 2.5% 47.5% 95% 2.5% FIGURE 5-7 Tail Probabilities for a 95 Percent Confidence Interval In working with confidence intervals, we specify the desired level of confidence and find the endpoints. We have given the formulas for important conventional intervals, but we may also have questions on other intervals, such as ‘‘How wide do I have to make the confidence interval to capture 75 percent of the returns on this portfolio?’’ We may also be interested in other probabilities. For example, we may ask, ‘‘What is the probability that the annual return on this equity index will be less than the one-year T-bill return?’’ There are as many different normal distributions as there are choices for mean (µ) and variance (σ2 ). We can answer all of the above questions in terms of any normal distribution. Spreadsheets, for example, have functions for the normal cdf for any specification of mean and variance. For the sake of efficiency, however, we would like to refer all probability statements to a single normal distribution. The standard normal distribution (the normal distribution with µ = 0 and σ = 1) fills that role. There are two steps in standardizing a random variable X : Subtract the mean of X from X , then divide that result by the standard deviation of X . If we have a list of observations on a normal random variable, X , we subtract the mean from each observation to get a list of deviations from the mean, then divide each deviation by the standard deviation. The result is the standard normal random variable, Z . (Z is the conventional symbol for a standard normal 195 Chapter 5 Common Probability Distributions TABLE 5-5 P(Z ≤ x) = N (x) for x ≥ 0 or P(Z ≤ z) = N (z) for z ≥ 0 x or z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00 0.10 0.20 0.30 0.40 0.50 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 random variable.) If we have X ∼ N (µ, σ2 ) (read ‘‘X follows the normal distribution with parameters µ and σ2 ’’), we standardize it using the formula Z = (X − µ)/σ (5-4) Suppose we have a normal random variable, X , with µ = 5 and σ = 1.5. We standardize X with Z = (X − 5)/1.5. For example, a value X = 9.5 corresponds to a standardized value of 3, calculated as Z = (9.5 − 5)/1.5 = 3. The probability that we will observe a value as small as or smaller than 9.5 for X ∼ N (5, 1.5) is exactly the same as the probability that we will observe a value as small as or smaller than 3 for Z ∼ N (0, 1). We can answer all probability questions about X using standardized values and probability tables for Z . We generally do not know the population mean and standard deviation, so we often use the sample mean X for µ and the sample standard deviation s for σ. Standard normal probabilities can also be computed with spreadsheets, statistical and econometric software, and programming languages. Tables of the cumulative distribution function for the standard normal random variable are in the back of this book. Table 5-5 shows an excerpt from those tables. N (x) is a conventional notation for the cdf of a standard normal variable.22 To find the probability that a standard normal variable is less than or equal to 0.24, for example, locate the row that contains 0.20, look at the 0.04 column, and find the entry 0.5948. Thus, P(Z ≤ 0.24) = 0.5948 or 59.48 percent. The following are some of the most frequently referenced values in the standard normal table: The 90th percentile point is 1.282: P(Z ≤ 1.282) = N (1.282) = 0.90 or 90 percent, and 10 percent of values remain in the right tail. • The 95th percentile point is 1.65: P(Z ≤ 1.65) = N (1.65) = 0.95 or 95 percent, and 5 percent of values remain in the right tail. Note the difference between the use of a percentile point when dealing with one tail rather than two tails. Earlier, we used 1.65 standard deviations for the 90 percent confidence interval, where 5 percent of values lie outside that interval on each of the two sides. Here we use 1.65 because we are concerned with the 5 percent of values that lie only on one side, the right tail. • The 99th percentile point is 2.327: P(Z ≤ 2.327) = N (2.327) = 0.99 or 99 percent, and 1 percent of values remain in the right tail. • 22 Another often-seen notation for the cdf of a standard normal variable is !(x). 196 Quantitative Investment Analysis The tables that we give for the normal cdf include probabilities for x ≤ 0. Many sources, however, give tables only for x ≥ 0. How would one use such tables to find a normal probability? Because of the symmetry of the normal distribution, we can find all probabilities using tables of the cdf of the standard normal random variable, P(Z ≤ x) = N (x), for x ≥ 0. The relations below are helpful for using tables for x ≥ 0, as well as in other uses: For a non-negative number x, use N (x) from the table. Note that for the probability to the right of x, we have P(Z ≥ x) = 1.0 − N (x). • For a negative number −x, N (−x) = 1.0 − N (x): Find N (x) and subtract it from 1. All the area under the normal curve to the left of x is N (x). The balance, 1.0 − N (x), is the area and probability to the right of x. By the symmetry of the normal distribution around its mean, the area and the probability to the right of x are equal to the area and the probability to the left of −x, N (−x). • For the probability to the right of −x, P(Z ≥ −x) = N (x). • EXAMPLE 5-9 Portfolio (2) Probabilities for a Common Stock Recall that in Example 5-8, the portfolio mean return estimate was 12 percent and the standard deviation of return estimate was 22 percent per year. Using these estimates, you want to calculate the following probabilities, assuming that a normal distribution describes returns. (You can use the excerpt from the table of normal probabilities to answer these questions.) 1. What is the probability that portfolio return will exceed 20 percent? 2. What is the probability that portfolio return will be between 12 percent and 20 percent? In other words, what is P(12% ≤ Portfolio return ≤ 20%)? 3. You can buy a one-year T-bill that yields 5.5 percent. This yield is effectively a one-year risk-free interest rate. What is the probability that your portfolio’s return will be equal to or less than the risk-free rate? If X is portfolio return, standardized portfolio return is Z = (X − X )/s = (X − 12%)/22%. We use this expression throughout the solutions. Solution to 1: For X = 20%, Z = (20% − 12%)/22% = 0.363636. You want to find P(Z > 0.363636). First note that P(Z > x) = P(Z ≥ x) because the normal is a continuous distribution. Recall that P(Z ≥ x) = 1.0 − P(Z ≤ x) or 1 − N (x). Rounding 0.363636 to 0.36, according to the table, N (0.36) = 0.6406. Thus, 1 − 0.6406 = 0.3594. The probability that portfolio return will exceed 20 percent is about 36 percent if your normality assumption is accurate. Solution to 2: P(12% ≤ Portfolio return ≤ 20%) = N (Z corresponding to 20%) − N (Z corresponding to 12%). For the first term, Z = (20% − 12%)/22% = 0.36 Chapter 5 Common Probability Distributions 197 approximately, and N (0.36) = 0.6406 (as in Solution 1). To get the second term immediately, note that 12 percent is the mean, and for the normal distribution 50 percent of the probability lies on either side of the mean. Therefore, N (Z corresponding to 12%) must equal 50 percent. So P(12% ≤ Portfolio return ≤ 20%) = 0.6406 − 0.50 = 0.1406 or approximately 14 percent. Solution to 3: If X is portfolio return, then we want to find P(Portfolio return ≤ 5.5%). This question is more challenging than Parts 1 or 2, but when you have studied the solution below, you will have a useful pattern for calculating other shortfall probabilities. There are three steps, which involve standardizing the portfolio return: First, subtract the portfolio mean return from each side of the inequality: P(Portfolio return − 12% ≤ 5.5% − 12%). Second, divide each side of the inequality by the standard deviation of portfolio return: P[(Portfolio return − 12%)/22% ≤ (5.5% − 12%)/22%] = P(Z ≤ −0.295455) = N (−0.295455). Third, recognize that on the left-hand side we have a standard normal variable, denoted by Z . As we pointed out above, N (−x) = 1 − N (x). Rounding −0.29545 to −0.30 for use with the excerpted table, we have N (−0.30) = 1 − N (0.30) = 1 − 0.6179 = 0.3821, roughly 38 percent. The probability that your portfolio will underperform the one-year risk-free rate is about 38 percent. We can get the answer above quickly by subtracting the mean portfolio return from 5.5 percent, dividing by the standard deviation of portfolio return, and evaluating the result (−0.295455) with the standard normal cdf. 3.3. Applications of the Normal Distribution Modern portfolio theory (MPT) makes wide use of the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return. In economic theory, mean–variance analysis holds exactly when investors are risk averse; when they choose investments so as to maximize expected utility, or satisfaction; and when either (1) returns are normally distributed or (2) investors have quadratic utility functions.23 Mean–variance analysis can still be useful, however—that is, it can hold approximately—when either assumption (1) or (2) is violated. Because practitioners prefer to work with observables such as returns, the proposition that returns are at least approximately normally distributed has played a key role in much of MPT. Mean–variance analysis generally considers risk symmetrically in the sense that standard deviation captures variability both above and below the mean.24 An alternative approach evaluates only downside risk. We discuss one such approach, safety-first rules, as it provides an excellent illustration of the application of normal distribution theory to practical investment problems. Safety-first rules focus on shortfall risk, the risk that portfolio value will fall below some minimum acceptable level over some time horizon. The risk that the assets in a defined benefit plan will fall below plan liabilities is an example of a shortfall risk. Suppose an investor views any return below a level of RL as unacceptable. Roy’s safety-first criterion states that the optimal portfolio minimizes the probability that portfolio return, RP , 23 Utility functions are mathematical representations of attitudes toward risk and return. shall discuss mean–variance analysis in detail in the chapter on portfolio concepts. 24 We 198 Quantitative Investment Analysis falls below the threshold level, RL .25 In symbols, the investor’s objective is to choose a portfolio that minimizes P(RP < RL ). When portfolio returns are normally distributed, we can calculate P(RP < RL ) using the number of standard deviations that RL lies below the expected portfolio return, E(RP ). The portfolio for which E(RP ) − RL is largest relative to standard deviation minimizes P(RP < RL ). Therefore, if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio): SFRatio = [E(RP ) − RL ]/σP The quantity E(RP ) − RL is the distance from the mean return to the shortfall level. Dividing this distance by σP gives the distance in units of standard deviation. There are two steps in choosing among portfolios using Roy’s criterion (assuming normality):26 1. Calculate each portfolio’s SFRatio. 2. Choose the portfolio with the highest SFRatio. For a portfolio with a given safety-first ratio, the probability that its return will be less than RL is N (−SFRatio), and the safety-first optimal portfolio has the lowest such probability. For example, suppose an investor’s threshold return, RL , is 2 percent. He is presented with two portfolios. Portfolio 1 has an expected return of 12 percent with a standard deviation of 15 percent. Portfolio 2 has an expected return of 14 percent with a standard deviation of 16 percent. The SFRatios are 0.667 = (12 − 2)/15 and 0.75 = (14 − 2)/16 for Portfolios 1 and 2, respectively. For the superior Portfolio 2, the probability that portfolio return will be less than 2 percent is N (−0.75) = 1 − N (0.75) = 1 − 0.7734 = 0.227 or about 23 percent, assuming that portfolio returns are normally distributed. You may have noticed the similarity of SFRatio to the Sharpe ratio. If we substitute the risk-free rate, RF , for the critical level RL , the SFRatio becomes the Sharpe ratio. The safety-first approach provides a new perspective on the Sharpe ratio: When we evaluate portfolios using the Sharpe ratio, the portfolio with the highest Sharpe ratio is the one that minimizes the probability that portfolio return will be less than the risk-free rate (given a normality assumption). EXAMPLE 5-10 The Safety-First Optimal Portfolio for a Client You are researching asset allocations for a client with an $800,000 portfolio. Although her investment objective is long-term growth, at the end of a year she may want to liquidate $30,000 of the portfolio to fund educational expenses. If that need arises, she 25 A.D. Roy (1952) introduced this criterion. If there is an asset offering a risk-free return over the time horizon being considered, and if RL is less than or equal to that risk-free rate, then it is optimal to be fully invested in the risk-free asset. Holding the risk-free asset in this case eliminates the chance that the threshold return is not met. 26 199 Chapter 5 Common Probability Distributions would like to be able to take out the $30,000 without invading the initial capital of $800,000. Table 5-6 shows three alternative allocations. TABLE 5-6 Mean and Standard Deviation for Three Allocations (in percent) Expected annual return Standard deviation of return A B C 25 27 11 8 14 20 Address these questions (assume normality for Parts 2 and 3): 1. Given the client’s desire not to invade the $800,000 principal, what is the shortfall level, RL ? Use this shortfall level to answer Part 2. 2. According to the safety-first criterion, which of the three allocations is the best? 3. What is the probability that the return on the safety-first optimal portfolio will be less than the shortfall level? Solution to 1: Because $30,000/$800,000 is 3.75 percent, for any return less than 3.75 percent the client will need to invade principal if she takes out $30,000. So RL = 3.75 percent. Solution to 2: To decide which of the three allocations is safety-first optimal, select the alternative with the highest ratio [E(RP ) − RL ]/σP : Allocation A : 0.787037 = (25 − 3.75)/27 Allocation B : 0.90625 = (11 − 3.75)/8 Allocation C : 0.5125 = (14 − 3.75)/20 Allocation B, with the largest ratio (0.90625), is the best alternative according to the safety-first criterion. Solution to 3: To answer this question, note that P(RB < 3.75) = N (−0.90625). We can round 0.90625 to 0.91 for use with tables of the standard normal cdf. First, we calculate N (−0.91) = 1 − N (0.91) = 1 − 0.8186 = 0.1814 or about 18.1 percent. Using a spreadsheet function for the standard normal cdf on −0.90625 without rounding, we get 18.24 percent or about 18.2 percent. The safety-first optimal portfolio has a roughly 18 percent chance of not meeting a 3.75 percent return threshold. Several points are worth noting. First, if the inputs were even slightly different, we could get a different ranking. For example, if the mean return on B were 10 rather than 11 percent, A would be superior to B. Second, if meeting the 3.75 percent return threshold were a necessity rather than a wish, $830,000 in one year could be modeled 200 Quantitative Investment Analysis as a liability. Fixed income strategies such as cash flow matching could be used to offset or immunize the $830,000 quasi-liability. Roy’s safety-first rule was the earliest approach to addressing shortfall risk. The standard mean–variance portfolio selection process can also accommodate a shortfall risk constraint.27 In many investment contexts besides Roy’s safety-first criterion, we use the normal distribution to estimate a probability. For example, Kolb, Gay, and Hunter (1985) developed an expression based on the standard normal distribution for the probability that a futures trader will exhaust his liquidity because of losses in a futures contract. Another arena in which the normal distribution plays an important role is financial risk management. Financial institutions such as investment banks, security dealers, and commercial banks have formal systems to measure and control financial risk at various levels, from trading positions to the overall risk for the firm.28 Two mainstays in managing financial risk are Value at Risk (VAR) and stress testing/scenario analysis. Stress testing/scenario analysis, a complement to VAR, refers to a set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios. Value at Risk (VAR) is a money measure of the minimum value of losses expected over a specified time period (for example, a day, a quarter, or a year) at a given level of probability (often 0.05 or 0.01). Suppose we specify a one-day time horizon and a level of probability of 0.05, which would be called a 95 percent one-day VAR.29 If this VAR equaled ¤5 million for a portfolio, there would be a 0.05 probability that the portfolio would lose ¤5 million or more in a single day (assuming our assumptions were correct). One of the basic approaches to estimating VAR, the variance–covariance or analytical method, assumes that returns follow a normal distribution. For more information on VAR, see Chance (2003). 3.4. The Lognormal Distribution Closely related to the normal distribution, the lognormal distribution is widely used for modeling the probability distribution of share and other asset prices. For example, the lognormal appears in the Black–Scholes–Merton option pricing model. The Black–Scholes–Merton model assumes that the price of the asset underlying the option is lognormally distributed. A random variable Y follows a lognormal distribution if its natural logarithm, ln Y , is normally distributed. The reverse is also true: If the natural logarithm of random variable Y , ln Y , is normally distributed, then Y follows a lognormal distribution. If you think of the term lognormal as ‘‘the log is normal,’’ you will have no trouble remembering this relationship. The two most noteworthy observations about the lognormal distribution are that it is bounded below by 0 and it is skewed to the right (it has a long right tail). Note these two properties in the graphs of the pdfs of two lognormal distributions in Figure 5-8. Asset prices are bounded from below by 0. In practice, the lognormal distribution has been found to be 27 See Leibowitz and Henriksson (1989), for example. risk is risk relating to asset prices and other financial variables. The contrast is to other, nonfinancial risks (for example, relating to operations and technology), which require different tools to manage. 29 In 95 percent one-day VAR, the 95 percent refers to the confidence in the value of VAR and is equal to 1 − 0.05; this is a traditional way to state VAR. 28 Financial 201 Chapter 5 Common Probability Distributions 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 FIGURE 5-8 Two Lognormal Distributions a usefully accurate description of the distribution of prices for many financial assets. On the other hand, the normal distribution is often a good approximation for returns. For this reason, both distributions are very important for finance professionals. Like the normal distribution, the lognormal distribution is completely described by two parameters. Unlike the other distributions we have considered, a lognormal distribution is defined in terms of the parameters of a different distribution. The two parameters of a lognormal distribution are the mean and standard deviation (or variance) of its associated normal distribution: the mean and variance of ln Y , given that Y is lognormal. Remember, we must keep track of two sets of means and standard deviations (or variances): the mean and standard deviation (or variance) of the associated normal distribution (these are the parameters), and the mean and standard deviation (or variance) of the lognormal variable itself. The expressions for the mean and variance of the lognormal variable itself are challenging. Suppose a normal random variable X has expected value µ and variance σ2 . Define Y = exp(X ). Remember that the operation indicated by exp(X ) or eX is the opposite operation from taking logs.30 Because ln Y = ln[exp(X )] = X is normal (we assume X is normal), Y is lognormal. What is the expected value of Y = exp(X )? A guess might be that the expected value of Y is exp(µ). The expected value is actually exp(µ + 0.50σ2 ), which is larger than exp(µ) by a factor of exp(0.50σ2 ) > 1.31 To get some insight into this concept, think of what happens if we increase σ2 . The distribution spreads out; it can spread upward, but it cannot spread downward past 0. As a result, the center of its distribution is pushed to the right—the distribution’s mean increases.32 The expressions for the mean and variance of a lognormal variable are summarized below, where µ and σ2 are the mean and variance of the associated normal distribution (refer to these expressions as needed, rather than memorizing them): 30 The quantity e ≈ 2.7182818. that exp(0.50σ2 ) > 1 because σ2 > 0. 32 Luenberger (1998) is the source of this explanation. 31 Note 202 Quantitative Investment Analysis Mean (µL ) of a lognormal random variable = exp(µ + 0.50σ2 ) • Variance (σL 2 ) of a lognormal random variable = exp(2µ + σ2 ) × [exp(σ2 ) − 1] • We now explore the relationship between the distribution of stock return and stock price. In the following we show that if a stock’s continuously compounded return is normally distributed, then future stock price is necessarily lognormally distributed.33 Furthermore, we show that stock price may be well described by the lognormal distribution even when continuously compounded returns do not follow a normal distribution. These results provide the theoretical foundation for using the lognormal distribution to model prices. To outline the presentation that follows, we first show that the stock price at some future time T , ST , equals the current stock price, S0 , multiplied by e raised to power r0,T , the continuously compounded return from 0 to T ; this relationship is expressed as ST = S0 exp(r0,T ). We then show that we can write r0,T as the sum of shorter-term continuously compounded returns and that if these shorter-period returns are normally distributed, then r0,T is normally distributed (given certain assumptions) or approximately normally distributed (not making those assumptions). As ST is proportional to the log of a normal random variable, ST is lognormal. To supply a framework for our discussion, suppose we have a series of equally spaced observations on stock price: S0 , S1 , S2 , . . . , ST . Current stock price, S0 , is a known quantity and so is nonrandom. The future prices (such as S1 ), however, are random variables. The price relative, S1 /S0 , is an ending price, S1 , over a beginning price, S0 ; it is equal to 1 plus the holding period return on the stock from t = 0 to t = 1: S1 /S0 = 1 + R0,1 For example, if S0 = $30 and S1 = $34.50, then S1 /S0 = $34.50/$30 = 1.15. Therefore, R0,1 = 0.15 or 15 percent. In general, price relatives have the form St+1 /St = 1 + Rt,t+1 where Rt,t+1 is the rate of return from t to t + 1. An important concept is the continuously compounded return associated with a holding period return such as R0,1 . The continuously compounded return associated with a holding period is the natural logarithm of 1 plus that holding period return, or equivalently, the natural logarithm of the ending price over the beginning price (the price relative).34 For example, if we observe a one-week holding period return of 0.04, the equivalent continuously compounded return, called the one-week continuously compounded return, is ln(1.04) = 0.039221; ¤1.00 invested for one week at 0.039221 continuously compounded gives ¤1.04, equivalent to a 4 percent one-week holding period return. The continuously compounded return from t to t + 1 is rt,t+1 = ln(St+1 /St ) = ln(1 + Rt,t+1 ) (5-5) 33 Continuous compounding treats time as essentially continuous or unbroken, in contrast to discrete compounding, which treats time as advancing in discrete finite intervals. Continuously compounded returns are the model for returns in so-called continuous time finance models such as the Black–Scholes–Merton option pricing model. See the chapter on the time value of money for more information on compounding. 34 In this chapter only, we use lowercase r to refer specifically to continuously compounded returns. Chapter 5 Common Probability Distributions 203 For our example, r0,1 = ln(S1 /S0 ) = ln(1+R0,1 ) = ln($34.50/$30) = ln(1.15) = 0.139762. Thus, 13.98 percent is the continuously compounded return from t = 0 to t = 1. The continuously compounded return is smaller than the associated holding period return. If our investment horizon extends from t = 0 to t = T , then the continuously compounded return to T is r0,T = ln (ST /S0 ) Applying the function exp to both sides of the equation, we have exp(r0,T ) = exp[ln(ST /S0 )] = ST /S0 , so ST = S0 exp(r0,T ) We can also express ST /S0 as the product of price relatives: ST /S0 = (ST /ST −1 )(ST −1 /ST −2 ) . . . (S1 /S0 ) Taking logs of both sides of this equation, we find that continuously compounded return to time T is the sum of the one-period continuously compounded returns: r0,T = rT −1,T + rT −2,T −1 + · · · + r0,1 (5-6) Using holding period returns to find the ending value of a $1 investment involves the multiplication of quantities (1 + holding period return). Using continuously compounded returns involves addition. A key assumption in many investment applications is that returns are independently and identically distributed (IID). Independence captures the proposition that investors cannot predict future returns using past returns (i.e., weak-form market efficiency). Identical distribution captures the assumption of stationarity, to which we will return in the chapter on time-series analysis.35 Assume that the one-period continuously compounded returns (such as r0,1 ) are IID random variables with mean µ and variance σ2 (but making no normality or other distributional assumption). Then E(r0,T ) = E(rT −1,T ) + E(rT −2,T −1 ) + · · · + E(r0,1 ) = µT (5-7) (we add up µ for a total of T times) and σ2 (r0,T ) = σ2 T (5-8) (as a consequence of the independence assumption). The variance of the T holding period continuously compounded return is T multiplied by the variance of the one-period continu√ ously compounded return; also, σ(r0,T ) = σ T . If the one-period continuously compounded returns on the right-hand side of Equation 5-6 are normally distributed, then the T holding 35 Stationarity implies that the mean and variance of return do not change from period to period. 204 Quantitative Investment Analysis period continuously compounded return, r0,T , is also normally distributed with mean µT and variance σ2 T . This relationship is so because a linear combination of normal random variables is also normal. But even if the one-period continuously compounded returns are not normal, their sum, r0,T , is approximately normal according to a result in statistics known as the central limit theorem.36 Now compare ST = S0 exp(r0,T ) to Y = exp(X ), where X is normal and Y is lognormal (as we discussed above). Clearly, we can model future stock price ST as a lognormal random variable because r0,T should be at least approximately normal. This assumption of normally distributed returns is the basis in theory for the lognormal distribution as a model for the distribution of prices of shares and other assets. Continuously compounded returns play a role in many option pricing models, as mentioned earlier. An estimate of volatility is crucial for using option pricing models such as the Black–Scholes–Merton model. Volatility measures the standard deviation of the continuously compounded returns on the underlying asset.37 In practice, we very often estimate volatility using a historical series of continuously compounded daily returns. We gather a set of daily holding period returns and then use Equation 5-5 to convert them into continuously compounded daily returns. We then compute the standard deviation of the continuously compounded daily returns and annualize that number using Equation 5-8.38 (By convention, volatility is stated as an annualized measure.)39 Example 5-11 illustrates the estimation of volatility for the shares of Michelin. EXAMPLE 5-11 Volatility as Used in Option Pricing Models Suppose you are researching Michelin (Euronext: MICP.PA) and are interested in Michelin’s price action in a week in which a number of international events affected stock markets. You decide to use volatility as a measure of the variability of Michelin shares during that week. Table 5-7 shows closing prices during that week. 36 We mentioned the central limit theorem earlier in our discussion of the normal distribution. To give a somewhat fuller statement of it, according to the central limit theorem the sum (as well as the mean) of a set of independent, identically distributed random variables with finite variances is normally distributed, whatever distribution the random variables follow. We discuss the central limit theorem in the chapter on sampling. 37 Volatility is also called the instantaneous standard deviation, and as such is denoted σ. The underlying asset, or simply the underlying, is the asset underlying the option. For more information on these concepts, see Chance (2003). 38 To compute the standard deviation of a set or sample of n returns, we sum the squared deviation of each return from the mean return and then divide that sum by n − 1. The result is the sample variance. Taking the square root of the sample variance gives the sample standard deviation. To review the calculation of standard deviation, see the chapter on statistical concepts and market returns. 39 Annualizing is often done on the basis of 250 days in a year, the approximate number of days markets are open for trading. The 250-day number may lead to a better estimate of volatility than the 365-day √ number. Thus if daily volatility were 0.01, we would state volatility (on an annual basis) as 0.01 250 = 0.1581. 205 Chapter 5 Common Probability Distributions TABLE 5-7 Michelin Daily Closing Prices Date 31 March 2003 01 April 2003 02 April 2003 03 April 2003 04 April 2003 Closing Price ¤25.20 ¤25.21 ¤25.52 ¤26.10 ¤26.14 Source: http://fr.finance.yahoo.com. Use the data in Table 5-7 to do the following: 1. Estimate the volatility of Michelin shares. (Annualize volatility based on 250 days in a year.) 2. Identify the probability distribution for Michelin share prices if continuously compounded daily returns follow the normal distribution. Solution to 1: First, use Equation 5-5 to calculate the continuously compounded daily returns; then find their standard deviation in the usual way. (In the calculation of sample variance to get sample standard deviation, use a divisor of 1 less than the sample size.) ln(25.21/25.20) = 0.000397, ln(25.52/25.21) = 0.012222 ln(26.10/25.52) = 0.022473, ln(26.14/26.10) = 0.001531 Sum = 0.036623, Mean = 0.009156, Variance = 0.000107, Standard Deviation = 0.010354 The standard deviation of continuously compounded daily returns is 0.010354. √ Equation 5-8 states that σ̂(r0,T ) = σ̂ T . In this example, σ̂ is the sample standard deviation of one-period continuously compounded returns. Thus, σ̂ refers to 0.010354. We want to annualize, so the horizon T corresponds to one year. As σ̂ is in days, we set T equal to the number of trading days in a year (250). We find that annualized volatility for Michelin stock that week was 16.4 percent, √ calculated as 0.010354 250 = 0.163711. Note that the sample mean, 0.009156, is a possible estimate of the mean, µ, of the continuously compounded one-period or daily returns. The sample mean can be translated into an estimate of the expected continuously compounded annual return using Equation 5-7: µ̂T = 0.009156(250) (using 250 to be consistent with the calculation of volatility). But four observations are far too few to estimate expected returns. The variability in the daily returns overwhelms any information about expected return in a series this short. Solution to 2: Michelin share prices should follow the lognormal distribution if the continuously compounded daily returns on Michelin shares follow the normal distribution. 206 Quantitative Investment Analysis We have shown that the distribution of stock price is lognormal, given certain assumptions. What are the mean and variance of ST if ST follows the lognormal distribution? Earlier in this section, we gave bullet-point expressions for the mean and variance of a lognormal random variable. In the bullet-point expressions, the µ̂ and σ̂2 would refer, in the context of this discussion, to the mean and variance of the T horizon (not the one-period) continuously compounded returns (assumed to follow a normal distribution), compatible with the horizon of ST .40 Related to the use of mean and variance (or standard deviation), earlier in this chapter we used those quantities to construct intervals in which we expect to find a certain percentage of the observations of a normally distributed random variable. Those intervals were symmetric about the mean. Can we state similar, symmetric intervals for a lognormal random variable? Unfortunately, we cannot. Because the lognormal distribution is not symmetric, such intervals are more complicated than for the normal distribution, and we will not discuss this specialist topic here.41 Finally, we have presented the relation between the mean and variance of continuously compounded returns associated with different time horizons (see Equations 5-7 and 5-8), but how are the means and variances of holding period returns and continuously compounded returns related? As analysts, we typically think in terms of holding period returns rather than continuously compounded returns, and we may desire to convert means and standard deviations of holding period returns to means and standard deviations of continuously compounded returns for an option application, for example. To effect such conversions (and those in the other direction, from a continuous compounding to a holding period basis), we can use the expressions in Ferguson (1993). 4. MONTE CARLO SIMULATION With an understanding of probability distributions, we are now prepared to learn about a computer-based technique in which probability distributions play an integral role. The technique is called Monte Carlo simulation. Monte Carlo simulation in finance involves the use of a computer to represent the operation of a complex financial system. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from a specified probability distribution or distributions to represent the role of risk in the system. Monte Carlo simulation has several quite distinct uses. One use is in planning. Stanford University researcher Sam Savage provided the following neat picture of that role: ‘‘What is the last thing you do before you climb on a ladder? You shake it, and that is Monte Carlo simulation.’’42 Just as shaking a ladder helps us assess the risks in climbing it, Monte Carlo simulation allows us to experiment with a proposed policy before actually implementing it. For example, investment performance can be evaluated with reference to a benchmark or a liability. Defined benefit pension plans often invest assets with reference to plan liabilities. Pension liabilities are a complex random process. In a Monte Carlo asset–liability financial planning study, the functioning of pension assets and liabilities is simulated over time, given assumptions 40 The expression for the mean is E(ST ) = S0 exp[E(r0,T ) + 0.5σ2 (r0,T )], for example. See Hull (2003) for a discussion of lognormal confidence intervals. 42 Business Week, 22 January 2001. 41 Chapter 5 Common Probability Distributions 207 about how assets are invested, the work force, and other variables. A key specification in this and all Monte Carlo simulations is the probability distributions of the various sources of risk (including interest rates and security market returns, in this case). The implications of different investment policy decisions on the plan’s funded status can be assessed through simulated time. The experiment can be repeated for another set of assumptions. We can view Example 5-12 below as coming under this heading. In that example, market return series are not long enough to address researchers’ questions on stock market timing, so the researchers simulate market returns to find answers to their questions. Monte Carlo simulation is also widely used to develop estimates of VAR. In this application, we simulate the portfolio’s profit and loss performance for a specified time horizon. Repeated trials within the simulation (each trial involving a draw of random observations from a probability distribution) produce a frequency distribution for changes in portfolio value. The point that defines the cutoff for the least favorable 5 percent of simulated changes is an estimate of 95 percent VAR, for example. In an extremely important use, Monte Carlo simulation is a tool for valuing complex securities, particularly European-style options, for which no analytic pricing formula is available.43 For other securities, such as mortgage-backed securities with complex embedded options, Monte Carlo simulation is also an important modeling resource. Researchers use Monte Carlo simulation to test their models and tools. How critical is a particular assumption to the performance of a model? Because we control the assumptions when we do a simulation, we can run the model through a Monte Carlo simulation to examine a model’s sensitivity to a change in our assumptions. To understand the technique of Monte Carlo simulation, let us present the process as a series of steps.44 To illustrate the steps, we take the case of using Monte Carlo simulation to value a type of option for which no analytic pricing formula is available, an Asian call option on a stock. An Asian call option is a European-style option with a value at maturity equal to the difference between the stock price at maturity and the average stock price during the life of the option, or $0, whichever is greater. For instance, if the final stock price is $34 with an average value of $31 over the life of the option, the value of the option at maturity is $3 (the greater of $34 − $31 = $3 and $0). Steps 1 through 3 of the process describe specifying the simulation; Steps 4 through 7 describe running the simulation. 1. Specify the quantities of interest (option value, for example, or the funded status of a pension plan) in terms of underlying variables. The underlying variable or variables could be stock price for an equity option, the market value of pension assets, or other variables relating to the pension benefit obligation for a pension plan. Specify the starting values of the underlying variables. To illustrate the steps, we are using the case of valuing an Asian call option on stock. We use CiT to represent the value of the option at maturity T . The subscript i in CiT indicates that CiT is a value resulting from the ith simulation trial, each simulation trial involving a drawing of random values (an iteration of Step 4 below). 43 A European-style option or European option is an option exercisable only at maturity. steps should be viewed as providing an overview of Monte Carlo simulation rather than as a detailed recipe for implementing a Monte Carlo simulation in its many varied applications. 44 The 208 Quantitative Investment Analysis 2. Specify a time grid. Take the horizon in terms of calendar time and split it into a number of subperiods, say K in total. Calendar time divided by the number of subperiods, K , is the time increment, "t. 3. Specify distributional assumptions for the risk factors that drive the underlying variables. For example, stock price is the underlying variable for the Asian call, so we need a model for stock price movement. Say we choose the following model for changes in stock price, where Zk stands for the standard normal random variable: "(Stock price) = (µ × Prior stock price × "t) + (σ × Prior stock price × Zk ) 4. 5. 6. 7. In the way that we are using the term, Zk is a risk factor in the simulation. Through our choice of µ and σ, we control the distribution of stock price. Although this example has one risk factor, a given simulation may have multiple risk factors. Using a computer program or spreadsheet function, draw K random values of each risk factor. In our example, the spreadsheet function would produce a draw of K values of the standard normal variable Zk : Z1 , Z2 , Z3 , . . . , ZK . Calculate the underlying variables using the random observations generated in Step 4. Using the above model of stock price dynamics, the result is K observations on changes in stock price. An additional calculation is needed to convert those changes into K stock prices (using initial stock price, which is given). Another calculation produces the average stock price during the life of the option (the sum of K stock prices divided by K ). Compute the quantities of interest. In our example, the first calculation is the value of an Asian call at maturity, CiT . A second calculation discounts this terminal value back to the present to get the call value as of today, Ci0 . We have completed one simulation trial. (The subscript i in Ci0 stands for the ith simulation trial, as it does in CiT .) In a Monte Carlo simulation, a running tabulation is kept of statistics relating to the distribution of the quantities of interest, including their mean value and standard deviation, over the simulation trials to that point. Iteratively go back to Step 4 until a specified number of trials, I , is completed. Finally, produce statistics for the simulation. The key value for our example is the mean value of Ci0 for the total number of simulation trials. This mean value is the Monte Carlo estimate of the value of the Asian call. How many simulation trials should be specified? In general, we need to increase the number of trials by a factor of 100 to get each extra digit of accuracy. Depending on the problem, tens of thousands of trials may be needed to obtain accuracy to two decimal places (as required for option value, for example). Conducting a large number of trials is not necessarily a problem, given today’s computing power. The number of trials needed can be reduced using variance reduction procedures, a topic outside the scope of this book.45 In Step 4 of our example, a computer function produced a set of random observations on a standard normal random variable. Recall that for a uniform distribution, all possible numbers are equally likely. The term random number generator refers to an algorithm that produces uniformly distributed random numbers between 0 and 1. In the context of computer 45 For details on this and other technical aspects of Monte Carlo simulation, see Hillier and Lieberman (2000). Chapter 5 Common Probability Distributions 209 simulations, the term random number refers to an observation drawn from a uniform distribution.46 For other distributions, the term ‘‘random observation’’ is used in this context. It is a remarkable fact that random observations from any distribution can be produced using the uniform random variable with endpoints 0 and 1. To see why this is so, consider the inverse transformation method of producing random observations. Suppose we are interested in obtaining random observations for a random variable, X , with cumulative distribution function F (x). Recall that F (x) evaluated at x is a number between 0 and 1. Suppose a random outcome of this random variable is 3.21 and that F (3.21) = 0.25 or 25 percent. Define an inverse of F , call it F −1 , that can do the following: Substitute the probability 0.25 into F −1 and it returns the random outcome 3.21. In other words, F −1 (0.25) = 3.21. To generate random observations on X , the steps are (1) generate a uniform random number, r, between 0 and 1 using the random number generator and (2) evaluate F −1 (r) to obtain a random observation on X . Random observation generation is a field of study in itself, and we have briefly discussed the inverse transformation method here just to illustrate a point. As a generalist you do not need to address the technical details of converting random numbers into random observations, but you do need to know that random observations from any distribution can be generated using a uniform random variable. In Examples 5-12 and 5-13, we give an application of Monte Carlo simulation to a question of great interest to investment practice: the potential gains from market timing. EXAMPLE 5-12 Potential Gains from Market Timing: A Monte Carlo Simulation (1) All active investors want to achieve superior performance. One possible source of superior performance is market timing ability. How accurate does an investor need to be as a bull- and bear-market forecaster for market timing to be profitable? What size gains compared with a buy-and-hold strategy accrue to a given level of accuracy? Because of the variability in asset returns, a huge amount of return data is needed to find statistically reliable answers to these questions. Chua, Woodward, and To (1987) thus selected Monte Carlo simulation to address the potential gains from market timing. They were interested in the perspective of a Canadian investor. To understand their study, suppose that at the beginning of a year, an investor predicts that the next year will see either a bull market or bear market. If the prediction is bull market, the investor puts all her money in stocks and earns the market return for that year. On the other hand, if the prediction is bear market, the investor holds T-bills and earns the T-bill return. After the fact, a market is categorized as bull market if the stock market return, RMt , minus T-bill return, RFt , is positive for the year; otherwise, the market is classed as bear market. The investment results of a market timer can be 46 The numbers that random number generators produce depend on a seed or initial value. If the same seed is fed to the same generator, it will produce the same sequence. All sequences eventually repeat. Because of this predictability, the technically correct name for the numbers produced by random number generators is pseudo-random numbers. Pseudo-random numbers have sufficient qualities of randomness for most practical purposes. 210 Quantitative Investment Analysis compared with those of a buy-and-hold investor. A buy-and-hold investor earns the market return every year. For Chua et al., one quantity of interest was the gain from market timing. They defined this quantity as the market timer’s average return minus the average return to a buy-and-hold investor. To simulate market returns, Chua et al. generated 10,000 random standard normal observations, Zt . At the time of the study, Canadian stocks had a historical mean annual return of 12.95 percent with a standard deviation of 18.30 percent. To reflect these parameters, the simulated market returns are RMt = 0.1830Zt + 0.1295, t = 1, 2, . . . , 10,000. Using a second set of 10,000 random standard normal observations, historical return parameters for Canadian T-bills, as well as the historical correlation of T-bill and stock returns, the authors generated 10,000 T-bill returns. An investor can have different skills in forecasting bull and bear markets. Chua et al. characterized market timers by accuracy in forecasting bull markets and accuracy in forecasting bear markets. For example, bull market forecasting accuracy of 50 percent means that when the timer forecasts bull market for the next year, she is right just half the time, indicating no skill. Suppose an investor has 60 percent accuracy in forecasting bull market and 80 percent accuracy in forecasting bear market (a 60–80 timer). We can simulate how an investor would fare. After generating the first observation on RMt − RFt , we know whether that observation is a bull or bear market. If the observation is bull market, then 0.60 (forecast accuracy for bull markets) is compared with a random number (between 0 and 1). If the random number is less than 0.60, which occurs with a 60 percent probability, then the market timer is assumed to have correctly predicted bull market and her return for that first observation is the market return. If the random number is greater than 0.60, then the market timer is assumed to have made an error and predicted bear market; her return for that observation is the risk-free rate. In a similar fashion, if that first observation is bear market, the timer has an 80 percent chance of being right in forecasting bear market based on a random number draw. In either case, her return is compared with the market return to record her gain versus a buyand-hold strategy. That process is one simulation trial. The simulated mean return earned by the timer is the average return earned by the timer over all trials in the simulation. To increase our understanding of the process, consider a hypothetical Monte Carlo simulation with four trials for the 60–80 timer (who, to reiterate, has 60 percent accuracy in forecasting bull markets and 80 percent accuracy in forecasting bear markets). Table 5-8 gives data for the simulation. Let us look at Trials 1 and 2. In Trial 1, the first random number drawn leads to a market return of 0.121. Because the market return, 0.121, exceeded the T-bill return, 0.050, we have a bull market. We generate a random number, 0.531, which we then compare with the timer’s bull market accuracy, 0.60. Because 0.531 is less than 0.60, the timer is assumed to have made a correct bull market forecast and thus to have invested in stocks. Thus the timer earns the stock market return, 0.121, for that trial. In the second trial we observe another bull market, but because the random number 0.725 is greater than 0.60, the timer is assumed to have made an error and predicted a bear market; therefore, the timer earned the T-bill return, 0.081, rather than the higher stock market return. 211 Chapter 5 Common Probability Distributions TABLE 5-8 Hypothetical Simulation for a 60–80 Market Timer After Draws for Zt and for the T-bill Return Trial 1 2 3 4 RMt 0.121 0.092 −0.020 0.052 Simulation Results RFt Bull or Bear Market? Value of X Timer’s Prediction Correct? Return Earned by Timer 0.050 0.081 0.034 0.055 Bull Bull Bear A 0.531 0.725 0.786 0.901 Yes No Yes B 0.121 0.081 0.034 C R=D Note: R is the mean return earned by the timer over the four simulation trials. Using the data in Table 5-8, determine the values of A, B, C, and D. Solution: The value of A is Bear because the stock market return was less than the T-bill return in Trial 4. The value of B is No. Because we observe a bear market, we compare the random number 0.901 with 0.80, the timer’s bear-market forecasting accuracy. Because 0.901 is greater than 0.8, the timer is assumed to have made an error. The value of C is 0.052, the return on the stock market, because the timer made an error and invested in the stock market and earned 0.052 rather than the higher T-bill return of 0.055. The value of D is R = (0.121 + 0.081 + 0.034 + 0.052) = 0.288/4 = 0.072. Note that we could calculate other statistics besides the mean, such as the standard deviation of the returns earned by the timer over the four trials in the simulation. EXAMPLE 5-13 Potential Gains from Market Timing: A Monte Carlo Simulation (2) Having discussed the plan of the Chua et al. study and illustrated the method for a hypothetical Monte Carlo simulation with four trials, we conclude our presentation of the study. The hypothetical simulation in Example 5-12 had four trials, far too few to reach statistically precise conclusions. The simulation of Chua et al. incorporated 10,000 trials. Chua et al. specified bull- and bear-market prediction skill levels of 50, 60, 70, 80, 90, and 100 percent. Table 5-9 presents a very small excerpt from their simulation results for the no transaction costs case (transaction costs were also examined). Reading across the row, the timer with 60 percent bull market and 80 percent bear market forecasting accuracy had a mean annual gain from market timing of −1.12 percent per year. On average, the buy-and-hold investor out-earned this skillful timer by 1.12 212 Quantitative Investment Analysis percentage points. There was substantial variability in gains across the simulation trials, however: The standard deviation of the gain was 14.77 percent, so in many trials (but not on average) the gain was positive. Row 3 (win/loss) is the ratio of profitable switches between stocks and T-bills to unprofitable switches. This ratio was a favorable 1.2070 for the 60–80 timer. (When transaction costs were considered, however, fewer switches are profitable: The win/loss ratio was 0.5832 for the 60–80 timer.) TABLE 5-9 Gains from Stock Market Timing (No Transaction Costs) Bull Market Accuracy (%) 60 Bear Market Accuracy (%) 50 60 Mean (%) −2.50 −1.99 S.D. (%) 13.65 14.11 Win/Loss 0.7418 0.9062 70 −1.57 14.45 1.0503 80 90 −1.12 −0.68 14.77 15.08 1.2070 1.3496 100 −0.22 15.42 1.4986 Source: Chua, Woodward, and To (1987), Table II (excerpt). The authors concluded that the cost of not being invested in the market during bull market years is high. Because a buy-and-hold investor never misses a bull market year, she has 100 percent forecast accuracy for bull markets (at the cost of 0 percent accuracy for bear markets). Given their definitions and assumptions, the authors also concluded that successful market timing requires a minimum accuracy of 80 percent in forecasting both bull and bear markets. Market timing is a continuing area of interest and study, and other perspectives exist. However, this example illustrates how Monte Carlo simulation is used to address important investment issues. The analyst chooses the probability distributions in Monte Carlo simulation. By contrast, historical simulation samples from a historical record of returns (or other underlying variables) to simulate a process. The concept underlying historical simulation (also called back simulation) is that the historical record provides the most direct evidence on distributions (and that the past applies to the future). For example, refer back to Step 2 in the outline of Monte Carlo simulation above and suppose the time increment is one day. Further, suppose we base the simulation on the record of daily stock returns over the last five years. In one type of historical simulation, we randomly draw K returns from that record to generate one simulation trial. We put back the observations into the sample, and in the next trial we again randomly sample with replacement. The simulation results directly reflect frequencies in the data. A drawback of this approach is that any risk not represented in the time period selected (for example, a stock market crash) will not be reflected in the simulation. Compared with Monte Carlo simulation, historical simulation does not lend itself to ‘‘what if’’ analyses. Nevertheless, historic simulation is an established alternative simulation methodology. Monte Carlo simulation is a complement to analytical methods. It provides only statistical estimates, not exact results. Analytical methods, where available, provide more insight into cause-and-effect relationships. For example, the Black–Scholes–Merton option pricing model for the value of a European call option is an analytical method, expressed as a formula. It Chapter 5 Common Probability Distributions 213 is a much more efficient method for valuing such a call than is Monte Carlo simulation. As an analytical expression, the Black–Scholes–Merton model permits the analyst to quickly gauge the sensitivity of call value to changes in current stock price and the other variables that determine call value. In contrast, Monte Carlo simulations do not directly provide such precise insights. However, only some types of options can be priced with analytical expressions. As financial product innovations proceed, the field of applications for Monte Carlo simulation continues to grow. CHAPTER 6 SAMPLING AND ESTIMATION 1. INTRODUCTION Each day, we observe the high, low, and close of stock market indexes from around the world. Indexes such as the S&P 500 Index and the Nikkei–Dow Jones Average are samples of stocks. Although the S&P 500 and the Nikkei do not represent the populations of U.S. or Japanese stocks, we view them as valid indicators of the whole population’s behavior. As analysts, we are accustomed to using this sample information to assess how various markets from around the world are performing. Any statistics that we compute with sample information, however, are only estimates of the underlying population parameters. A sample, then, is a subset of the population—a subset studied to infer conclusions about the population itself. This chapter explores how we sample and use sample information to estimate population parameters. In the next section, we discuss sampling —the process of obtaining a sample. In investments, we continually make use of the mean as a measure of central tendency of random variables, such as return and earnings per share. Even when the probability distribution of the random variable is unknown, we can make probability statements about the population mean using the central limit theorem. In Section 3, we discuss and illustrate this key result. Following that discussion, we turn to statistical estimation. Estimation seeks precise answers to the question ‘‘What is this parameter’s value?’’ The central limit theorem and estimation are the core of the body of methods presented in this chapter. In investments, we apply these and other statistical techniques to financial data; we often interpret the results for the purpose of deciding what works and what does not work in investments. We end this chapter with a discussion of the interpretation of statistical results based on financial data and the possible pitfalls in this process. 2. SAMPLING In this section, we present the various methods for obtaining information on a population (all members of a specified group) through samples (part of the population). The information on a population that we try to obtain usually concerns the value of a parameter, a quantity computed from or used to describe a population of data. When we use a sample to estimate a parameter, we make use of sample statistics (statistics, for short). A statistic is a quantity computed from or used to describe a sample of data. 215 216 Quantitative Investment Analysis We take samples for one of two reasons. In some cases, we cannot possibly examine every member of the population. In other cases, examining every member of the population would not be economically efficient. Thus, savings of time and money are two primary factors that cause an analyst to use sampling to answer a question about a population. In this section, we discuss two methods of random sampling: simple random sampling and stratified random sampling. We then define and illustrate the two types of data an analyst uses: cross-sectional data and time-series data. 2.1. Simple Random Sampling Suppose a telecommunications equipment analyst wants to know how much major customers will spend on average for equipment during the coming year. One strategy is to survey the population of telecom equipment customers and inquire what their purchasing plans are. In statistical terms, the characteristics of the population of customers’ planned expenditures would then usually be expressed by descriptive measures such as the mean and variance. Surveying all companies, however, would be very costly in terms of time and money. Alternatively, the analyst can collect a representative sample of companies and survey them about upcoming telecom equipment expenditures. In this case, the analyst will compute the sample mean expenditure, X , a statistic. This strategy has a substantial advantage over polling the whole population because it can be accomplished more quickly and at lower cost. Sampling, however, introduces error. The error arises because not all the companies in the population are surveyed. The analyst who decides to sample is trading time and money for sampling error. When an analyst chooses to sample, he must formulate a sampling plan. A sampling plan is the set of rules used to select a sample. The basic type of sample from which we can draw statistically sound conclusions about a population is the simple random sample (random sample, for short). • Definition of Simple Random Sample. A simple random sample is a subset of a larger population created in such a way that each element of the population has an equal probability of being selected to the subset. The procedure of drawing a sample to satisfy the definition of a simple random sample is called simple random sampling. How is simple random sampling carried out? We need a method that ensures randomness—the lack of any pattern—in the selection of the sample. For a finite (limited) population, the most common method for obtaining a random sample involves the use of random numbers (numbers with assured properties of randomness). First, we number the members of the population in sequence. For example, if the population contains 500 members, we number them in sequence with three digits, starting with 001 and ending with 500. Suppose we want a simple random sample of size 50. In that case, using a computer random-number generator or a table of random numbers, we generate a series of three-digit random numbers. We then match these random numbers with the number codes of the population members until we have selected a sample of size 50. Sometimes we cannot code (or even identify) all the members of a population. We often use systematic sampling in such cases. With systematic sampling, we select every kth member until we have a sample of the desired size. The sample that results from this procedure should be approximately random. Real sampling situations may require that we take an approximately random sample. Chapter 6 Sampling and Estimation 217 Suppose the telecommunications equipment analyst polls a random sample of telecom equipment customers to determine the average equipment expenditure. The sample mean will provide the analyst with an estimate of the population mean expenditure. Any difference between the sample mean and the population mean is called sampling error. • Definition of Sampling Error. Sampling error is the difference between the observed value of a statistic and the quantity it is intended to estimate. A random sample reflects the properties of the population in an unbiased way, and sample statistics, such as the sample mean, computed on the basis of a random sample are valid estimates of the underlying population parameters. A sample statistic is a random variable. In other words, not only do the original data from the population have a distribution but so does the sample statistic. This distribution is the statistic’s sampling distribution. • Definition of Sampling Distribution of a Statistic. The sampling distribution of a statistic is the distribution of all the distinct possible values that the statistic can assume when computed from samples of the same size randomly drawn from the same population. In the case of the sample mean, for example, we refer to the ‘‘sampling distribution of the sample mean’’ or the distribution of the sample mean. We will have more to say about sampling distributions later in this chapter. Next, however, we look at another sampling method that is useful in investment analysis. 2.2. Stratified Random Sampling The simple random sampling method just discussed may not be the best approach in all situations. One frequently used alternative is stratified random sampling. • Definition of Stratified Random Sampling. In stratified random sampling, the population is divided into subpopulations (strata) based on one or more classification criteria. Simple random samples are then drawn from each stratum in sizes proportional to the relative size of each stratum in the population. These samples are then pooled to form a stratified random sample. In contrast to simple random sampling, stratified random sampling guarantees that population subdivisions of interest are represented in the sample. Another advantage is that estimates of parameters produced from stratified sampling have greater precision—that is, smaller variance or dispersion—than estimates obtained from simple random sampling. Bond indexing is one area in which stratified sampling is frequently applied. Indexing is an investment strategy in which an investor constructs a portfolio to mirror the performance of a specified index. In pure bond indexing, also called the full-replication approach, the investor attempts to fully replicate an index by owning all the bonds in the index in proportion to their market value weights. Many bond indexes consist of thousands of issues, however, so pure bond indexing is difficult to implement. In addition, transaction costs would be high because many bonds do not have liquid markets. Although a simple random sample could be a solution to the cost problem, the sample would probably not match the index’s major risk factors—interest rate sensitivity, for example. Because the major risk factors of fixed-income 218 Quantitative Investment Analysis portfolios are well known and quantifiable, stratified sampling offers a more effective approach. In this approach, we divide the population of index bonds into groups of similar duration (interest rate sensitivity), cash flow distribution, sector, credit quality, and call exposure. We refer to each group as a stratum or cell (a term frequently used in this context).1 Then, we choose a sample from each stratum proportional to the relative market weighting of the stratum in the index to be replicated. EXAMPLE 6-1 Bond Indexes and Stratified Sampling Suppose you are the manager of a mutual fund indexed to the Lehman Brothers Government Index. You are exploring several approaches to indexing, including a stratified sampling approach. You first distinguish agency bonds from U.S. Treasury bonds. For each of these two groups, you define 10 maturity intervals—1 to 2 years, 2 to 3 years, 3 to 4 years, 4 to 6 years, 6 to 8 years, 8 to 10 years, 10 to 12 years, 12 to 15 years, 15 to 20 years, and 20 to 30 years—and also separate the bonds with coupons (annual interest rates) of 6 percent or less from the bonds with coupons of more than 6 percent. 1. How many cells or strata does this sampling plan entail? 2. If you use this sampling plan, what is the minimum number of issues the indexed portfolio can have? 3. Suppose that in selecting among the securities that qualify for selection within each cell, you apply a criterion concerning the liquidity of the security’s market. Is the sample obtained random? Explain your answer. Solution to 1: We have 2 issuer classifications, 10 maturity classifications, and 2 coupon classifications. So, in total, this plan entails 2(10)(2) = 40 different strata or cells. (This answer is an application of the multiplication rule of counting discussed in the chapter on probability concepts.) Solution to 2: You cannot have fewer than one issue for each cell, so the portfolio must include at least 40 issues. Solution to 3: If you apply any additional criteria to the selection of securities for the cells, not every security that might be included has an equal probability of being selected. As a result, the sampling is not random. In practice, indexing using stratified sampling usually does not strictly involve random sampling because the selection of bond issues within cells is subject to various additional criteria. Because the purpose of sampling in this application is not to make an inference about a population parameter but rather to index a portfolio, lack of randomness is not in itself a problem in this application of stratified sampling. 1 See Fabozzi (2004b). Chapter 6 Sampling and Estimation 219 In the next section, we discuss the kinds of data used by financial analysts in sampling and practical issues that arise in selecting samples. 2.3. Time-Series and Cross-Sectional Data Investment analysts commonly work with both time-series and cross-sectional data. A time series is a sequence of returns collected at discrete and equally spaced intervals of time (such as a historical series of monthly stock returns). Cross-sectional data are data on some characteristic of individuals, groups, geographical regions, or companies at a single point in time. The 2003 year-end book value per share for all New York Stock Exchange–listed companies is an example of cross-sectional data. Economic or financial theory offers no basis for determining whether a long or short time period should be selected to collect a sample. As analysts, we might have to look for subtle clues. For example, combining data from a period of fixed exchange rates with data from a period of floating exchange rates would be inappropriate. The variance of exchange rates when exchange rates were fixed would certainly be less than when rates were allowed to float. As a consequence, we would not be sampling from a population described by a single set of parameters.2 Tight versus loose monetary policy also influences the distribution of returns to stocks; thus, combining data from tight-money and loose-money periods would be inappropriate. Example 6-2 illustrates the problems that can arise when sampling from more than one distribution. EXAMPLE 6-2 Calculating Sharpe Ratios: One or Two Years of Quarterly Data? Analysts often use the Sharpe ratio to evaluate the performance of a managed portfolio. The Sharpe ratio is the average return in excess of the risk-free rate divided by the standard deviation of returns. This ratio measures the excess return earned per unit of standard deviation of return. To compute the Sharpe ratio, suppose that an analyst collects eight quarterly excess returns (i.e., total return in excess of the risk-free rate). During the first year, the investment manager of the portfolio followed a low-risk strategy, and during the second year, the manager followed a high-risk strategy. For each of these years, the analyst also tracks the quarterly excess returns of some benchmark against which the manager will be evaluated. For each of the two years, the Sharpe ratio for the benchmark is 0.21. Table 6-1 gives the calculation of the Sharpe ratio of the portfolio. For the first year, during which the manager followed a low-risk strategy, the average quarterly return in excess of the risk-free rate was 1 percent with a standard deviation of 4.62 percent. The Sharpe ratio is thus 1/4.62 = 0.22. The second year’s results mirror the first year except for the higher average return and volatility. The Sharpe ratio for the second year is 4/18.48 = 0.22. The Sharpe ratio for the benchmark is 0.21 during 2 When the mean or variance of a time series is not constant through time, the time series is not stationary. We discuss stationarity in more detail in the chapter on time-series analysis. 220 Quantitative Investment Analysis TABLE 6-1 Calculation of Sharpe Ratios: Low-Risk and High-Risk Strategies Quarter/Measure Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarterly average Quarterly standard deviation Sharpe ratio = 0.22 = 1/4.62 = 4/18.48 Year 1 Excess Returns Year 2 Excess Returns −3% 5 −3 5 −12% 20 −12 20 1% 4.62% 4% 18.48% the first and second years. Because larger Sharpe ratios are better than smaller ones (providing more return per unit of risk), the manager appears to have outperformed the benchmark. Now, suppose the analyst believes a larger sample to be superior to a small one. She thus decides to pool the two years together and calculate a Sharpe ratio based on eight quarterly observations. The average quarterly excess return for the two years is the average of each year’s average excess return. For the two-year period, the average excess return is (1 + 4)/2 = 2.5 percent per quarter. The standard deviation for all eight quarters measured from the sample mean of 2.5 percent is 12.57 percent. The portfolio’s Sharpe ratio for the two-year period is now 2.5/12.57 = 0.199; the Sharpe ratio for the benchmark remains 0.21. Thus, when returns for the two-year period are pooled, the manager appears to have provided less return per unit of risk than the benchmark and less when compared with the separate yearly results. The problem with using eight quarters of return data is that the analyst has violated the assumption that the sampled returns come from the same population. As a result of the change in the manager’s investment strategy, returns in Year 2 followed a different distribution than returns in Year 1. Clearly, during Year 1, returns were generated by an underlying population with lower mean and variance than the population of the second year. Combining the results for the first and second years yielded a sample that was representative of no population. Because the larger sample did not satisfy model assumptions, any conclusions the analyst reached based on the larger sample are incorrect. For this example, she was better off using a smaller sample than a larger sample because the smaller sample represented a more homogeneous distribution of returns. The second basic type of data is cross-sectional data.3 With cross-sectional data, the observations in the sample represent a characteristic of individuals, groups, geographical 3 The reader may also encounter two types of datasets that have both time-series and cross-sectional aspects. Panel data consist of observations through time on a single characteristic of multiple observational units. For example, the annual inflation rate of the Eurozone countries over a five-year period would represent panel data. Longitudinal data consist of observations on characteristic(s) of the same observational unit through time. Observations on a set of financial ratios for a single company over a 10-year period would be an example of longitudinal data. Both panel and longitudinal data may be represented by arrays (matrixes) in which successive rows represent the observations for successive time periods. Chapter 6 Sampling and Estimation 221 regions, or companies at a single point in time. The telecommunications analyst discussed previously is essentially collecting a cross-section of planned capital expenditures for the coming year. Whenever we sample cross-sectionally, certain assumptions must be met if we wish to summarize the data in a meaningful way. Again, a useful approach is to think of the observation of interest as a random variable that comes from some underlying population with a given mean and variance. As we collect our sample and begin to summarize the data, we must be sure that all the data do, in fact, come from the same underlying population. For example, an analyst might be interested in how efficiently companies use their inventory assets. Some companies, however, turn over their inventory more quickly than others because of differences in their operating environments (e.g., grocery stores turn over inventory more quickly than automobile manufacturers, in general). So the distribution of inventory turnover rates may not be characterized by a single distribution with a given mean and variance. Therefore, summarizing inventory turnover across all companies might be inappropriate. If random variables are generated by different underlying distributions, the sample statistics computed from combined samples are not related to one underlying population parameter. The size of the sampling error in such cases is unknown. In instances such as these, analysts often summarize company-level data by industry. Attempting to summarize by industry partially addresses the problem of differing underlying distributions, but large corporations are likely to be in more than one industrial sector, so analysts should be sure they understand how companies are assigned to the industry groups. Whether we deal with time-series data or cross-sectional data, we must be sure to have a random sample that is representative of the population we wish to study. With the objective of inferring information from representative samples, we now turn to the next part of this chapter, which focuses on the central limit theorem as well as point and interval estimates of the population mean. 3. DISTRIBUTION OF THE SAMPLE MEAN Earlier in this chapter, we presented a telecommunications equipment analyst who decided to sample in order to estimate mean planned capital expenditures by his customers. Supposing that the sample is representative of the underlying population, how can the analyst assess the sampling error in estimating the population mean? Viewed as a formula that takes a function of the random outcomes of a random variable, the sample mean is itself a random variable with a probability distribution. That probability distribution is called the statistic’s sampling distribution.4 To estimate how closely the sample mean can be expected to match the underlying population mean, the analyst needs to understand the sampling distribution of the mean. Fortunately, we have a result, the central limit theorem, that helps us understand the sampling distribution of the mean for many of the estimation problems we face. 4 Sometimes confusion arises because ‘‘sample mean’’ is also used in another sense. When we calculate the sample mean for a particular sample, we obtain a definite number, say 8. If we state that ‘‘the sample mean is 8,’’ we are using ‘‘sample mean’’ in the sense of a particular outcome of sample mean as a random variable. The number 8 is of course a constant and does not have a probability distribution. In this discussion, we are not referring to ‘‘sample mean’’ in the sense of a constant number related to a particular sample. 222 Quantitative Investment Analysis 3.1. The Central Limit Theorem One of the most practically useful theorems in probability theory, the central limit theorem has important implications for how we construct confidence intervals and test hypotheses. Formally, it is stated as follows: • The Central Limit Theorem. Given a population described by any probability distribution having mean µ and finite variance σ2 , the sampling distribution of the sample mean X computed from samples of size n from this population will be approximately normal with mean µ (the population mean) and variance σ2 /n (the population variance divided by n) when the sample size n is large. The central limit theorem allows us to make quite precise probability statements about the population mean by using the sample mean, whatever the distribution of the population, because the sample mean follows an approximate normal distribution for large-size samples. The obvious question is, ‘‘When is a sample’s size large enough that we can assume the sample mean is normally distributed?’’ In general, when sample size n is greater than or equal to 30, we can assume that the sample mean is approximately normally distributed.5 The central limit theorem states that the variance of the distribution of the sample mean is σ2 /C. The positive square root of variance is standard deviation. The standard deviation of a sample statistic is known as the standard error of the statistic. The standard error of the sample mean is an important quantity in applying the central limit theorem in practice. • Definition of the Standard Error of the Sample Mean. For sample mean X calculated from a sample generated by a population with standard deviation σ, the standard error of the sample mean is given by one of two expressions: σ σX = √ n (6-1) when we know σ, the population standard deviation, or by s sX = √ n (6-2) when we do not know the population standard deviation and need to use the sample standard deviation, s, to estimate it.6 5 When the underlying population is very nonnormal, a sample size well in excess of 30 may be required for the normal distribution to be a good description of the sampling distribution of the mean. 6 We need to note a technical point: When we take a sample of size n from a finite population of size N , we apply a shrinkage factor to the estimate of the standard error of the sample mean that is called the finite population correction factor (fpc). The fpc is equal to [(N − n)/(N − 1)]1/2 . Thus, if N = 100 and n = 20, [(100 − 20)/(100 − 1)]1/2 = 0.898933. If we have estimated a standard error of, say, 20, according to Equation 6-1 or Equation 6-2, the new estimate is 20(0.898933) = 17.978663. The fpc applies only when we sample from a finite population without replacement; most practitioners also do not apply the fpc if sample size n is very small relative to N (say, less than 5 percent of N ). For more information on the finite population correction factor, see Daniel and Terrell (1995). 223 Chapter 6 Sampling and Estimation In practice, we almost always need to use Equation 6-2. The estimate of s is given by the square root of the sample variance, s2 , calculated as follows: s2 = n ! i=1 (Xi − X )2 n−1 (6-3) We will soon see how we can use the sample mean and its standard error to make probability statements about the population mean by using the technique of confidence intervals. First, however, we provide an illustration of the central limit theorem’s force. EXAMPLE 6-3 The Central Limit Theorem It is remarkable that the sample mean for large sample sizes will be distributed normally regardless of the distribution of the underlying population. To illustrate the central limit theorem in action, we specify in this example a distinctly nonnormal distribution and use it to generate a large number of random samples of size 100. We then calculate the sample mean for each sample. The frequency distribution of the calculated sample means is an approximation of the sampling distribution of the sample mean for that sample size. Does that sampling distribution look like a normal distribution? We return to the telecommunications analyst studying the capital expenditure plans of telecom businesses. Suppose that capital expenditures for communications equipment form a continuous uniform random variable with a lower bound equal to $0 and an upper bound equal to $100—for short, call this a uniform (0, 100) random variable. The probability function of this continuous uniform random variable has a rather simple shape that is anything but normal. It is a horizontal line with a vertical intercept equal to 1/100. Unlike a normal random variable, for which outcomes close to the mean are most likely, all possible outcomes are equally likely for a uniform random variable. To illustrate the power of the central limit theorem, we conduct a Monte Carlo simulation to study the capital expenditure plans of telecom businesses.7 In this simulation, we collect 200 random samples of the capital expenditures of 100 companies (200 random draws, each consisting of the capital expenditures of 100 companies with n = 100). In each simulation trial, 100 values for capital expenditure are generated from the uniform (0, 100) distribution. For each random sample, we then compute the sample mean. We conduct 200 simulation trials in total. Because we have specified the distribution generating the samples, we know that the population mean capital expenditure is equal to ($0 + $100 million)/2 = $50 million; the population variance of capital expenditures is equal to (100 − 0)2 /12 = 833.33; thus, the standard deviation 7 Monte Carlo simulation involves the use of a computer to represent the operation of a system subject to risk. An integral part of Monte Carlo simulation is the generation of a large number of random samples from a specified probability distribution or distributions. 224 Quantitative Investment Analysis √ is $28.87 million and the standard error is 28.87/ 100 = 2.887 under the central limit theorem.8 The results of this Monte Carlo experiment are tabulated in Table 6-2 in the form of a frequency distribution. This distribution is the estimated sampling distribution of the sample mean. The frequency distribution can be described as bell-shaped and centered close to the population mean of 50. The most frequent, or modal, range, with 41 observations, is 48.5 to 50. The overall average of the sample means is $49.92, with a standard error equal to $2.80. The calculated standard error is close to the value of 2.887 given by the central limit theorem. The discrepancy between calculated and expected values of the mean and standard deviation under the central limit theorem is a result of random chance (sampling error). TABLE 6-2 Frequency Distribution: 200 Random Samples of a Uniform (0,100) Random Variable Range of Sample Means ($ million) 42.5 ≤ X < 44 44 ≤ X < 45.5 45.5 ≤ X < 47 47 ≤ X < 48.5 48.5 ≤ X < 50 50 ≤ X < 51.5 51.5 ≤ X < 53 53 ≤ X < 54.5 54.5 ≤ X < 56 56 ≤ X < 57.5 Absolute Frequency 1 6 22 39 41 39 23 12 12 5 Note: X is the mean capital expenditure for each sample. In summary, although the distribution of the underlying population is very nonnormal, the simulation has shown that a normal distribution well describes the estimated sampling distribution of the sample mean, with mean and standard error consistent with the values predicted by the central limit theorem. To summarize, according to the central limit theorem, when we sample from any distribution, the distribution of the sample mean will have the following properties as long as our sample size is large: • • 8 The distribution of the sample mean X will be approximately normal. The mean of the distribution of X will be equal to the mean of the population from which the samples are drawn. If a is the lower limit of a uniform random variable and b is the upper limit, then the random variable’s mean is given by (a + b)/2 and its variance is given by (b − a)2 /12. The chapter on common probability distributions fully describes continuous uniform random variables. Chapter 6 Sampling and Estimation • 225 The variance of the distribution of X will be equal to the variance of the population divided by the sample size. With the central limit theorem in hand, we next discuss the concepts and tools related to estimating the population parameters, with a special focus on the population mean. We focus on the population because analysts are more likely to meet interval estimates for the population mean than any other type of interval estimate. 4. POINT AND INTERVAL ESTIMATES OF THE POPULATION MEAN Statistical inference traditionally consists of two branches, hypothesis testing and estimation. Hypothesis testing addresses the question ‘‘Is the value of this parameter (say, a population mean) equal to some specific value (0, for example)?’’ In this process, we have a hypothesis concerning the value of a parameter, and we seek to determine whether the evidence from a sample supports or does not support that hypothesis. We discuss hypothesis testing in detail in the chapter on hypothesis testing. The second branch of statistical inference, and the focus of this chapter, is estimation. Estimation seeks an answer to the question ‘‘What is this parameter’s (for example, the population mean’s) value?’’ In estimating, unlike in hypothesis testing, we do not start with a hypothesis about a parameter’s value and seek to test it. Rather, we try to make the best use of the information in a sample to form one of several types of estimates of the parameter’s value. With estimation, we are interested in arriving at a rule for best calculating a single number to estimate the unknown population parameter (a point estimate). Together with calculating a point estimate, we may also be interested in calculating a range of values that brackets the unknown population parameter with some specified level of probability (a confidence interval). In Section 4.1, we discuss point estimates of parameters and then, in Section 4.2, the formulation of confidence intervals for the population mean. 4.1. Point Estimators An important concept introduced in this chapter is that sample statistics viewed as formulas involving random outcomes are random variables. The formulas that we use to compute the sample mean and all the other sample statistics are examples of estimation formulas or estimators. The particular value that we calculate from sample observations using an estimator is called an estimate. An estimator has a sampling distribution; an estimate is a fixed number pertaining to a given sample and thus has no sampling distribution. To take the example of the mean, the calculated value of the sample mean in a given sample, used as an estimate of the population mean, is called a point estimate of the population mean. As Example 6-3 illustrated, the formula for the sample mean can and will yield different results in repeated samples as different samples are drawn from the population. In many applications, we have a choice among a number of possible estimators for estimating a given parameter. How do we make our choice? We often select estimators because they have one or more desirable statistical properties. Following is a brief description of three desirable properties of estimators: unbiasedness (lack of bias), efficiency, and consistency.9 9 See Daniel and Terrell (1995) or Greene (2003) for a thorough treatment of the properties of estimators. 226 • Quantitative Investment Analysis Definition of Unbiasedness. An unbiased estimator is one whose expected value (the mean of its sampling distribution) equals the parameter it is intended to estimate. For example, the expected value of the sample mean, X , equals µ, the population mean, so we say that the sample mean is an unbiased estimator (of the population mean). The sample variance, s2 , which is calculated using a divisor of n − 1 (Equation 6-3), is an unbiased estimator of the population variance, σ2 . If we were to calculate the sample variance using a divisor of n, the estimator would be biased: Its expected value would be smaller than the population variance. We would say that sample variance calculated with a divisor of n is a biased estimator of the population variance. Whenever one unbiased estimator of a parameter can be found, we can usually find a large number of other unbiased estimators. How do we choose among alternative unbiased estimators? The criterion of efficiency provides a way to select from among unbiased estimators of a parameter. • Definition of Efficiency. An unbiased estimator is efficient if no other unbiased estimator of the same parameter has a sampling distribution with smaller variance. To explain the definition, in repeated samples we expect the estimates from an efficient estimator to be more tightly grouped around the mean than estimates from other unbiased estimators. Efficiency is an important property of an estimator.10 Sample mean X is an efficient estimator of the population mean; sample variance s2 is an efficient estimator of σ2 . Recall that a statistic’s sampling distribution is defined for a given sample size. Different sample sizes define different sampling distributions. For example, the variance of sampling distribution of the sample mean is smaller for larger sample sizes. Unbiasedness and efficiency are properties of an estimator’s sampling distribution that hold for any size sample. An unbiased estimator is unbiased equally in a sample of size 10 and in a sample of size 1,000. In some problems, however, we cannot find estimators that have such desirable properties as unbiasedness in small samples.11 In this case, statisticians may justify the choice of an estimator based on the properties of the estimator’s sampling distribution in extremely large samples, the estimator’s so-called asymptotic properties. Among such properties, the most important is consistency. • Definition of Consistency. A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increases. Somewhat more technically, we can define a consistent estimator as an estimator whose sampling distribution becomes concentrated on the value of the parameter it is intended to estimate as the sample size approaches infinity. The sample mean, in addition to being an efficient estimator, is also a consistent√estimator of the population mean: As sample size n goes to infinity, its standard error, σ/ n, goes to 0 and its sampling distribution becomes concentrated right over the value of population mean, µ. To summarize, we can think of a consistent estimator as one that tends to produce more and more accurate estimates of the population parameter as we increase the sample’s size. If an estimator is consistent, we may attempt to increase the accuracy of estimates of a population parameter by calculating 10 An efficient estimator is sometimes referred to as the best unbiased estimator. 11 Such problems frequently arise in regression and time-series analyses, which we discuss in later chapters. Chapter 6 Sampling and Estimation 227 estimates using a larger sample. For an inconsistent estimator, however, increasing sample size does not help to increase the probability of accurate estimates. 4.2. Confidence Intervals for the Population Mean When we need a single number as an estimate of a population parameter, we make use of a point estimate. However, because of sampling error, the point estimate is not likely to equal the population parameter in any given sample. Often, a more useful approach than finding a point estimate is to find a range of values that we expect to bracket the parameter with a specified level of probability—an interval estimate of the parameter. A confidence interval fulfills this role. • Definition of Confidence Interval. A confidence interval is a range for which one can assert with a given probability 1 − α, called the degree of confidence, that it will contain the parameter it is intended to estimate. This interval is often referred to as the (1 − α)% confidence interval for the parameter. The endpoints of a confidence limit are referred to as the lower and upper confidence limits. In this chapter, we are concerned only with two-sided confidence intervals—confidence intervals for which we calculate both lower and upper limits.12 Confidence intervals are frequently given either a probabilistic interpretation or a practical interpretation. In the probabilistic interpretation, we interpret a 95 percent confidence interval for the population mean as follows: In repeated sampling, 95 percent of such confidence intervals will, in the long run, include or bracket the population mean. For example, suppose we sample from the population 1,000 times, and based on each sample, we construct a 95 percent confidence interval using the calculated sample mean. Because of random chance, these confidence intervals will vary from each other, but we expect 95 percent, or 950, of these intervals to include the unknown value of the population mean. In practice, we generally do not carry out such repeated sampling. Therefore, in the practical interpretation, we assert that we are 95 percent confident that a single 95 percent confidence interval contains the population mean. We are justified in making this statement because we know that 95 percent of all possible confidence intervals constructed in the same manner will contain the population mean. The confidence intervals that we discuss in this chapter have structures similar to the following basic structure: • Construction of Confidence Intervals. A (1 − α)% confidence interval for a parameter has the following structure: Point estimate ± Reliability factor × Standard error 12 It is also possible to define two types of one-sided confidence intervals for a population parameter. A lower one-sided confidence interval establishes a lower limit only. Associated with such an interval is an assertion that with a specified degree of confidence the population parameter equals or exceeds the lower limit. An upper one-sided confidence interval establishes an upper limit only; the related assertion is that the population parameter is less than or equal to that upper limit, with a specified degree of confidence. Investment researchers rarely present one-sided confidence intervals, however. 228 Quantitative Investment Analysis where Point estimate = a point estimate of the parameter (a value of a sample statistic) Reliability factor = a number based on the assumed distribution of the point estimate and the degree of confidence (1 − α) for the confidence interval Standard error = the standard error of the sample statistic providing the point estimate13 The most basic confidence interval for the population mean arises when we are sampling from a normal distribution with known variance. The reliability factor in this case is based on the standard normal distribution, which has a mean of 0 and a variance of 1. A standard normal random variable is conventionally denoted by Z . The notation zα denotes the point of the standard normal distribution such that α of the probability remains in the right tail. For example, 0.05 or 5 percent of the possible values of a standard normal random variable are larger than z0.05 = 1.65. Suppose we want to construct a 95 percent confidence interval for the population mean and, for this purpose, we have taken a sample of size 100 from a normally distributed population with known variance of σ2 = 400 (so, σ = 20). We calculate a sample mean of X = 25. Our point estimate of the population mean is, therefore, 25. If we move 1.96 standard deviations above the mean of a normal distribution, 0.025 or 2.5 percent of the probability remains in the right tail; by symmetry of the normal distribution, if we move 1.96 standard deviations below the mean, 0.025 or 2.5 percent of the probability remains in the left tail. In total, 0.05 or 5 percent of the probability is in the two tails and 0.95 or 95 percent lies in between. So, z0.025 = 1.96 is the reliability factor for this 95 percent confidence interval. Note the relationship (1 − α)% for the confidence interval and the zα/2 for the reliability √ factor. The standard error of the sample mean, given by Equation 6-1, is σX = 20/ 100 = 2. The confidence interval, therefore, has a lower limit of X − 1.96σX = 25 − 1.96(2) = 25 − 3.92 = 21.08. The upper limit of the confidence interval is X + 1.96σX = 25 + 1.96(2) = 25 + 3.92 = 28.92. The 95 percent confidence interval for the population mean spans 21.08 to 28.92. • Confidence Intervals for the Population Mean (Normally Distributed Population with Known Variance). A (1 − α)% confidence interval for population mean µ when we are sampling from a normal distribution with known variance σ2 is given by σ X ± zα/2 √ n (6-4) The reliability factors for the most frequently used confidence intervals are as follows: • Reliability Factors for Confidence Intervals Based on the Standard Normal Distribution. We use the following reliability factors when we construct confidence intervals based on the standard normal distribution:14 13 The quantity (Reliability factor) × (Standard error) is sometimes called the precision of the estimator; larger values of the product imply lower precision in estimating the population parameter. 14 Most practitioners use values for z0.05 and z0.005 that are carried to two decimal places. For reference, more exact values for z0.05 and z0.005 are 1.645 and 2.575, respectively. For a quick calculation of a 95 percent confidence interval, z0.025 is sometimes rounded from 1.96 to 2. 229 Chapter 6 Sampling and Estimation • • • 90 percent confidence intervals: Use z0.05 = 1.65 95 percent confidence intervals: Use z0.025 = 1.96 99 percent confidence intervals: Use z0.005 = 2.58 These reliability factors highlight an important fact about all confidence intervals. As we increase the degree of confidence, the confidence interval becomes wider and gives us less precise information about the quantity we want to estimate. ‘‘The surer we want to be, the less we have to be sure of.’’15 In practice, the assumption that the sampling distribution of the sample mean is at least approximately normal is frequently reasonable, either because the underlying distribution is approximately normal or because we have a large sample and the central limit theorem applies. However, rarely do we know the population variance in practice. When the population variance is unknown but the sample mean is at least approximately normally distributed, we have two acceptable ways to calculate the confidence interval for the population mean. We will soon discuss the more conservative approach, which is based on Student’s t-distribution (the t-distribution, for short).16 In investment literature, it is the most frequently used approach in both estimation and hypothesis tests concerning the mean when the population variance is not known, whether sample size is small or large. A second approach to confidence intervals for the population mean, based on the standard normal distribution, is the z-alternative. It can be used only when sample size is large. (In general, a sample size of 30 or larger may be considered large.) In contrast to the confidence interval given in Equation 6-4, this confidence interval uses the sample standard deviation, s, in computing the standard error of the sample mean (Equation 6-2). • Confidence Intervals for the Population Mean—The z-Alternative (Large Sample, Population Variance Unknown). A (1 − α)% confidence interval for population mean µ when sampling from any distribution with unknown variance and when sample size is large is given by s X ± zα/2 √ n (6-5) Because this type of confidence interval appears quite often, we illustrate its calculation in Example 6-4. EXAMPLE 6-4 Confidence Interval for the Population Mean of Sharpe Ratios—z-Statistic Suppose an investment analyst takes a random sample of U.S. equity mutual funds and calculates the average Sharpe ratio. The sample size is 100, and the average Sharpe ratio is 0.45. The sample has a standard deviation of 0.30. Calculate and interpret the 90 15 Freund and Williams (1977), p. 266. distribution of the statistic t is called Student’s t-distribution after the pen name ‘‘Student’’ used by W. S. Gosset, who published his work in 1908. 16 The 230 Quantitative Investment Analysis percent confidence interval for the population mean of all U.S. equity mutual funds by using a reliability factor based on the standard normal distribution. The reliability factor for a 90 percent confidence interval, as given earlier, is z0.05 = 1.65. The confidence interval will be 0.30 s X ± z0.05 √ = 0.45 ± 1.65 √ = 0.45 ± 1.65(0.03) = 0.45 ± 0.0495 n 100 The confidence interval spans 0.4005 to 0.4995, or 0.40 to 0.50, carrying two decimal places. The analyst can say with 90 percent confidence that the interval includes the population mean. In this example, the analyst makes no specific assumption about the probability distribution describing the population. Rather, the analyst relies on the central limit theorem to produce an approximate normal distribution for the sample mean. As Example 6-4 shows, even if we are unsure of the underlying population distribution, we can still construct confidence intervals for the population mean as long as the sample size is large because we can apply the central limit theorem. We now turn to the conservative alternative, using the t-distribution, for constructing confidence intervals for the population mean when the population variance is not known. For confidence intervals based on samples from normally distributed populations with unknown variance, the theoretically correct reliability factor is based on the t-distribution. Using a reliability factor based on the t-distribution is essential for a small sample size. Using a t reliability factor is appropriate when the population variance is unknown, even when we have a large sample and could use the central limit theorem to justify using a z reliability factor. In this large sample case, the t-distribution provides more-conservative (wider) confidence intervals. The t-distribution is a symmetrical probability distribution defined by a single parameter known as degrees of freedom (df). Each value for the number of degrees of freedom defines one distribution in this family of distributions. We will shortly compare t-distributions with the standard normal distribution, but first we need to understand the concept of degrees of freedom. We can do so by examining the calculation of the sample variance. Equation 6-3 gives the unbiased estimator of the sample variance that we use. The term in the denominator, n − 1, which is the sample size minus 1, is the number of degrees of freedom in estimating the population variance when using Equation 6-3. We also use n − 1 as the number of degrees of freedom for determining reliability factors based on the t-distribution. The term ‘‘degrees of freedom’’ is used because in a random sample, we assume that observations are selected independently of each other. The numerator of the sample variance, however, uses the sample mean. How does the use of the sample mean affect the number of observations collected independently for the sample variance formula? With a sample of size 10 and a mean of 10 percent, for example, we can freely select only 9 observations. Regardless of the 9 observations selected, we can always find the value for the 10th observation that gives a mean equal to 10 percent. From the standpoint of the sample variance formula, then, there are 9 degrees of freedom. Given that we must first compute the sample mean from the total of n independent observations, only n − 1 observations can be chosen independently for the calculation of the sample variance. The concept of degrees of freedom comes up frequently in statistics, and you will see it often in later chapters. 231 Chapter 6 Sampling and Estimation Normal Distribution t (df = 2) t (df = 8) !6 !4 !2 0 2 4 6 FIGURE 6-1 Student’s t-Distribution Versus the Standard Normal Distribution √ Suppose we sample from a normal distribution. The ratio z = (X − µ)/(σ/ n) is distributed normally √ with a mean of 0 and standard deviation of 1; however, the ratio t = (X − µ)/(s/ n) follows the t-distribution with a mean of 0 and n − 1 degrees of freedom. The ratio represented by t is not normal because t is the ratio of two random variables, the sample mean and the sample standard deviation. The definition of the standard normal random variable involves only one random variable, the sample mean. As degrees of freedom increase, however, the t-distribution approaches the standard normal distribution. Figure 6-1 shows the standard normal distribution and two t-distributions, one with df = 2 and one with df = 8. Of the three distributions shown in Figure 6-1, the standard normal distribution is clearly the most peaked; it has tails that approach zero faster than the tails of the two t-distributions. The t-distribution is also symmetrically distributed around its mean value of zero, just like the normal distribution. The t-distribution with df = 2 is the least peaked of the three distributions, and its tails lie above the tails for the normal and t with df = 8. The t-distribution with df = 8 has an intermediate degree of peakedness, and its tails lie above the tails for the normal but below those for t with df = 2. As the degrees of freedom increase, the t-distribution approaches the standard normal. The t-distribution with df = 8 is closer to the standard normal than the t-distribution with df = 2. Beyond plus and minus four standard deviations from the mean, the area under the standard normal distribution appears to approach 0; both t-distributions continue to show some area under each curve beyond four standard deviations, however. The t-distributions have fatter tails, but the tails of the t-distribution with df = 8 more closely resemble the normal distribution’s tails. As the degrees of freedom increase, the tails of the t-distribution become less fat. Frequently referred to values for the t-distribution are presented in tables at the end of the book. For each degree of freedom, five values are given: t0.10 , t0.05 , t0.025 , t0.01 , and t0.005 . The values for t0.10 , t0.05 , t0.025 , t0.01 , and t0.005 are such that, respectively, 0.10, 0.05, 0.025, 0.01, and 0.005 of the probability remains in the right tail, for the specified number of 232 Quantitative Investment Analysis degrees of freedom.17 For example, for df = 30, t0.10 = 1.310, t0.05 = 1.697, t0.025 = 2.042, t0.01 = 2.457, and t0.005 = 2.750. We now give the form of confidence intervals for the population mean using the t-distribution. • Confidence Intervals for the Population Mean (Population Variance Unknown)— t-Distribution. If we are sampling from a population with unknown variance and either of the conditions below holds: • • the sample is large, or the sample is small but the population is normally distributed, or approximately normally distributed, then a (1 − α)% confidence interval for the population mean µ is given by s X ± tα/2 √ n (6-6) where the number of degrees of freedom for tα/2 is n − 1 and n is the sample size. Example 6-5 reprises the data of Example 6-4 but uses the t-statistic rather than the z-statistic to calculate a confidence interval for the population mean of Sharpe ratios. EXAMPLE 6-5 Confidence Interval for the Population Mean of Sharpe Ratios—t-Statistic As in Example 6-4, an investment analyst seeks to calculate a 90 percent confidence interval for the population mean Sharpe ratio of U.S. equity mutual funds based on a random sample of 100 U.S. equity mutual funds. The sample mean Sharpe ratio is 0.45, and the sample standard deviation of the Sharpe ratios is 0.30. Now recognizing that the population variance of the distribution of Sharpe ratios is unknown, the analyst decides to calculate the confidence interval using the theoretically correct t-statistic. Because the sample size is 100, df = 99. In the tables in the back of the book, the closest value is df = 100. Using df = 100 and reading down the 0.05 column, we find that t0.05 = 1.66. This reliability factor is slightly larger than the reliability factor z0.05 = 1.65 that was used in Example 6-4. The confidence interval will be 0.30 s X ± t0.05 √ = 0.45 ± 1.66 √ = 0.45 ± 1.66(0.03) = 0.45 ± 0.0498 n 100 17 The values t0.10 , t0.05 , t0.025 , t0.01 , and t0.005 are also referred to as one-sided critical values of t at the 0.10, 0.05, 0.025, 0.01, and 0.005 significance levels, for the specified number of degrees of freedom. 233 Chapter 6 Sampling and Estimation The confidence interval spans 0.4002 to 0.4998, or 0.40 to 0.50, carrying two decimal places. To two decimal places, the confidence interval is unchanged from the one computed in Example 6-4. Table 6-3 summarizes the various reliability factors that we have used. TABLE 6-3 Basis of Computing Reliability Factors Sampling from: Statistic for Small Sample Size Normal distribution with known variance Normal distribution with unknown variance Nonnormal distribution with known variance Nonnormal distribution with unknown variance ∗ Statistic for Large Sample Size z t not available not available z t∗ z t∗ Use of z also acceptable. 4.3. Selection of Sample Size What choices affect the width of a confidence interval? To this point we have discussed two factors that affect width: the choice of statistic (t or z) and the choice of degree of confidence (affecting which specific value of t or z we use). These two choices determine the reliability factor. (Recall that a confidence interval has the structure Point estimate ± Reliability factor × Standard error.) The choice of sample size also affects the width of a confidence interval. All else equal, a larger sample size decreases the width of a confidence interval. Recall the expression for the standard error of the sample mean: Standard error of the sample mean = Sample standard deviation " Sample size We see that the standard error varies inversely with the square root of sample size. As we increase sample size, the standard error decreases and consequently the width of the confidence interval also decreases. The larger the sample size, the greater precision with which we can estimate the population parameter.18 All else equal, larger samples are good, in that sense. In practice, however, two considerations may operate against increasing sample size. First, as we saw in Example 6-2 concerning the Sharpe ratio, increasing the size of a sample may result in sampling from more than one population. Second, increasing sample size may involve additional expenses that outweigh the value of additional precision. Thus three issues that the analyst should weigh in selecting sample size are the need for precision, the risk of sampling from more than one population, and the expenses of different sample sizes. 18 A formula exists for determining the sample size needed to obtain a desired width for a confidence interval. Define E = Reliability factor × Standard error. The smaller E is, the smaller the width of the confidence interval, because 2E is the confidence interval’s width. The sample size to obtain a desired value of E at a given degree of confidence (1 − α) is n = [(tα/2 s)/E]2 . 234 Quantitative Investment Analysis EXAMPLE 6-6 A Money Manager Estimates Net Client Inflows A money manager wants to obtain a 95 percent confidence interval for fund inflows and outflows over the next six months for his existing clients. He begins by calling a random sample of 10 clients and inquiring about their planned additions to and withdrawals from the fund. The manager then computes the change in cash flow for each client sampled as a percentage change in total funds placed with the manager. A positive percentage change indicates a net cash inflow to the client’s account, and a negative percentage change indicates a net cash outflow from the client’s account. The manager weights each response by the relative size of the account within the sample and then computes a weighted average. As a result of this process, the money manager computes a weighted average of 5.5 percent. Thus, a point estimate is that the total amount of funds under management will increase by 5.5 percent in the next six months. The standard deviation of the observations in the sample is 10 percent. A histogram of past data looks fairly close to normal, so the manager assumes the population is normal. 1. Calculate a 95 percent confidence interval for the population mean and interpret your findings. The manager decides to see what the confidence interval would look like if he had used a sample size of 20 or 30 and found the same mean (5.5 percent) and standard deviation (10 percent). 2. Using the sample mean of 5.5 percent and standard deviation of 10 percent, compute the confidence interval for sample sizes of 20 and 30. For the sample size of 30, use Equation 6-6. 3. Interpret your results from Parts 1 and 2. Solution to 1: Because the population is unknown and the sample size is small, the manager must use the t-statistic in Equation 6-6 to calculate the confidence interval. Based on the sample size of 10, df = n − 1 = 10 − 1 = 9. For a 95 percent confidence interval, he needs to use the value of t0.025 for df = 9. According to the tables in the back of the book, this value is 2.262. Therefore, a 95 percent confidence interval for the population mean is 10% s X ± t0.025 √ = 5.5% ± 2.262 √ n 10 = 5.5% ± 2.262(3.162) = 5.5% ± 7.15% 235 Chapter 6 Sampling and Estimation The confidence interval for the population mean spans −1.65 percent to +12.65 percent.19 The manager can be confident at the 95 percent level that this range includes the population mean. Solution to 2: Table 6-4 gives the calculations for the three sample sizes. TABLE 6-4 The 95 Percent Confidence Interval for Three Sample Sizes Distribution t(n = 10) t(n = 20) t(n = 30) 95% Confidence Interval Lower Bound Upper Bound Relative Size 5.5% ± 2.262(3.162) 5.5% ± 2.093(2.236) 5.5% ± 2.045(1.826) −1.65% 0.82% 1.77% 12.65% 10.18% 9.23% 100.0% 65.5% 52.2% Solution to 3: The width of the confidence interval decreases as we increase the sample size. This decrease is a function of the standard error becoming smaller as n increases. The reliability factor also becomes smaller as the number of degrees of freedom increases. The last column of Table 6-4 shows the relative size of the width of confidence intervals based on n = 10 to be 100 percent. Using a sample size of 20 reduces the confidence interval’s width to 65.5 percent of the interval width for a sample size of 10. Using a sample size of 30 cuts the width of the interval almost in half. Comparing these choices, the money manager would obtain the most precise results using a sample of 30. Having covered many of the fundamental concepts of sampling and estimation, we are in a good position to focus on sampling issues of special concern to analysts. The quality of inferences depends on the quality of the data as well as on the quality of the sampling plan used. Financial data pose special problems, and sampling plans frequently reflect one or more biases. The next section of this chapter discusses these issues. 5. MORE ON SAMPLING We have already seen that the selection of sample period length may raise the issue of sampling from more than one population. There are, in fact, a range of challenges to valid sampling that arise in working with financial data. In this section we discuss four such sampling-related issues: data-mining bias, sample selection bias, look-ahead bias, and time-period bias. All of these issues are important for point and interval estimation and hypothesis testing. As we will see, if the sample is biased in any way, then point and interval estimates and any other conclusions that we draw from the sample will be in error. 19 We assumed in this example that sample size is sufficiently small compared with the size of the client base that we can disregard the finite population correction factor (mentioned in Footnote 6). 236 Quantitative Investment Analysis 5.1. Data-Mining Bias Data mining relates to overuse of the same or related data in ways that we shall describe shortly. Data-mining bias refers to the errors that arise from such misuse of data. Investment strategies that reflect data-mining biases are often not successful in the future. Nevertheless, both investment practitioners and researchers have frequently engaged in data mining. Analysts thus need to understand and guard against this problem. Data-mining is the practice of determining a model by extensive searching through a dataset for statistically significant patterns (that is, repeatedly ‘‘drilling’’ in the same data until finding something that appears to work).20 In exercises involving statistical significance we set a significance level, which is the probability of rejecting the hypothesis we are testing when the hypothesis is in fact correct.21 Because rejecting a true hypothesis is undesirable, the investigator often sets the significance level at a relatively small number such as 0.05 or 5 percent.22 Suppose we test the hypothesis that a variable does not predict stock returns, and we test in turn 100 different variables. Let us also suppose that in truth none of the 100 variables has the ability to predict stock returns. Using a 5 percent significance level in our tests, we would still expect that 5 out of 100 variables would appear to be significant predictors of stock returns because of random chance alone. We have mined the data to find some apparently significant variables. In essence, we have explored the same data again and again until we found some after-the-fact pattern or patterns in the dataset. This is the sense in which data mining involves overuse of data. If we were to just report the significant variables, without also reporting the total number of variables that we tested that were unsuccessful as predictors, we would be presenting a very misleading picture of our findings. Our results would appear to be far more significant than they actually were, because a series of tests such as the one just described invalidates the conventional interpretation of a given significance level (such as 5 percent), according to the theory of inference. How can we investigate the presence of data-mining bias? With most financial data, the most ready means is to conduct out-of-sample tests of the proposed variable or strategy. An out-of-sample test uses a sample that does not overlap the time period(s) of the sample(s) on which a variable, strategy, or model, was developed. If a variable or investment strategy is the result of data mining, it should generally not be significant in out-of-sample tests. A variable or investment strategy that is statistically and economically significant in out-of-sample tests, and that has a plausible economic basis, may be the basis for a valid investment strategy. Caution is still warranted, however. The most crucial out-of-sample test is future investment success. If the strategy becomes known to other investors, prices may adjust so that the strategy, however well tested, does not work in the future. To summarize, the analyst should be aware that many apparently profitable investment strategies may reflect data-mining bias and thus should be cautious about the future applicability of published investment research results. Untangling the extent of data mining can be complex. To assess the significance of an investment strategy, we need to know how many unsuccessful strategies were tried not only 20 Some researchers use the term ‘‘data snooping’’ instead of data mining. convey an understanding of data mining, it is very helpful to introduce some basic concepts related to hypothesis testing. The chapter on hypothesis testing contains further discussion of significance levels and tests of significance. 22 In terms of our previous discussion of confidence intervals, significance at the 5 percent level corresponds to a hypothesized value for a population statistic falling outside a 95 percent confidence interval based on an appropriate sample statistic (e.g., the sample mean, when the hypothesis concerns the population mean). 21 To Chapter 6 Sampling and Estimation 237 by the current investigator but also by previous investigators using the same or related datasets. Much research, in practice, closely builds on what other investigators have done, and so reflects intergenerational data mining, to use the terminology of McQueen and Thorley (1999). Intergenerational data mining involves using information developed by previous researchers using a dataset to guide current research using the same or a related dataset.23 Analysts have accumulated many observations about the peculiarities of many financial datasets, and other analysts may develop models or investment strategies that will tend to be supported within a dataset based on their familiarity with the prior experience of other analysts. As a consequence, the importance of those new results may be overstated. Research has suggested that the magnitude of this type of data-mining bias may be considerable.24 With the background of the above definitions and explanations, we can understand McQueen and Thorley’s (1999) cogent exploration of data mining in the context of the popular Motley Fool ‘‘Foolish Four’’ investment strategy. The Foolish Four strategy, first presented in 1996, was a version of the Dow Dividend Strategy that was tuned by its developers to exhibit an even higher arithmetic mean return than the Dow Dividend Strategy over 1973 to 1993.25 From 1973 to 1993, the Foolish Four portfolio had an average annual return of 25 percent, and the claim was made in print that the strategy should have similar returns in the future. As McQueen and Thorley discussed, however, the Foolish Four strategy was very much subject to data-mining bias, including bias from intergenerational data mining, as the strategy’s developers exploited observations about the dataset made by earlier workers. McQueen and Thorley highlighted the data-mining issues by taking the Foolish Four portfolio one step further. They mined the data to create a ‘‘Fractured Four’’ portfolio that earned nearly 35 percent over 1973 to 1996, beating the Foolish Four strategy by almost 8 percentage points. Observing that all of the Foolish Four stocks did well in even years but not odd years and that the second-to-lowest-priced high-yielding stock was relatively the best performing stock in odd years, the strategy of the Fractured Four portfolio was to hold the Foolish Four stocks with equal weights in even years and hold only the second-to-lowest-priced stock in odd years. How likely is it that a performance difference between even and odd years reflected underlying economic forces, rather than a chance pattern of the data over the particular time period? Probably, very unlikely. Unless an investment strategy reflected underlying economic forces, we would not expect it to have any value in a forward-looking sense. Because the Foolish Four strategy also partook of data mining, the same issues applied to it. McQueen and Thorley found that in an out-of-sample test over the 1949–72 period, the Foolish Four strategy had about the same mean return as buying and holding the DJIA, but with higher 23 The term ‘‘intergenerational’’ comes from viewing each round of researchers as a generation. Campbell, Lo, and MacKinlay (1997) have called intergenerational data mining ‘‘data snooping.’’ The latter phrase, however, is commonly used as a synonym of data mining; thus McQueen and Thorley’s terminology is less ambiguous. The term ‘‘intragenerational data mining’’ is available when we want to highlight that the reference is to an investigator’s new or independent data mining. 24 For example, Lo and MacKinlay (1990) concluded that the magnitude of this type of bias on tests of the capital asset pricing model was considerable. 25 The Dow Dividend Strategy, also known as Dogs of the Dow Strategy, consists of holding an equally weighted portfolio of the 10 highest-yielding DJIA stocks as of the beginning of a year. At the time of McQueen and Thorley’s research, the Foolish Four strategy was as follows: At the beginning of each year, the Foolish Four portfolio purchases a 4-stock portfolio from the 5 lowest-priced stocks of the 10 highest-yielding DJIA stocks. The lowest-priced stock of the 5 is excluded, and 40 percent is invested in the second-to-lowest-priced stock, with 20 percent weights in the remaining 3. 238 Quantitative Investment Analysis risk. If the higher taxes and transaction costs of the Foolish Four strategy were accounted for, the comparison would have been even more unfavorable. McQueen and Thorley presented two signs that can warn analysts about the potential existence of data mining: Too much digging/too little confidence. The testing of many variables by the researcher is the ‘‘too much digging’’ warning sign of a data-mining problem. Unfortunately, many researchers do not disclose the number of variables examined in developing a model. Although the number of variables examined may not be reported, we should look closely for verbal hints that the researcher searched over many variables. The use of terms such as ‘‘we noticed (or noted) that’’ or ‘‘someone noticed (or noted) that,’’ with respect to a pattern in a dataset, should raise suspicions that the researchers were trying out variables based on their own or others’ observations of the data. • No story/no future. The absence of an explicit economic rationale for a variable or trading strategy is the ‘‘no story’’ warning sign of a data-mining problem. Without a plausible economic rationale or story for why a variable should work, the variable is unlikely to have predictive power. In a demonstration exercise using an extensive search of variables in an international financial database, Leinweber (1997) found that butter production in a particular country remote from the United States explained 75 percent of the variation in U.S. stock returns as represented by the S&P 500. Such a pattern, with no plausible economic rationale, is highly likely to be a random pattern particular to a specific time period.26 What if we do have a plausible economic explanation for a significant variable? McQueen and Thorley caution that a plausible economic rationale is a necessary but not a sufficient condition for a trading strategy to have value. As we mentioned earlier, if the strategy is publicized, market prices may adjust to reflect the new information as traders seek to exploit it; as a result, the strategy may no longer work. • 5.2. Sample Selection Bias When researchers look into questions of interest to analysts or portfolio managers, they may exclude certain stocks, bonds, portfolios, or time periods from the analysis for various reasons—perhaps because of data availability. When data availability leads to certain assets being excluded from the analysis, we call the resulting problem sample selection bias. For example, you might sample from a database that tracks only companies currently in existence. Many mutual fund databases, for instance, provide historical information about only those funds that currently exist. Databases that report historical balance sheet and income statement information suffer from the same sort of bias as the mutual fund databases: Funds or companies that are no longer in business do not appear there. So, a study that uses these types of databases suffers from a type of sample selection bias known as survivorship bias. Dimson, Marsh, and Staunton (2002) raised the issue of survivorship bias in international indexes: An issue that has achieved prominence is the impact of market survival on estimated long-run returns. Markets can experience not only disappointing performance but also total loss of value through confiscation, hyperinflation, nationalization, and 26 In the finance literature, such a random but irrelevant-to-the-future pattern is sometimes called an artifact of the dataset. Chapter 6 Sampling and Estimation 239 market failure. By measuring the performance of markets that survive over long intervals, we draw inferences that are conditioned on survival. Yet, as pointed out by Brown, Goetzmann, and Ross (1995) and Jorion and Goetzmann (1999), one cannot determine in advance which markets will survive and which will perish. (p. 41) Survivorship bias sometimes appears when we use both stock price and accounting data. For example, many studies in finance have used the ratio of a company’s market price to book equity per share (i.e., the price-to-book ratio, P/B) and found that P/B is inversely related to a company’s returns [see Fama and French (1992, 1993)]. P/B is also used to create many popular value and growth indexes. If the database that we use to collect accounting data excludes failing companies, however, a survivorship bias might result. Kothari, Shanken, and Sloan (1995) investigated just this question and argued that failing stocks would be expected to have low returns and low P/Bs. If we exclude failing stocks, then those stocks with low P/Bs that are included will have returns that are higher on average than if all stocks with low P/Bs were included. Kothari, Shanken, and Sloan suggested that this bias is responsible for the previous findings of an inverse relationship between average return and P/B.27 The only advice we can offer at this point is to be aware of any biases potentially inherent in a sample. Clearly, sample selection biases can cloud the results of any study. A sample can also be biased because of the removal (or delisting) of a company’s stock from an exchange.28 For example, the Center for Research in Security Prices at the University of Chicago is a major provider of return data used in academic research. When a delisting occurs, CRSP attempts to collect returns for the delisted company, but many times, it cannot do so because of the difficulty involved; CRSP must simply list delisted company returns as missing. A study in the Journal of Finance by Shumway and Warther (1999) documented the bias caused by delisting for CRSP Nasdaq return data. The authors showed that delistings associated with poor company performance (e.g., bankruptcy) are missed more often than delistings associated with good or neutral company performance (e.g., merger or moving to another exchange). In addition, delistings occur more frequently for small companies. Sample selection bias occurs even in markets where the quality and consistency of the data are quite high. Newer asset classes such as hedge funds may present even greater problems of sample selection bias. Hedge funds are a heterogeneous group of investment vehicles typically organized so as to be free from regulatory oversight. In general, hedge funds are not required to publicly disclose performance (in contrast to, say, mutual funds). Hedge funds themselves decide whether they want to be included in one of the various databases of hedge fund performance. Hedge funds with poor track records clearly may not wish to make their records public, creating a problem of self-selection bias in hedge fund databases. Further, as pointed out by Fung and Hsieh (2002), because only hedge funds with good records will volunteer to enter a database, in general, overall past hedge fund industry performance will tend to appear better than it really is. Furthermore, many hedge fund databases drop funds that go out of business, creating survivorship bias in the database. Even if the database does not drop defunct 27 See Fama and French (1996, p. 80) for discussion of data snooping and survivorship bias in their tests. occur for a variety of reasons: merger, bankruptcy, liquidation, or migration to another exchange. 28 Delistings 240 Quantitative Investment Analysis hedge funds, in the attempt to eliminate survivorship bias, the problem remains of hedge funds that stop reporting performance because of poor results.29 5.3. Look-Ahead Bias A test design is subject to look-ahead bias if it uses information that was not available on the test date. For example, tests of trading rules that use stock market returns and accounting balance sheet data must account for look-ahead bias. In such tests, a company’s book value per share is commonly used to construct the P/B variable. Although the market price of a stock is available for all market participants at the same point in time, fiscal year-end book equity per share might not become publicly available until sometime in the following quarter. 5.4. Time-Period Bias A test design is subject to time-period bias if it is based on a time period that may make the results time-period specific. A short time series is likely to give period-specific results that may not reflect a longer period. A long time series may give a more accurate picture of true investment performance; its disadvantage lies in the potential for a structural change occurring during the time frame that would result in two different return distributions. In this situation, the distribution that would reflect conditions before the change differs from the distribution that would describe conditions after the change. EXAMPLE 6-7 Biases in Investment Research An analyst is reviewing the empirical evidence on historical U.S. equity returns. She finds that value stocks (i.e., those with low P/Bs) outperformed growth stocks (i.e., those with high P/Bs) in some recent time periods. After reviewing the U.S. market, the analyst wonders whether value stocks might be attractive in the United Kingdom. She investigates the performance of value and growth stocks in the U.K. market from January 1990 to December 2003. To conduct this research, the analyst does the following: • • • • • 29 obtains the current composition of the Financial Times Stock Exchange (FTSE) All Share Index, which is a market-capitalization-weighted index; eliminates the few companies that do not have December fiscal year-ends; uses year-end book values and market prices to rank the remaining universe of companies by P/Bs at the end of the year; based on these rankings, divides the universe into 10 portfolios, each of which contains an equal number of stocks; calculates the equal-weighted return of each portfolio and the return for the FTSE All Share Index for the 12 months following the date each ranking was made; and See Ackerman, McEnally, and Ravenscraft (1999) and Fung and Hsieh (2002) for more details on the problems of interpreting hedge fund performance. Note that an offsetting type of bias may occur if successful funds stop reporting performance because they no longer want new cash inflows. Chapter 6 Sampling and Estimation • 241 subtracts the FTSE returns from each portfolio’s returns to derive excess returns for each portfolio. Describe and discuss each of the following biases introduced by the analyst’s research design: survivorship bias, look-ahead bias, and • time-period bias. • • Survivorship bias. A test design is subject to survivorship bias if it fails to account for companies that have gone bankrupt, merged, or otherwise departed the database. In this example, the analyst used the current list of FTSE stocks rather than the actual list of stocks that existed at the start of each year. To the extent that the computation of returns excluded companies removed from the index, the performance of the portfolios with the lowest P/B is subject to survivorship bias and may be overstated. At some time during the testing period, those companies not currently in existence were eliminated from testing. They would probably have had low prices (and low P/Bs) and poor returns. Look-ahead bias. A test design is subject to look-ahead bias if it uses information unavailable on the test date. In this example, the analyst conducted the test under the assumption that the necessary accounting information was available at the end of the fiscal year. For example, the analyst assumed that book value per share for fiscal 1990 was available on 31 December 1990. Because this information is not released until several months after the close of a fiscal year, the test may have contained look-ahead bias. This bias would make a strategy based on the information appear successful, but it assumes perfect forecasting ability. Time-period bias. A test design is subject to time-period bias if it is based on a time period that may make the results time-period specific. Although the test covered a period extending more than 10 years, that period may be too short for testing an anomaly. Ideally, an analyst should test market anomalies over several business cycles to ensure that results are not period specific. This bias can favor a proposed strategy if the time period chosen was favorable to the strategy. CHAPTER 7 HYPOTHESIS TESTING 1. INTRODUCTION Analysts often confront competing ideas about how financial markets work. Some of these ideas develop through personal research or experience with markets; others come from interactions with colleagues; and many others appear in the professional literature on finance and investments. In general, how can an analyst decide whether statements about the financial world are probably true or probably false? When we can reduce an idea or assertion to a definite statement about the value of a quantity, such as an underlying or population mean, the idea becomes a statistically testable statement or hypothesis. The analyst may want to explore questions such as the following: Is the underlying mean return on this mutual fund different from the underlying mean return on its benchmark? • Did the volatility of returns on this stock change after the stock was added to a stock market index? • Are a security’s bid-ask spreads related to the number of dealers making a market in the security? • Do data from a national bond market support a prediction of an economic theory about the term structure of interest rates (the relationship between yield and maturity)? • To address these questions, we use the concepts and tools of hypothesis testing. Hypothesis testing is part of statistical inference, the process of making judgments about a larger group (a population) on the basis of a smaller group actually observed (a sample). The concepts and tools of hypothesis testing provide an objective means to gauge whether the available evidence supports the hypothesis. After a statistical test of a hypothesis we should have a clearer idea of the probability that a hypothesis is true or not, although our conclusion always stops short of certainty. Hypothesis testing has been a powerful tool in the advancement of investment knowledge and science. As Robert L. Kahn of the Institute for Social Research (Ann Arbor, Michigan) has written, ‘‘The mill of science grinds only when hypothesis and data are in continuous and abrasive contact.’’ The main emphases of this chapter are the framework of hypothesis testing and tests concerning mean and variance, two quantities frequently used in investments. We give an overview of the procedure of hypothesis testing in the next section. We then address testing hypotheses about the mean and hypotheses about the differences between means. In the fourth section of this chapter, we address testing hypotheses about a single variance and hypotheses about the differences between variances. We end the chapter with an overview of some other important issues and techniques in statistical inference. 243 244 Quantitative Investment Analysis 2. HYPOTHESIS TESTING Hypothesis testing, as we have mentioned, is part of the branch of statistics known as statistical inference. Traditionally, the field of statistical inference has two subdivisions: estimation and hypothesis testing. Estimation addresses the question ‘‘What is this parameter’s (e.g., the population mean’s) value?’’ The answer is in the form of a confidence interval built around a point estimate. Take the case of the mean: We build a confidence interval for the population mean around the sample mean as a point estimate. For the sake of specificity, suppose the sample mean is 50 and a 95 percent confidence interval for the population mean is 50 ± 10 (the confidence interval runs from 40 to 60). If this confidence interval has been properly constructed, there is a 95 percent probability that the interval from 40 to 60 contains the population mean’s value.1 The second branch of statistical inference, hypothesis testing, has a somewhat different focus. A hypothesis testing question is ‘‘Is the value of the parameter (say, the population mean) 45 (or some other specific value)?’’ The assertion ‘‘the population mean is 45’’ is a hypothesis. A hypothesis is defined as a statement about one or more populations. This section focuses on the concepts of hypothesis testing. The process of hypothesis testing is part of a rigorous approach to acquiring knowledge known as the scientific method. The scientific method starts with observation and the formulation of a theory to organize and explain observations. We judge the correctness of the theory by its ability to make accurate predictions—for example, to predict the results of new observations.2 If the predictions are correct, we continue to maintain the theory as a possibly correct explanation of our observations. When risk plays a role in the outcomes of observations, as in finance, we can only try to make unbiased, probability-based judgments about whether the new data support the predictions. Statistical hypothesis testing fills that key role of testing hypotheses when chance plays a role. In an analyst’s day-to-day work, he may address questions to which he might give answers of varying quality. When an analyst correctly formulates the question into a testable hypothesis and carries out and reports on a hypothesis test, he has provided an element of support to his answer consistent with the standards of the scientific method. Of course, the analyst’s logic, economic reasoning, information sources, and perhaps other factors also play a role in our assessment of the answer’s quality.3 We organize this introduction to hypothesis testing around the following list of seven steps. • Steps in Hypothesis Testing. The steps in testing a hypothesis are as follows:4 1. 2. 3. 4. 5. 6. 7. 1 Stating the hypotheses. Identifying the appropriate test statistic and its probability distribution. Specifying the significance level. Stating the decision rule. Collecting the data and calculating the test statistic. Making the statistical decision. Making the economic or investment decision. We discussed the construction and interpretation of confidence intervals in the chapter on sampling. be testable, a theory must be capable of making predictions that can be shown to be wrong. 3 See Freeley and Steinberg (1999) for a discussion of critical thinking applied to reasoned decision making. 4 This list is based on one in Daniel and Terrell (1986). 2 To Chapter 7 Hypothesis Testing 245 We will explain each of these steps using as illustration a hypothesis test concerning the sign of the risk premium on Canadian stocks. The steps above constitute a traditional approach to hypothesis testing. We will end the section with a frequently used alternative to those steps, the p-value approach. The first step in hypothesis testing is stating the hypotheses. We always state two hypotheses: the null hypothesis (or null), designated H0 , and the alternative hypothesis, designated Ha . • Definition of Null Hypothesis. The null hypothesis is the hypothesis to be tested. For example, we could hypothesize that the population mean risk premium for Canadian equities is less than or equal to zero. The null hypothesis is a proposition that is considered true unless the sample we use to conduct the hypothesis test gives convincing evidence that the null hypothesis is false. When such evidence is present, we are led to the alternative hypothesis. • Definition of Alternative Hypothesis. The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. Our alternative hypothesis is that the population mean risk premium for Canadian equities is greater than zero. Suppose our question concerns the value of a population parameter, θ, in relation to one possible value of the parameter, θ0 (these are read, respectively, ‘‘theta’’ and ‘‘theta sub zero’’).5 Examples of a population parameter include the population mean, µ, and the population variance, σ2 . We can formulate three different sets of hypotheses, which we label according to the assertion made by the alternative hypothesis. • Formulations of Hypotheses. We can formulate the null and alternative hypotheses in three different ways: 1. H0 : θ = θ0 versus Ha : θ ̸= θ0 (a ‘‘not equal to’’ alternative hypothesis) 2. H0 : θ ≤ θ0 versus Ha : θ > θ0 (a ‘‘greater than’’ alternative hypothesis) 3. H0 : θ ≥ θ0 versus Ha : θ < θ0 (a ‘‘less than’’ alternative hypothesis) In our Canadian example, θ = µRP and represents the population mean risk premium on Canadian equities. Also, θ0 = 0 and we are using the second of the above three formulations. The first formulation is a two-sided hypothesis test (or two-tailed hypothesis test): We reject the null in favor of the alternative if the evidence indicates that the population parameter is either smaller or larger than θ0 . In contrast, Formulations 2 and 3 are each a one-sided hypothesis test (or one-tailed hypothesis test). For Formulations 2 and 3, we reject the null only if the evidence indicates that the population parameter is respectively greater than or less than θ0 . The alternative hypothesis has one side. Notice that in each case above, we state the null and alternative hypotheses such that they account for all possible values of the parameter. With Formulation 1, for example, the parameter is either equal to the hypothesized value θ0 (under the null hypothesis) or not equal to the hypothesized value θ0 (under the alternative hypothesis). Those two statements logically exhaust all possible values of the parameter. 5 Greek letters, such as σ, are reserved for population parameters; Roman letters in italics, such as s, are used for sample statistics. 246 Quantitative Investment Analysis Despite the different ways to formulate hypotheses, we always conduct a test of the null hypothesis at the point of equality, θ = θ0 . Whether the null is H0 : θ = θ0 , H0 : θ ≤ θ0 , or H0 : θ ≥ θ0 , we actually test θ = θ0 . The reasoning is straightforward. Suppose the hypothesized value of the parameter is 5. Consider H0 : θ ≤ 5, with a ‘‘greater than’’ alternative hypothesis, Ha : θ > 5. If we have enough evidence to reject H0 : θ = 5 in favor of Ha : θ > 5, we definitely also have enough evidence to reject the hypothesis that the parameter, θ, is some smaller value, such as 4.5 or 4. To review, the calculation to test the null hypothesis is the same for all three formulations. What is different for the three formulations, we will see shortly, is how the calculation is evaluated to decide whether or not to reject the null. How do we choose the null and alternative hypotheses? Probably most common are ‘‘not equal to’’ alternative hypotheses. We reject the null because the evidence indicates that the parameter is either larger or smaller than θ0 . Sometimes, however, we may have a ‘‘suspected’’ or ‘‘hoped for’’ condition for which we want to find supportive evidence.6 In that case, we can formulate the alternative hypothesis as the statement that this condition is true; the null hypothesis that we test is the statement that this condition is not true. If the evidence supports rejecting the null and accepting the alternative, we have statistically confirmed what we thought was true. For example, economic theory suggests that investors require a positive risk premium on stocks (the risk premium is defined as the expected return on stocks minus the risk-free rate). Following the principle of stating the alternative as the ‘‘hoped for’’ condition, we formulate the following hypotheses: H0 : The population mean risk premium on Canadian stocks is less than or equal to 0. Ha : The population mean risk premium on Canadian stocks is positive. Note that ‘‘greater than’’ and ‘‘less than’’ alternative hypotheses reflect the beliefs of the researcher more strongly than a ‘‘not equal to’’ alternative hypothesis. To emphasize an attitude of neutrality, the researcher may sometimes select a ‘‘not equal to’’ alternative hypothesis when a one-sided alternative hypothesis is also reasonable. The second step in hypothesis testing is identifying the appropriate test statistic and its probability distribution. • Definition of Test Statistic. A test statistic is a quantity, calculated based on a sample, whose value is the basis for deciding whether or not to reject the null hypothesis. The focal point of our statistical decision is the value of the test statistic. Frequently (in all the cases that we examine in this chapter), the test statistic has the form Test statistic = Sample statistic − Value of the population parameter under H0 Standard error of the sample statistic (7-1) For our risk premium example, the population parameter of interest is the population mean risk premium, µRP . We label the hypothesized value of the population mean under H0 as µ0 . Restating the hypotheses using symbols, we test H0 : µRP ≤ µ0 versus Ha : µRP > µ0 . However, because under the null we are testing µ0 = 0, we write H0 : µRP ≤ 0 versus Ha : µRP > 0. 6 Part of this discussion of the selection of hypotheses follows Bowerman and O’Connell (1997, p. 386). 247 Chapter 7 Hypothesis Testing The sample mean provides an estimate of the population mean. Therefore, we can use the sample mean risk premium calculated from historical data, X RP , as the sample statistic in Equation 7-1. The standard deviation of the sample statistic, known as the ‘‘standard error’’ of the statistic, is the denominator in Equation 7-1. For this example, the sample statistic is a sample mean. For a sample mean, X , calculated from a sample generated by a population with standard deviation σ, the standard error is given by one of two expressions: σ σX = √ n (7-2) when we know σ (the population standard deviation), or s sX = √ n (7-3) when we do not know the population standard deviation and need to use the sample standard deviation s to estimate it. For this example, because we do not know the population standard deviation of the process generating the return, we use Equation 7-3. The test statistic is thus X RP − µ0 X RP − 0 = √ sX s/ n In making the substitution of 0 for µ0 , we use the fact already highlighted that we test any null hypothesis at the point of equality, as well as the fact that µ0 = 0 here. We have identified a test statistic to test the null hypothesis. What probability distribution does it follow? We will encounter four distributions for test statistics in this chapter: the t-distribution (for a t-test), the standard normal or z-distribution (for a z-test), • the chi-square (χ2 ) distribution (for a chi-square test), and • the F -distribution (for an F -test). • • We will discuss the details later, but assume we can conduct a z-test based on the cen-tral limit theorem because our Canadian sample has many observations.7 To summarize, the test statistic for the hypothesis test concerning the mean risk premium is X RP /s X . We can conduct a z-test because we can plausibly assume that the test statistic follows a standard normal distribution. The third step in hypothesis testing is specifying the significance level. When the test statistic has been calculated, two actions are possible: (1) We reject the null hypothesis or (2) we do not reject the null hypothesis. The action we take is based on comparing the calculated test statistic to a specified possible value or values. The comparison values we choose are based on the level of significance selected. The level of significance reflects how much sample evidence we require to reject the null. Analogous to its counterpart in a court of law, the required standard of proof can change according to the nature of the hypotheses and the seriousness of the consequences of making a mistake. There are four possible outcomes when we test a null hypothesis: 7 The central limit theorem says that the sampling distribution of the sample mean will be approximately normal with mean µ and variance σ2 /n when the sample size is large. The sample we will use for this example has 103 observations. 248 Quantitative Investment Analysis TABLE 7-1 Type I and Type II Errors in Hypothesis Testing Decision Do not reject H0 Reject H0 (accept Ha ) 1. 2. 3. 4. True Situation H0 True H0 False Correct Decision Type I Error Type II Error Correct Decision We reject a false null hypothesis. This is a correct decision. We reject a true null hypothesis. This is called a Type I error. We do not reject a false null hypothesis. This is called a Type II error. We do not reject a true null hypothesis. This is a correct decision. We illustrate these outcomes in Table 7-1. When we make a decision in a hypothesis test, we run the risk of making either a Type I or a Type II error. These are mutually exclusive errors: If we mistakenly reject the null, we can only be making a Type I error; if we mistakenly fail to reject the null, we can only be making a Type II error. The probability of a Type I error in testing a hypothesis is denoted by the Greek letter alpha, α. This probability is also known as the level of significance of the test. For example, a level of significance of 0.05 for a test means that there is a 5 percent probability of rejecting a true null hypothesis. The probability of a Type II error is denoted by the Greek letter beta, β. Controlling the probabilities of the two types of errors involves a trade-off. All else equal, if we decrease the probability of a Type I error by specifying a smaller significance level (say, 0.01 rather than 0.05), we increase the probability of making a Type II error because we will reject the null less frequently, including when it is false. The only way to reduce the probabilities of both types of errors simultaneously is to increase the sample size, n. Quantifying the trade-off between the two types of error in practice is usually impossible because the probability of a Type II error is itself hard to quantify. Consider H0 : θ ≤ 5 versus Ha : θ > 5. Because every true value of θ greater than 5 makes the null hypothesis false, each value of θ greater than 5 has a different β (Type II error probability). In contrast, it is sufficient to state a Type I error probability for θ = 5, the point at which we conduct the test of the null hypothesis. Thus, in general, we specify only α, the probability of a Type I error, when we conduct a hypothesis test. Whereas the significance level of a test is the probability of incorrectly rejecting the null, the power of a test is the probability of correctly rejecting the null—that is, the probability of rejecting the null when it is false.8 When more than one test statistic is available to conduct a hypothesis test, we should prefer the most powerful, all else equal.9 To summarize, the standard approach to hypothesis testing involves specifying a level of significance (probability of Type I error) only. It is most appropriate to specify this significance level prior to calculating the test statistic. If we specify it after calculating the test statistic, we may be influenced by the result of the calculation, which detracts from the objectivity of the test. We can use three conventional significance levels to conduct hypothesis tests: 0.10, 0.05, and 0.01. Qualitatively, if we can reject a null hypothesis at the 0.10 level of significance, we have some evidence that the null hypothesis is false. If we can reject a null hypothesis at the 8 The power of a test is, in fact, 1 minus the probability of a Type II error. 9 We do not always have information on the relative power of the test for competing test statistics, however. Chapter 7 Hypothesis Testing 249 0.05 level, we have strong evidence that the null hypothesis is false. And if we can reject a null hypothesis at the 0.01 level, we have very strong evidence that the null hypothesis is false. For the risk premium example, we will specify a 0.05 significance level. The fourth step in hypothesis testing is stating the decision rule. The general principle is simply stated. When we test the null hypothesis, if we find that the calculated value of the test statistic is as extreme or more extreme than a given value or values determined by the specified level of significance, α, we reject the null hypothesis. We say the result is statistically significant. Otherwise, we do not reject the null hypothesis and we say the result is not statistically significant. The value or values with which we compare the calculated test statistic to make our decision are the rejection points (critical values) for the test.10 • Definition of a Rejection Point (Critical Value) for the Test Statistic. A rejection point (critical value) for a test statistic is a value with which the computed test statistic is compared to decide whether to reject or not reject the null hypothesis. For a one-tailed test, we indicate a rejection point using the symbol for the test statistic with a subscript indicating the specified probability of a Type I error, α; for example, zα . For a two-tailed test, we indicate zα/2 . To illustrate the use of rejection points, suppose we are using a z-test and have chosen a 0.05 level of significance. For a test of H0 : θ = θ0 versus Ha : θ ̸= θ0 , two rejection points exist, one negative and one positive. For a two-sided test at the 0.05 level, the total probability of a Type I error must sum to 0.05. Thus, 0.05/2 = 0.025 of the probability should be in each tail of the distribution of the test statistic under the null. Consequently, the two rejection points are z0.025 = 1.96 and −z0.025 = −1.96. Let z represent the calculated value of the test statistic. We reject the null if we find that z < −1.96 or z > 1.96. We do not reject if −1.96 ≤ z ≤ 1.96. • For a test of H0 : θ ≤ θ0 versus Ha : θ > θ0 at the 0.05 level of significance, the rejection point is z0.05 = 1.645. We reject the null hypothesis if z > 1.645. The value of the standard normal distribution such that 5 percent of the outcomes lie to the right is z0.05 = 1.645. • For a test of H0 : θ ≥ θ0 versus Ha : θ < θ0 , the rejection point is −z0.05 = −1.645. We reject the null hypothesis if z < −1.645. • Figure 7-1 illustrates a test H0 : µ = µ0 versus Ha : µ ̸= µ0 at the 0.05 significance level using a z-test. The ‘‘acceptance region’’ is the traditional name for the set of values of the test statistic for which we do not reject the null hypothesis. (The traditional name, however, is inaccurate. We should avoid using phrases such as ‘‘accept the null hypothesis’’ because such a statement implies a greater degree of conviction about the null than is warranted when we fail to reject it.11 ) On either side of the acceptance region is a rejection region (or critical region). If the null hypothesis that µ = µ0 is true, the test statistic has a 2.5 percent chance of falling in the left rejection region and a 2.5 percent chance of falling in the right rejection region. Any calculated value of the test statistic that falls in either of these two regions causes us to reject the null hypothesis at the 0.05 significance level. The rejection points 10 ‘‘Rejection point’’ is a descriptive synonym for the more traditional term ‘‘critical value.’’ The analogy in some courts of law (for example, in the United States) is that if a jury does not return a verdict of guilty (the alternative hypothesis), it is most accurate to say that the jury has failed to reject the null hypothesis, namely, that the defendant is innocent. 11 250 Quantitative Investment Analysis 0.95 0.025 –1.96 Rejection Region z < –1.96 0 Acceptance Region 0.025 1.96 Rejection Region z > 1.96 FIGURE 7-1 Rejection Points (Critical Values), 0.05 Significance Level, Two-Sided Test of the Population Mean Using a z-Test of 1.96 and −1.96 are seen to be the dividing lines between the acceptance and rejection regions. Figure 7-1 affords a good opportunity to highlight the relationship between confidence intervals and hypothesis tests. A 95 percent confidence interval for the population mean, µ, based on sample mean, X , is given by X − 1.96 s X to X + 1.96 s X , where s X is the standard error of the sample mean (Equation 7-3).12 Now consider one of the conditions for rejecting the null hypothesis: X − µ0 > 1.96 sX Here, µ0 is the hypothesized value of the population mean. The condition states that rejection is warranted if the test statistic exceeds 1.96. Multiplying both sides by s X , we have X − µ0 > 1.96 s X or, after rearranging, X − 1.96 s X > µ0 , which we can also write as µ0 < X − 1.96 s X . This expression says that if the hypothesized population mean, µ0 , is less than the lower limit of the 95 percent confidence interval based on the sample mean, we must reject the null hypothesis at the 5 percent significance level (the test statistic falls in the rejection region to the right). Now, we can take the other condition for rejecting the null hypothesis X − µ0 < −1.96 sX 12 Just as with the hypothesis test, we can use this confidence interval, based on the standard normal distribution, when we have large samples. An alternative hypothesis test and confidence interval uses the t-distribution, which requires concepts that we introduce in the next section. 251 Chapter 7 Hypothesis Testing and, using algebra as before, rewrite it as µ0 > X + 1.96 s X . If the hypothesized population mean is larger than the upper limit of the 95 percent confidence interval, we reject the null hypothesis at the 5 percent level (the test statistic falls in the rejection region to the left). Thus, an α significance level in a two-sided hypothesis test can be interpreted in exactly the same way as a (1 − α) confidence interval. In summary, when the hypothesized value of the population parameter under the null is outside the corresponding confidence interval, the null hypothesis is rejected. We could use confidence intervals to test hypotheses; practitioners, however, usually do not. Computing a test statistic (one number, versus two numbers for the usual confidence interval) is more efficient. Also, analysts encounter actual cases of one-sided confidence intervals only rarely. Furthermore, only when we compute a test statistic can we obtain a p-value, a useful quantity relating to the significance of our results (we will discuss p-values shortly). To return to our risk premium test, we stated hypotheses H0: µRP ≤ 0 versus Ha : µRP > 0. We identified the test statistic as X RP /s X and stated that it follows a standard normal distribution. We are, therefore, conducting a one-sided z-test. We specified a 0.05 significance level. For this one-sided z-test, the rejection point at the 0.05 level of significance is 1.645. We will reject the null if the calculated z-statistic is larger than 1.645. Figure 7-2 illustrates this test. The fifth step in hypothesis testing is collecting the data and calculating the test statistic. The quality of our conclusions depends not only on the appropriateness of the statistical model but also on the quality of the data we use in conducting the test. We first need to check for measurement errors in the recorded data. Some other issues to be aware of include sample selection bias and time-period bias. Sample selection bias refers to bias introduced by systematically excluding some members of the population according to a particular attribute. One type of sample selection bias is survivorship bias. For example, if we define our sample as U.S. bond mutual funds currently operating and we collect returns for just these funds, we will systematically exclude funds that have not survived to the present date. Nonsurviving funds 0.95 0 Acceptance Region a = 0.05 1.645 Rejection Region z > 1.645 FIGURE 7-2 Rejection Point (Critical Value), 0.05 Significance Level, One-Sided Test of the Population Mean Using a z-Test 252 Quantitative Investment Analysis are likely to have underperformed surviving funds, on average; as a result the performance reflected in the sample may be biased upward. Time-period bias refers to the possibility that when we use a time-series sample, our statistical conclusion may be sensitive to the starting and ending dates of the sample.13 To continue with the risk premium hypothesis, we focus on Canadian equities. According to Dimson, Marsh, and Staunton (2002) as updated to the end of 2002,14 for the period 1900 to 2002 inclusive (103 annual observations), the arithmetic mean equity risk premium for Canadian stocks relative to bond returns, X RP , was 5.1 percent per year. The sample standard deviation of the annual risk premiums was 18.6 percent. √ Using Equation 7-3, the standard √ error of the sample mean is s X = s/ n = 18.6%/ 103 = 1.833%. The test statistic is z = X RP /s X = 5.1%/1.833% = 2.78. The sixth step in hypothesis testing is making the statistical decision. For our example, because the test statistic z = 2.78 is larger than the rejection point of 1.645, we reject the null hypothesis in favor of the alternative hypothesis that the risk premium on Canadian stocks is positive. The first six steps are the statistical steps. The final decision concerns our use of the statistical decision. The seventh and final step in hypothesis testing is making the economic or investment decision. The economic or investment decision takes into consideration not only the statistical decision but also all pertinent economic issues. In the sixth step, we found strong statistical evidence that the Canadian risk premium is positive. The magnitude of the estimated risk premium, 5.1 percent a year, is economically very meaningful as well. Based on these considerations, an investor might decide to commit funds to Canadian equities. A range of nonstatistical considerations, such as the investor’s tolerance for risk and financial position, might also enter the decision-making process. The preceding discussion raises an issue that often arises in this decision-making step. We frequently find that slight differences between a variable and its hypothesized value are statistically significant but not economically meaningful. For example, we may be testing an investment strategy and reject a null hypothesis that the mean return to the strategy is zero based on a large sample. Equation 7-1 shows that the smaller the standard error of the sample statistic (the divisor in the formula), the larger the value of the test statistic and the greater the chance the null will be rejected, all else equal. The standard error decreases as the sample size, n, increases, so that for very large samples, we can reject the null for small departures from it. We may find that although a strategy provides a statistically significant positive mean return, the results are not economically significant when we account for transaction costs, taxes, and risk. Even if we conclude that a strategy’s results are economically meaningful, we should explore the logic of why the strategy might work in the future before actually implementing it. Such considerations cannot be incorporated into a hypothesis test. Before leaving the subject of the process of hypothesis testing, we should discuss an important alternative approach called the p-value approach to hypothesis testing. Analysts and researchers often report the p-value (also called the marginal significance level) associated with hypothesis tests. • 13 Definition of p-Value. The p-value is the smallest level of significance at which the null hypothesis can be rejected. These issues are discussed further in the chapter on sampling. by communication dated 19 May 2003 to the authors. 14 Updated Chapter 7 Hypothesis Testing 253 For the value of the test statistic of 2.78 in the risk premium hypothesis test, using a spreadsheet function for the standard normal distribution, we calculate a p-value of 0.002718. We can reject the null hypothesis at that level of significance. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value for a two-sided test that a parameter equals zero is frequently generated automatically by statistical and econometric software programs.15 We can use p-values in the hypothesis testing framework presented above as an alternative to using rejection points. If the p-value is less than our specified level of significance, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis. Using the p-value in this fashion, we reach the same conclusion as we do using rejection points. For example, because 0.002718 is less than 0.05, we would reject the null hypothesis in the risk premium test. The p-value, however, provides more precise information on the strength of the evidence than does the rejection points approach. The p-value of 0.002718 indicates that the null is rejected at a far smaller level of significance than 0.05. If one researcher examines a question using a 0.05 significance level and another researcher uses a 0.01 significance level, the reader may have trouble comparing the findings. This concern has given rise to an approach to presenting the results of hypothesis tests that features p-values and omits specification of the significance level (Step 3). The interpretation of the statistical results is left to the consumer of the research. This has sometimes been called the p-value approach to hypothesis testing.16 3. HYPOTHESIS TESTS CONCERNING THE MEAN Hypothesis tests concerning the mean are among the most common in practice. In this section we discuss such tests for several distinct types of problems. In one type (discussed in Section 3.1), we test whether the population mean of a single population is equal to (or greater or less than) some hypothesized value. Then, in Sections 3.2 and 3.3, we address inference on means based on two samples. Is an observed difference between two sample means due to chance or to different underlying (population) means? When we have two random samples that are independent of each other—no relationship exists between the measurements in one sample and the measurements in the other—the techniques of Section 3.2 apply. When the samples are dependent, the methods of Section 3.3 are appropriate.17 15 We can use spreadsheets to calculate p-values as well. In Microsoft Excel, for example, we may use the worksheet functions TTEST, NORMSDIST, CHIDIST, and FDIST to calculate p-values for t-tests, z-tests, chi-square tests, and F -tests, respectively. 16 Davidson and MacKinnon (1993) argued the merits of this approach: ‘‘The P value approach does not necessarily force us to make a decision about the null hypothesis. If we obtain a P value of, say, 0.000001, we will almost certainly want to reject the null. But if we obtain a P value of, say, 0.04, or even 0.004, we are not obliged to reject it. We may simply file the result away as information that casts some doubt on the null hypothesis, but that is not, by itself, conclusive. We believe that this somewhat agnostic attitude toward test statistics, in which they are merely regarded as pieces of information that we may or may not want to act upon, is usually the most sensible one to take.’’ (p. 80). 17 When we want to test whether the population means of more than two populations are equal, we use analysis of variance (ANOVA). We introduce ANOVA in its most common application, regression analysis, in the chapter on correlation and regression analysis. 254 Quantitative Investment Analysis 3.1. Tests Concerning a Single Mean An analyst who wants to test a hypothesis concerning the value of an underlying or population mean will conduct a t-test in the great majority of cases. A t-test is a hypothesis test using a statistic (t-statistic) that follows a t-distribution. The t-distribution is a probability distribution defined by a single parameter known as degrees of freedom (df). Each value of degrees of freedom defines one distribution in this family of distributions. The t-distribution is closely related to the standard normal distribution. Like the standard normal distribution, a t-distribution is symmetrical with a mean of zero. However, the t-distribution is more spread out: It has a standard deviation greater than 1 (compared to 1 for the standard normal)18 and more probability for outcomes distant from the mean (it has fatter tails than the standard normal distribution). As the number of degrees of freedom increases with sample size, the spread decreases and the t-distribution approaches the standard normal distribution as a limit. Why is the t-distribution the focus for the hypothesis tests of this section? In practice, investment analysts need to estimate the population standard deviation by calculating a sample standard deviation; that is, the population variance (or standard deviation) is unknown. For hypothesis tests concerning the population mean of a normally distributed population with unknown variance, the theoretically correct test statistic is the t-statistic. What if a normal distribution does not describe the population? The t-test is robust to moderate departures from normality, except for outliers and strong skewness.19 When we have large samples, departures of the underlying distribution from the normal are of increasingly less concern. The sample mean is approximately normally distributed in large samples according to the central limit theorem, whatever the distribution describing the population. In general, a sample size of 30 or more usually can be treated as a large sample and a sample size of 29 or less is treated as a small sample.20 • Test Statistic for Hypothesis Tests of the Population Mean (Practical Case—Population Variance Unknown). If the population sampled has unknown variance and either of the conditions below holds: 1. the sample is large, or 2. the sample is small but the population sampled is normally distributed, or approximately normally distributed, then the test statistic for hypothesis tests concerning a single population mean, µ, is tn−1 = X − µ0 √ s/ n (7-4) where tn−1 = t-statistic with n − 1 degrees of freedom (n is the sample size) X = the sample mean 18 The formula for the variance of a t-distribution is df/(df − 2). Moore and McCabe (1998). A statistic is robust if the required probability calculations are insensitive to violations of the assumptions. 20 Although this generalization is useful, we caution that the sample size needed to obtain an approximately normal sampling distribution for the sample mean depends on how nonnormal the original population is. For some populations, ‘‘large’’ may be a sample size well in excess of 30. 19 See Chapter 7 Hypothesis Testing 255 µ0 = the hypothesized value of the population mean s = the sample standard deviation The denominator of the t-statistic is an estimate of the sample mean standard error, √ s X = s/ n.21 In Example 7-1, because the sample size is small, the test is called a small sample test concerning the population mean. EXAMPLE 7-1 Risk and Return Characteristics of an Equity Mutual Fund (1) You are analyzing Sendar Equity Fund, a midcap growth fund that has been in existence for 24 months. During this period, it has achieved a mean monthly return of 1.50 percent with a sample standard deviation of monthly returns of 3.60 percent. Given its level of systematic (market) risk and according to a pricing model, this mutual fund was expected to have earned a 1.10 percent mean monthly return during that time period. Assuming returns are normally distributed, are the actual results consistent with an underlying or population mean monthly return of 1.10 percent? 1. Formulate null and alternative hypotheses consistent with the verbal description of the research goal. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Identify the rejection point or points for the hypothesis tested in Part 1 at the 0.10 level of significance. 4. Determine whether the null hypothesis is rejected or not rejected at the 0.10 level of significance. (Use the tables in the back of this book.) Solution to 1: We have a ‘‘not equal to’’ alternative hypothesis, where µ is the underlying mean return on Sendar Equity Fund—H0 : µ = 1.10 versus Ha : µ ̸= 1.10. Solution to 2: Because the population variance is not known, we use a t-test with 24 − 1 = 23 degrees of freedom. Solution to 3: Because this is a two-tailed test, we have the rejection point tα/2,n−1 = t0.05,23 . In the table for the t-distribution, we look across the row for 23 degrees of freedom to the 0.05 column, to find 1.714. The two rejection points for this two-sided 21 A technical note, for reference, is required. When the sample comes from a finite population, estimates of the standard error of the mean, whether from Equation 7-2 or Equation 7-3, overestimate the true standard error. To address this, the computed standard!error is multiplied by a shrinkage factor called the finite population correction factor (fpc), equal to (N − n)/(N − 1), where N is the population size and n is the sample size. When the sample size is small relative to the population size (less than 5 percent of the population size), the fpc is usually ignored. The overestimation problem arises only in the usual situation of sampling without replacement (after an item is selected, it cannot be picked again), as opposed to sampling with replacement. 256 Quantitative Investment Analysis test are 1.714 and −1.714. We will reject the null if we find that t > 1.714 or t < −1.714. Solution to 4: t23 = 1.50 − 1.10 0.40 = 0.544331 or 0.544 √ = 0.734847 3.60/ 24 Because 0.544 does not satisfy either t > 1.714 or t < −1.714, we do not reject the null hypothesis. The confidence interval approach provides another perspective on this hypothesis test. The theoretically correct 100(1 − α)% confidence interval for the population mean of a normal distribution with unknown variance, based on a sample of size n, is X − tα/2 s X to X + tα/2 s X where tα/2 is the value of t such that α/2 of the probability remains in the right tail and where −tα/2 is the value of t such that α/2 of the probability remains in the left tail, for n − 1 degrees of freedom. Here, the 90 percent confidence interval runs from 1.5 − (1.714)(0.734847) = 0.240 to 1.5 + (1.714)(0.734847) = 2.760, compactly [0.240, 2.760]. The hypothesized value of mean return, 1.10, falls within this confidence interval, and we see from this perspective also that the null hypothesis is not rejected. At a 10 percent level of significance, we conclude that a population mean monthly return of 1.10 percent is consistent with the 24-month observed data series. Note that 10 percent is a relatively high probability of rejecting the hypothesis of a 1.10 percent population mean monthly return when it is true. EXAMPLE 7-2 A Slowdown in Payments of Receivables FashionDesigns, a supplier of casual clothing to retail chains, is concerned about a possible slowdown in payments from its customers. The controller’s office measures the rate of payment by the average number of days in receivables.22 FashionDesigns has generally maintained an average of 45 days in receivables. Because it would be too costly to analyze all of the company’s receivables frequently, the controller’s office uses sampling to track customers’ payment rates. A random sample of 50 accounts shows a mean number of days in receivables of 49 with a standard deviation of 8 days. 1. Formulate null and alternative hypotheses consistent with determining whether the evidence supports the suspected condition that customer payments have slowed. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 22 This measure represents the average length of time that the business must wait after making a sale before receiving payment. The calculation is (Accounts receivable)/(Average sales per day). Chapter 7 Hypothesis Testing 257 3. Identify the rejection point or points for the hypothesis tested in Part 1 at the 0.05 and 0.01 levels of significance. 4. Determine whether the null hypothesis is rejected or not rejected at the 0.05 and 0.01 levels of significance. Solution to 1: The suspected condition is that the number of days in receivables has increased relative to the historical rate of 45 days, which suggests a ‘‘greater than’’ alternative hypothesis. With µ as the population mean number of days in receivables, the hypotheses are H0 : µ ≤ 45 versus Ha : µ > 45. Solution to 2: Because the population variance is not known, we use a t-test with 50 − 1 = 49 degrees of freedom. Solution to 3: The rejection point is found across the row for degrees of freedom of 49. To find the one-tailed rejection point for a 0.05 significance level, we use the 0.05 column: The value is 1.677. To find the one-tailed rejection point for a 0.01 level of significance, we use the 0.01 column: The value is 2.405. To summarize, at a 0.05 significance level, we reject the null if we find that t > 1.677; at a 0.01 significance level, we reject the null if we find that t > 2.405. 4 49 − 45 = 3.536 = √ 1.131371 8/ 50 Because 3.536 > 1.677, the null hypothesis is rejected at the 0.05 level. Because 3.536 > 2.405, the null hypothesis is also rejected at the 0.01 level. We can say with a high level of confidence that FashionDesigns has experienced a slowdown in customer payments. The level of significance, 0.01, is a relatively low probability of rejecting the hypothesized mean of 45 days or less. Rejection gives us confidence that the mean has increased above 45 days. Solution to 4: t49 = We stated above that when population variance is not known, we use a t-test for tests concerning a single population mean. Given at least approximate normality, the t-test is always called for when we deal with small samples and do not know the population variance. For large samples, the central limit theorem states that the sample mean is approximately normally distributed, whatever the distribution of the population. So the t-test is still appropriate, but an alternative test may be more useful when sample size is large. For large samples, practitioners sometimes use a z-test in place of a t-test for tests concerning a mean.23 The justification for using the z-test in this context is twofold. First, in large samples, the sample mean should follow the normal distribution at least approximately, as we have already stated, fulfilling the normality assumption of the z-test. Second, the difference between the rejection points for the t-test and z-test becomes quite small when sample size is large. For a two-sided test at the 0.05 level of significance, the rejection points for a z-test are 1.96 and −1.96. For a t-test, the rejection points are 2.045 and −2.045 for df = 29 (about a 4 percent difference between the z and t rejection points) and 2.009 and −2.009 23 These practitioners choose between t-tests and z-tests based on sample size. For small samples (n they use a t-test, and for large samples, a z-test. < 30), 258 Quantitative Investment Analysis for df = 50 (about a 2.5 percent difference between the z and t rejection points). Because the t-test is readily available as statistical program output and theoretically correct for unknown population variance, we present it as the test of choice. In a very limited number of cases, we may know the population variance; in such cases, the z-test is theoretically correct.24 • The z-Test Alternative. 1. If the population sampled is normally distributed with known variance σ2 , then the test statistic for a hypothesis test concerning a single population mean, µ, is z= X − µ0 √ σ/ n (7-5) 2. If the population sampled has unknown variance and the sample is large, in place of a t-test, an alternative test statistic (relying on the central limit theorem) is z= X − µ0 √ s/ n (7-6) In the above equations, σ = the known population standard deviation s = the sample standard deviation µ0 = the hypothesized value of the population mean When we use a z-test, we most frequently refer to a rejection point in the list below. • Rejection Points for a z-Test. A. Significance level of α = 0.10. 1. H0 : θ = θ0 versus Ha : θ ̸= θ0 . The rejection points are z0.05 = 1.645 and −z0.05 = −1.645. Reject the null hypothesis if z > 1.645 or if z < −1.645. 2. H0 : θ ≤ θ0 versus Ha : θ > θ0 . The rejection point is z0.10 = 1.28. Reject the null hypothesis if z > 1.28. 3. H0 : θ ≥ θ0 versus Ha : θ < θ0 . The rejection point is −z0.10 = −1.28. Reject the null hypothesis if z < −1.28. B. Significance level of α = 0.05. 1. H0 : θ = θ0 versus Ha : θ ̸= θ0 . The rejection points are z0.025 = 1.96 and −z0.025 = −1.96. Reject the null hypothesis if z > 1.96 or if z < −1.96. 24 For example, in Monte Carlo simulation, we prespecify the probability distributions for the risk factors. If we use a normal distribution, we know the true values of mean and variance. Monte Carlo simulation involves the use of a computer to represent the operation of a system subject to risk; we discuss Monte Carlo simulation in the chapter on common probability distributions. Chapter 7 Hypothesis Testing 259 2. H0 : θ ≤ θ0 versus Ha : θ > θ0 . The rejection point is z0.05 = 1.645. Reject the null hypothesis if z > 1.645. 3. H0 : θ ≥ θ0 versus Ha : θ < θ0 . The rejection point is −z0.05 = −1.645. Reject the null hypothesis if z < −1.645. C. Significance level of α = 0.01. 1. H0 : θ = θ0 versus Ha : θ ̸= θ0 . The rejection points are z0.005 = 2.575 and −z0.005 = −2.575. Reject the null hypothesis if z > 2.575 or if z < −2.575. 2. H0 : θ ≤ θ0 versus Ha : θ > θ0 . The rejection point is z0.01 = 2.33. Reject the null hypothesis if z > 2.33. 3. H0 : θ ≥ θ0 versus Ha : θ < θ0 . The rejection point is −z0.01 = −2.33. Reject the null hypothesis if z < −2.33. EXAMPLE 7-3 The Effect of Commercial Paper Issuance on Stock Prices Commercial paper (CP) is unsecured short-term corporate debt that, like U.S. Treasury bills, is characterized by a single payment at maturity. When a company enters the CP market for the first time, how do stock market participants react to the announcement of the CP ratings? Nayar and Rozeff (1994) addressed this question using data for the period October 1981 to December 1985. During this period, 132 CP issues (96 industrial and 36 non-industrial) received an initial rating in Standard & Poor’s CreditWeek or Moody’s Investors Service Bond Survey. Nayar and Rozeff categorized ratings as superior or inferior. Superior CP ratings were A1+ or A1 from Standard & Poor’s and Prime-1 (P1) from Moody’s. Inferior CP ratings were A2 or lower from Standard & Poor’s and Prime-2 (P2) or lower from Moody’s. The publication day of the initial ratings was designated t = 0. The researchers found, however, that companies themselves often disseminate the rating information prior to publication in CreditWeek or the Bond Survey. The reaction of stock price was studied on the day before publication, t − 1, because that date was closer to the actual date of information release. If CP ratings provide new information useful for equity valuation, the information should cause a change in stock prices and returns once it is available. Only one component of stock returns is of interest: the return in excess of that predicted given a stock’s market risk or beta, called the abnormal return. Positive (negative) abnormal returns indicate that investors perceive favorable (unfavorable) corporate news in the ratings announcement. Although Nayar and Rozeff examined abnormal returns for various time horizons or event windows, we report a selection of their findings for the day prior to rating publication (t − 1): All CP Issues (n = 132 issues). The null hypothesis was that the average abnormal stock return on day t − 1 was 0. The null would be true if stock investors did not find either positive or negative information in the announcement. Mean abnormal return = 0.39 percent 260 Quantitative Investment Analysis Sample standard error of the mean of abnormal returns = 0.1336 percent25 Industrial CP Issues with Superior Ratings (n = 72 issues). The null hypothesis was that the average abnormal stock return on day t − 1 was 0. The null would be true if stock investors did not find either positive or negative information in the announcement. Mean abnormal return = 0.79 percent Sample standard error of the mean of abnormal returns = 0.197 percent Industrial CP Issues with Inferior Ratings (n = 24 issues). The null hypothesis was that the average abnormal stock return on day t − 1 was 0. The null would be true if stock investors did not find either positive or negative information in the announcement. Mean abnormal return = −0.57 percent Sample standard error of the mean of abnormal returns = 0.38 percent The researchers chose to use z-tests. 1. With respect to each of the three cases, suppose that the null hypothesis reflects the belief that investors do not, on average, perceive either positive or negative information in initial ratings. State one set of hypotheses (a null hypothesis and an alternative hypothesis) that covers all three cases. 2. Determine whether the null hypothesis formulated in Part 1 is rejected or not rejected at the 0.05 and 0.01 levels of significance for the All CP Issues case. Interpret the results. 3. Determine whether the null hypothesis formulated in Part 1 is rejected or not rejected at the 0.05 and 0.01 levels of significance for the Industrial CP Issues with Superior Ratings case. Interpret the results. 4. Determine whether the null hypothesis formulated in Part 1 is rejected or not rejected at the 0.05 and 0.01 levels of significance for the Industrial CP Issues with Inferior Ratings case. Interpret the results. Solution to 1: A set of hypotheses consistent with no information in CP credit ratings relevant to stock investors is H0 : The population mean abnormal return on day t − 1 equals 0 Ha : The population mean abnormal return on day t − 1 does not equal 0 Solution to 2: From the information on rejection points for z-tests, we know that we reject the null hypothesis at the 0.05 significance level if z > 1.96 or if z < −1.96, and at the 0.01 significance level if z > 2.575 or if z < −2.575. Using the z-test, z = (0.39% − 0%)/0.1336% = 2.92 is significant at the 0.05 and 0.01 levels. The null is rejected. The fact of CP issuance itself appears to be viewed as favorable news. Because it is possible that significant results could be due to outliers, the researchers also reported the number of cases of positive and negative abnormal returns. The ratio of cases of positive to negative abnormal returns was 80:52, which tends to support the conclusion of positive abnormal returns from the z-test. 25 This standard error was calculated as a sample standard deviation over the 132 issues (a cross-sectional standard deviation) divided by the square root of 132. Other standard errors were calculated similarly. 261 Chapter 7 Hypothesis Testing Solution to 3: Using the z-test, z = (0.79% − 0%)/0.197% = 4.01 is significant at the 0.05 and 0.01 levels. Stocks earned clearly positive abnormal returns in response to the news of a superior initial CP rating. Investors may view rating agencies as certifying through a superior rating that a company’s future prospects are strong. The ratio of cases of positive to negative abnormal returns was 48:24, which tends to support the conclusion of positive abnormal returns from the z-test. Solution to 4: Using the z-test, z = (−0.57% − 0%)/0.38% = −1.50 is not significant at either the 0.01 or 0.05 levels. In the case of inferior ratings, we cannot conclude that investors found either positive or negative information in the announcements of initial CP ratings. The ratio of cases of positive to negative abnormal returns was 11:13 and tends to support the conclusion of the z-test, which did not reject the null hypothesis. Nearly all practical situations involve an unknown population variance. Table 7-2 summarizes our discussion for tests concerning the population mean when the population variance is unknown. TABLE 7-2 Test Concerning the Population Mean (Population Variance Unknown) Population normal Population nonnormal Large Sample (n ≥ 30) Small Sample (n < 30) t-Test (z-Test alternative) t-Test (z-Test alternative) t-Test Not Available 3.2. Tests Concerning Differences between Means We often want to know whether a mean value—for example, a mean return—differs between two groups. Is an observed difference due to chance or to different underlying values for the mean? We have two samples, one for each group. When it is reasonable to believe that the samples are from populations at least approximately normally distributed and that the samples are also independent of each other, the techniques of this section apply. We discuss two t-tests for a test concerning differences between the means of two populations. In one case, the population variances, although unknown, can be assumed to be equal. Then, we efficiently combine the observations from both samples to obtain a pooled estimate of the common but unknown population variance. A pooled estimate is an estimate drawn from the combination of two different samples. In the second case, we do not assume that the unknown population variances are equal, and an approximate t-test is then available. Letting µ1 and µ2 stand, respectively, for the population means of the first and second populations, we most often want to test whether the population means are equal or whether one is larger than the other. Thus we usually formulate the following hypotheses: 1. H0 : µ1 − µ2 = 0 versus Ha : µ1 − µ2 ̸= 0 (the alternative is that µ1 ̸= µ2 ) 2. H0 : µ1 − µ2 ≤ 0 versus Ha : µ1 − µ2 > 0 (the alternative is that µ1 > µ2 ) 3. H0 : µ1 − µ2 ≥ 0 versus Ha : µ1 − µ2 < 0 (the alternative is that µ1 < µ2 ) We can, however, formulate other hypotheses, such as H0 : µ1 − µ2 = 2 versus Ha : µ1 − µ2 ̸= 2. The procedure is the same. 262 Quantitative Investment Analysis The definition of the t-test follows. • Test Statistic for a Test of the Difference between Two Population Means (Normally Distributed Populations, Population Variances Unknown but Assumed Equal). When we can assume that the two populations are normally distributed and that the unknown population variances are equal, a t-test based on independent random samples is given by t= (X 1 − X 2 ) − (µ1 − µ2 ) " 2 #1/2 sp sp2 + n1 n2 (7-7) (n1 − 1)s12 + (n2 − 1)s22 is a pooled estimator of the common variance. n1 + n2 − 2 The number of degrees of freedom is n1 + n2 − 2. where sp2 = EXAMPLE 7-4 Mean Returns on the S&P 500: A Test of Equality across Decades The realized mean monthly return on the S&P 500 Index in the 1980s appears to have been substantially different from the mean return in the 1970s. Was the difference statistically significant? The data, shown in Table 7-3, indicate that assuming equal population variances for returns in the two decades is not unreasonable. TABLE 7-3 S&P 500 Monthly Return and Standard Deviation for Two Decades Decade Number of Months (n) Mean Monthly Return (%) Standard Deviation 1970s 1980s 120 120 0.580 1.470 4.598 4.738 1. Formulate null and alternative hypotheses consistent with a two-sided hypothesis test. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Identify the rejection point or points for the hypothesis tested in Part 1 at the 0.10, 0.05, and 0.01 levels of significance. 4. Determine whether the null hypothesis is rejected or not rejected at the 0.10, 0.05, and 0.01 levels of significance. Solution to 1: Letting µ1 stand for the population mean return for the 1970s and µ2 stand for the population mean return for the 1980s, we formulate the following hypotheses: H0 : µ1 − µ2 = 0 versus Ha : µ1 − µ2 ̸= 0 Solution to 2: Because the two samples are drawn from different decades, they are independent samples. The population variances are not known but can be assumed 263 Chapter 7 Hypothesis Testing to be equal. Given all these considerations, the t-test given in Equation 7-7 has 120 + 120 − 2 = 238 degrees of freedom. Solution to 3: In the tables (Appendix B), the closest degree of freedom to 238 is 200. For a two-sided test, the rejection points are ±1.653, ± 1.972, and ±2.601 for, respectively, the 0.10, 0.05, and 0.01 levels for df = 200. To summarize, at the 0.10 level, we will reject the null if t < −1.653 or t > 1.653; at the 0.05 level, we will reject the null if t < −1.972 or t > 1.972; and at the 0.01 level, we will reject the null if t < −2.601 or t > 2.601. Solution to 4: In calculating the test statistic, the first step is to calculate the pooled estimate of variance: (120 − 1)(4.598)2 + (120 − 1)(4.738)2 (n1 − 1)s12 + (n2 − 1)s22 = n1 + n2 − 2 120 + 120 − 2 5,187.239512 = 21.795124 = 238 sp2 = t= = (X 1 − X 2 ) − (µ1 − µ2 ) (0.580 − 1.470) − 0 =$ % " 2 #1/2 2 21.795124 1/2 21.795124 sp sp + + 120 120 n1 n2 −0.89 = −1.477 0.602704 The t value of −1.477 is not significant at the 0.10 level, so it is also not significant at the 0.05 and 0.01 levels. Therefore, we do not reject the null hypothesis at any of the three levels. In many cases of practical interest, we cannot assume that population variances are equal. The following test statistic is often used in the investment literature in such cases: • Test Statistic for a Test of the Difference between Two Population Means (Normally Distributed Populations, Unequal and Unknown Population Variances). When we can assume that the two populations are normally distributed but do not know the population variances and cannot assume that they are equal, an approximate t-test based on independent random samples is given by t= (X 1 − X 2 ) − (µ1 − µ2 ) $ 2 %1/2 s2 s1 + 2 n1 n2 (7-8) where we use tables of the t-distribution using ‘‘modified’’ degrees of freedom computed with the formula $ 2 %2 s22 s1 + n1 n2 df = 2 (7-9) 2 (s22 /n2 )2 (s1 /n1 ) + n1 n2 264 Quantitative Investment Analysis A practical tip is to compute the t-statistic before computing the degrees of freedom. Whether or not the t-statistic is significant will sometimes be obvious. EXAMPLE 7-5 Recovery Rates on Defaulted Bonds: A Hypothesis Test How are the required yields on risky corporate bonds determined? Two key factors are the expected probability of default and the expected amount that will be recovered in the event of default, or the recovery rate. Altman and Kishore (1996) documented for the first time the average recovery rates on defaulted bonds stratified by industry and seniority. For their study period, 1971 to 1995, Altman and Kishore discovered that defaulted bonds of public utilities and chemicals, petroleum, and plastics manufacturers experienced much higher recovery rates than did other industrial sectors. Could the differences be explained by a greater preponderance of senior debt in the higher-recovery sectors? They studied this by examining recovery rates stratified by seniority. We discuss their results for senior secured bonds. With µ1 denoting the population mean recovery rate for the senior secured bonds of utilities and µ2 denoting the population mean recovery rate for the senior secured bonds of other sectors (non-utilities), the hypotheses are H0 : µ1 − µ2 = 0 versus Ha : µ1 − µ2 ̸= 0. Table 7-4 excerpts from their findings. TABLE 7-4 Recovery Rates by Seniority Industry Group Industry Group/ Number of Average Standard Seniority Observations Price∗ Deviation Public Utilities Senior Secured 21 $64.42 $14.03 Ex-Utilities Sample Number of Average Standard Observations Price∗ Deviation 64 $55.75 $25.17 Source: Altman and Kishore (1996), Table 5. ∗ This is the average price at default and is a measure of recovery rate. Following the researchers, assume that the populations (recovery rates of utilities, recovery rates of non-utilities) are normally distributed and that the samples are independent. Based on the data in the table, address the following: 1. Discuss why Altman and Kishore would choose a test based on Equation 7-8 rather than Equation 7-7. 2. Calculate the test statistic to test the null hypothesis given above. 3. What is the value of the test’s modified degrees of freedom? 4. Determine whether to reject the null hypothesis at the 0.10 level. Solution to 1: The sample standard deviation for the recovery rate on the senior secured bonds of utilities ($14.03) appears much smaller than the sample standard deviation of 265 Chapter 7 Hypothesis Testing the comparable bonds for non-utilities ($25.17). Properly choosing not to assume equal variances, Altman and Kishore employed the approximate t-test given in Equation 7-8. Solution to 2: The test statistic is t=$ (X 1 − X 2 ) %1/2 s2 s12 + 2 n1 n2 where X 1 = sample mean recovery rate for utilities = 64.42 X 2 = sample mean recovery rate for non-utility sectors = 55.75 s12 = sample variance for utilities = 14.032 = 196.8409 s22 = sample variance for non-utilities = 25.172 = 633.5289 n1 = sample size of the utility sample = 21 n2 = sample size of the non-utility sample = 64 Thus, t = (64.42 − 55.75)/[(196.8409/21) + (633.5289/64)]1/2 = 8.67/(9.373376 + 9.898889)1/2 = 8.67/4.390019 = 1.975. The calculated t-statistic is thus 1.975. Solution to 3: % %2 $ s2 s12 196.8409 633.5289 2 + + 2 n1 n2 21 64 df = 2 = (633.5289/64)2 (196.8409/21)2 (s22 /n2 )2 (s1 /n1 )2 + + 21 64 n1 n2 $ = 371.420208 = 64.99 or 65 degrees of freedom 5.714881 Solution to 4: The closest entry to df = 65 in the tables for the t-distribution is df = 60. For α = 0.10, we find tα/2 = 1.671. Thus, we reject the null if t < −1.671 or t > 1.671. Based on the computed value of 1.975, we reject the null hypothesis at the 0.10 level. Some evidence exists that recovery rates differ between utilities and other industries. Why? Altman and Kishore suggest that the differing nature of the companies’ assets and industry competitive structures may explain the different recovery rates. 3.3. Tests Concerning Mean Differences In the previous section, we presented two t-tests for discerning differences between population means. The tests were based on two samples. An assumption for those tests’ validity was that the samples were independent—that is, unrelated to each other. When we want to conduct tests on two means based on samples that we believe are dependent, the methods of this section apply. The t-test in this section is based on data arranged in paired observations, and the test itself is sometimes called a paired comparisons test. Paired observations are observations that are dependent because they have something in common. A paired comparisons test is a statistical 266 Quantitative Investment Analysis test for differences in dependent items. For example, we may be concerned with the dividend policy of companies before and after a change in the tax law affecting the taxation of dividends. We then have pairs of ‘‘before’’ and ‘‘after’’ observations for the same companies. We may test a hypothesis about the mean of the differences (mean differences) that we observe across companies. In other cases, the paired observations are not on the same units. For example, we may be testing whether the mean returns earned by two investment strategies were equal over a study period. The observations here are dependent in the sense that there is one observation for each strategy in each month, and both observations depend on underlying market risk factors. Because the returns to both strategies are likely to be related to some common risk factors, such as the market return, the samples are dependent. By calculating a standard error based on differences, the t-test presented below takes account of correlation between the observations. Letting A represent ‘‘after’’ and B ‘‘before,’’ suppose we have observations for the random variables XA and XB and that the samples are dependent. We arrange the observations in pairs. Let di denote the difference between two paired observations. We can use the notation di = xAi − xBi , where xAi and xBi are the ith pair of observations, i = 1, 2, . . . , n on the two variables. Let µd stand for the population mean difference. We can formulate the following hypotheses, where µd0 is a hypothesized value for the population mean difference: 1. H0 : µd = µd0 versus Ha : µd ̸= µd0 2. H0 : µd ≤ µd0 versus Ha : µd > µd0 3. H0 : µd ≥ µd0 versus Ha : µd < µd0 In practice, the most commonly used value for µd0 is 0. As usual, we are concerned with the case of normally distributed populations with unknown population variances, and we will formulate a t-test. To calculate the t-statistic, we first need to find the sample mean difference: n 1& d= di n i=1 (7-10) where n is the number of pairs of observations. The sample variance, denoted by sd2 , is sd2 = n & i=1 (di − d)2 n−1 (7-11) Taking the square root of this quantity, we have the sample standard deviation, sd , which then allows us to calculate the standard error of the mean difference as follows:26 sd sd = √ n • (7-12) Test Statistic for a Test of Mean Differences (Normally Distributed Populations, Unknown Population Variances). When we have data consisting of paired observations 26 We can also use the ' following equivalent expression, which makes use of the correlation between the two variables: s d = sA2 + sB2 − 2r(XA , XB )sA sB , where sA2 is the sample variance of XA , sB2 is the sample variance of XB , and r(XA , XB ) is the sample correlation between XA and XB . 267 Chapter 7 Hypothesis Testing from samples generated by normally distributed populations with unknown variances, a t-test is based on d − µd0 t= (7-13) sd with n − 1 degrees of freedom, where n is the number of paired observations, d is the sample mean difference (as given by Equation 7-10), and s d is the standard error of d (as given by Equation 7-12). Table 7-5 reports the quarterly returns from 1997 to 2002 for two managed portfolios specializing in precious metals. The two portfolios were closely similar in risk (as measured by standard deviation of return and other measures) and had nearly identical expense ratios. A major investment services company rated Portfolio B more highly than Portfolio A in early 2003. In investigating the portfolios’ relative performance, suppose we want to test the hypothesis that the mean quarterly return on Portfolio A equaled the mean quarterly return on Portfolio B from 1997 to 2002. Because the two portfolios shared essentially the same set of risk factors, their returns were not independent, so a paired comparisons test is appropriate. Let µd stand for the population mean value of difference between the returns TABLE 7-5 Quarterly Returns on Two Managed Portfolios: 1997–2002 Quarter 4Q:2002 3Q:2002 2Q:2002 1Q:2002 4Q:2001 3Q:2001 2Q:2001 1Q:2001 Portfolio A (%) 11.40 −2.17 10.72 38.91 4.36 5.13 26.36 −5.53 Portfolio B (%) 14.64 0.44 19.51 50.40 1.01 10.18 17.77 4.76 4Q:2000 5.27 −5.36 3Q:2000 −7.82 −1.54 2Q:2000 2.34 0.19 1Q:2000 −14.38 −12.07 4Q:1999 −9.80 −9.98 3Q:1999 19.03 26.18 2Q:1999 4.11 −2.39 1Q:1999 −4.12 −2.51 4Q:1998 −0.53 −11.32 3Q:1998 5.06 0.46 2Q:1998 −14.01 −11.56 1Q:1998 12.50 3.52 4Q:1997 −29.05 −22.45 3Q:1997 3.60 0.10 2Q:1997 −7.97 −8.96 1Q:1997 −8.62 −0.66 Mean 1.87 2.52 Sample standard deviation of differences = 6.71 Difference (Portfolio A − Portfolio B) −3.24 −2.61 −8.79 −11.49 3.35 −5.05 8.59 −10.29 10.63 −6.28 2.15 −2.31 0.18 −7.15 6.50 −1.61 10.79 4.60 −2.45 8.98 −6.60 3.50 0.99 −7.96 −0.65 268 Quantitative Investment Analysis on the two portfolios during this period. We test H0 : µd = 0 versus Ha : µd ̸= 0 at a 0.05 significance level. The sample mean difference, d, between Portfolio A and Portfolio√ B is −0.65 percent per quarter. The standard error of the sample mean difference is s d = 6.71/ 24 = 1.369673. The calculated test statistic is t = (−0.65 − 0)/1.369673 = −0.475 with n − 1 = 24 − 1 = 23 degrees of freedom. At the 0.05 significance level, we reject the null if t > 2.069 or if t < −2.069. Because −0.475 is not less than −2.069, we fail to reject the null. At the 0.10 significance level, we reject the null if t > 1.714 or if t < −1.714. Thus the difference in mean quarterly returns is not significant at any conventional significance level. The following example illustrates the application of this test to evaluate two competing investment strategies. EXAMPLE 7-6 The Dow-10 Investment Strategy McQueen, Shields, and Thorley (1997) examined the popular investment strategy of investing in the 10 stocks with the highest yields (rebalancing annually) in the Dow Jones Industrial Average, compared with a buy-and-hold strategy in all 30 stocks of the DJIA. Their study period was the 50 years from 1946 to 1995. From Table 7-6 we have d = 3.06% and sd = 6.62%. TABLE 7-6 Annual Return Summary for Dow-10 and Dow-30 Portfolios: 1946 to 1995 (n = 50) Strategy Dow-10 Dow-30 Difference Mean Return Standard Deviation 16.77% 13.71 3.06 19.10% 16.64 6.62∗ Source: McQueen, Shields, and Thorley (1997), Table 1. ∗ Sample standard deviation of differences. 1. Formulate null and alternative hypotheses consistent with a two-sided test that the mean difference between the Dow-10 and Dow-30 strategies equals 0. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Identify the rejection point or points for the hypothesis tested in Part 1 at the 0.01 level of significance. 4. Determine whether the null hypothesis is rejected or not rejected at the 0.01 level of significance. (Use the tables in the back of this book.) 5. Discuss the choice of a paired comparisons test. Solution to 1: With µd as the underlying mean difference between the Dow-10 and Dow-30 strategies, we have H0 : µd = 0 versus Ha : µd ̸= 0. Solution to 2: Because the population variance is unknown, the test statistic is a t-test with 50 − 1 = 49 degrees of freedom. Chapter 7 Hypothesis Testing 269 Solution to 3: In the table for the t-distribution, we look across the row for 49 degrees of freedom to the 0.005 column, to find 2.68. We will reject the null if we find that t > 2.68 or t < −2.68. Solution to 4: t49 = 3.06 3.06 = 3.2685 or 3.27 √ = 0.936209 6.62/ 50 Because 3.27 > 2.68, we reject the null hypothesis. The authors concluded that the difference in mean returns was clearly statistically significant. However, after adjusting for the Dow-10’s higher risk, extra transaction costs, and unfavorable tax treatment, they found that the Dow-10 portfolio did not beat the Dow-30 economically. Solution to 5: The Dow-30 includes the Dow-10. As a result, they are not independent samples; in general, the correlation of returns on the Dow-10 and Dow-30 should be positive. Because the samples are dependent, a paired comparisons test was appropriate. 4. HYPOTHESIS TESTS CONCERNING VARIANCE Because variance and standard deviation are widely used quantitative measures of risk in investments, analysts should be familiar with hypothesis tests concerning variance. The tests discussed in this section make regular appearances in investment literature. We examine two types: tests concerning the value of a single population variance and tests concerning the differences between two population variances. 4.1. Tests Concerning a Single Variance In this section, we discuss testing hypotheses about the value of the variance, σ2 , of a single population. We use σ20 to denote the hypothesized value of σ2 . We can formulate hypotheses as follows: 1. H0 : σ2 = σ20 versus Ha : σ2 ̸= σ20 (a ‘‘not equal to’’ alternative hypothesis) 2. H0 : σ2 ≤ σ20 versus Ha : σ2 > σ20 (a ‘‘greater than’’ alternative hypothesis) 3. H0 : σ2 ≥ σ20 versus Ha : σ2 < σ20 (a ‘‘less than’’ alternative hypothesis) In tests concerning the variance of a single normally distributed population, we make use of a chi-square test statistic, denoted χ2 . The chi-square distribution, unlike the normal and t-distributions, is asymmetrical. Like the t-distribution, the chi-square distribution is a family of distributions. A different distribution exists for each possible value of degrees of freedom, n − 1 (n is sample size). Unlike the t-distribution, the chi-square distribution is bounded below by 0; χ2 does not take on negative values. 270 • Quantitative Investment Analysis Test Statistic for Tests Concerning the Value of a Population Variance (Normal Population). If we have n independent observations from a normally distributed population, the appropriate test statistic is (n − 1)s2 χ2 = (7-14) σ20 with n − 1 degrees of freedom. In the numerator of the expression is the sample variance, calculated as n & (Xi − X )2 s2 = i=1 n−1 (7-15) In contrast to the t-test, for example, the chi-square test is sensitive to violations of its assumptions. If the sample is not actually random or if it does not come from a normally distributed population, inferences based on a chi-square test are likely to be faulty. If we choose a level of significance, α, the rejection points for the three kinds of hypotheses are as follows: • Rejection Points for Hypothesis Tests on the Population Variance. 1. ‘‘Not equal to’’ Ha : Reject the null hypothesis if the test statistic is greater than the upper α/2 point (denoted χ2α/2 ) or less than the lower α/2 point (denoted χ21−α/2 ) of the chi-square distribution with df = n − 1.27 2. ‘‘Greater than’’ Ha : Reject the null hypothesis if the test statistic is greater than the upper α point of the chi-square distribution with df = n − 1. 3. ‘‘Less than’’ Ha : Reject the null hypothesis if the test statistic is less than the lower α point of the chi-square distribution with df = n − 1. EXAMPLE 7-7 Risk and Return Characteristics of an Equity Mutual Fund (2) You continue with your analysis of Sendar Equity Fund, a midcap growth fund that has been in existence for only 24 months. Recall that during this period, Sendar Equity achieved a sample standard deviation of monthly returns of 3.60 percent. You now want to test a claim that the particular investment disciplines followed by Sendar result in a standard deviation of monthly returns of less than 4 percent. 1. Formulate null and alternative hypotheses consistent with the verbal description of the research goal. 27 Just as with other hypothesis tests, the chi-square test can be given a confidence interval interpretation. Unlike confidence intervals based on z- or t-statistics, however, chi-square confidence intervals for variance are asymmetric. A two-sided confidence interval for population variance, based on a sample of size n, has a lower limit L = (n − 1)s2 /χ2α/2 and an upper limit U = (n − 1)s2 /χ21−α/2 . Under the null hypothesis, the hypothesized value of the population variance should fall within these two limits. Chapter 7 Hypothesis Testing 271 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Identify the rejection point or points for the hypothesis tested in Part 1 at the 0.05 level of significance. 4. Determine whether the null hypothesis is rejected or not rejected at the 0.05 level of significance. (Use the tables in the back of this book.) Solution to 1: We have a ‘‘less than’’ alternative hypothesis, where σ is the underlying standard deviation of return on Sendar Equity Fund. Being careful to square standard deviation to obtain a test in terms of variance, the hypotheses are H0 : σ2 ≥ 16.0 versus Ha : σ2 < 16.0. Solution to 2: The test statistic is chi-square with 24 − 1 = 23 degrees of freedom. Solution to 3: The lower 0.05 rejection point is found on the line for df = 23, under the 0.95 column (95 percent probability in the right tail, to give 0.95 probability of getting a test statistic this large or larger). The rejection point is 13.091. We will reject the null if we find that chi-square is less than 13.091. Solution to 4: χ2 = (n − 1)s2 23 × 3.602 298.08 = 18.63 = = 2 2 4 16 σ0 Because 18.63 (the calculated value of the test statistic) is not less than 13.091, we do not reject the null hypothesis. We cannot conclude that Sendar’s investment disciplines result in a standard deviation of monthly returns of less than 4 percent. 4.2. Tests Concerning the Equality (Inequality) of Two Variances Suppose we have a hypothesis about the relative values of the variances of two normally distributed populations with means µ1 and µ2 and variances σ21 and σ22 . We can formulate all hypotheses as one of the choices below: 1. H0 : σ21 = σ22 versus Ha : σ21 ̸= σ22 2. H0 : σ21 ≤ σ22 versus Ha : σ21 > σ22 3. H0 : σ21 ≥ σ22 versus Ha : σ21 < σ22 Note that at the point of equality, the null hypothesis σ21 = σ22 implies that the ratio of population variances equals 1: σ21 /σ22 = 1. Given independent random samples from these populations, tests related to these hypotheses are based on an F -test, which is the ratio of sample variances. Suppose we use n1 observations in calculating the sample variance s12 and n2 observations in calculating the sample variance s22 . Tests concerning the difference between the variances of two populations make use of the F -distribution. Like the chi-square distribution, the F -distribution is a family of asymmetrical distributions bounded from below by 0. Each F -distribution is defined by two values of degrees of freedom, called the numerator 272 Quantitative Investment Analysis and denominator degrees of freedom.28 The F -test, like the chi-square test, is not robust to violations of its assumptions. • Test Statistic for Tests Concerning Differences between the Variances of Two Populations (Normally Distributed Populations). Suppose we have two samples, the first with n1 observations and sample variance s12 , the second with n2 observations and sample variance s22 . The samples are random, independent of each other, and generated by normally distributed populations. A test concerning differences between the variances of the two populations is based on the ratio of sample variances F= s12 s22 (7-16) with df1 = n1 − 1 numerator degrees of freedom and df2 = n2 − 1 denominator degrees of freedom. Note that df1 and df2 are the divisors used in calculating s12 and s22 , respectively. A convention, or usual practice, is to use the larger of the two ratios s12 /s22 or s22 /s12 as the actual test statistic. When we follow this convention, the value of the test statistic is always greater than or equal to 1; tables of critical values of F then need include only values greater than or equal to 1. Under this convention, the rejection point for any formulation of hypotheses is a single value in the right-hand side of the relevant F -distribution. Note that the labeling of populations as ‘‘1’’ or ‘‘2’’ is arbitrary in any case. • Rejection Points for Hypothesis Tests on the Relative Values of Two Population Variances. Follow the convention of using the larger of the two ratios s12 /s22 and s22 /s12 and consider two cases: 1. A ‘‘not equal to’’ alternative hypothesis: Reject the null hypothesis at the α significance level if the test statistic is greater than the upper α/2 point of the F -distribution with the specified numerator and denominator degrees of freedom. 2. A ‘‘greater than’’ or ‘‘less than’’ alternative hypothesis: Reject the null hypothesis at the α significance level if the test statistic is greater than the upper α point of the F -distribution with the specified number of numerator and denominator degrees of freedom. Thus, if we conduct a two-sided test at the α = 0.01 level of significance, we need to find the rejection point in F -tables at the α/2 = 0.01/2 = 0.005 significance level for a one-sided test (Case 1). But a one-sided test at 0.01 uses rejection points in F -tables for α = 0.01 (Case 2). As an example, suppose we are conducting a two-sided test at the 0.05 significance level. We calculate a value of F of 2.77 with 12 numerator and 19 denominator degrees of freedom. Using the F -tables for 0.05/2 = 0.025 in the back of the book, we find that the rejection point is 2.72. Because the value 2.77 is greater than 2.72, we reject the null hypothesis at the 0.05 significance level. If the convention stated above is not followed and we are given a calculated value of F less than 1, can we still use F -tables? The answer is yes; using a reciprocal property of F -statistics, we can calculate the needed value. The easiest way to present this property is to 28 The relationship between the chi-square and F -distributions is as follows: If χ2 is one chi-square 1 random variable with m degrees of freedom and χ22 is another chi-square random variable with n degrees of freedom, then F = (χ21 /m)/(χ22 /n) follows an F -distribution with m numerator and n denominator degrees of freedom. 273 Chapter 7 Hypothesis Testing show a calculation. Suppose our chosen level of significance is 0.05 for a two-tailed test and we have a value of F of 0.11, with 7 numerator degrees of freedom and 9 denominator degrees of freedom. We take the reciprocal, 1/0.11 = 9.09. Then we look up this value in the F -tables for 0.025 (because it is a two-tailed test) with degrees of freedom reversed: F for 9 numerator and 7 denominator degrees of freedom. In other words, F9,7 = 1/F7,9 and 9.09 exceeds the critical value of 4.82, so F7,9 = 0.11 is significant at the 0.05 level. EXAMPLE 7-8 Volatility and the Crash of 1987 You are investigating whether the population variance of returns on the S&P 500 changed subsequent to the October 1987 market crash. You gather the data in Table 7-7 for 120 months of returns before October 1987 and 120 months of returns after October 1987. You have specified a 0.01 level of significance. TABLE 7-7 S&P 500 Returns and Variance before and after October 1987 Before October 1987 After October 1987 n Mean Monthly Return (%) Variance of Returns 120 120 1.498 1.392 18.776 13.097 1. Formulate null and alternative hypotheses consistent with the verbal description of the research goal. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Determine whether or not to reject the null hypothesis at the 0.01 level of significance. (Use the F -tables in the back of this book.) Solution to 1: We have a ‘‘not equal to’’ alternative hypothesis: 2 2 2 2 H0 : σBefore = σAfter versus Ha : σBefore ̸= σAfter Solution to 2: To test a null hypothesis of the equality of two variances, we use F = s12 /s22 with 120 − 1 = 119 numerator and denominator degrees of freedom. Solution to 3: The ‘‘before’’ sample variance is larger, so following a convention for calculating F -statistics, the ‘‘before’’ sample variance goes in the numerator: F = 18.776/13.097 = 1.434. Because this is a two-tailed test, we use F -tables for the 0.005 level (= 0.01/2) to give a 0.01 significance level. In the tables in the back of the book, the closest value to 119 degrees of freedom is 120 degrees of freedom. At the 0.01 level, the rejection point is 1.61. Because 1.434 is less than the critical value 1.61, we cannot reject the null hypothesis that the population variance of returns is the same in the preand postcrash periods. 274 Quantitative Investment Analysis EXAMPLE 7-9 The Volatility of Derivatives Expiration Days In the 1980s concern arose in the United States about the triple occurrence of stock option, index option, and futures expirations on the same day during four months of the year. Such days were known as ‘‘triple witching days.’’ Table 7-8 presents evidence on the daily standard deviation of return for normal days and options/futures expiration days during the period 1 July 1983 to 24 October 1986. The tabled data refer to options and futures on the S&P 100, a subset of the S&P 500 that includes 100 of the most liquid S&P 500 stocks on which there are traded options. TABLE 7-8 Standard Deviation of Return: 1 July 1983 to 24 October 1986 Type of Day Normal trading Options/futures expiration n Standard Deviation (%) 115 12 0.786 1.178 Source: Based on Edwards (1988), Table I. 1. Formulate null and alternative hypotheses consistent with the belief that triple witching days displayed above-normal volatility. 2. Identify the test statistic for conducting a test of the hypotheses in Part 1. 3. Determine whether or not to reject the null hypothesis at the 0.05 level of significance. (Use the F -tables in the back of this book.) Solution to 1: We have a ‘‘greater than’’ alternative hypothesis: 2 2 2 2 H0 : σExpirations ≤ σNormal versus Ha : σExpirations > σNormal Solution to 2: Let σ21 represent the variance of triple witching days, and σ22 represent the variance of normal days, following the convention for the selection of the numerator and the denominator stated earlier. To test the null hypothesis, we use F = s12 /s22 with 12 − 1 = 11 numerator and 115 − 1 = 114 denominator degrees of freedom. Solution to 3: F = (1.178)2 /(0.786)2 = 1.388/0.618 = 2.25. Because this is a onetailed test at the 0.05 significance level, we use F -tables for the 0.05 level directly. In the tables in the back of the book, the closest value to 114 degrees of freedom is 120 degrees of freedom. At the 0.05 level, the rejection point is 1.87. Because 2.25 is greater than 1.87, we reject the null hypothesis. It appears that triple witching days had above-normal volatility. Chapter 7 Hypothesis Testing 275 5. OTHER ISSUES: NONPARAMETRIC INFERENCE The hypothesis-testing procedures we have discussed to this point have two characteristics in common. First, they are concerned with parameters, and second, their validity depends on a definite set of assumptions. Mean and variance, for example, are two parameters, or defining quantities, of a normal distribution. The tests also make specific assumptions—in particular, assumptions about the distribution of the population producing the sample. Any test or procedure with either of the above two characteristics is a parametric test or procedure. In some cases, however, we are concerned about quantities other than parameters of distributions. In other cases, we may believe that the assumptions of parametric tests do not hold for the particular data we have. In such cases, a nonparametric test or procedure can be useful. A nonparametric test is a test that is not concerned with a parameter, or a test that makes minimal assumptions about the population from which the sample comes.29 We primarily use nonparametric procedures in three situations: when the data we use do not meet distributional assumptions, when the data are given in ranks, or when the hypothesis we are addressing does not concern a parameter. The first situation occurs when the data available for analysis suggest that the distributional assumptions of the parametric test are not satisfied. For example, we may want to test a hypothesis concerning the mean of a population but believe that neither a t-test nor a z-test is appropriate because the sample is small and may come from a markedly nonnormally distributed population. In that case, we may use a nonparametric test. The nonparametric test will frequently involve the conversion of observations (or a function of observations) into ranks according to magnitude, and sometimes it will involve working with only ‘‘greater than’’ or ‘‘less than’’ relationships (using the signs + and − to denote those relationships). Characteristically, one must refer to specialized statistical tables to determine the rejection points of the test statistic, at least for small samples.30 Such tests, then, typically interpret the null hypothesis as a thesis about ranks or signs. In Table 7-9, we give examples of nonparametric alternatives to the parametric tests we have discussed in this chapter.31 The reader should consult a comprehensive business statistics textbook for an introduction to such tests and a specialist textbook for details.32 We pointed out that when we use nonparametric tests, we often convert the original data into ranks. In some cases, the original data are already ranked. In those cases, we also use nonparametric tests because parametric tests generally require a stronger measurement scale than ranks. For example, if our data were the rankings of investment managers, hypotheses concerning those rankings would be tested using nonparametric procedures. Ranked data also appear in many other finance contexts. For example, Heaney, Koga, Oliver, and Tran (1999) studied the relationship between the size of Japanese companies (as measured by revenue) and their use of derivatives. The companies studied used derivatives to hedge one or more 29 Some writers make a distinction between ‘‘nonparametric’’ and ‘‘distribution-free’’ tests. They refer to procedures that do not concern the parameters of a distribution as nonparametric and to procedures that make minimal assumptions about the underlying distribution as distribution free. We follow a commonly accepted, inclusive usage of the term nonparametric. 30 For large samples, there is often a transformation of the test statistic that permits the use of tables for the standard normal or t-distribution. 31 In some cases, there are several nonparametric alternatives to a parametric test. 32 See, for example, Hettmansperger and McKean (1998) or Siegel (1956). 276 Quantitative Investment Analysis TABLE 7-9 Nonparametric Alternatives to Parametric Tests Tests concerning a single mean Tests concerning differences between means Tests concerning mean differences (paired comparisons tests) Parametric Nonparametric t-Test z-Test t-Test Approximate t-test t-Test Wilcoxon signed-rank test Mann–Whitney U test Wilcoxon signed-rank test Sign test of five types of risk exposure: interest rate risk, foreign exchange risk, commodity price risk, marketable security price risk, and credit risk. The researchers gave a ‘‘perceived scope of risk exposure’’ score to each company that was equal to the number of types of risk exposure that the company reported hedging. Although revenue is measured on a strong scale (a ratio scale), scope of risk exposure is measured on only an ordinal scale.33 The researchers thus employed nonparametric statistics to explore the relationship between derivatives usage and size. A third situation in which we use nonparametric procedures occurs when our question does not concern a parameter. For example, if the question concerns whether a sample is random or not, we use the appropriate nonparametric test (a so-called runs test). Another type of question nonparametrics can address is whether a sample came from a population following a particular probability distribution (using the Kolmogorov–Smirnov test, for example). We end this chapter by describing in some detail a nonparametric statistic that has often been used in investment research, the Spearman rank correlation. 5.1. Tests Concerning Correlation: The Spearman Rank Correlation Coefficient In many contexts in investments, we want to assess the strength of the linear relationship between two variables—the correlation between them. In a majority of cases, we use the correlation coefficient described in the chapters on probability concepts and on correlation and regression. However, the t-test of the hypothesis that two variables are uncorrelated, based on the correlation coefficient, relies on fairly stringent assumptions.34 When we believe that the population under consideration meaningfully departs from those assumptions, we can employ a test based on the Spearman rank correlation coefficient, rS . The Spearman rank correlation coefficient is essentially equivalent to the usual correlation coefficient calculated on the ranks of the two variables (say, X and Y ) within their respective samples. Thus it is a number between −1 and +1, where −1 (+1) denotes a perfect inverse (positive) straight-line 33 We discussed scales of measurement in the chapter on statistical concepts and market returns. t-test is described in the chapter on correlation and regression. The assumption of the test is that each observation (x, y) on the two variables (X , Y ) is a random observation from a bivariate normal distribution. Informally, in a bivariate or two-variable normal distribution, each individual variable is normally distributed and their joint relationship is completely described by the correlation, ρ, between them. For more details, see, for example, Daniel and Terrell (1986). 34 The 277 Chapter 7 Hypothesis Testing relationship between the variables and 0 represents the absence of any straight-line relationship (no correlation). The calculation of rS requires the following steps: 1. Rank the observations on X from largest to smallest. Assign the number 1 to the observation with the largest value, the number 2 to the observation with second-largest value, and so on. In case of ties, we assign to each tied observation the average of the ranks that they jointly occupy. For example, if the third- and fourth-largest values are tied, we assign both observations the rank of 3.5 (the average of 3 and 4). Perform the same procedure for the observations on Y . 2. Calculate the difference, di , between the ranks of each pair of observations on X and Y . 3. Then, with n the sample size, the Spearman rank correlation is given by35 6 rS = 1 − n & di2 i=1 n(n2 − 1) (7-17) Suppose an investor wants to invest in a U.S. large-cap growth mutual fund. He has narrowed the field to 10 funds. In examining the funds, a question arises as to whether the funds’ reported three-year Sharpe ratios are related to their most recent reported expense ratios. Because the assumptions of the t-test on the correlation coefficient may not be met, it is appropriate to conduct a test on the rank correlation coefficient.36 Table 7-10 presents the calculation of rS .37 The first two rows contain the original data. The row of X ranks converts the Sharpe ratios to ranks; the row of Y ranks converts the expense ratios to ranks. We want to test H0 : ρ = 0 versus Ha : ρ ̸= 0, where ρ is defined in this context as the population correlation of X and Y after ranking. For small samples, the rejection points for the test based on rS must be looked up in Table 7-11. For large samples (say, n > 30), we can conduct a t-test using t= (n − 2)1/2 rs (1 − rs2 )1/2 (7-18) based on n − 2 degrees of freedom. In the example at hand, a two-tailed test with a 0.05 significance level, Table 7-11 gives the upper-tail rejection point for n = 10 as 0.6364 (we use the 0.025 column for a two-tailed test at a 0.05 significance level). Accordingly, we reject the null hypothesis if rS is less than −0.6364 or greater than 0.6364. With rS equal to 0.2545, we do not reject the null hypothesis. In the mutual fund example, we converted observations on two variables into ranks. If one or both of the original variables were in the form of ranks, we would need to use rS to investigate correlation. 35 Calculating the usual correlation coefficient on the ranks would yield approximately the same result as Equation 7-17. 36 The expense ratio (the ratio of a fund’s operating expenses to average net assets) is bounded both from below (by zero) and from above. The Sharpe ratio is also observed within a limited range, in practice. Thus neither variable can be normally distributed, and hence jointly they cannot follow a bivariate normal distribution. In short, the assumptions of a t-test are not met. 37 The data for the table are based on statistics reported in Standard & Poor’s Mutual Fund Reports for actual large-cap growth funds for the three-year period ending in the first quarter of 2003. The negative Sharpe ratios reflect in part declining U.S. equity markets during this period. 278 Quantitative Investment Analysis TABLE 7-10 The Spearman Rank Correlation: An Example 1 2 3 4 Mutual Fund 5 6 7 8 9 10 Sharpe Ratio (X ) −1.08 −0.96 −1.13 −1.16 −0.91 −1.08 −1.18 −1.00 −1.06 −1.00 Expense Ratio (Y ) 1.34 0.92 1.02 1.45 1.35 0.50 1.00 1.50 1.45 1.50 X Rank 6.5 2 8 9 1 6.5 10 3.5 5 3.5 Y Rank 6 9 7 3.5 5 10 8 1.5 3.5 1.5 0.5 −7 1 5.5 −4 −3.5 2 2 1. 5 2 di 0.25 49 1 30.25 16 12.25 4 4 2.25 4 di2 6!di2 6(123) rS = 1 − =1− = 0.2545 n(n2 − 1) 10(100 − 1) TABLE 7-11 Spearman Rank Correlation Distribution Approximate Upper-Tail Rejection Points Sample size: n α = 0.05 α = 0.025 α = 0.01 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.8000 0.7714 0.6786 0.6190 0.5833 0.5515 0.5273 0.4965 0.4780 0.4593 0.4429 0.4265 0.4118 0.3994 0.3895 0.3789 0.3688 0.3597 0.3518 0.3435 0.3362 0.3299 0.3236 0.3175 0.3113 0.3059 0.9000 0.8286 0.7450 0.7143 0.6833 0.6364 0.6091 0.5804 0.5549 0.5341 0.5179 0.5000 0.4853 0.4716 0.4579 0.4451 0.4351 0.4241 0.4150 0.4061 0.3977 0.3894 0.3822 0.3749 0.3685 0.3620 0.9000 0.8857 0.8571 0.8095 0.7667 0.7333 0.7000 0.6713 0.6429 0.6220 0.6000 0.5824 0.5637 0.5480 0.5333 0.5203 0.5078 0.4963 0.4852 0.4748 0.4654 0.4564 0.4481 0.4401 0.4320 0.4251 Note: The corresponding lower tail critical value is obtained by changing the sign of the upper-tail critical value. Chapter 7 Hypothesis Testing 279 5.2. Nonparametric Inference: Summary Nonparametric statistical procedures extend the reach of inference because they make few assumptions, can be used on ranked data, and may address questions unrelated to parameters. Quite frequently, nonparametric tests are reported alongside parametric tests. The reader can then assess how sensitive the statistical conclusion is to the assumptions underlying the parametric test. However, if the assumptions of the parametric test are met, the parametric test (where available) is generally preferred to the nonparametric test because the parametric test usually permits us to draw sharper conclusions.38 For complete coverage of all the nonparametric procedures that may be encountered in the finance and investment literature, it is best to consult a specialist textbook.39 38 To use a concept introduced in an earlier section, the parametric test is often more powerful. for example, Hettmansperger and McKean (1998) or Siegel (1956). 39 See, CHAPTER 8 CORRELATION AND REGRESSION 1. INTRODUCTION As a financial analyst, you will often need to examine the relationship between two or more financial variables. For example, you might want to know whether returns to different stock market indexes are related and, if so, in what way. Or you might hypothesize that the spread between a company’s return on invested capital and its cost of capital helps to explain the company’s value in the marketplace. Correlation and regression analysis are tools for examining these issues. This chapter is organized as follows. In Section 2, we present correlation analysis, a basic tool in measuring how two variables vary in relation to each other. Topics covered include the calculation, interpretation, uses, limitations, and statistical testing of correlations. Section 3 introduces basic concepts in regression analysis, a powerful technique for examining the ability of one or more variables (independent variables) to explain or predict another variable (the dependent variable). 2. CORRELATION ANALYSIS We have many ways to examine how two sets of data are related. Two of the most useful methods are scatter plots and correlation analysis. We examine scatter plots first. 2.1. Scatter Plots A scatter plot is a graph that shows the relationship between the observations for two data series in two dimensions. Suppose, for example, that we want to graph the relationship between long-term money growth and long-term inflation in six industrialized countries to see how strongly the two variables are related. Table 8-1 shows the average annual growth rate in the money supply and the average annual inflation rate from 1970 to 2001 for the six countries. To translate the data in Table 8-1 into a scatter plot, we use the data for each country to mark a point on a graph. For each point, the x-axis coordinate is the country’s annual average money supply growth from 1970–2001 and the y-axis coordinate is the country’s annual average inflation rate from 1970–2001. Figure 8-1 shows a scatter plot of the data in Table 8-1. 281 282 Quantitative Investment Analysis TABLE 8-1 Annual Money Supply Growth Rate and Inflation Rate by Country, 1970–2001 Country Money Supply Growth Rate Inflation Rate 11.66% 9.15% 10.60% 5.75% 12.58% 6.34% 9.35% 6.76% 5.19% 8.15% 3.39% 7.58% 5.09% 6.03% Australia Canada New Zealand Switzerland United Kingdom United States Average Source: International Monetary Fund. Inflation (%) 10 8 6 4 2 0 0 2 4 6 8 10 12 14 Money Supply Growth (%) FIGURE 8-1 Scatter Plot of Annual Money Supply Growth Rate and Inflation Rate by Country: 1970–2001 Source: International Monetary Fund. Note that each observation in the scatter plot is represented as a point, and the points are not connected. The scatter plot does not show which observation comes from which country; it shows only the actual observations of both data series plotted as pairs. For example, the rightmost point shows the data for the United Kingdom. The data plotted in Figure 8-1 show a fairly strong linear relationship with a positive slope. Next we examine how to quantify this linear relationship. 2.2. Correlation Analysis In contrast to a scatter plot, which graphically depicts the relationship between two data series, correlation analysis expresses this same relationship using a single number. The correlation coefficient is a measure of how closely related two data series are. In particular, the correlation 283 Chapter 8 Correlation and Regression Variable B 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0 1 2 3 4 5 6 7 8 Variable A FIGURE 8-2 Variables with a Correlation of 1 coefficient measures the direction and extent of linear association between two variables. A correlation coefficient can have a maximum value of 1 and a minimum value of −1. A correlation coefficient greater than 0 indicates a positive linear association between the two variables: When one variable increases (or decreases), the other also tends to increase (or decrease). A correlation coefficient less than 0 indicates a negative linear association between the two variables: When one increases (or decreases), the other tends to decrease (or increase). A correlation coefficient of 0 indicates no linear relation between the two variables.1 Figure 8-2 shows the scatter plot of two variables with a correlation of 1. Note that all the points on the scatter plot in Figure 8-2 lie on a straight line with a positive slope. Whenever variable A increases by one unit, variable B increases by half a unit. Because all of the points in the graph lie on a straight line, an increase of one unit in A is associated with exactly the same half-unit increase in B, regardless of the level of A. Even if the slope of the line in the figure were different (but positive), the correlation between the two variables would be 1 as long as all the points lie on that straight line. Figure 8-3 shows a scatter plot for two variables with a correlation coefficient of −1. Once again, the plotted observations fall on a straight line. In this graph, however, the line has a negative slope. As A increases by one unit, B decreases by half a unit, regardless of the initial value of A. Figure 8-4 shows a scatter plot of two variables with a correlation of 0; they have no linear relation. This graph shows that the value of A tells us absolutely nothing about the value of B. 2.3. Calculating and Interpreting the Correlation Coefficient To define and calculate the correlation coefficient, we need another measure of linear association: covariance. In the chapter on probability concepts, we defined covariance as the 1 Later, we show that variables with a correlation of 0 can have a strong nonlinear relation. 284 Quantitative Investment Analysis Variable B 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0 1 2 3 4 5 6 7 8 Variable A FIGURE 8-3 Variables with a Correlation of −1 expected value of the product of the deviations of two random variables from their respective population means. That was the definition of population covariance, which we would also use in a forward-looking sense. To study historical or sample correlations, we need to use sample covariance. The sample covariance of X and Y , for a sample of size n, is Cov (X , Y ) = n ! i=1 (Xi − X )(Yi − Y )/(n − 1) (8-1) The sample covariance is the average value of the product of the deviations of observations on two random variables from their sample means.2 If the random variables are returns, the unit of covariance would be returns squared. The sample correlation coefficient is much easier to explain than the sample covariance. To understand the sample correlation coefficient, we need the expression for the sample standard deviation of a random variable X . We need to calculate the sample variance of X to obtain its sample standard deviation. The variance of a random variable is simply the covariance of the random variable with itself. The expression for the sample variance of X , sX2 , is sX2 = n ! i=1 (Xi − X )2 /(n − 1) The sample standard deviation is the positive square root of the sample variance: sX = 2 The " sX2 use of n − 1 in the denominator is a technical point; it ensures that the sample covariance is an unbiased estimate of population covariance. 285 Chapter 8 Correlation and Regression Variable B 1.2 1.0 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2 Variable A FIGURE 8-4 Variables with a Correlation of 0 Both the sample variance and the sample standard deviation are measures of the dispersion of observations about the sample mean. Standard deviation uses the same units as the random variable; variance is measured in the units squared. The formula for computing the sample correlation coefficient is r= Cov (X , Y ) sX sY (8-2) The correlation coefficient is the covariance of two variables (X and Y ) divided by the product of their sample standard deviations (sX and sY ). Like covariance, the correlation coefficient is a measure of linear association. The correlation coefficient, however, has the advantage of being a simple number, with no unit of measurement attached. It has no units because it results from dividing the covariance by the product of the standard deviations. Because we will be using sample variance, standard deviation, and covariance in this chapter, we will repeat the calculations for these statistics. 286 Quantitative Investment Analysis TABLE 8-2 Sample Covariance and Sample Standard Deviations: Annual Money Supply Growth Rate and Inflation Rate by Country, 1970–2001 Country Money Supply Growth Rate Xi Inflation Rate Yi 0.1166 0.0915 0.1060 0.0575 0.1258 0.0634 0.5608 0.0935 0.0676 0.0519 0.0815 0.0339 0.0758 0.0509 0.3616 0.0603 Australia Canada New Zealand Switzerland United Kingdom United States Sum Average Covariance Variance Standard deviation Cross-Product (Xi − X )(Yi − Y ) Squared Deviations (Xi − X )2 Squared Deviations (Yi − Y )2 0.000169 0.000017 0.000265 0.000950 0.000501 0.000283 0.002185 0.000534 0.000004 0.000156 0.001296 0.001043 0.000906 0.003939 0.000053 0.000071 0.000449 0.000697 0.000240 0.000088 0.001598 0.000788 0.028071 0.000320 0.017889 0.000437 Source: International Monetary Fund. Notes: 1. Divide the cross-product sum by n − 1 (with n = 6) to obtain the covariance of X and Y . 2. Divide the squared deviations sums by n − 1 (with n = 6) to obtain the variances of X and Y . Table 8-2 shows how to compute the various components of the correlation equation (Equation 8-2) from the data in Table 8-1.3 The individual observations on countries’ annual average money supply growth from 1970–2001 are denoted Xi , and individual observations on countries’ annual average inflation rate from 1970–2001 are denoted Yi . The remaining columns show the calculations for the inputs to correlation: the sample covariance and the sample standard deviations. Using the data shown in Table 8-2, we can compute the sample correlation coefficient for these two variables as follows: r= Cov (X , Y ) 0.000437 = 0.8702 = sX sY (0.028071)(0.017889) The correlation coefficient of approximately 0.87 indicates a strong linear association between long-term money supply growth and long-term inflation for the countries in the sample. The correlation coefficient captures this strong association numerically, whereas the scatter plot in Figure 8-1 shows the information graphically. What assumptions are necessary to compute the correlation coefficient? Correlation coefficients can be computed validly if the means and variances of X and Y , as well as the covariance of X and Y , are finite and constant. Later, we will show that when these 3 We have not used full precision in the table’s calculations. We used the average value of the money supply growth rate of 0.5608/6 = 0.0935, rounded to four decimal places, in the cross-product and squared deviation calculations, and similarly, we used the mean inflation rate as rounded to 0.0603 in those calculations. We computed standard deviation as the square root of variance rounded to six decimal places, as shown in the table. Had we used full precision in all calculations, some of the table’s entries would be different and the computed value of correlation would be 0.8709 rather than 0.8702, not materially affecting our conclusions. 287 Chapter 8 Correlation and Regression assumptions are not true, correlations between two different variables can depend greatly on the sample that is used. 2.4. Limitations of Correlation Analysis Correlation measures the linear association between two variables, but it may not always be reliable. Two variables can have a strong nonlinear relation and still have a very low correlation. For example, the relation B = (A − 4)2 is a nonlinear relation contrasted to the linear relation B = 2A − 4. The nonlinear relation between variables A and B is shown in Figure 8-5. Below a level of 4 for A, variable B decreases with increasing values of A. When A Variable B 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 Variable A FIGURE 8-5 Variables with a Strong Non-Linear Association 288 Quantitative Investment Analysis Monthly Total Returns (S&P 500) (%) 15 10 5 0 !5 !10 !15 !20 !0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 Monthly Inflation (%) FIGURE 8-6 U.S. Inflation and Stock Returns in the 1990s Source: Ibbotson Associates. is 4 or greater, however, B increases whenever A increases. Even though these two variables are perfectly associated, the correlation between them is 0.4 Correlation also may be an unreliable measure when outliers are present in one or both of the series. Outliers are small numbers of observations at either extreme (small or large) of a sample. Figure 8-6 shows a scatter plot of the monthly returns to the Standard & Poor’s 500 Index and the monthly inflation rate in the United States during the 1990s (January 1990 through December 1999). In the scatter plot in Figure 8-6, most of the data lie clustered together with little discernible relation between the two variables. In three cases, however (the three circled observations), inflation was greater than 0.8 percent in a particular month and stock returns were strongly negative. These observations are outliers. If we compute the correlation coefficient for the entire data sample, that correlation is −0.2997. If we eliminate the three outliers, however, the correlation is −0.1347. The correlation in Figure 8-6 is quite sensitive to excluding only three observations. Does it make sense to exclude those observations? Are they noise or news? One possible partial explanation of Figure 8-6 is that during the 1990s, whenever inflation was very high during a month, market participants became concerned that the Federal Reserve would raise interest rates, which would cause the value of stocks to decline. This story offers one plausible explanation for how investors reacted to large inflation announcements. Consequently, the outliers may provide important information about market reactions during this period. Therefore, the correlation that includes the outliers may make more sense than the correlation that excludes them. 4 The perfect association is the quadratic relationship B = (A − 4)2 . Chapter 8 Correlation and Regression 289 As a general rule, we must determine whether a computed sample correlation changes greatly by removing a few outliers. But we must also use judgment to determine whether those outliers contain information about the two variables’ relationship (and should thus be included in the correlation analysis) or contain no information (and should thus be excluded). Keep in mind that correlation does not imply causation. Even if two variables are highly correlated, one does not necessarily cause the other in the sense that certain values of one variable bring about the occurrence of certain values of the other. Furthermore, correlations can be spurious in the sense of misleadingly pointing towards associations between variables. The term spurious correlation has been used to refer to (1) correlation between two variables that reflects chance relationships in a particular data set, (2) correlation induced by a calculation that mixes each of two variables with a third, and (3) correlation between two variables arising not from a direct relation between them but from their relation to a third variable. As an example of the second kind of spurious correlation, two variables that are uncorrelated may be correlated if divided by a third variable. As an example of the third kind of spurious correlation, height may be positively correlated with the extent of a person’s vocabulary, but the underlying relationships are between age and height and between age and vocabulary. Investment professionals must be cautious in basing investment strategies on high correlations. Spurious correlation may suggest investment strategies that appear profitable but actually would not be so, if implemented. 2.5. Uses of Correlation Analysis In this section, we give examples of correlation analysis for investment. Because investors’ expectations about inflation are important in determining asset prices, inflation forecast accuracy will serve as our first example. EXAMPLE 8-1 Evaluating Economic Forecasts (1) Investors closely watch economists’ forecasts of inflation, but do these forecasts contain useful information? In the United States, the Survey of Professional Forecasters (SPF) gathers professional forecasters’ predictions about many economic variables.5 Since the early 1980s, SPF has gathered predictions on the U.S. inflation rate using the change in the U.S. consumer price index (CPI) for all urban consumers and all items to measure inflation. If these forecasts of inflation could perfectly predict actual inflation, the correlation between forecasts and inflation would be 1; that is, predicted and actual inflation would always be the same. Figure 8-7 shows a scatter plot of the mean forecast of current-quarter percentage change in CPI from previous quarter and actual percentage change in CPI, on an 5 The survey was originally developed by Victor Zarnowitz for the American Statistical Association and the National Bureau of Economic Research. Starting in 1990, the survey has been directed by Dean Croushire of the Federal Reserve Bank of Philadelphia. 290 Quantitative Investment Analysis Actual Change (Annualized %) 10 8 6 4 2 Correlation = 0.7138 0 !2 !2 0 2 4 6 8 10 Predicted Change (Annualized %) FIGURE 8-7 Actual Change in CPI vs. Predicted Change Source: Federal Reserve Banks of Philadelphia and St. Louis. annualized basis, from the first quarter of 1983 to the last quarter of 2002.6 In this scatter plot, the forecast for each quarter is plotted on the x-axis and the actual change in the CPI is plotted on the y-axis. As Figure 8-7 shows, a fairly strong linear association exists between the forecast and the actual inflation rate, suggesting that professional forecasts of inflation might be useful in investment decision-making. In fact, the correlation between the two series is 0.7138. Although there is no causal relation here, there is a direct relation because forecasters assimilate information to forecast inflation. One important issue in evaluating a portfolio manager’s performance is determining an appropriate benchmark for the manager. In recent years, style analysis has been an important component of benchmark selection.7 6 In this scatter plot, the actual change in CPI is from the Federal Reserve’s economic and financial database, available at the Web site of the Federal Reserve Bank of St. Louis. 7 See, for example, Sharpe (1992) and Buetow, Johnson, and Runkle (2000). Chapter 8 Correlation and Regression EXAMPLE 8-2 291 Style Analysis Correlations Suppose a portfolio manager uses small-cap stocks in an investment portfolio. By applying style analysis, we can try to determine whether the portfolio manager uses a small-cap growth style or a small-cap value style. In the United States, the Russell 2000 Growth Index and the Russell 2000 Value Index are often used as benchmarks for small-cap growth and small-cap value managers, respectively. Correlation analysis shows, however, that the returns to these two indexes are very closely associated with each other. For the 20 years ending in 2002 (January 1983 to December 2002), the correlation between the monthly returns to the Russell 2000 Growth Index and the Russell 2000 Value Index was 0.8526. If the correlation between the returns to the two indexes were 1, there would be absolutely no difference in equity management style between small-cap value and small-cap growth. If we knew the return to one style, we could be certain about the return to the other style. Because the returns to the two indexes are highly correlated, we can say that very little difference exists between the two return series, and therefore, we may not be able to justify distinguishing between small-cap growth and small-cap value as different investment styles. The previous examples in this chapter have examined the correlation between two variables. Often, however, investment managers need to understand the correlations among many asset returns. For example, investors who have any exposure to movements in exchange rates must understand the correlations of the returns to different foreign currencies and other assets in order to determine their optimal portfolios and hedging strategies.8 In the following example, we see how a correlation matrix shows correlation between pairs of variables when we have more than two variables. We also see one of the main challenges to investment managers: Investment return correlations can change substantially over time. EXAMPLE 8-3 Exchange Rate Return Correlations The exchange rate return measures the periodic domestic currency return to holding foreign currency. Suppose a change in inflation rates in the United Kingdom and the United States results in the U.S. dollar price of a pound changing from $1.50 to $1.25. If this change occurred in one month, the return in that month to holding pounds would be (1.25 − 1.50)/1.50 = −16.67 percent, in terms of dollars. Table 8-3 shows a correlation matrix of monthly returns in U.S. dollars to holding Canadian, Japanese, Swedish, or British currencies.9 To interpret a correlation matrix, we first examine the top panel of this table. The first column of numbers of that panel 8 See, for example, Clarke and Kritzman (1996). for the 1980s run from January 1980 through December 1989. Data for the 1990s run from January 1990 through December 1999. 9 Data 292 Quantitative Investment Analysis shows the correlations between USD returns to holding the Canadian dollar and USD returns to holding Canadian, Japanese, Swedish, and British currencies. Of course, any variable is perfectly correlated with itself, and so the correlation between USD returns to holding the Canadian dollar and USD returns to holding the Canadian dollar is 1. The second row of this column shows that the correlation between USD returns to holding the Canadian dollar and USD returns to holding the Japanese yen was 0.2593 from 1980 to 1989. The remaining correlations in the panel show how the USD returns to other combinations of currency holdings were correlated during this period. Note that Table 8-3 omits many of the correlations. For example, Column 2 of the panel omits the correlation between USD returns to holding yen and USD returns TABLE 8-3 Correlations of Monthly U.S. Dollar Returns to Selected Foreign Currency Returns 1980–1989 Canada Japan Sweden United Kingdom Canada Japan Sweden United Kingdom 1.0000 0.2593 0.2834 0.3925 1.0000 0.6576 0.6068 1.0000 0.6840 1.0000 1990–1999 Canada Japan Sweden United Kingdom Canada Japan Sweden United Kingdom 1.0000 −0.0734 0.1640 0.0475 1.0000 0.2860 0.2906 1.0000 0.6444 1.0000 Source: Ibbotson Associates. to holding Canadian dollars. This correlation is omitted because it is identical to the correlation between USD returns to holding Canadian dollars and USD returns to holding yen shown in Row 2 of Column 1. Other omitted correlations would also have been duplicative. In fact, correlations are always symmetrical: The correlation between X and Y is always the same as the correlation between Y and X . If you compare the two panels of this table, you will find that many of the currency return correlations changed dramatically between the 1980s and the 1990s. In the 1980s, for example, the correlation between the return to holding Japanese yen and the return to holding either Swedish kronor (0.6576) or British pounds (0.6068) was almost as high as the correlation between the return to holding kronor and the return to holding pounds (0.6840). In the 1990s, however, the correlation between yen returns and either krona or pound returns dropped by more than half (to 0.2860 and 0.2906, respectively), but the correlation between krona and pound returns hardly changed at all (0.6444). Some of the correlations between returns to the Canadian dollar and returns to other currencies dropped even more dramatically. In the 1980s, the correlation between Canadian dollar 293 Chapter 8 Correlation and Regression returns and Japanese yen returns was 0.2593. By the 1990s, that correlation actually became negative (−0.0734). The correlation between the Canadian dollar returns and British pound returns dropped from 0.3925 in the 1980s to 0.0475 in the 1990s. Optimal asset allocation depends on expectations of future correlations. With less than perfect positive correlation between two assets’ returns, there are potential riskreduction benefits to holding both assets. Expectations of future correlation may be based on historical sample correlations, but the variability in historical sample correlations poses challenges. We discuss these issues in detail in the chapter on portfolio concepts. In the next example, we extend the discussion of the correlations of stock market indexes begun in Example 8-2 to indexes representing large-cap, small-cap, and broad-market returns. This type of analysis has serious diversification and asset allocation consequences because the strength of the correlations among the assets tells us how successfully the assets can be combined to diversify risk. EXAMPLE 8-4 Correlations among Stock Return Series Table 8-4 shows the correlation matrix of monthly returns to three U.S. stock indexes during the period January 1971 to December 1999 and in three subperiods (the 1970s, 1980s, and 1990s).10 The large-cap style is represented by the return to the S&P 500 Index, the small-cap style is represented by the return to the Dimensional Fund Advisors U.S. Small-Stock Index, and the broad-market returns are represented by the return to the Wilshire 5000 Index. TABLE 8-4 Correlations of Monthly Returns to Various U.S. Stock Indexes 1971–1999 S&P 500 U.S. Small-Stock Wilshire 5000 1971–1979 S&P 500 U.S. Small-Stock Wilshire 5000 1980–1989 S&P 500 U.S. Small-Stock Wilshire 5000 S&P 500 U.S. Small-Stock Wilshire 5000 1.0000 0.7615 0.9894 1.0000 0.8298 1.0000 S&P 500 U.S. Small-Stock Wilshire 5000 1.0000 0.7753 0.9906 1.0000 0.8375 1.0000 S&P 500 U.S. Small-Stock Wilshire 5000 1.0000 0.8440 0.9914 1.0000 0.8951 1.0000 (continued ) 10 The 1970s data have an initiation date of January 1971 because that is the starting date of the Wilshire 5000 total return series. 294 Quantitative Investment Analysis TABLE 8-4 (continued ) 1990–1999 S&P 500 U.S. Small-Stock Wilshire 5000 S&P 500 U.S. Small-Stock Wilshire 5000 1.0000 0.6843 0.9858 1.0000 0.7768 1.0000 Source: Ibbotson Associates. The first column of numbers in the top panel of Table 8-4 shows nearly perfect positive correlation between returns to the S&P 500 and returns to the Wilshire 5000: The correlation between the two return series is 0.9894. This result should not be surprising, because both the S&P 500 and the Wilshire 5000 are value-weighted indexes, and large-cap stock returns receive most of the weight in both indexes. In fact, the companies that make up the S&P 500 have about 80 percent of the total market value of all companies included in the Wilshire 5000. Small stocks also have a reasonably high correlation with large stocks. In the total sample, the correlation between the S&P 500 returns and the U.S. Small-Stock returns is 0.7615. The correlation between U.S. Small-Stock returns and returns to the Wilshire 5000 is slightly higher (0.8298). This result is also not too surprising because the Wilshire 5000 contains small-cap stocks and the S&P 500 does not. The second, third, and fourth panels of Table 8-4 show that correlations among the various stock market return series show some variation from decade to decade. For example, the correlation between returns to the S&P 500 and U.S. small-cap stocks dropped from 0.8440 in the 1980s to 0.6843 in the 1990s.11 For asset allocation purposes, correlations among asset classes are studied carefully with a view towards maintaining appropriate diversification based on forecasted correlations. EXAMPLE 8-5 Correlations of Debt and Equity Returns Table 8-5 shows the correlation matrix for various U.S. debt returns and S&P 500 returns using monthly data from January 1926 to December 2002. The first column of numbers, in particular, shows the correlations of S&P 500 returns with various debt returns. Note that S&P 500 returns are almost completely uncorrelated (−0.0174) with 30-day Treasury bill returns for this period. Long-term corporate debt returns are somewhat more correlated (0.2143) with S&P 500 returns. Returns on high-yield corporate bonds have the highest correlation (0.6471) with S&P 500 total returns. This high correlation is understandable; high-yield debt securities behave partially as equities because of their high default risk. If a company defaults, holders of high-yield debt typically lose most of their investment. 11 The correlation coefficient for the 1990s was less than that for the 1980s at the 0.01 significance level. A test for this type of hypothesis on the correlation coefficient can be conducted using Fisher’s z-transformation. See Daniel and Terrell (1995) for information on this method. 295 Chapter 8 Correlation and Regression TABLE 8-5 Correlations among U.S. Stock and Debt Returns, 1926–2002 All S&P 500 U.S. LongTerm Corp. S&P 500 U.S. Long-Term Corp. U.S. Long-Term Govt. U.S. 30-Day T-bill High-Yield Corp. 1.0000 0.2143 0.1466 −0.0174 0.6471 1.0000 0.8480 0.0970 0.4274 U.S. LongTerm Govt. U.S. 30-Day T-Bill HighYield Corp. 1.0000 0.1119 0.3131 1.0000 0.0174 1.0000 Source: Ibbotson Associates. Long-term government bonds, however, have a low correlation (0.1466) with S&P 500 returns. We expect some correlation between these variables because interest rate increases reduce the present value of future cash flows for both bonds and stocks. The relatively low correlation between these two return series, however, shows that other factors affect the returns on stocks besides interest rates. Without these other factors, the correlation between bond and stock returns would be higher. The second column of numbers in Table 8-5 shows that the correlation between long-term government bond and corporate bond returns is quite high (0.8480) for this time period. Although this correlation is the highest in the entire matrix, it is not 1. The correlation is less than 1 because the default premium for long-term corporate bonds changes, whereas U.S. government bonds do not incorporate a default premium. As a result, changes in required yields for government bonds have a correlation less than 1 with changes in required yields for corporate bonds, and return correlations between government bonds and corporate bonds are also below 1. Note also that the correlation of high-yield corporate bond returns with long-term government bond returns (0.3131), indicated in the third column of numbers, is less than half the correlation of high-yield corporate bond returns with S&P 500 returns. This relatively low correlation is another indicator that high-yield bond returns behave more similarly to equity returns than to debt returns. Note finally that 30-day T-bill returns have a very low correlation with all other return series. In fact, the correlations between T-bill returns and other return series are lower than any of the other correlations in this table. In the final example of this section, correlation is used in a financial statement setting to show that net income is an inadequate proxy for cash flow. EXAMPLE 8-6 Correlations among Net Income, Cash Flow from Operations, and Free Cash Flow to the Firm Net income (NI), cash flow from operations (CFO), and free cash flow to the firm (FCFF) are three measures of company performance that analysts often use to value companies. 296 Quantitative Investment Analysis Differences in these measures for given companies would not cause differences in the relative valuation if the measures were highly correlated. CFO equals net income plus the net noncash charges that were subtracted to obtain net income, minus the company’s investment in working capital during the same time period. FCFF equals CFO plus net-of-tax interest expense, minus the company’s investment in fixed capital over the time period. FCFF may be interpreted as the cash flow available to the company’s suppliers of capital (debtholders and shareholders) after all operating expenses have been paid and necessary investments in working and fixed capital have been made.12 Some analysts base their valuations only on NI, ignoring CFO and FCFF. If the correlations among NI, CFO, and FCFF were very high, then an analyst’s decision to ignore CFO and FCFF would be easy to understand because NI would then appear to capture everything one needs to know about cash flow. TABLE 8-6 Correlations among Performance Measures: U.S. Women’s Clothing Stores, 2001 NI CFO FCFF NI CFO FCFF 1.0000 0.6959 0.4045 1.0000 0.8217 1.0000 Source: Compustat. Table 8-6 shows the correlations among NI, CFO, and FCFF for a group of six publicly traded U.S. companies involved in retailing women’s clothing for 2001. Before computing the correlations, we normalized all of the data by dividing each company’s three performance measures by the company’s revenue for the year.13 Because CFO and FCFF include NI as a component (in the sense that CFO and FCFF can be obtained by adding and subtracting various quantities from NI), we might expect that the correlations between NI and CFO and between NI and FCFF would be positive. Table 8-6 supports that conclusion. These correlations with NI, however, are much smaller than the correlation between CFO and FCFF (0.8217). The lowest correlation in the table is between NI and FCFF (0.4045). This relatively low correlation shows that NI contained some but far from all the information in FCFF for these companies in 2001. Later in this chapter, we will test whether the correlation between NI and FCFF is significantly different from zero. 12 For more on these three measures and their use in equity valuation, see Stowe, Robinson, Pinto, and McLeavey (2002). The statements in the footnoted paragraph explain the relationships among these measures according to U.S. GAAP. Stowe et al. also discuss the relationships among these measures according to international accounting standards. 13 The results in this table are based on data for all women’s clothing stores (U.S. Occupational Health and Safety Administration Standard Industrial Classification 5621) with a market capitalization of more than $250 million at the end of 2001. The market-cap criterion was used to eliminate microcap firms, whose performance-measure correlations may be different from those of higher-valued firms. We will discuss in detail the data normalization used in this example (dividing by firm revenue) in the section on misspecified regressions in Chapter 9. 297 Chapter 8 Correlation and Regression 2.6. Testing the Significance of the Correlation Coefficient Significance tests allow us to assess whether apparent relationships between random variables are the result of chance. If we decide that the relationships do not result from chance, we will be inclined to use this information in predictions because a good prediction of one variable will help us predict the other variable. Using the data in Table 8-2, we calculated 0.8702 as the sample correlation between long-term money growth and long-term inflation in six industrialized countries between 1970 and 2001. That estimated correlation seems high, but is it significantly different from 0? Before we can answer this question, we must know some details about the distribution of the underlying variables themselves. For purposes of simplicity, let us assume that both of the variables are normally distributed.14 We propose two hypotheses: the null hypothesis, H0 , that the correlation in the population is 0 (ρ = 0); and the alternative hypothesis, Ha , that the correlation in the population is different from 0 (ρ ̸= 0). The alternative hypothesis is a test that the correlation is not equal to 0; therefore, a twotailed test is appropriate.15 As long as the two variables are distributed normally, we can test to determine whether the null hypothesis should be rejected using the sample correlation, r. The formula for the t-test is √ r n−2 t= √ 1 − r2 (8-3) This test statistic has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true. One practical observation concerning Equation 8-3 is that the magnitude of r needed to reject the null hypothesis H0: ρ = 0 decreases as sample size n increases, for two reasons. First, as n increases, the number of degrees of freedom increases and the absolute value of the critical value tc decreases. Second, the absolute value of the numerator increases with larger n, resulting in larger-magnitude t-values. For example, with sample size n = 12, r = 0.58 results in a t-statistic of 2.252 that is just significant at the 0.05 level (tc = 2.228). With a sample size n = 32, a smaller sample correlation r = 0.35 yields a t-statistic of 2.046 that is just significant at the 0.05 level (tc = 2.042); the r = 0.35 would not be significant with a sample size of 12 even at the 0.10 significance level. Another way to make this point is that sampling from the same population, a false null hypothesis H0: ρ = 0 is more likely to be rejected as we increase sample size, all else equal. EXAMPLE 8-7 Testing the Correlation between Money Supply Growth and Inflation Earlier in this chapter, we showed that the sample correlation between long-term money supply growth and long-term inflation in six industrialized countries was 0.8702 during 14 Actually, we must assume that the variables come from a bivariate normal distribution. If two variables, X and Y , come from a bivariate normal distribution, then for each value of X the distribution of Y is normal. See, for example, Ross (1997) or Greene (2003). 15 See the chapter on hypothesis testing for a more in-depth discussion of two-tailed tests. 298 Quantitative Investment Analysis the 1970–2001 period. Suppose we want to test the null hypothesis, H0 , that the true correlation in the population is 0 (ρ = 0) against the alternative hypothesis, Ha , that the correlation in the population is different from 0 (ρ ̸= 0). Recalling that this sample has six observations, we can compute the statistic for testing the null hypothesis as follows: √ 0.8702 6 − 2 t= √ = 3.532 1 − 0.87022 The value of the test statistic is 3.532. As the table of critical values of the t-distribution for a two-tailed test shows, for a t-distribution with n − 2 = 6 − 2 = 4 degrees of freedom at the 0.05 level of significance, we can reject the null hypothesis (that the population correlation is equal to 0) if the value of the test statistic is greater than 2.776 or less than −2.776. The fact that we can reject the null hypothesis of no correlation based on only six observations is quite unusual; it further demonstrates the strong relation between long-term money supply growth and long-term inflation in these six countries. EXAMPLE 8-8 Testing the Krona–Yen Return Correlation The data in Table 8-3 showed that the sample correlation between the USD monthly returns to Swedish kronor and Japanese yen was 0.2860 for the period from January 1990 through December 1999. If we observe this sample correlation, can we reject a null hypothesis that the underlying or population correlation equals 0? With 120 months from January 1990 through December 1999, we use the following statistic to test the null hypothesis, H0 , that the true correlation in the population is 0, against the alternative hypothesis, Ha , that the correlation in the population is different from 0: √ 0.2860 120 − 2 t= √ = 3.242 1 − 0.28602 At the 0.05 significance level, the critical level for this test statistic is 1.98 (n = 120, degrees of freedom = 118). When the test statistic is either larger than 1.98 or smaller than −1.98, we can reject the hypothesis that the correlation in the population is 0. The test statistic is 3.242, so we can reject the null hypothesis. Note that the sample correlation coefficient in this case is significantly different from 0 at the 0.05 level, even though the coefficient is much smaller than that in the previous example. The correlation coefficient, though smaller, is still significant because the sample is much larger (120 observations instead of 6 observations). Chapter 8 Correlation and Regression 299 The above example shows the importance of sample size in tests of the significance of the correlation coefficient. The following example also shows the importance of sample size and examines the relationship at the 0.01 level of significance as well as at the 0.05 level. EXAMPLE 8-9 T-Bill Returns The Correlation Between Bond Returns and Table 8-5 showed that the sample correlation between monthly returns to U.S. government bonds and monthly returns to 30-day T-bills was 0.1119 from January 1926 through December 2002. Suppose we want to test whether the correlation coefficient is statistically significantly different from zero. There are 924 months during the period January 1926 to December 2002. Therefore, to test the null hypothesis, H0 (that the true correlation in the population is 0), against the alternative hypothesis, Ha (that the correlation in the population is different from 0), we use the following test statistic: √ 0.1119 924 − 2 t= √ = 3.4193 1 − 0.11192 At the 0.05 significance level, the critical value for the test statistic is approximately 1.96. At the 0.01 significance level, the critical value for the test statistic is approximately 2.58. The test statistic is 3.4193, so we can reject the null hypothesis of no correlation in the population at both the 0.05 and 0.01 levels. This example shows that, in large samples, even relatively small correlation coefficients can be significantly different from zero. In the final example of this section, we explore another situation of small sample size. EXAMPLE 8-10 Testing the Correlation Between Net Income and Free Cash Flow to the Firm Earlier in this chapter, we showed that the sample correlation between NI and FCFF for six women’s clothing stores was 0.4045 in 2001. Suppose we want to test the null hypothesis, H0 , that the true correlation in the population is 0 (ρ = 0) against the alternative hypothesis, Ha , that the correlation in the population is different from 0 (ρ ̸= 0). Recalling that this sample has six observations, we can compute the statistic for testing the null hypothesis as follows: √ 0.4045 6 − 2 t= √ = 0.8846 1 − 0.40452 With n − 2 = 6 − 2 = 4 degrees of freedom and a 0.05 significance level, we reject the null hypothesis that the population correlation equals 0 for values of the test statistic 300 Quantitative Investment Analysis greater than 2.776 or less than −2.776. In this case, however, the t-statistic is 0.8846, so we cannot reject the null hypothesis. Therefore, for this sample of women’s clothing stores, there is no statistically significant correlation between NI and FCFF, when each is normalized by dividing by sales for the company.16 The scatter plot creates a visual picture of the relationship between two variables, while the correlation coefficient quantifies the existence of any linear relationship. Large absolute values of the correlation coefficient indicate strong linear relationships. Positive coefficients indicate a positive relationship and negative coefficients indicate a negative relationship between two data sets. In Examples 8-8 and 8-9, we saw that relatively small sample correlation coefficients (0.2860 and 0.1119) can be statistically significant and thus might provide valuable information about the behavior of economic variables. Next we will introduce linear regression, another tool useful in examining the relationship between two variables. 3. LINEAR REGRESSION 3.1. Linear Regression with One Independent Variable As a financial analyst, you will often want to understand the relationship between financial or economic variables, or to predict the value of one variable using information about the value of another variable. For example, you may want to know the impact of changes in the 10-year Treasury bond yield on the earnings yield of the S&P 500 (the earnings yield is the reciprocal of the price-to-earnings ratio). If the relationship between those two variables is linear, you can use linear regression to summarize it. Linear regression allows us to use one variable to make predictions about another, test hypotheses about the relation between two variables, and quantify the strength of the relationship between the two variables. The remainder of this chapter focuses on linear regression with a single independent variable. In the next chapter, we will examine regression with more than one independent variable. Regression analysis begins with the dependent variable (denoted Y ), the variable that you are seeking to explain. The independent variable (denoted X ) is the variable you are using to explain changes in the dependent variable. For example, you might try to explain small-stock returns (the dependent variable) based on returns to the S&P 500 (the independent variable). Or you might try to explain inflation (the dependent variable) as a function of growth in a country’s money supply (the independent variable). Linear regression assumes a linear relationship between the dependent and the independent variables. The following regression equation describes that relation: Yi = b0 + b1 Xi + εi , i = 1, . . . , n 16 (8-4) It is worth repeating that the smaller the sample, the greater the evidence in terms of the magnitude of the sample correlation needed to reject the null hypothesis of zero correlation. With a sample size of 6, the absolute value of the sample correlation would need to be greater than 0.81 (carrying two decimal places) for us to reject the null hypothesis. Viewed another way, the value of 0.4045 in the text would be significant if the sample size were 24, because 0.4045(24 − 2)1/2 /(1 − 0.40452 )1/2 = 2.075, which is greater than the critical t-value of 2.074 at the 0.05 significance level with 22 degrees of freedom. 301 Chapter 8 Correlation and Regression This equation states that the dependent variable, Y , is equal to the intercept, b0 , plus a slope coefficient, b1 , times the independent variable, X , plus an error term, ε. The error term represents the portion of the dependent variable that cannot be explained by the independent variable. We refer to the intercept b0 and the slope coefficient b1 as the regression coefficients. Regression analysis uses two principal types of data: cross-sectional and time series. Cross-sectional data involve many observations on X and Y for the same time period. Those observations could come from different companies, asset classes, investment funds, people, countries, or other entities, depending on the regression model. For example, a cross-sectional model might use data from many companies to test whether predicted earnings-per-share growth explains differences in price-to-earnings ratios (P/Es) during a specific time period. The word ‘‘explain’’ is frequently used in describing regression relationships. One estimate of a company’s P/E that does not depend on any other variable is the average P/E. If a regression of a P/E on an independent variable tends to give more-accurate estimates of P/E than just assuming that the company’s P/E equals the average P/E, we say that the independent variable helps explain P/Es because using that independent variable improves our estimates. Finally, note that if we use cross-sectional observations in a regression, we usually denote the observations as i = 1, 2, . . . , n. Time-series data use many observations from different time periods for the same company, asset class, investment fund, person, country, or other entity, depending on the regression model. For example, a time-series model might use monthly data from many years to test whether U.S. inflation rates determine U.S. short-term interest rates.17 If we use time-series data in a regression, we usually denote the observations as t = 1, 2, . . . , T .18 Exactly how does linear regression estimate b0 and b1 ? Linear regression, also known as linear least squares, computes a line that best fits the observations; it chooses values for the intercept, b0 , and slope, b1 , that minimize the sum of the squared vertical distances between the observations and the regression line. Linear regression chooses the estimated or fitted parameters b̂0 and b̂1 in Equation 8-4 to minimize19 n ! i=1 (Yi − b̂0 − b̂1 Xi )2 (8-5) In this equation, the term (Yi − b̂0 − b̂1 Xi )2 means (Dependent variable–Predicted value of dependent variable)2 . Using this method to estimate the values of b̂0 and b̂1 , we can fit a line through the observations on X and Y that best explains the value that Y takes for any particular value of X .20 Note that we never observe the population parameter values b0 and b1 in a regression model. Instead, we observe only b̂0 and b̂1 , which are estimates of the population parameter values. Thus predictions must be based on the parameters’ estimated values, and testing is based on estimated values in relation to the hypothesized population values. 17 A mix of time-series and cross-sectional data, also known as panel data, is now frequently used in financial analysis. The analysis of panel data is an advanced topic that Greene (2003) discusses in detail. 18 In this chapter, we primarily use the notation i = 1, 2, . . . , n even for time series to prevent confusion that would be caused by switching back and forth between different notations. 19 Hats over the symbols for coefficients indicate estimated values. 20 For a discussion of the precise statistical sense in which the estimates of b and b are optimal, see 0 1 Greene (2003). 302 Quantitative Investment Analysis Inflation (%) 10 8 6 4 2 0 0 2 4 6 8 10 12 14 Money Growth (%) FIGURE 8-8 Fitted Regression Line Explaining the Inflation Rate Using Growth in the Money Supply by Country: 1970–2001 Source: International Monetary Fund. Figure 8-8 gives a visual example of how linear regression works. The figure shows the linear regression that results from estimating the regression relation between the annual rate of inflation (the dependent variable) and annual rate of money supply growth (the independent variable) for six industrialized countries from 1970 to 2001 (n = 6).21 The equation to be estimated is Long-term rate of inflation = b0 + b1 (Long-term rate of money supply growth) +ε. The distance from each of the six data points to the fitted regression line is the regression residual, which is the difference between the actual value of the dependent variable and the predicted value of the dependent variable made by the regression equation. Linear regression chooses the estimated coefficients b̂0 and b̂1 in Equation 8-4 such that the sum of the squared vertical distances is minimized. The estimated regression equation is Long-term inflation = 0.0084 + 0.5545 (Long-term money supply growth).22 According to this regression equation, if the long-term money supply growth is 0 for any particular country, the long-term rate of inflation in that country will be 0.84 percent. For every 1-percentage-point increase in the long-term rate of money supply growth for a country, the long-term inflation rate is predicted to increase by 0.5545 percentage points. In a regression such as this one, which contains one independent variable, the slope coefficient equals Cov(Y , X )/Var(X ). We can solve for the slope coefficient using data from Table 8-2, excerpted here: 21 These 22 We data appear in Table 8-2. entered the monthly returns as decimals. 303 Chapter 8 Correlation and Regression TABLE 8-2 (excerpted) Sum Average Covariance Variance Standard deviation Money Supply Growth Rate Xi Inflation Rate Yi 0.5608 0.0935 0.3616 0.0603 Cross-Product (Xi − X )(Yi − Y ) Squared Deviations (Xi − X )2 Squared Deviations (Yi − Y )2 0.002185 0.003939 0.001598 0.000788 0.028071 0.000320 0.017889 0.000437 Cov (Y , X ) = 0.000437 Var (X ) = 0.000788 Cov (Y , X )/Var (X ) = 0.000437/0.000788 b̂1 = 0.5545 In a linear regression, the regression line fits through the point corresponding to the means of the dependent and the independent variables. As shown in Table 8-1 (excerpted below), from 1970 to 2001, the mean long-term growth rate of the money supply for these six countries was 9.35 percent, whereas the mean long-term inflation rate was 6.03 percent. TABLE 8-1 (excerpted) Average Money Supply Growth Rate Inflation Rate 9.35% 6.03% Because the point (9.35, 6.03) lies on the regression line b̂0 = Y − b̂1 X , we can solve for the intercept using this point as follows: b̂0 = 0.0603 − 0.5545(0.0935) = 0.0084 We are showing how to solve the linear regression equation step by step to make the source of the numbers clear. Typically, an analyst will use the data analysis function on a spreadsheet or a statistical package to perform linear regression analysis. Later, we will discuss how to use regression residuals to quantify the uncertainty in a regression model. 3.2. Assumptions of the Linear Regression Model We have discussed how to interpret the coefficients in a linear regression model. Now we turn to the statistical assumptions underlying this model. Suppose that we have n observations on both the dependent variable, Y , and the independent variable, X , and we want to estimate Equation 8-4: Yi = b0 + b1 Xi + εi , i = 1, . . . , n 304 Quantitative Investment Analysis To be able to draw valid conclusions from a linear regression model with a single independent variable, we need to make the following six assumptions, known as the classic normal linear regression model assumptions: 1. The relationship between the dependent variable, Y , and the independent variable, X is linear in the parameters b0 and b1 . This requirement means that b0 and b1 are raised to the first power only and that neither b0 nor b1 is multiplied or divided by another regression parameter (as in b0 /b1 , for example). The requirement does not exclude X from being raised to a power other than 1. 2. The independent variable, X , is not random.23 3. The expected value of the error term is 0: E(ε) = 0. 4. The variance of the error term is the same for all observations: E(ε2i ) = σε 2 , i = 1, . . . , n. 5. The error term, ε, is uncorrelated across observations. Consequently, E(εi εj ) = 0 for all i not equal to j.24 6. The error term, ε, is normally distributed.25 Now we can take a closer look at each of these assumptions. Assumption 1 is critical for a valid linear regression. If the relationship between the independent and dependent variables is nonlinear in the parameters, then estimating that relation with a linear regression model will produce invalid results. For example, Yi = b0 eb1 Xi + εi is nonlinear in b1 , so we could not apply the linear regression model to it.26 Even if the dependent variable is nonlinear, linear regression can be used as long as the regression is linear in the parameters. So, for example, linear regression can be used to estimate the equation Yi = b0 + b1 Xi2 + εi . Assumptions 2 and 3 ensure that linear regression produces the correct estimates of b0 and b1 . Assumptions 4, 5, and 6 let us use the linear regression model to determine the distribution of the estimated parameters b̂0 and b̂1 and thus test whether those coefficients have a particular value. • 23 Assumption 4, that the variance of the error term is the same for all observations, is also known as the homoskedasticity assumption. The chapter on regression analysis discusses how to test for and correct violations of this assumption. Although we assume that the independent variable in the regression model is not random, that assumption is clearly often not true. For example, it is unrealistic to assume that the monthly returns to the S&P 500 are not random. If the independent variable is random, then is the regression model incorrect? Fortunately, no. Econometricians have shown that even if the independent variable is random, we can still rely on the results of regression models given the crucial assumption that the error term is uncorrelated with the independent variable. The mathematics underlying this reliability demonstration, however, are quite difficult. See, for example, Greene (2003) or Goldberger (1998). 24 Var(εi ) = E[εi − E(εi )]2 = E(εi − 0)2 = E(εi )2 . Cov(εi , εj ) = E{[εi − E(εi )][εj − E(εj )]} = E[(εi − 0)(εj − 0)] = E(εi εj ) = 0. 25 If the regression errors are not normally distributed, we can still use regression analysis. Econometricians who dispense with the normality assumption use chi-square tests of hypotheses rather than F -tests. This difference usually does not affect whether the test will result in a particular null hypothesis being rejected. 26 For more information on nonlinearity in the parameters, see Gujarati (2003). Chapter 8 Correlation and Regression 305 Assumption 5, that the errors are uncorrelated across observations, is also necessary for correctly estimating the variances of the estimated parameters b̂0 and b̂1 . The chapter on multiple regression discusses violations of this assumption. • Assumption 6, that the error term is normally distributed, allows us to easily test a particular hypothesis about a linear regression model.27 • EXAMPLE 8-11 Evaluating Economic Forecasts (2) If economic forecasts were completely accurate, every prediction of change in an economic variable in a quarter would exactly match the actual change that occurs in that quarter. Even though forecasts can be inaccurate, we hope at least that they are unbiased—that is, that the expected value of the forecast error is zero. An unbiased forecast can be expressed as E(Actual change − Predicted change) = 0. In fact, most evaluations of forecast accuracy test whether forecasts are unbiased.28 Figure 8-9 repeats Figure 8-7 in showing a scatter plot of the mean forecast of current-quarter percentage change in CPI from the previous quarter and actual percentage change in CPI, on an annualized basis, from the first quarter of 1983 to the last quarter of 2002, but it adds the fitted regression line for the equation Actual percentage change = b0 + b1 (Predicted percentage change) + ε. If the forecasts are unbiased, the intercept, b0 , should be 0 and the slope, b1 , should be 1. We should also find E(Actual change − Predicted change) = 0. If forecasts are actually unbiased, as long as b0 = 0 and b1 = 1, the error term [Actual change − b0 − b1 (Predicted change)] will have an expected value of 0, as required by Assumption 3 of the linear regression model. With unbiased forecasts, any other values of b0 and b1 would yield an error term with an expected value different from 0. If b0 = 0 and b1 = 1, our best guess of actual change in CPI would be 0 if professional forecasters’ predictions of change in CPI were 0. For every 1-percentage-point increase in the prediction of change by the professional forecasters, the regression model would predict a 1-percentage-point increase in actual change. The fitted regression line in Figure 8-9 comes from the equation Actual change = −0.0140 + 0.9637(Predicted change). Note that the estimated values of both b0 and b1 are close to the values b0 = 0 and b1 = 1 that are consistent with unbiased forecasts. Later in this chapter, we discuss how to test the hypotheses that b0 = 0 and b1 = 1. 27 For large sample sizes, we may be able to drop the assumption of normality by appeal to the central limit theorem, which was discussed in the chapter on sampling; see Greene (2003). Asymptotic theory shows that, in many cases, the test statistics produced by standard regression programs are valid even if the error term is not normally distributed. As illustrated in the chapter on statistical concepts and market returns, however, non-normality of some financial time series can be quite severe. With severe non-normality, even with a relatively large number of observations, invoking asymptotic theory to justify using test statistics from linear regression models may be inappropriate. 28 See, for example, Keane and Runkle (1990). 306 Quantitative Investment Analysis Actual Change (Annualized %) 10 8 6 4 2 Correlation = 0.7138 0 !2 !2 0 2 4 6 8 10 Predicted Change (Annualized %) FIGURE 8-9 Actual Change in CPI vs. Predicted Change Source: Federal Reserve Banks of Philadelphia and St. Louis. 3.3. The Standard Error of Estimate The linear regression model sometimes describes the relationship between two variables quite well, but sometimes it does not. We must be able to distinguish between these two cases in order to use regression analysis effectively. Therefore, in this section and the next, we discuss statistics that measure how well a given linear regression model captures the relationship between the dependent and independent variables. Figure 8-9, for example, shows a strong relation between predicted inflation and actual inflation. If we knew professional forecasters’ predictions for inflation in a particular quarter, we would be reasonably certain that we could use this regression model to forecast actual inflation relatively accurately. In other cases, however, the relation between the dependent and independent variables is not strong. Figure 8-10 adds a fitted regression line to the data on inflation and stock returns in the 1990s from Figure 8-6. In this figure, the actual observations are generally much farther 307 Chapter 8 Correlation and Regression from the fitted regression line than in Figure 8-9. Using the estimated regression equation to predict monthly stock returns assuming a particular level of inflation might result in an inaccurate forecast. As noted, the regression relation in Figure 8-10 is less precise than that in Figure 8-9. The standard error of estimate (sometimes called the standard error of the regression) measures this uncertainty. This statistic is very much like the standard deviation for a single variable, except that it measures the standard deviation of ε̂i , the residual term in the regression. The formula for the standard error of estimate (SEE) for a linear regression model with one independent variable is SEE = # n $1/2 ! (Yi − b̂0 − b̂1 Xi )2 i=1 n−2 $1/2 # n ! (ε̂i )2 = n−2 i=1 (8-6) In the numerator of this equation, we are computing the difference between the dependent variable’s actual value for each observation and its predicted value (b̂0 + b̂1 Xi ) for each observation. The difference between the actual and predicted values of the dependent variable is the regression residual, ε̂i . Equation 8-6 looks very much like the formula for computing a standard deviation, except that n − 2 appears in the denominator instead of n − 1. We use n − 2 because the sample includes n observations and the linear regression model estimates two parameters (b̂0 and b̂1 ); the difference between the number of observations and the number of parameters is n − 2. This difference is also called the degrees of freedom; it is the denominator needed to ensure that the estimated standard error of estimate is unbiased. Monthly Total Returns (S&P 500) (%) 15 10 5 0 !5 !10 !15 !20 !0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 Monthly Inflation (%) FIGURE 8-10 Fitted Regression Line Explaining Stock Returns by Inflation During the 1990s Source: Ibbotson Associates. 308 Quantitative Investment Analysis EXAMPLE 8-12 Computing the Standard Error of Estimate Recall that the estimated regression equation for the inflation and money supply growth data shown in Figure 8-8 was Yi = 0.0084 + 0.5545Xi . Table 8-7 uses this estimated equation to compute the data needed for the standard error of estimate. TABLE 8-7 Computing the Standard Error of Estimate Country Money Supply Growth Rate Xi Inflation Rate Yi Predicted Inflation Rate Ŷi Australia Canada New Zealand Switzerland United Kingdom United States 0.1166 0.0915 0.1060 0.0575 0.1258 0.0634 0.0676 0.0519 0.0815 0.0339 0.0758 0.0509 0.0731 0.0591 0.0672 0.0403 0.0782 0.0436 Regression Residual Yi − Ŷi Squared Residual (Yi − Ŷi )2 −0.0055 −0.0072 0.0143 −0.0064 −0.0024 0.0073 0.000030 0.000052 0.000204 0.000041 0.000006 0.000053 Sum 0.000386 Source: International Monetary Fund. The first and second columns of numbers in Table 8-7 show the long-term money supply growth rates, Xi , and long-term inflations rates, Yi , for the six countries. The third column of numbers shows the predicted value of the dependent variable from the fitted regression equation for each observation. For the United States, for example, the predicted value of long-term inflation is 0.0084 + 0.5545(0.0634) = 0.0436 or 4.36 percent. The next-to-last column contains the regression residual, which is the difference between the actual value of the dependent variable, Yi , and the predicted value of the dependent variable, (Ŷi = b̂0 + b̂1 Xi ). So for the United States, the residual is equal to 0.0509 − 0.0436 = 0.0073 or 0.73 percent. The last column contains the squared regression residual. The sum of the squared residuals is 0.000386. Applying the formula for the standard error of estimate, we obtain % 0.000386 6−2 &1/2 = 0.009823 Thus the standard error of estimate is about 0.98 percent. Later, we will combine this estimate with estimates of the uncertainty about the parameters in this regression to determine confidence intervals for predicting inflation rates from money supply growth. We will see that smaller standard errors result in more-accurate predictions. 309 Chapter 8 Correlation and Regression 3.4. The Coefficient of Determination Although the standard error of estimate gives some indication of how certain we can be about a particular prediction of Y using the regression equation, it still does not tell us how well the independent variable explains variation in the dependent variable. The coefficient of determination does exactly this: It measures the fraction of the total variation in the de-pendent variable that is explained by the independent variable. We can compute the coefficient of determination in two ways. The simpler method, which can be used in a linear regression with one independent variable, is to square the correlation coefficient between the dependent and independent variables. For example, recall that the correlation coefficient between the long-term rate of money growth and the long-term rate of inflation between 1970 and 2001 for six industrialized countries was 0.8702. Thus the coefficient of determination in the regression shown in Figure 8-8 is (0.8702)2 = 0.7572. So in this regression, the long-term rate of money supply growth explains approximately 76 percent of the variation in the long-term rate of inflation across the countries between 1970 and 2001. The problem with this method is that it cannot be used when we have more than one independent variable.29 Therefore, we need an alternative method of computing the coefficient of determination for multiple independent variables. We now present the logic behind that alternative. If we did not know the regression relationship, our best guess for the value of any particular observation of the dependent variable would simply be Y , the mean of the dependent variable. One measure of accuracy in predicting Yi based on Y is the sample n ! (Yi − Y )2 variance of Yi , . An alternative to using Y to predict a particular observation Yi n−1 i=1 is using the regression relationship to make that prediction. In that case, our predicted value would be Ŷi = b̂0 + b̂1 Xi . If the regression relationship works well, the error in predicting Yi using Ŷi should be much smaller than the error in predicting Yi using Y . If we call n n ! ! 2 (Yi − Y ) the total variation of Y and (Yi − Ŷi )2 the unexplained variation from the i=1 i=1 regression, then we can measure the explained variation from the regression using the following equation: Total variation = Unexplained variation + Explained variation (8-7) The coefficient of determination is the fraction of the total variation that is explained by the regression. This gives us the relationship Total variation − Unexplained variation Explained variation = (8-8) Total variation Total variation Unexplained variation =1− Total variation Note that total variation equals explained variation plus unexplained variation, as shown in Equation 8-7. Most regression programs report the coefficient of determination as R 2 .30 R2 = 29 We will discuss such models in the chapter on multiple regression. As we illustrate in the tables of regression output later in this chapter, regression programs also report multiple R, which is the correlation between the actual values and the forecast values of Y . The coefficient of determination is the square of multiple R. 30 310 Quantitative Investment Analysis EXAMPLE 8-13 Inflation Rate and Growth in the Money Supply Using the data in Table 8-7, we can see that the unexplained variation from the regression, which is the sum of the squared residuals, equals 0.000386. Table 8-8 shows the computation of total variation in the dependent variable, the long-term rate of inflation. TABLE 8-8 Computing Total Variation Country Australia Canada New Zealand Switzerland United Kingdom United States Money Supply Growth Rate Xi 0.1166 0.0915 0.1060 0.0575 0.1258 0.0634 Average: Inflation Rate Yi 0.0676 0.0519 0.0815 0.0339 0.0758 0.0509 0.0603 Deviation from Mean Yi − Y 0.0073 −0.0084 0.0212 −0.0264 0.0155 −0.0094 Sum: Squared Deviation (Yi − Y )2 0.000053 0.000071 0.000449 0.000697 0.000240 0.000088 0.001598 Source: International Monetary Fund. The average inflation rate for this period is 6.03 percent. The next-to-last column shows the amount each country’s long-term inflation rate deviates from that average; the last column shows the square of that deviation. The sum of those squared deviations is the total variation in Y for the sample (0.001598), shown in Table 8-8. The coefficient of determination for the regression is Total variation − Unexplained variation 0.001598 − 0.000386 = = 0.7584 Total variation 0.001598 Note that this method gives the same result rounded to two decimal places, 0.76, that we obtained earlier (the difference at greater decimal places results from rounding). We will use this method again in the chapter on multiple regression; when we have more than one independent variable, this method is the only way to compute the coefficient of determination. 3.5. Hypothesis Testing In this section, we address testing hypotheses concerning the population values of the intercept or slope coefficient of a regression model. This topic is critical in practice. For example, we may want to check a stock’s valuation using the capital asset pricing model; we hypothesize that the stock has a market-average beta or level of systematic risk. Or we may want to test the hypothesis that economists’ forecasts of the inflation rate are unbiased (not overestimates or 311 Chapter 8 Correlation and Regression underestimates, on average). In each case, does the evidence support the hypothesis? Questions such as these can be addressed with hypothesis tests within a regression model. Such tests are often t-tests of the value of the intercept or slope coefficient(s). To understand the concepts involved in this test, it is useful to first review a simple, equivalent approach based on confidence intervals. We can perform a hypothesis test using the confidence interval approach if we know three things: (1) the estimated parameter value, b̂0 or b̂1 , (2) the hypothesized value of the parameter, b0 or b1 , and (3) a confidence interval around the estimated parameter. A confidence interval is an interval of values that we believe includes the true parameter value, b1 , with a given degree of confidence. To compute a confidence interval, we must select the significance level for the test and know the standard error of the estimated coefficient. Suppose we regress a stock’s returns on a stock market index’s returns and find that the slope coefficient (b̂1 ) is 1.5 with a standard error (sb̂1 ) of 0.200. Assume we used 62 monthly observations in our regression analysis. The hypothesized value of the parameter (b1 ) is 1.0, the market average slope coefficient. The estimated and the population slope coefficients are often called beta, because the population coefficient is often represented by the Greek symbol beta (β) rather than the b1 we use in this text. Our null hypothesis is that b1 = 1.0 and b̂1 is the estimate for b1 . We will use a 95 percent confidence interval for our test, or we could say that the test has a significance level of 0.05. Our confidence interval will span the range b̂1 − tc sb̂1 to b̂1 + tc sb̂1 , or b̂1 ± tc sb̂1 (8-9) where tc is the critical t value.31 The critical value for the test depends on the number of degrees of freedom for the t-distribution under the null hypothesis. The number of degrees of freedom equals the number of observations minus the number of parameters estimated. In a regression with one independent variable, there are two estimated parameters, the intercept term and the coefficient on the independent variable. For 62 observations and two parameters estimated in this example, we have 60 degrees of freedom (62 − 2). For 60 degrees of freedom, the table of critical values in the back of the book shows that the critical t-value at the 0.05 significance level is 2.00. Substituting the values from our example into Equation 8-9 gives us the interval b̂1 ± tc sb̂1 = 1.5 ± 2.00(0.200) = 1.5 ± 0.400 = 1.10 to 1.90 Under the null hypothesis, the probability that the confidence interval includes b1 is 95 percent. Because we are testing b1 = 1.0 and because our confidence interval does not include 1.0, we can reject the null hypothesis. Therefore, we can be 95 percent confident that the stock’s beta is different from 1.0. 31 We use the t-distribution for this test because we are using a sample estimate of the standard error, sb , rather than its true (population) value. In the chapter on sampling and estimation, we discussed the concept of degrees of freedom. 312 Quantitative Investment Analysis In practice, the most common way to test a hypothesis using a regression model is with a t-test of significance. To test the hypothesis, we can compute the statistic t= b̂1 − b1 sb̂1 (8-10) This test statistic has a t-distribution with n − 2 degrees of freedom because two parameters were estimated in the regression. We compare the absolute value of the t-statistic to tc . If the absolute value of t is greater than tc , then we can reject the null hypothesis. Substituting the values from the above example into this relationship gives the t-statistic associated with the probability that the stock’s beta equals 1.0 (b1 = 1.0). t= b̂1 − b1 sb̂1 = (1.5 − 1.0)/0.200 = 2.50 Because t > tc , we reject the null hypothesis that b1 = 1.0. The t-statistic in the example above is 2.50, and at the 0.05 significance level, tc = 2.00; thus we reject the null hypothesis because t > tc . This statement is equivalent to saying that we are 95 percent confident that the interval for the slope coefficient does not contain the value 1.0. If we were performing this test at the 0.01 level, however, tc would be 2.66 and we would not reject the hypothesis because t would not be greater than tc at this significance level. A 99 percent confidence interval for the slope coefficient does contain the value 1.0. The choice of significance level is always a matter of judgment. When we use higher levels of confidence, the tc increases. This choice leads to wider confidence intervals and to a decreased likelihood of rejecting the null hypothesis. Analysts often choose the 0.05 level of significance, which indicates a 5 percent chance of rejecting the null hypothesis when, in fact, it is true (a Type I error). Of course, decreasing the level of significance from 0.05 to 0.01 decreases the probability of Type I error, but it increases the probability of Type II error—failing to reject the null hypothesis when, in fact, it is false.32 Often, financial analysts do not simply report whether or not their tests reject a particular hypothesis about a regression parameter. Instead, they report the p-value or probability value for a particular hypothesis. The p-value is the smallest level of significance at which the null hypothesis can be rejected. It allows the reader to interpret the results rather than be told that a certain hypothesis has been rejected or accepted. In most regression software packages, the p-values printed for regression coefficients apply to a test of null hypothesis that the true parameter is equal to 0 against the alternative that the parameter is not equal to 0, given the estimated coefficient and the standard error for that coefficient. For example, if the p-value is 0.005, we can reject the hypothesis that the true parameter is equal to 0 at the 0.5 percent significance level (99.5 percent confidence). The standard error of the estimated coefficient is an important input for a hypothesis test concerning the regression coefficient (and for a confidence interval for the estimated coefficient). Stronger regression results lead to smaller standard errors of an estimated parameter and result 32 For a full discussion of Type I and Type II errors, see the chapter on hypothesis testing. 313 Chapter 8 Correlation and Regression in tighter confidence intervals. If the standard error (sb̂1 ) in the above example were 0.100 instead of 0.200, the confidence interval range would be half as large and the t-statistic twice as large. With a standard error this small, we would reject the null hypothesis even at the 0.01 significance level because we would have t = (1.5 − 1)/0.1 = 5.00 and tc = 2.66. With this background, we can turn to hypothesis tests using actual regression results. The next three examples illustrate hypothesis tests in a variety of typical investment contexts. EXAMPLE 8-14 Estimating Beta for General Motors Stock You are an investor in General Motors stock and want an estimate of its beta. As in the text example, you hypothesize that GM has an average level of market risk and that its required return in excess of the risk-free rate is the same as the market’s required excess return. One regression that summarizes these statements is (R − RF ) = α + β(RM − RF ) + ε (8-11) where RF is the periodic risk-free rate of return (known at the beginning of the period), RM is the periodic return on the market, R is the periodic return to the stock of the company, and β is the covariance of stock and market return divided by the variance of the market return, Cov(R, RM )/σM 2 . Estimating this equation with linear regression provides an estimate of β, β̂, which tells us the size of the required return premium for the security, given expectations about market returns.33 Suppose we want to test the null hypothesis, H0 , that β = 1 for GM stock to see whether GM stock has the same required return premium as the market as a whole. We need data on returns to GM stock, a risk-free interest rate, and the returns to the market index. For this example, we use data from January 1998 through December 2002 (n = 60). The return to GM stock is R. The monthly return to 30-day Treasury bills is RF . The return to the S&P 500 is RM .34 We are estimating two parameters, so the number of degrees of freedom is n − 2 = 60 − 2 = 58. Table 8-9 shows the results from the regression (R − RF ) = α + β(RM − RF ) + ε. We are testing the null hypothesis, H0 , that β for GM equals 1 (β = 1) against the alternative hypothesis that β does not equal 1 (β ̸= 1). The estimated β̂ from the regression is 1.1958. The estimated standard error for that coefficient in the regression, sβ̂ is 0.2354. The regression equation has 58 degrees of freedom (60 − 2), so the critical value for the test statistic is approximately tc = 2.00 at the 0.05 significance 33 Beta (β) is typically estimated using 60 months of historical data, but the data-sample length sometimes varies. Although monthly data is typically used, some financial analysts estimate β using daily data. For more information on methods of estimating β, see Reilly and Brown (2003). The expected excess return for GM stock above the risk-free rate (R − RF ) is β(RM − RF ), given a particular excess return to the market above the risk-free rate (RM − RF ). This result holds because we regress (R − RF ) against (RM − RF ). For example, if a stock’s beta is 1.5, its expected excess return is 1.5 times that of the market portfolio. 34 Data on GM stock returns came from Bloomberg. Data on T-bill returns and S&P 500 returns came from Ibbotson Associates. 314 Quantitative Investment Analysis level. Therefore, the 95 percent confidence interval for the data for any particular hypothesized value of β is shown by the range β̂ ± tc sβ̂ 1.1958 ± 2.00(0.2354) 0.7250 to 1.6666 In this case, the hypothesized parameter value is β = 1, and the value 1 falls inside this confidence interval, so we cannot reject the hypothesis at the 0.05 significance level. This means that we cannot reject the hypothesis that GM stock has the same systematic risk as the market as a whole. TABLE 8-9 Estimating Beta for GM Stock Regression Statistics Multiple R R-squared Standard error of estimate Observations Alpha Beta 0.5549 0.3079 0.0985 60 Coefficients Standard Error 0.0036 1.1958 0.0127 0.2354 t-Statistic 0.2840 5.0795 Source: Ibbotson Associates and Bloomberg L.P. Another way of looking at this issue is to compute the t-statistic for the GM beta hypothesized parameter using Equation 8-10: t= β̂ − β 1.1958 − 1.0 = 0.8318 = sβ̂ 0.2354 This t-statistic is less than the critical t-value of 2.00. Therefore, neither approach allows us to reject the null hypothesis. Note that the t-statistic associated with β̂ in the regression results in Table 8-9 is 5.0795. Given the significance level we are using, we cannot reject the null hypothesis that β = 1, but we can reject the hypothesis that β = 0.35 Note also that the R 2 in this regression is only 0.3079. This result suggests that only about 31 percent of the total variation in the excess return to GM stock (the return to GM above the risk-free rate) can be explained by excess return to the market portfolio. The remaining 69 percent of GM stock’s excess return variation is the nonsystematic component, which can be attributed to company-specific risk. 35 The t-statistics for a coefficient automatically reported by statistical software programs assume that the null hypothesis states that the coefficient is equal to 0. If you have a different null hypothesis, as we do in this example (β = 1), then you must either construct the correct test statistic yourself or instruct the program to compute it. 315 Chapter 8 Correlation and Regression In the next example, we show a regression hypothesis test with a one-sided alternative. EXAMPLE 8-15 Explaining Company Value Based on Returns to Invested Capital Some financial analysts have argued that one good way to measure a company’s ability to create wealth is to compare the company’s return on invested capital (ROIC) to its weighted-average cost of capital (WACC). If a company has an ROIC greater than its cost of capital, the company is creating wealth; if its ROIC is less than its cost of capital, it is destroying wealth.36 Enterprise value (EV) is a market-price-based measure of company value defined as the market value of equity and debt minus the value of cash and investments. Invested capital (IC) is an accounting measure of company value defined as the sum of the book values of equity and debt. Higher ratios of EV to IC should reflect greater success at wealth creation in general. Mauboussin (1996) argued that the spread between ROIC and WACC helps explains the ratio of EV to IC. Using data on companies in the foodprocessing industry, we can test the relationship between EV/IC and (ROIC–WACC) using the regression model given in Equation 8-12. EVi /ICi = b0 + b1 (ROICi − WACCi ) + εi (8-12) where the subscript i is an index to identify the company. Our null hypothesis is H0: b1 ≤ 0, and we specify a significance level of 0.05. If we reject the null hypothesis, we have evidence of a statistically significant relationship between EV/IC and (ROIC–WACC). We estimate Equation 8-12 using data from nine food-processing companies for 2001.37 The results of this regression are displayed in Table 8-10 and Figure 8-11. TABLE 8-10 Explaining Enterprise Value/Invested Capital by the ROIC–WACC Spread Regression Statistics Multiple R R-squared Standard error of estimate Observations Intercept Spread 0.9469 0.8966 0.7422 9 Coefficients Standard Error t-Statistic 1.3478 30.0169 0.3511 3.8519 3.8391 7.7928 Source: Nelson (2003). 36 See, for example, Stewart (1991) and Mauboussin (1996). data come from Nelson (2003). Many sell-side analysts use this type of regression. It is one of the most frequently used cross-sectional regressions in published analyst reports. 37 Our 316 Quantitative Investment Analysis EV/IC 9 8 7 6 5 4 3 2 1 0 −5 0 5 10 15 20 25 ROIC-WACC (%) FIGURE 8-11 Fitted Regression Line Explaining Enterprise Value/Invested Capital Using ROIC–WACC Spread for the Food Industry Source: CSFB Food Investors Handbook 2003. We reject the null hypothesis based on the t-statistic of approximately 7.79 on estimated slope coefficient. There is a strong positive relationship between the return spread (ROIC–WACC) and the ratio of EV to IC in our sample of companies. Figure 8-11 illustrates the strong positive relationship. The R 2 of 0.8966 indicates that the return spread explains about 90 percent of the variation in the ratio of EV to IC among the food-processing companies in the sample in 2001. The coefficient on the return spread of 30.0169 implies that the predicted increase in EV/IC is 0.01(30.0169) = 0.3002 or about 30 percent for a 1-percentage-point increase in the return spread, for our sample of companies. In the final example of this section, we show that the null hypothesis can involve a slope coefficient of 1 just as well as a slope of 0. EXAMPLE 8-16 Testing Whether Inflation Forecasts Are Unbiased Example 8-11 introduced the concept of testing for bias in forecasts. That example showed that if a forecast is unbiased, its expected error is 0. We can examine whether a time-series of forecasts for a particular economic variable is unbiased by comparing the forecast at each date with the actual value of the economic variable announced after the forecast. If the forecasts are unbiased, then, by definition, the average realized forecast 317 Chapter 8 Correlation and Regression error should be close to 0. In that case, the value of b0 (the intercept) should be 0 and the value of b1 (the slope) should be 1, as discussed in Example 8-11. Refer once again to Figure 8-9, which shows the current-quarter predictions of percentage change in CPI made by professional economic forecasters and the actual percentage change from the first quarter of 1983 through the fourth quarter of 2002 (n = 80). To test whether the forecasts are unbiased, we must estimate the regression shown in Example 8-11. We report the results of this regression in Table 8-11. The equation to be estimated is Actual percentage change in CPIt = b0 + b1 (Predicted changet ) + εt This regression estimates two parameters (the intercept and the slope); therefore, the regression has n − 2 = 80 − 2 = 78 degrees of freedom. TABLE 8-11 Testing Whether Forecasts of CPI Are Unbiased (Dependent Variable: CPI Change Expressed in Percent) Regression Statistics Multiple R R-squared Standard error of estimate Observations Intercept Forecast (slope) 0.7138 0.5095 1.0322 80 Coefficients Standard Error t-Statistic −0.0140 0.9637 0.3657 0.1071 −0.0384 9.0008 Sources: Federal Reserve Banks of Philadelphia and St. Louis. We can now test two null hypotheses about the parameters in this regression. Our first null hypothesis is that the intercept in this regression is 0 (H0: b0 = 0). The alternative hypothesis is that the intercept does not equal 0 (Ha : b0 ̸= 0). Our second null hypothesis is that the slope coefficient in this regression is 1 (H0: b1 = 1). The alternative hypothesis is that the slope coefficient does not equal 1 (Ha : b1 ̸= 1). To test the hypotheses about b0 and b1 , we must first decide on a critical value based on a particular significance level and then construct the confidence intervals for each parameter. If we choose the 0.05 significance level, with 78 degrees of freedom, the critical value, tc , is approximately 1.99. The estimated value of the parameter b̂0 is −0.0140, and the estimated value of the standard error for b̂0 (sb̂0 ) is 0.3657. Let B0 stand for any particular hypothesized value. Therefore, under the null hypothesis that b0 = B0 , a 95 percent confidence interval for b0 is b̂0 ± tc sb̂0 −0.0140 ± 1.99(0.3657) −0.7417 to 0.7137 318 Quantitative Investment Analysis In this case, B0 is 0. The value of 0 falls within this confidence interval, so we cannot reject the first null hypothesis that b0 = 0. We will explain how to interpret this result shortly. Our second null hypothesis is based on the same sample as our first null hypothesis. Therefore, the critical value for testing that hypothesis is the same as the critical value for testing the first hypothesis (tc = 1.99). The estimated value of the parameter b̂1 is 0.9637, and the estimated value of the standard error for b̂1 , sb̂1 , is 0.1071. Therefore, the 95 percent confidence interval for any particular hypothesized value of b1 can be constructed as follows: b̂1 ± tc sb̂1 0.9637 ± 1.99(0.1071) 0.7506 to 1.1768 In this case, our hypothesized value of b1 is 1. The value 1 falls within this confidence interval, so we cannot reject the null hypothesis that b1 = 1 at the 0.05 significance level. Because we did not reject either of the null hypotheses (b0 = 0, b1 = 1) about the parameters in this model, we cannot reject the hypothesis that the forecasts of CPI change were unbiased.38 As an analyst, you often will need forecasts of economic growth to help you make recommendations about asset allocation, expected returns, and other investment decisions. The hypothesis tests just conducted suggest that you cannot reject the hypothesis that the CPI predictions in the Survey of Professional Forecasters are unbiased. If you need an unbiased forecast of future percentage change in CPI for your asset-allocation decision, you might want to use these forecasts. 3.6. Analysis of Variance in a Regression with One Independent Variable Analysis of variance (ANOVA) is a statistical procedure for dividing the total variability of a variable into components that can be attributed to different sources.39 In regression analysis, we use ANOVA to determine the usefulness of the independent variable or variables in explaining variation in the dependent variable. An important statistical test conducted in analysis of variance is the F -test. The F -statistic tests whether all the slope coefficients in a linear regression are equal to 0. In a regression with one independent variable, this is a test of the null hypothesis H0: b1 = 0 against the alternative hypothesis Ha : b1 ̸= 0. To correctly determine the test statistic for the null hypothesis that the slope coefficient equals 0, we need to know the following: 38 Jointly testing the hypothesis b0 = 0 and b1 = 1 would require us to take into account the covariance of b̂0 and b̂1 . For information on testing joint hypotheses of this type, see Greene (2003). 39 In this chapter, we focus on regression applications of ANOVA, the most common context in which financial analysts will encounter this tool. In this context, ANOVA is used to test whether all the regression slope coefficients are equal to 0. Analysts also use ANOVA to test a hypothesis that the means of two or more populations are equal. See Daniel and Terrell (1995) for details. 319 Chapter 8 Correlation and Regression the total number of observations (n); the total number of parameters to be estimated (in a one-independent-variable regression, this number is two: the intercept and the slope coefficient); n ! • the sum of squared errors or residuals, (Yi − Ŷi )2 , abbreviated SSE. This value is also • • i=1 known as the residual sum of squares; and n ! • the regression sum of squares, (Ŷi − Y )2 , abbreviated RSS. This value is the amount of i=1 total variation in Y that is explained in the regression equation. Total variation (TSS) is the sum of SSE and RSS. The F -test for determining whether the slope coefficient equals 0 is based on an F -statistic, constructed using these four values. The F -statistic measures how well the regression equation explains the variation in the dependent variable. The F -statistic is the ratio of the average regression sum of squares to the average sum of the squared errors. The average regression sum of squares is computed by dividing the regression sum of squares by the number of slope parameters estimated (in this case, one). The average sum of squared errors is computed by dividing the sum of squared errors by the number of observations, n, minus the total number of parameters estimated (in this case, two: the intercept and the slope). These two divisors are the degrees of freedom for an F -test. If there are n observations, the F -test for the null hypothesis that the slope coefficient is equal to 0 is here denoted F# slope parameters, n−# parameters = F1,n−2 , and the test has 1 and n − 2 degrees of freedom. Suppose, for example, that the independent variable in a regression model explains none of the variation in the dependent variable. Then the predicted value for the regression model, Ŷi , is the average value of the dependent variable Y . In this case, the regression sum of squares n ! (Ŷi − Y )2 is 0. Therefore, the F -statistic is 0. If the independent variable explains little of i=1 the variation in the dependent variable, the value of the F -statistic will be very small. The formula for the F -statistic in a regression with one independent variable is F= Mean regression sum of squares RSS/1 = SSE/(n − 2) Mean squared error (8-13) If the regression model does a good job of explaining variation in the dependent variable, then this ratio should be high. The explained regression sum of squares per estimated parameter will be high relative to the unexplained variation for each degree of freedom. A table of critical values for this F -statistic is given in the back of this book. Even though the F -statistic is commonly computed by regression software packages, analysts typically do not use ANOVA and F -tests in regressions with just one independent variable. Why not? In such regressions, the F -statistic is the square of the t-statistic for the slope coefficient. Therefore, the F -test duplicates the t-test for the significance of the slope coefficient. This relation is not true for regressions with two or more slope coefficients. Nevertheless, the oneslope coefficient case gives a foundation for understanding the multiple-slope coefficient cases. Often, mutual fund performance is evaluated based on whether the fund has positive alpha—significantly positive excess risk-adjusted returns.40 One commonly used 40 Note that the Greek letter alpha, α, is traditionally used to represent the intercept in Equation 8-14 and should not be confused with another traditional usage of α to represent a significance level. 320 Quantitative Investment Analysis method of risk adjustment is based on the capital asset pricing model. Consider the regression (Ri − RF ) = αi + βi (RM − RF ) + εi (8-14) where RF is the periodic risk-free rate of return (known at the beginning of the period), RM is the periodic return on the market, Ri is the periodic return to Mutual Fund i, and βi is the fund’s beta. A fund has zero risk-adjusted excess return if αi = 0. If αi = 0, then (Ri − RF ) = βi (RM − RF ) + εi and taking expectations, E(Ri ) = RF + βi (RM − RF ), implying that βi completely explains the fund’s mean excess returns. If, for example, αi > 0, the fund is earning higher returns than expected given its beta. In summary, to test whether a fund has a positive alpha, we must test the null hypothesis that the fund has no risk-adjusted excess returns (H0: α = 0) against the alternative hypothesis of nonzero risk-adjusted returns (Ha : α ̸= 0). EXAMPLE 8-17 Performance Evaluation: The Dreyfus Appreciation Fund Table 8-12 presents results evaluating the excess return to the Dreyfus Appreciation Fund from January 1998 through December 2002. Note that the estimated beta in this regression, β̂i , is 0.7902. The Dreyfus Appreciation Fund was estimated to be about 0.8 times as risky as the market as a whole. TABLE 8-12 Performance Evaluation of Dreyfus Appreciation Fund, January 1998 to December 2002 Regression Statistics Multiple R R-squared Standard error of estimate Observations ANOVA Regression Residual Total Alpha Beta 0.9280 0.8611 0.0174 60 Degrees of Freedom (df ) Sum of Squares (SS) Mean Sum of Squares (MSS) 1 58 59 0.1093 0.0176 0.1269 0.1093 0.0003 F 359.64 Coefficients Standard Error t-Statistic 0.0009 0.7902 0.0023 0.0417 0.4036 18.9655 Source: Center for Research in Security Prices, University of Chicago. Note also that the estimated alpha (α̂) in this regression is positive (0.0009). The value of the coefficient is only a little more than one-third the size of the standard error for that coefficient (0.0023), so the t-statistic for the coefficient is only 0.4036. Therefore, we cannot reject the null hypothesis (α = 0) that the fund did not have Chapter 8 Correlation and Regression 321 a significant excess return beyond the return associated with the market risk of the fund. This result means that the returns to the fund were explained by the market risk of the fund and there was no additional statistical significance to the excess returns to the fund during this period.41 Because the t-statistic for the slope coefficient in this regression is 18.9655, the p-value for that coefficient is less than 0.0001 and is approximately zero. Therefore, the probability that the true value of this coefficient is actually 0 is microscopic. How can we use an F -test to determine whether the slope coefficient in this regression is equal to 0? The ANOVA portion of Table 8-12 provides the data we need. In this case, • • • • the total number of observations (n) is 60; the total number of parameters to be estimated is 2 (intercept and slope); the sum of squared errors or residuals, SSE, is 0.0176; and the regression sum of squares, RSS, is 0.1093. Therefore, the F -statistic to test whether the slope coefficient is equal to 0 is 0.1093/1 = 360.19 0.0176/(60 − 2) (The slight difference from the F -statistic in Table 8-12 is due to rounding.) The ANOVA output would show that the p-value for this F -statistic is less than 0.0001 and is exactly the same as the p-value for the t-statistic for the slope coefficient. Therefore, the F -test tells us nothing more than we already knew from the t-test. Note also that the F -statistic (359.64) is the square of the t-statistic (18.9655). 3.7. Prediction Intervals Financial analysts often want to use regression results to make predictions about a dependent variable. For example, we might ask, ‘‘How fast will the sales of XYZ Corporation grow this year if real GDP grows by 4 percent?’’ But we are not merely interested in making these forecasts; we also want to know how certain we should be about the forecasts’ results. For example, if we predicted that sales for XYZ Corporation would grow by 6 percent this year, our prediction would mean more if we were 95 percent confident that sales growth would fall in the interval from 5 percent to 7 percent, rather than only 25 percent confident that this outcome would occur. Therefore, we need to understand how to compute confidence intervals around regression forecasts. We must take into account two sources of uncertainty when using the regression model Yi = b0 + b1 Xi + εi , i = 1, . . . , n and the estimated parameters, b̂0 and b̂1 , to make a 41 This example introduces a well-known investment use of regression involving the capital asset pricing model. Researchers, however, recognize qualifications to the interpretation of alpha from a linear regression. The systematic risk of a managed portfolio is controlled by the portfolio manager. If, as a consequence, portfolio beta is correlated with the return on the market (as could result from market timing), inferences on alpha based on least-squares beta, as here, can be mistaken. This advanced subject is discussed in Dybvig and Ross (1985a) and (1985b). 322 Quantitative Investment Analysis prediction. First, the error term itself contains uncertainty. The standard deviation of the error term, σε , can be estimated from the standard error of estimate for the regression equation. A second source of uncertainty in making predictions about Y , however, comes from uncertainty in the estimated parameters b̂0 and b̂1 . If we knew the true values of the regression parameters, b0 and b1 , then the variance of our prediction of Y , given any particular predicted (or assumed) value of X , would simply be s2 , the squared standard error of estimate. The variance would be s2 because the prediction, Ŷ , would come from the equation Ŷ = b0 + b1 X and (Y − Ŷ ) = ε. Because we must estimate the regression parameters b̂0 and b̂1 however, our prediction of Y , Ŷ , given any particular predicted value of X , is actually Ŷ = b̂0 + b̂1 X . The estimated variance of the prediction error, sf2 of Y , given X , is sf2 =s 2 ' 1 (X − X )2 1+ + n (n − 1)sx2 ( (8-15) This estimated variance depends on the squared standard error of estimate, s2 ; the number of observations, n; the value of the independent variable, X , used to predict the dependent variable; • the estimated mean, X ; and • variance, sx2 of the independent variable.42 • • • Once we have this estimate of the variance of the prediction error, determining a prediction interval around the prediction is very similar to estimating a confidence interval around an estimated parameter, as shown earlier in this chapter. We need to take the following four steps to determine the prediction interval for the prediction: 1. Make the prediction. 2. Compute the variance of the prediction error using Equation 8-15. 3. Choose a significance level, α, for the forecast. For example, the 0.05 level, given the degrees of freedom in the regression, determines the critical value for the forecast interval, tc . 4. Compute the (1 − α) percent prediction interval for the prediction, namely Ŷ ± tc sf . EXAMPLE 8-18 Predicting the Ratio of Enterprise Value to Invested Capital We continue with the example of explaining the ratio of enterprise value to invested capital among food-processing companies by the spread between the return to invested capital and the weighted-average cost of capital (ROIC–WAAC). In Example 8-15, we estimated the regression given in Table 8-10. 42 For a derivation of this equation, see Pindyck and Rubinfeld (1998). 323 Chapter 8 Correlation and Regression You are interested in predicting the ratio of enterprise value to invested capital for a company if the return spread between ROIC and WACC is 10 percentage points. TABLE 8-10 (repeated) Explaining Enterprise Value/Invested Capital by the ROIC–WACC Spread Regression Statistics Multiple R R-squared Standard error of estimate Observations Intercept Spread 0.9469 0.8966 0.7422 9 Coefficients Standard Error t-Statistic 1.3478 30.0169 0.3511 3.8519 3.8391 7.7928 Source: Nelson (2003). What is the 95 percent confidence interval for the ratio of enterprise value to invested capital for that company? Using the data provided in Table 8-10, take the following steps: 1. Make the prediction: Expected EV/IC = 1.3478 + 30.0169(0.10) = 4.3495. This regression suggests that if the return spread between ROIC and WACC (Xi ) is 10 percent, the EV/IC ratio will be 4.3495. 2. Compute the variance of the prediction error. To compute the variance of the forecast error, we must know the standard error of the estimate of the equation, s = 0.7422 (as shown in Table 8-10); • the mean return spread, X = 0.0647 (this computation is not shown in the table); and • the variance of the mean return spread in the sample, sx2 = 0.004641 (this computation is not shown in the table). • Using these data, you can compute the variance of the forecast error (sf2 ) for predicting EV/IC for a company with a 10 percent spread between ROIC and WACC. * ) 1 (0.10 − 0.0647)2 2 2 sf = 0.7422 1 + + 9 (9 − 1)0.004641 = 0.630556 In this example, the variance of the forecast error is 0.630556, and the standard deviation of the forecast error is sf = (0.630556)1/2 = 0.7941. 324 Quantitative Investment Analysis 3. Determine the critical value of the t-statistic. Given a 95 percent confidence interval and 9 − 2 = 7 degrees of freedom, the critical value of the t-statistic, tc , is 2.365 using the tables in the back of the book. 4. Compute the prediction interval. The 95 percent confidence interval for EV/IC extends from 4.3495 − 2.365(0.7941) to 4.3495 + 2.365(0.7941), or 2.4715 to 6.2275. In summary, if the spread between the ROIC and the WACC is 10 percent, the 95 percent prediction interval for EV/IC will extend from 2.4715 to 6.2275. The small sample size is reflected in the relatively large prediction interval. 3.8. Limitations of Regression Analysis Although this chapter has shown many of the uses of regression models for financial analysis, regression models do have limitations. First, regression relations can change over time, just as correlations can. This fact is known as the issue of parameter instability, and its existence should not be surprising as the economic, tax, regulatory, political, and institutional contexts in which financial markets operate change. Whether considering cross-sectional or timeseries regression, the analyst will probably face this issue. As one example, cross-sectional regression relationships between stock characteristics may differ between growth-led and value-led markets. As a second example, the time-series regression estimating the beta often yields significantly different estimated betas depending on the time period selected. In both cross-sectional and time-series contexts, the most common problem is sampling from more than one population, with the challenge of identifying when doing so is an issue. A second limitation to the use of regression results specific to investment contexts is that public knowledge of regression relationships may negate their future usefulness. Suppose, for example, an analyst discovers that stocks with a certain characteristic have had historically very high returns. If other analysts discover and act upon this relationship, then the prices of stocks with that characteristic will be bid up. The knowledge of the relationship may result in the relation no longer holding in the future. Finally, if the regression assumptions listed in Section 3.2 are violated, hypothesis tests and predictions based on linear regression will not be valid. Although there are tests for violations of regression assumptions, often uncertainty exists as to whether an assumption has been violated. This limitation will be discussed in detail in the chapter on multiple regression. CHAPTER 9 MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS 1. INTRODUCTION As financial analysts, we often need to use more-sophisticated statistical methods than correlation analysis or regression involving a single independent variable. For example, a trading desk interested in the costs of trading Nasdaq stocks might want information on the determinants of the bid–ask spread on the Nasdaq. A mutual fund analyst might want to know whether returns to a technology mutual fund behaved more like the returns to a growth stock index or like the returns to a value stock index. An investor might be interested in the factors that determine whether analysts cover a stock. We can answer these questions using linear regression with more than one independent variable—multiple linear regression. In Sections 2 and 3, we introduce and illustrate the basic concepts and models of multiple regression analysis. These models rest on assumptions that are sometimes violated in practice. In Section 4, we discuss three major violations of a regression assumption. We address practical concerns such as how to diagnose an assumption violation and what remedial steps to take when a model assumption has been violated. Section 5 outlines some guidelines for building good regression models and discusses ways that analysts sometimes go wrong in this endeavor. In a number of investment applications, we are in-terested in the probability that one of two outcomes occurs: For example, we may be interested in whether a stock has analyst coverage or not. Section 6 discusses a class of models, qualitative dependent variable models, that addresses such questions. 2. MULTIPLE LINEAR REGRESSION As investment analysts, we often hypothesize that more than one variable explains the behavior of a variable in which we are interested. The variable we seek to explain is called the dependent variable. The variables that we believe explain the dependent variable are called the independent variables.1 A tool that permits us to examine the relationship (if any) between the two types of variables is multiple linear regression. Multiple linear regression allows us to determine the effect of more than one independent variable on a particular dependent variable. 1 Independent variables are also called explanatory variables or regressors. 325 326 Quantitative Investment Analysis To give an example of how we might use this tool, suppose we want to know whether the bid–ask spread for stocks trading in a dealer market is affected by the number of market makers (dealers) for that stock and the market capitalization of the stock. We can address this question using the following multiple linear regression model: Yi = b0 + b1 X1i + b2 X2i + ϵi where Yi X1i X2i ϵi = the natural logarithm of the bid-ask spread for stock i (the dependent variable) = the natural logarithm of the number of market makers for stock i = the natural logarithm of the market capitalization of company i = the error term Of course, linear regression models can use more than two independent variables to explain the dependent variable. A multiple linear regression model has the general form Yi = b0 + b1 X1i + b2 X2i + · · · + bk Xki + ϵi , i = 1, 2, . . . , n (9-1) where Yi = the ith observation of the dependent variable Y Xji = the ith observation of the independent variable Xj , j = 1, 2, . . . , k b0 = the intercept of the equation b1 , . . . , bk = the slope coefficients for each of the independent variables ϵi = the error term n = the number of observations A slope coefficient, bj , measures how much the dependent variable, Y , changes when the independent variable, Xj , changes by one unit, holding all other independent variables constant. For example, if b1 = 1 and all of the other independent variables remain constant, then we predict that if X1 increases by one unit, Y will also increase by one unit. If b1 = −1 and all of the other independent variables are held constant, then we predict that if X1 increases by one unit, Y will decrease by one unit. Multiple linear regression estimates b0 , . . . , bk . In this chapter, we will refer to both the intercept, b0 , and the slope co-efficients, b1 , . . . , bk , as regression coefficients. As we proceed with our discussion, keep in mind that a regression equation has k slope coefficients and k + 1 regression coefficients. In practice, we use software to estimate a multiple regression model. Example 9-1 presents an application of multiple regression analysis in investment practice. In the course of discussing a hypothesis test, Example 9-1 presents typical regression output and its interpretation. EXAMPLE 9-1 Explaining the Bid–Ask Spread As the manager of the trading desk at an investment management firm, you have noticed that the average bid–ask spreads of different Nasdaq-listed stocks can vary widely. Chapter 9 Multiple Regression and Issues in Regression Analysis 327 When the ratio of a stock’s bid–ask spread to its price is higher than for another stock, your firm’s costs of trading in that stock tend to be higher. You have formulated the hypothesis that Nasdaq stocks’ percentage bid–ask spreads are related to the number of market makers and the company’s stock market capitalization. You have decided to investigate your hypothesis using multiple regression analysis. You specify a regression model in which the dependent variable measures the percentage bid–ask spread and the independent variables measure the number of market makers and the company’s stock market capitalization. The regression is estimated using data from December 2002 for 1,819 Nasdaq-listed stocks. Based on earlier published research exploring bid–ask spreads, you express the dependent and independent variables as natural logarithms, a so-called log-log regression model. A log-log regression model may be appropriate when one believes that proportional changes in the dependent variable bear a constant relationship to proportional changes in the independent variable(s), as we illustrate below. You formulate the multiple regression: Yi = b0 + b1 X1i + b2 X2i + ϵi (9-2) where Yi = the natural logarithm of (bid–ask spread/stock price) for stock i X1i = the natural logarithm of the number of Nasdaq market makers for stock i X2i = the natural logarithm of the market capitalization (measured in millions of dollars) of company i In a log-log regression such as Equation 9-2, the slope coefficients are interpreted as elasticities, assumed to be constant. For example, b2 = −0.75 means that for a 1 percent increase in the market capitalization, we expect bid–ask spread/stock price to decrease by 0.75 percent, holding all other independent variables constant.2 Reasoning that greater competition tends to lower costs, you suspect that the greater the number of market makers, the smaller the percentage bid–ask spread. Therefore, you formulate a first null hypothesis and alternative hypothesis: H0 : b1 ≥ 0 Ha : b1 < 0 The null hypothesis is the hypothesis that the ‘‘suspected’’ condition is not true. If the evidence supports rejecting the null hypothesis and accepting the alternative hypothesis, you have statistically confirmed your suspicion.3 You also believe that the stocks of companies with higher market capitalization may have more-liquid markets, tending to lower percentage bid–ask spreads. Therefore, you formulate a second null hypothesis and alternative hypothesis: 2 Note that !(ln X ) ≈ !X /X , where ! represents ‘‘change in’’ and !X /X is a proportional change in X . We discuss the model further in Example 9-11. 3 An alternative valid formulation is a two-sided test H0 : b1 = 0, versus Ha : b1 ̸= 0 which reflects the beliefs of the researcher less strongly. See the chapter on hypothesis testing. A two-sided test could also be conducted for the hypothesis on market capitalization that we discuss next. 328 Quantitative Investment Analysis H0 : b2 ≥ 0 Ha : b2 < 0 For both tests, we use a t-test, rather than a z-test, because we do not know the population variance of b1 and b2 .4 Suppose that you choose a 0.01 significance level for both tests. Table 9-1 shows the results of estimating this linear regression using data from December 2002. TABLE 9-1 Results from Regressing ln(Bid–Ask Spread/Price) on ln(Number of Market Makers) and ln(Market Cap) Intercept ln(Number of Nasdaq market makers) ln(Company’s market cap) ANOVA Coefficient Standard Error t-Statistic −0.7586 −0.2790 −0.6635 0.1369 0.0673 0.0246 −5.5416 −4.1427 −27.0087 df SS MSS F Significance F 2 1,816 1,818 2,681.6482 2,236.2820 4,917.9302 1,340.8241 1.2314 1,088.8325 0.00 Residual standard error Multiple R-squared Observations 1.1097 0.5453 1,819 Regression Residual Total Source: FactSet, Nasdaq. If the regression result is not significant, we follow the useful principle of not proceeding to interpret the individual regression coefficients. Thus the analyst might look first at the ANOVA section, which addresses the regression’s overall significance. • The ANOVA (analysis of variance) section reports quantities related to the overall explanatory power and significance of the regression. SS stands for sum of squares, and MSS stands for mean sum of squares (SS divided by df). The F -test reports the overall significance of the regression. For example, an entry of 0.01 for the significance of F means that the regression is significant at the 0.01 level. In Table 9-1, the regression is even more significant becausethe significance of F is 0 at two decimal places. Later in the chapter, we will present more information on the F -test. Having ascertained that the overall regression is highly significant, an analyst might turn to the first listed column in the first section of the regression output. 4 The use of t-tests and z-tests is discussed in the chapter on hypothesis testing. Chapter 9 Multiple Regression and Issues in Regression Analysis 329 The Coefficients column gives the estimates of the intercept, b0 , and the slope coefficients, b1 and b2 . These estimates are all negative, but are they significantly negative? The Standard Error column gives the standard error (the standard deviation) of the estimated regression coefficients. The test statistic for hypotheses concerning the population value of a regression coefficient has the form (Estimated regression coefficient—Hypothesized population value of the regression coefficient)/(Standard error of the regression coefficient). This is a t-test. Under the null hypothesis, the hypothesized population value of the regression coefficient is 0. Thus (Estimated regression coefficient)/(Standard error of the regression coefficient) is the t-statistic given in the third column. For example, the t-statistic for the intercept is −0.7586/0.1369 = −5.5416, ignoring the effects of rounding errors. To evaluate the significance of the t-statistic we need to determine a quantity called degrees of freedom (df).5 The calculation is Degrees of freedom = Number of observations—(Number of independent variables + 1) = n − (k + 1). • The final section of Table 9-1 presents two measures of how well the estimated regression fits or explains the data. The first is the standard deviation of the regression residual, the residual standard error. This standard deviation is called the standard error of estimate (SEE). The second measure quantifies the degree of linear association between the dependent variable and all of the independent variables jointly. This measure is known as multiple R 2 or simply R 2 (the square of the correlation between predicted and actual values of the dependent variable).6 A value of 0 for R 2 indicates no linear association; a value of 1 indicates perfect linear association. The final item in Table 9-1 is the number of observations in the sample (1,819). • Having reviewed the meaning of typical regression output, we can return to complete the hypothesis tests. The estimated regression supports the hypothesis that the greater the number of market makers, the smaller the percentage bid–ask spread: We reject H0 : b1 ≥ 0 in favor of Ha : b1 < 0. The results also support the belief that the stocks of companies with higher market capitalization have lower percentage bid–ask spreads: We reject H0 : b2 ≥ 0 in favor of Ha : b2 < 0. To see that the null hypothesis is rejected for both tests, we can use tables in the back of the book.7 For both tests, df = 1, 819 − 3 = 1, 816. The tables do not give critical values for degrees of freedom that large. The critical value for a one-tailed test with df = 200 at the 0.01 significance level is 2.345; for a larger number of degrees of freedom, the critical value would be even smaller in magnitude. Therefore, in our one-sided tests, we reject the null hypothesis in favor of the alternative hypothesis if b̂j − bj b̂j − 0 t= = < −2.345 sb̂j sb̂j 5 To calculate the degrees of freedom lost in the regression, we add 1 to the number of independent variables to account for the intercept term. The t-test and the concept of degrees of freedom are discussed in the chapter on sampling. 6 Multiple R 2 is also known as the multiple coefficient of determination, or simply the coefficient of determination. 7 See Appendix B for t-test values. 330 Quantitative Investment Analysis where b̂j = the regression estimate of bj , j = 1, 2 bj = the hypothesized value8 of the coefficient (0) sb̂j = the estimated standard error of b̂j The t-values of −4.1427 and −27.0087 for the estimates of b1 and b2 , respectively, are both less than −2.345. Before proceeding further, we should address the interpretation of a prediction stated in natural logarithm terms. We can convert a natural logarithm to the original units by taking the antilogarithm. To illustrate this conversion, suppose that a particular stock has five Nasdaq market makers and a market capitalization of $100 million. The natural logarithm of the number of Nasdaq market makers is equal to ln 5 = 1.6094, and the natural logarithm of the company’s market cap (in millions) is equal to ln 100 = 4.6052. With these values, the regression model predicts that the natural log of the ratio of the bid–ask spread to the stock price will be −0.7586 + (−0.2790 × 1.6094) + (−0.6635 × 4.6052) = −4.2632. We take the antilogarithm of −4.2632 by raising e to that power: e−4.2632 = 0.0141. The predicted bid–ask spread will be 1.41 percent of the stock price.9 Later we state the assumptions of the multiple regression model; before using an estimated regression to make predictions in actual practice, we should assure ourselves that those assumptions are satisfied. In Table 9-1, we presented output common to most regression software programs. Many software programs also report p-values for the regression coefficients.10 For each regression coefficient, the p-value would be the smallest level of significance at which we can reject a null hypothesis that the population value of the coefficient is 0, in a two-sided test. The lower the p-value, the stronger the evidence against that null hypothesis. A p-value quickly allows us to determine if an independent variable is significant at a conventional significance level such as 0.05, or at any other standard we believe is appropriate. Having estimated Equation 9-1, we can write Ŷi = b̂0 + b̂1 X1i + b̂2 X2i = −0.7586 − 0.2790X1i − 0.6635X2i where Ŷi stands for the predicted value of Yi , and b̂0 , b̂1 , and b̂2 , stand for the estimated values of b0 , b1 , and b2 , respectively. How should we interpret the estimated slope coefficients −0.2790 and −0.6635? Interpreting the slope coefficients in a multiple linear regression model is different than doing so in the one-independent-variable regressions explored in the chapter on correlation 8 To economize on notation in stating test statistics, in this context we use bj to represent the hypothesized value of the parameter (elsewhere we use it to represent the unknown population parameter). 9 The operation illustrated (taking the antilogarithm) recovers the value of a variable in the original units as eln X = X . 10 The entry 0.00 for the significance of F was a p-value for the F -test. See the chapter on hypothesis testing for more information on the p-value. Chapter 9 Multiple Regression and Issues in Regression Analysis 331 and regression. Suppose we have a one-independent-variable regression that we estimate as Ŷi = 0.50 + 0.75X1i . The interpretation of the slope estimate 0.75 is that for every 1 unit increase in X1 , we expect Y to increase by 0.75 units. If we were to add a second independent variable to the equation, we would generally find that the estimated coefficient on X1 is not 0.75 unless the second independent variable were uncorrelated with X1 . The slope coefficients in a multiple regression are known as partial regression coefficients or partial slope coefficients and need to be interpreted with care.11 Suppose the coefficient on X1 in a regression with the second independent variable was 0.60. Can we say that for every 1-unit increase in X1 , we expect Y to increase by 0.60 units? Not without qualification. For every 1-unit increase in X1 , we still expect Y to increase by 0.75 units when X2 is not held constant. We would interpret 0.60 as the expected increase in Y for a 1-unit increase X1 holding the second independent variable constant. To explain what the shorthand reference ‘‘holding the second independent constant’’ refers to, if we were to regress X1 on X2 , the residuals from that regression would represent the part of X1 that is uncorrelated with X2 . We could then regress Y on those residuals in a 1-independent-variable regression. We would find that the slope coefficient on the residuals would be 0.60; by construction, 0.60 would represent the expected effect on Y of a 1-unit increase in X1 after removing the part of X1 that is correlated with X2 . Consistent with this explanation, we can view 0.60 as the expected net effect on Y of a 1-unit increase in X1 , after accounting for any effects of the other independent variables on the expected value of Y . To reiterate, a partial regression coefficient measures the expected change in the dependent variable for a one-unit increase in an independent variable, holding all the other independent variables constant. To apply this process to the regression in Table 9-1, we see that the estimated coefficient on the natural logarithm of market capitalization is −0.6635. Therefore, the model predicts that an increase of 1 in the natural logarithm of the company’s market capitalization is associated with a −0.6635 change in the natural logarithm of the ratio of the bid–ask spread to the stock price, holding the natural logarithm of the number of market makers constant. We need to be careful not to expect that the natural logarithm of the ratio of the bid-ask spread to the stock price would differ by −0.6635 if we compared two stocks for which the natural logarithm of the company’s market capitalization differed by 1, because in all likelihood the number of market makers for the two stocks would differ as well, which would affect the dependent variable. The value −0.6635 is the expected net effect of difference in log market capitalizations, net of the effect of the log number of market makers on the expected value of the dependent variable. 2.1. Assumptions of the Multiple Linear Regression Model Before we can conduct correct statistical inference on a multiple linear regression model (a model with more than one independent variable estimated using ordinary least squares), we need to know the assumptions underlying that model.12 Suppose we have n observations on 11 The terminology comes from the fact that they correspond to the partial derivatives of Y with respect to the independent variables. Note that in this usage, the term ‘‘regression coefficients’’ refers just to the slope coefficients. 12 Ordinary least squares (OLS) is an estimation method based on the criterion of minimizing the sum of the squared residuals of a regression. 332 Quantitative Investment Analysis the dependent variable, Y , and the independent variables, X1 , X2 , . . . , Xk , and we want to estimate the equation Yi = b0 + b1 X1i + b2 X2i + · · · + bk Xki + ϵi . In order to make a valid inference from a multiple linear regression model, we need to make the following six assumptions, which as a group define the classical normal multiple linear regression model: 1. The relationship between the dependent variable, Y , and the independent variables, X1 , X2 , . . . , Xk , is linear as described in Equation 9-1. 2. The independent variables (X1 , X2 , . . . , Xk ) are not random.13 Also, no exact linear relation exists between two or more of the independent variables.14 3. The expected value of the error term, conditioned on the independent variables, is 0: E(ϵ|X1 , X2 , . . . , Xk ) = 0. 4. The variance of the error term is the same for all observations:15 E(ϵi 2 ) = σϵ2 . 5. The error term is uncorrelated across observations: E(ϵi ϵj ) = 0, j ̸= i. 6. The error term is normally distributed. Note that these assumptions are almost exactly the same as those for the single-variable linear regression model presented in the chapter on linear regression. Assumption 2 is modified such that no exact linear relation exists between two or more independent variables or combinations of independent variables. If this part of Assumption 2 is violated, then we cannot compute linear regression estimates.16 Also, even if no exact linear relationship exists between two or more independent variables, or combinations of independent variables, linear regression may encounter problems if two or more of the independent variables or combinations thereof are highly correlated. Such a high correlation is known as multicollinearity, which we will discuss later in this chapter. We will also discuss the consequences of supposing that Assumptions 4 and 5 are met if, in fact, they are violated. Although Equation 9-1 may seem to apply only to cross-sectional data because the notation for the observations is the same (i = 1, . . . , n), all of these results apply to time-series data as well. For example, if we analyze data from many time periods for one company, we would typically use the notation Yt , X1t , X2t , . . . , Xkt , in which the first subscript denotes the variable and the second denotes the tth time period. 13 As discussed in the chapter on correlation and regression, even though we assume that independent variables in the regression model are not random, often that assumption is clearly not true. For example, the monthly returns to the S&P 500 are not random. If the independent variable is random, then is the regression model incorrect? Fortunately, no. Even if the independent variable is random but uncorrelated with the error term, we can still rely on the results of regression models. See, for example, Greene (2003) or Goldberger (1998). 14 No independent variable can be expressed as a linear combination of any set of the other independent variables. Technically, a constant equal to 1 is included as an independent variable associated with the intercept in this condition. 15 Var(ϵ) = E(ϵ2 ) and Cov(ϵ ϵ ) = E(ϵ ϵ ) because E(ϵ) = 0. i j i j 16 When we encounter this kind of linear relationship (called perfect collinearity), we cannot compute the matrix inverse needed to compute the linear regression estimates. See Greene (2003) for a further description of this issue. 333 Chapter 9 Multiple Regression and Issues in Regression Analysis EXAMPLE 9-2 Factors Explaining Pension Fund Performance Ambachtsheer, Capelle, and Scheibelhut (1998) tested to see which factors affect the performance of pension funds. Specifically, they wanted to know whether the risk-adjusted net value added (RANVA) of 80 U.S. and Canadian pension funds depended on the size of the individual fund and the proportion of the fund’s assets that were passively managed (indexed). Using data from 80 funds for four years (1993 to 1996), the authors regressed RANVA on the size of the pension fund and the fraction of pension fund assets that were passively managed.17 They used the equation where RANVAi = b0 + b1 Sizei + b2 Passivei + ϵi RANVAi = the average RANVA (in percent) for fund i from 1993 to 1996 Sizei = the log10 of average assets under management for fund i Passivei = the fraction (decimal) of passively managed assets in fund i Table 9-2 shows the results of their analysis.18 TABLE 9-2 Results from Regressing RANVA on Size and Passive Management Coefficients Standard Error t-Statistic Intercept Size Passive management −2.1 0.4 0.8 0.45 0.14 0.42 −4.7 2.8 1.9 Source: Ambachtsheer, Capelle, and Scheibelhut (1998). Suppose we use the results in Table 9-2 to test the null hypothesis that a pension fund’s size had no effect on its RANVA. Our null hypothesis is that the coefficient on the size variable equals 0 (H0 : b1 = 0), and our alternative hypothesis is that the coefficient does not equal 0 (Ha : b1 ̸= 0). The t-statistic for testing that hypothesis is 0.4 − 0 b̂1 − b1 = 2.8 = sb̂1 0.14 17 As mentioned in an earlier footnote, technically a constant equal to 1 is included as an independent variable associated with the intercept term in a regression. Because all the regressions reported in this chapter include an intercept term, we will not separately mention a constant as an independent variable in the remainder of this chapter. 18 Size is the log base 10 of average assets. A log transformation is commonly used for independent variables that can take a wide range of values; company size and fund size are two such variables. One reason to use the log transformation is to improve the statistical properties of the residuals. If the authors had not taken the log of assets and instead used assets as the independent variable, the regression model probably would not have explained RANVA as well. 334 Quantitative Investment Analysis With 80 observations and three coefficients, the t-statistic has 80 − 3 = 77 degrees of freedom. At the 0.05 significance level, the critical value for t is about 1.99. The computed t-statistic on the size coefficient is 2.8, which suggests strongly that we can reject the null hypothesis that size is unrelated to RANVA. The estimated coefficient of 0.4 implies that every 10-fold increase in fund size (an increase of 1 in Sizei ) is associated with an expected 0.4 percentage point increase (40 basis points) in RANVAi holding constant the fraction of passively managed assets. Because Sizei is the base 10 log of average assets, an increase of 1 in Size is the same as a 10-fold increase in fund assets. Of course, no causal relation between size and RANVA is clear: Funds that are more successful may attract more assets. This regression equation is consistent with that result, as well as the result that larger funds perform better. On one hand, we could argue that larger funds are more successful. On the other hand, we could argue that more successful funds attract more assets and become larger. Now suppose we want to test the null hypothesis that passive management is not related to RANVA; we want to test whether the coefficient on the fraction of assets under passive management equals 0 (H0 : b2 = 0) against the alternative hypothesis that the coefficient on the fraction of assets under passive management does not equal 0 (Ha : b2 ̸= 0). The t-statistic to test this hypothesis is 0.8 − 0 b̂2 − b2 = 1.9 = sb̂2 0.42 The critical value of the t-test is 1.99 at the 0.05 significance level and about 1.66 at the 0.10 level. Therefore, at the 0.10 significance level, we can reject the null hypothesis that passive management has no effect on fund returns; however, we cannot do so at the 0.05 significance level. Although researchers typically use a significance level of 0.05 or smaller, these results and others like them are strong enough that many pension plan sponsors have increased the use of passive management for pension fund assets. We can interpret the coefficient on passive man-agement of 0.8 as implying that an increase of 0.10 in the proportion of a fund’s passively managed assets is associated with an expected 0.08 percentage point increase (8 basis points) in RANVA for the fund, holding Size constant. EXAMPLE 9-3 Explaining Returns to the Fidelity Select Technology Fund Suppose you are considering an investment in the Fidelity Select Technology Fund (FSPTX), a U.S. mutual fund specializing in technology stocks. You want to know whether the fund behaves more like a large-cap growth fund or a large-cap value fund.19 You decide to estimate the regression 19 This regression is related to return-based style analysis, one of the most frequent applications of regression analysis in the investment profession. For more information, see Sharpe (1988), who pioneered this field, and Buetow, Johnson, and Runkle (2000). 335 Chapter 9 Multiple Regression and Issues in Regression Analysis Yt = b0 + b1 X1t + b2 X2t + ϵt where Yt = the monthly return to the FSPTX X1t = the monthly return to the S&P 500/BARRA Growth Index X2t = the monthly return to the S&P 500/BARRA Value Index The S&P 500/BARRA Growth and Value indexes represent predominantly large-cap growth and value stocks, respectively. Table 9-3 shows the results of this linear regression using monthly data from January 1998 through December 2002. The estimated intercept in the regression is 0.0079. Thus, if both the return to the S&P 500/BARRA Growth Index and the return to the S&P 500/BARRA Value Index equal 0 in a specific month, the regression model predicts that the return to the FSPTX will be 0.79 percent. The coefficient on the large-cap growth index is 2.2308, and the coefficient on the large-cap value index return is −0.4143. Therefore, if in a given month the return to the S&P 500/BARRA Growth Index was 1 percent and the return to the S&P 500/BARRA Value Index was −2 percent, the model predicts that the return to the FSPTX would be 0.0079 + 2.2308(0.01) − 0.4143(−0.02) = 3.85 percent. TABLE 9-3 Results from Regressing the FSPTX Returns on the S&P 500/BARRA Growth and Value Indexes Intercept S&P 500/BARRA Growth Index S&P 500/BARRA Value Index Coefficient Standard Error t-Statistic 0.0079 2.2308 −0.4143 0.0091 0.2299 0.2597 0.8635 9.7034 −1.5953 ANOVA df SS MSS F Significance F Regression Residual Total 2 57 59 0.8649 0.2851 1.1500 0.4324 0.0050 86.4483 5.48E-18 Residual standard error Multiple R-squared Observations 0.0707 0.7521 60 Source: Ibbotson Associates. We may want to know whether the coefficient on the returns to the S&P 500/BARRA Value Index is statistically significant. Our null hypothesis states that the coefficient equals 0 (H0 : b2 = 0); our alternative hypothesis states that the coefficient does not equal 0 (Ha : b2 ̸= 0). Our test of the null hypothesis uses a t-test constructed as follows: t= b̂2 − b2 −0.4143 − 0 = −1.5953 = sb̂2 0.2597 336 Quantitative Investment Analysis where b̂2 = the regression estimate of b2 b2 = the hypothesized value20 of the coefficient (0) sb̂2 = the estimated standard error of b̂2 This regression has 60 observations and three coefficients (two independent variables and the intercept); therefore, the t-test has 60 − 3 = 57 degrees of freedom. At the 0.05 significance level, the critical value for the test statistic is about 2.00.21 The absolute value of the test statistic is 1.5953. Because the test statistic’s absolute value is less than the critical value (1.5953 < 2.00), we fail to reject the null hypothesis that b2 = 0. (Note that the t-tests reported in Table 9-3, as well as the other regression tables, are tests of the null hypothesis that the population value of a regression coefficient equals 0.) Similar analysis shows that at the 0.05 significance level, we cannot reject the null hypothesis that the intercept equals 0 (H0 : b0 = 0) in favor of the alternative hypothesis that the intercept does not equal 0 (Ha : b0 ̸= 0). Table 9-3 shows that the t-statistic for testing that hypothesis is 0.8635, a result smaller in absolute value than the critical value of 2.00. However, at the 0.05 significance level we can reject the null hypothesis that the coefficient on the S&P 500/BARRA Growth Index equals 0 (H0 : b1 = 0) in favor of the alternative hypothesis that the coefficient does not equal 0 (Ha : b1 ̸= 0). As Table 9-3 shows, the t-statistic for testing that hypothesis is 9.70, a result far above the critical value of 2.00. Thus multiple regression analysis suggests that returns to the FSPTX are very closely associated with the returns to the S&P 500/BARRA Growth Index, but they are not related to S&P 500/BARRA Value Index (the t-statistic of −1.60 is not statistically significant). 2.2. Predicting the Dependent Variable in a Multiple Regression Model Financial analysts often want to predict the value of the dependent variable in a multiple regression based on assumed values of the independent variables. We have previously discussed how to make such a prediction in the case of only one independent variable. The process for making that prediction with multiple linear regression is very similar. To predict the value of a dependent variable using a multiple linear regression model, we follow these three steps: 1. Obtain estimates b̂0 , b̂1 , b̂2 , . . . , b̂k of the regression parameters b0 , b1 , b2 , . . . , bk . 2. Determine the assumed values of the independent variables, X̂1i , X̂2i , . . . , X̂ki . 3. Compute the predicted value of the dependent variable, Ŷi , using the equation Ŷi = b̂0 + b̂1 X̂1i + b̂2 X̂2i + · · · + b̂k X̂ki (9-3) Two practical points concerning using an estimated regression to predict the dependent variable are in order. First, we should be confident that the assumptions of the regression 20 To economize on notation in stating test statistics, in this context we use b 2 to represent the hypothesized value of the parameter (elsewhere we use it to represent the unknown population parameter). 21 See Appendix B for t-test values. Chapter 9 Multiple Regression and Issues in Regression Analysis 337 model are met. Second, we should be cautious about predictions based on values of the independent variables that are outside the range of the data on which the model was estimated; such predictions are often unreliable. EXAMPLE 9-4 Predicting a Pension Fund’s RANVA In Example 9-2, we explained the RANVA for U.S. and Canadian pension funds based on the log base 10 of the assets under management for a fund (Sizei ) and the fraction of assets in the fund that were passively managed (Passivei ). RANVAi = b0 + b1 Sizei + b2 Passivei + ϵi Now we can use the results of the regression reported in Table 9-2 (excerpted here) to predict the performance (RANVA) for a pension fund. TABLE 9-2 (excerpt) Coefficients Intercept Size Passive management −2.1 0.4 0.8 Suppose that a particular fund has assets under management of $10 million, and 25 percent of the assets are passively managed. The log base 10 of the assets under management equals log(10, 000, 000) = 7. The fraction of assets in the fund that are passively managed is 0.25. Accordingly, the predicted RANVA for that fund, based on the regression, is −2.1 + (0.4 × 7) + (0.8 × 0.25) = 0.9 percent (90 basis points). The regression predicts that the RANVA will be 90 basis points for a pension fund with assets under management of $10 million, 25 percent of which are passively managed. When predicting the dependent variable using a linear regression model, we encounter two types of uncertainty: uncertainty in the regression model itself, as reflected in the standard error of estimate, and uncertainty about the estimates of the regression model’s parameters. In the chapter on correlation and regression, we presented procedures for constructing a prediction interval for linear regression with one independent variable. For multiple regression, however, computing a prediction interval to properly incorporate both types of uncertainty requires matrix algebra, which is outside the scope of this book.22 22 For more information, see Greene (2003). 338 Quantitative Investment Analysis 2.3. Testing Whether All Population Regression Coefficients Equal Zero Earlier, we illustrated how to conduct hypothesis tests on regression coefficients individually. But what about the significance of the regression as a whole? As a group, do the independent variables help explain the dependent variable? To address this question, we test the null hypothesis that all the slope coefficients in a regression are simultaneously equal to 0. In this section, we discuss analysis of variance (ANOVA), which provides information about a regression’s explanatory power and the inputs for an F -test of the above null hypothesis. If none of the independent variables in a regression model helps explain the dependent variable, the slope coefficients should all equal 0. In a multiple regression, however, we cannot test the null hypothesis that all slope coefficients equal 0 based on t-tests that each individual slope coefficient equals 0, because the individual tests do not account for the effects of interactions among the independent variables. For example, a classic symptom of multicollinearity is that we can reject the hypothesis that all the slope coefficients equal 0 even though none of the t-statistics for the individual estimated slope coefficients is significant. Conversely, we can construct unusual examples in which the estimated slope coefficients are significantly different from 0 although jointly they are not. To test the null hypothesis that all of the slope coefficients in the multiple regression model are jointly equal to 0 (H0 : b1 = b2 = · · · = bk = 0) against the alternative hypothesis that at least one slope coefficient is not equal to 0 we must use an F -test. The F -test is viewed as a test of the regression’s overall significance. To correctly calculate the test statistic for the null hypothesis, we need four inputs: total number of observations, n; total number of regression coefficients to be estimated, k + 1, where k is the number of slope coefficients; n n ! ! • sum of squared errors or residuals, ϵ̂2i , abbreviated SSE, also known as (Yi − Ŷi )2 = • • i=1 i=1 the residual sum of squares (unexplained variation);23 and n ! • regression sum of squares, (Ŷi − Y )2 , abbreviated RSS.24 This amount is the variation i=1 in Y from its mean that the regression equation explains (explained variation). The F -test for determining whether the slope coefficients equal 0 is based on an F -statistic calculated using the four values listed above.25 The F -statistic measures how well the regression equation explains the variation in the dependent variable; it is the ratio of the mean regression sum of squares to the mean squared error. We compute the mean regression sum of squares by dividing the regression sum of squares by the number of slope coefficients estimated, k. We compute the mean squared error by dividing the sum of squared errors by the number of observations, n, minus (k + 1). The two divisors in these computations are the degrees of freedom for calculating an F -statistic. For n observations and k slope coefficients, the F -test for the null hypothesis that the slope coefficients are all equal to 0 is denoted Fk,n−(k+1) . The subscript indicates that the test should have k degrees of freedom in the numerator (numerator degrees of freedom) and n − (k + 1) degrees of freedom in the denominator (denominator degrees of freedom). 23 In a table of regression output, this is the number under the ‘‘SS’’ column in the row ‘‘Residual.’’ a table of regression output, this is the number under the ‘‘SS’’ column in the row ‘‘Regression.’’ 25 F -tests are described in further detail in the chapter on hypothesis testing. 24 In 339 Chapter 9 Multiple Regression and Issues in Regression Analysis The formula for the F -statistic is RSS F= k SSE [n−(k+1)] = MSR Mean regression sum of squares = Mean squared error MSE (9-4) where MSR is the mean regression sum of squares and MSE is the mean squared error. In our regression output tables, MSR and MSE are the first and second quantities under the MSS (mean sum of squares) column in the ANOVA section of the output. If the regression model does a good job of explaining variation in the dependent variable, then the ratio MSR/MSE will be large. What does this F -test tell us when the independent variables in a regression model explain none of the variation in the dependent variable? In this case, each predicted value in the regression model, Ŷi , has the average value of the dependent variable, Y , and the regression n ! sum of squares, (Ŷi − Y )2 is 0. Therefore, the F -statistic for testing the null hypothesis i=1 (that all the slope coefficients are equal to 0) has a value of 0 when the independent variables do not explain the dependent variable at all. To specify the details of making the statistical decision when we have calculated F , we reject the null hypothesis at the α significance level if the calculated value of F is greater than the upper α critical value of the F distribution with the specified numerator and denominator degrees of freedom. Note that we use a one-tailed F -test.26 Appendix D provides the critical values for the F -test. We can illustrate the test using Example 9-1, in which we investigated whether the natural log of the number of Nasdaq market makers and the natural log of the stock’s market capitalization explained the natural log of the bid–ask spread divided by price. Assume that we set the significance level for this test to α = 0.05 (i.e., a 5 percent probabilitythat we will mistakenly reject the null hypothesis if it is true). Table 9-1 (excerpted here) presents the results of variance computations for this regression. TABLE 9-1 (excerpt) ANOVA df SS MSS F Regression 2 2,681.6482 1,340.8241 1,088.8325 Residual 1,816 2,236.2820 1.2314 Total 1,818 4,917.9302 Significance F 0.00 This model has two slope coefficients (k = 2), so there are two degrees of freedom in the numerator of this F -test. With 1,819 observations in the sample, the number of degrees of freedom in the denominator of the F -test is n − (k + 1) = 1, 819 − 3 = 1, 816. The sum of the squared errors is 2,236.2820. The regression sum of squares is 2,681.6482. Therefore, the F -test for the null hypothesis that the two slope coefficients in this model equal 0 is 2681.6482/2 = 1, 088.8325 2236.2820/1, 816 This test statistic is distributed as an F2,1,816 random variable under the null hypothesis that the slope coefficients are equal to 0. In the table for the 0.05 significance level, we look at the 26 We use a one-tailed test because MSR necessarily increases relative to MSE as the explanatory power of the regression increases. 340 Quantitative Investment Analysis second column, which shows F -distributions with two degrees of freedom in the numerator. Near the bottom of the column, we find that the critical value of the F -test needed to reject the null hypothesis is between 3.00 and 3.07.27 The actual value of the F -test statistic at 1,088.83 is much greater, so we reject the null hypothesis that coefficients of both independent variables equal 0. In fact, Table 9-1, under ‘‘Significance F ,’’ reports a p-value of 0. This p-value means that the smallest level of significance at which the null hypothesis can be rejected is practically 0. The large value for this F -statistic implies a minuscule probability of incorrectly rejecting the null hypothesis (a mistake known as a Type I error). 2.4. Adjusted R 2 In the chapter on correlation and regression, we presented the coefficient of variation, R 2 , as a measure of the goodness of fit of an estimated regression to the data. In a multiple linear regression, however, R 2 is less appropriate as a measure of whether a regression model fits the data well (goodness of fit). Recall that R 2 is defined as Total variation − Unexplained variation Total variation The numerator equals the regression sum of squares, RSS. Thus R 2 states RSS as a fraction of n ! the total sum of squares, (Yi − Y )2 . If we add regression variables to the model, the amount i=1 of unexplained variation will decrease, and RSS will increase, if the new independent variable explains any of the unexplained variation in the model. Such a reduction occurs when the new independent variable is even slightly correlated with the dependent variable and is not a linear combination of other independent variables in the regression.28 Consequently, we can increase R 2 simply by including many additional independent variables that explain even a slight amount of the previously unexplained variation, even if the amount they explain is not statistically significant. Some financial analysts use an alternative measure of goodness of fit called adjusted R 2 , 2 or R . This measure of fit does not automatically increase when another variable is added to a regression; it is adjusted for degrees of freedom. Adjusted R 2 is typically part of the multiple regression output produced by statistical software packages. 2 The relation between R 2 and R is # " n−1 2 (1 − R 2 ) R =1− n−k−1 where n is the number of observations and k is the number of independent variables (the number of slope coefficients). Note that if k ≥ 1, then R 2 is strictly greater than adjusted R 2 . When 2 a new independent variable is added, R can decrease if adding that variable results in only a 2 small increase in R 2 . In fact, R can be negative, although R 2 is always nonnegative.29 If we use 2 R to compare regression models, it is important that the dependent variable be defined the same way in both models and that the sample sizes used to estimate the models are the same.30 27 We see a range of values because the denominator has more than 120 degrees of freedom but less than an infinite number of degrees of freedom. 28 We say that variable y is a linear combination of variables x and z if y = ax + bz for some constants a and b. A variable can also be a linear combination of more than two variables. 29 When R 2 is negative, we can effectively consider its value to be 0. 30 See Gujarati (2003). The value of adjusted R 2 depends on sample size. These points hold if we are using R 2 to compare two regression models. Chapter 9 Multiple Regression and Issues in Regression Analysis 341 2 For example, it makes a difference for the value of R if the dependent variable is GDP (gross domestic product) or ln(GDP), even if the independent variables are identical. Furthermore, 2 we should be aware that a high R does not necessarily indicate that the regression is well specified in the sense of including the correct set of variables.31 One reason for caution is that 2 a high R may reflect peculiarities of the dataset used to estimate the regression. To evaluate a regression model, we need to take many other factors into account, as we discuss in Section 5.1. 3. USING DUMMY VARIABLES IN REGRESSIONS Often, financial analysts need to use qualitative variables as independent variables in a regression. One type of qualitative variable, called a dummy variable, takes on a value of 1 if a particular condition is true and 0 if that condition is false.32 For example, suppose we want to test whether stock returns were different in January than during the remaining months of a particular year. We include one independent variable in the regression, X1t , that has a value of 1 for each January and a value of 0 for every other month of the year. We estimate the regression model Yt = b0 + b1 X1t + ϵt In this equation, the coefficient b0 is the average value of Yt in months other than January, and b1 is the difference between the average value of Yt in January and the average value of Yt in months other than January. We need to exercise care in choosing the number of dummy variables in a regression. The rule is that if we want to distinguish among n categories, we need n − 1 dummy variables. For example, to distinguish between during January and not during January above (n = 2 categories), we used one dummy variable (n − 1 = 2 − 1 = 1). If we want to distinguish between each of the four quarters in a year, we would include dummy variables for three of the four quarters in a year. If we make the mistake of including dummy variables for four rather than three quarters, we have violated Assumption 2 of the multiple regression model and cannot estimate the regression. The next example illustrates the use of dummy variables in a regression with monthly data. EXAMPLE 9-5 Month-of-the-Year Effects on Small-Stock Returns For many years, financial analysts have been concerned about seasonality in stock returns.33 In particular, analysts have researched whether returns to small stocks differ during various months of the year. Suppose we want to test whether total 31 See Mayer (1975, 1980). all qualitative variables are simple dummy variables. For example, in a trinomial choice model (a model with three choices), a qualitative variable might have the value 0, 1, or 2. 33 For a discussion of this issue, see Siegel (1998). 32 Not 342 Quantitative Investment Analysis returns to one small-stock index, the Russell 2000 Index, differ by month. Using data from January 1979 (the first available date for the Russell 2000 data) through the end of 2002, we can estimate a regression including an intercept and 11 dummy variables, one for each of the first 11 months of the year. The equation that we estimate is Returnst = b0 + b1 Jant + b2 Febt + · · · + b11 Novt + ϵt where each monthly dummy variable has a value of 1 when the month occurs (e.g., Jan1 = Jan13 = 1, as the first observation is a January) and a value of 0 for the other months. Table 9-4 shows the results of this regression. TABLE 9-4 Results from Regressing Russell 2000 Returns on Monthly Dummy Variables Intercept January February March April May June July August September October November ANOVA Regression Residual Total Coefficient Standard Error t-Statistic 0.0301 0.0003 −0.0111 −0.0211 −0.0141 −0.0137 −0.0200 −0.0405 −0.0230 −0.0375 −0.0393 −0.0059 0.0116 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 0.0164 2.5902 0.0176 −0.6753 −1.2846 −0.8568 −0.8320 −1.2164 −2.4686 −1.4025 −2.2864 −2.3966 −0.3565 df SS MSS F Significance F 11 276 287 0.0543 0.8924 0.9467 0.0049 0.0032 1.5270 0.1213 Residual standard error Multiple R-squared Observations 0.0569 0.0574 288 Source: Ibbotson Associates. The intercept, b0 , measures the average return for stocks in December because there is no dummy variable for December.34 This equation estimates that the average return in December is 3.01 percent (b̂0 = 0.0301). Each of the estimated coefficients for the dummy variables shows the estimated difference between returns in that month and returns for December. So, for example, the estimated additional return in January is 0.03 percent higher than December (b̂1 = 0.0003). This gives a January return prediction of 3.04 percent (3.01 December + 0.03 additional). 34 When Jan = Feb = · · · = Nov = 0, the return is not associated with January through t t t November so the month is December and the regression equation simplifies to Returnst = b0 + ϵt . Because E(Returnst ) = b0 + E(ϵt ) = b0 , the intercept b0 represents the mean return for December. Chapter 9 Multiple Regression and Issues in Regression Analysis 343 The low R 2 in this regression (0.0574), however, suggests that a month-of-the-year effect in small-stock returns may not be very important for explaining small-stock returns. We can use the F -test to analyze the null hypothesis that jointly, the monthly dummy variables all equal 0 (H0 : b1 = b2 = · · · = b11 = 0). We are testing for significant monthly variation in small-stock returns. Table 9-4 shows the data needed to perform an analysis of variance. The number of degrees of freedom in the numerator of the F -test is 11; the number of degrees of freedom in the denominator is [288 − (11 + 1)] = 276. The regression sum of squares equals 0.0543, and the sum of squared errors equals 0.8924. Therefore, the F -statistic to determine whether all of the regression slope coefficients are jointly equal to 0 is 0.0543/11 = 1.53 0.8924/276 Appendix D shows the critical values for this F -test. If we choose a significance level of 0.05 and look in Column 11 (because the numerator has 11 degrees of freedom), we see that the critical value is 1.87 when the denominator has 120 degrees of freedom. The denominator actually has 276 degrees of freedom, so the critical value of the F -statistic is smaller than 1.87 (for df = 120) but larger than 1.79 (for an infinite number of degrees of freedom). The value of the test statistic is 1.53, so we clearly cannot reject the null hypothesis that all of the coefficients jointly are equal to 0. The p-value of 0.1213 shown for the F -test in Table 9-4 means that the smallest level of significance at which we can reject the null hypothesis is roughly 0.12, or 12 percent—above the conventional level of 5 percent. Among the 11 monthly dummy variables, July, September, and October have a t-statistic with an absolute value greater than 2. Although the coefficients for these dummy variables are statistically significant, we have so many insignificant estimated coefficients that we cannot reject the null hypothesis that returns are equal across the months. This test suggests that the significance of a few coefficients in this regression model may be the result of random variation. We may thus want to avoid portfolio strategies calling for differing investment weights for small stocks in different months. EXAMPLE 9-6 Determinants of Spreads on New High-Yield Bonds Fridson and Garman (1998) used data from 1995 and 1996 to examine variables that may explain the initial yield spread between a newly issued high-yield bond and a Treasury bond with similar maturity. They built a model of yield spreads using variables that affect the creditworthiness and interest-rate risk of the bond. Their model included the following factors: • • • Rating: Moody’s senior-equivalent rating Zero-coupon status: Dummy variable (0 = no, 1 = yes) BB-B spread: Yield differential (Merrill Lynch Single-B Index minus Double-B Index, in basis points) 344 Quantitative Investment Analysis Seniority: Dummy variable (0 = senior, 1 = subordinated) Callability: Dummy variable (0 = noncallable, 1 = callable) Term: Maturity (years) First-time issuer: Dummy variable (0 = no, 1 = yes) • Underwriter type: Dummy variable (0 = investment bank, 1 = commercial bank) • Interest rate change • • • • Table 9-5 shows the authors’ results. TABLE 9-5 Multiple Regression Model of New High-Yield Issue Spread, 1995–96 Coefficient Standard Error t-Statistic Intercept −213.67 Rating 66.19 Zero-coupon status 136.54 BB-B spread 95.31 Seniority 41.46 Callability 51.65 Term −8.51 First-time issuer 25.23 Underwriter type 28.13 Interest rate change 40.44 R-squared Observations 63.03 4.13 32.82 24.82 11.95 15.42 2.71 10.97 12.67 19.08 −3.39 16.02 4.16 3.84 3.47 3.35 −3.14 2.30 2.22 2.12 0.56 428 Source: Fridson and Garman (1998). We can summarize Fridson and Garman’s findings as follows: • • • • • • • • Bond rating has the highest significance level of any coefficient in the regression. This result should not be surprising, because the rating captures rating agencies’ estimates of the risk involved with the bond. Zero-coupon status increases the yield spread because zero-coupon bonds have more interest rate risk than coupon bonds of a similar maturity. The BB-B spread affects yields because it captures the market’s evaluation of how much influence rating differentials have on credit risk. Seniority affects yields because subordinated debt has a much lower recovery rate in the case of default. Callability increases yields because it limits upside potential on the bond if yields decline. Term actually reduces the yield spread. Perhaps term enters with a negative coefficient because the market is willing to buy long-term debt only from high-quality companies; lower-quality companies must issue shorter-term debt. First-time issuers must pay a premium because the market does not know much about them. Bonds underwritten by commercial banks have a premium over bonds underwritten by investment banks, most likely because the market believes that investment banks have a competitive edge in attracting high-quality corporate clients. Chapter 9 Multiple Regression and Issues in Regression Analysis • 345 Interest-rate increases in Treasuries during the previous month cause yield spreads to widen, presumably because the market believes that increasing interest rates will worsen the economic prospects of companies issuing high-yield debt. Note that all of the coefficients in this regression model are statistically significant at the 0.05 level. The smallest absolute value of a t-statistic in this table is 2.12. 4. VIOLATIONS OF REGRESSION ASSUMPTIONS In Section 2.1, we presented the assumptions of the multiple linear regression model. Inference based on an estimated regression model rests on those assumptions being satisfied. In applying regression analysis to financial data, analysts need to be able to diagnose violations of regression assumptions, understand the consequences of violations, and know the remedial steps to take. In the following sections we discuss three regression violations: heteroskedasticity, serial correlation, and multicollinearity. 4.1. Heteroskedasticity So far, we have made an important assumption that the variance of error in a regression is constant across observations. In statistical terms, we assumed that the errors were homoskedastic. Errors in financial data, however, are often heteroskedastic: the variance of the errors differs across observations. In this section, we discuss how heteroskedasticity affects statistical analysis, how to test for heteroskedasticity, and how to correct for it. We can see the difference between homoskedastic and heteroskedastic errors by comparing two graphs. Figure 9-1 shows the values of the dependent and independent variables and a fitted regression line for a model with homoskedastic errors. There is no systematic relationship between the value of the independent variable and the regression residuals (the vertical distance between a plotted point and the fitted regression line). Figure 9-2 shows the values of the dependent and independent variables and a fitted regression line for a model with heteroskedastic errors. Here, a systematic relationship is visually apparent: On average, the regression residuals grow much larger as the size of the independent variable increases. 4.1.1. The Consequences of Heteroskedasticity What are the consequences when the assumption of constant error variance is violated? Although heteroskedasticity does not affect the consistency35 of the regression parameter estimators, it can lead to mistakes in inference. 35 Informally, an estimator of a regression parameter is consistent if the probability that estimates of a regression parameter differ from the true value of the parameter decreases as the number of observations used in the regression increases. The regression parameter estimates from ordinary least squares are consistent regardless of whether the errors are heteroskedastic or homoskedastic. See the chapter on sampling and, for a more advanced discussion, see Greene (2003). 346 Quantitative Investment Analysis Dependent Variable 30 25 20 15 10 5 0 –5 0 5 10 15 20 25 30 20 25 30 Independent Variable FIGURE 9-1 Regression with Homoskedasticity Dependent Variable 60 50 40 30 20 10 0 0 5 10 15 Independent Variable FIGURE 9-2 Regression with Heteroskedasticity When errors are heteroskedastic, the F -test for the overall significance of the regression is unreliable.36 Furthermore, t-tests for the significance of individual regression coefficients are unreliable because heteroskedasticity introduces bias into estimators of the standard error of regression coefficients. If a regression shows significant heteroskedasticity, the standard errors 36 This unreliability occurs because the mean squared error is a biased estimator of the true population variance given heteroskedasticity. Chapter 9 Multiple Regression and Issues in Regression Analysis 347 and test statistics computed by regression programs will be incorrect unless they are adjusted for heteroskedasticity. In regressions with financial data, the most likely result of heteroskedasticity is that the estimated standard errors will be underestimated and the t-statistics will be inflated. When we ignore heteroskedasticity, we tend to find significant relationships where none actually exist.37 The consequences in practice may be serious if we are using regression analysis in the development of investment strategies. As Example 9-7 shows, the issue impinges even on our understanding of financial models. EXAMPLE 9-7 Heteroskedasticity and Tests of an Asset Pricing Model MacKinlay and Richardson (1991) examined how heteroskedasticity affects tests of the capital asset pricing model (CAPM).38 These authors argued that if the CAPM is correct, they should find no significant differences between the risk-adjusted returns for holding small stocks versus large stocks. To implement their test, MacKinlay and Richardson grouped all stocks on the New York and American exchanges by marketvalue decile with annual reassignment. They then tested for systematic differences in risk-adjusted returns across market-capitalization-based stock portfolios. They estimated the following regression: ri,t = αi + βi rm,t + ϵi,t where ri,t = excess return (return above the risk-free rate) to portfolio i in period t rm,t = excess return to the market as a whole in period t The CAPM formulation hypothesizes that excess returns on a portfolio are explained by excess returns on the market as a whole. That hypothesis implies that αi = 0 for every i; on average, no excess return accrues to any portfolio after taking into account its systematic (market) risk. Using data from January 1926 to December 1988 and a market index based on equal-weighted returns, MacKinlay and Richardson failed to reject the CAPM at the 0.05 level when they assumed that the errors in the regression model are normally distributed and homoskedastic. They found, however, that they could reject the CAPM when they corrected their test statistics to account for heteroskedasticity. They rejected the hypothesis that there are no size-based, risk-adjusted excess returns in historical data.39 We have stated that effects of heteroskedasticity on statistical inference can be severe. To be more precise about this concept, we should distinguish between two broad kinds of heteroskedasticity, unconditional and conditional. 37 Sometimes, however, failure to adjust for heteroskedasticity results in standard errors that are too large (and t-statistics that are too small). 38 For more on the CAPM, see Bodie, Kane, and Marcus (2001), for example. 39 MacKinlay and Richardson also show that when using value-weighted returns, one can reject the CAPM whether or not one assumes normally distributed returns and homoskedasticity. 348 Quantitative Investment Analysis Unconditional heteroskedasticity occurs when heteroskedasticity of the error variance is not correlated with the independent variables in the multiple regression. Although this form of heteroskedasticity violates Assumption 4 of the linear regression model, it creates no major problems for statistical inference. The type of heteroskedasticity that causes the most problems for statistical inference is conditional heteroskedasticity —heteroskedasticity in the error variance that is correlated with (conditional on) the values of the independent variables in the regression. Fortunately, many statistical software packages easily test and correct for conditional heteroskedasticity. 4.1.2. Testing for Heteroskedasticity Because of conditional heteroskedasticity’s consequences on inference, the analyst must be able to diagnose its presence. The Breusch–Pagan test is widely used in finance research because of its generality.40 Breusch and Pagan (1979) suggested the following test for conditional heteroskedasticity: Regress the squared residuals from the estimated regression equation on the independent variables in the regression. If no conditional heteroskedasticity exists, the independent variables will not explain much of the variation in the squared residuals. If conditional heteroskedasticity is present in the original regression, however, the independent variables will explain a significant portion of the variation in the squared residuals. The independent variables can explain the variation because each observation’s squared residual will be correlated with the independent variables if the independent variables affect the variance of the errors. Breusch and Pagan showed that under the null hypothesis of no conditional heteroskedasticity, nR 2 (from the regression of the squared residuals on the independent variables from the original regression) will be a χ2 random variable with the number of degrees of freedom equal to the number of independent variables in the regression.41 Therefore, the null hypothesis states that the regression’s squared error term is uncorrelated with the independent variables. The alternative hypothesis states that the squared error term is correlated with the independent variables. Example 9-8 illustrates the Breusch–Pagan test for conditional heteroskedasticity. EXAMPLE 9-8 Testing for Conditional Heteroskedasticity in the Relation between Interest Rates and Expected Inflation Suppose an analyst wants to know how closely nominal interest rates are related to expected inflation to determine how to allocate assets in a fixed income portfolio. The analyst wants to test the Fisher effect, the hypothesis suggested by Irving Fisher that nominal interest rates increase by 1 percentage point for every 1 percentage point increase in expected inflation.42 The Fisher effect assumes the following relation between nominal interest rates, real interest rates, and expected inflation: 40 Some other tests require more-specific assumptions about the functional form of the heteroskedasticity. For more information, see Greene (2003). 41 The Breusch–Pagan test is distributed as a χ2 random variable in large samples. The constant 1 technically associated with the intercept term in a regression is not counted here in computing the number of independent variables. For more on the Breusch–Pagan test, see Greene (2003). 42 For more on the Fisher effect, see, for example, Mankiw (2000). 349 Chapter 9 Multiple Regression and Issues in Regression Analysis i = r + πe where i = the nominal interest rate r = the real interest rate (assumed constant) πe = the expected rate of inflation To test the Fisher effect using time-series data, we could specify the following regression model for the nominal interest rate: it = b0 + b1 πt e + ϵt (9-5) Noting that the Fisher effect predicts that the coefficient on the inflation variable is 1, we can state the null and alternative hypotheses as H0 : b1 = 1 Ha : b1 ̸= 1 We might also specify a 0.05 significance level for the test. Before we estimate Equation 9-5, we must decide how to measure expected inflation (πt e ) and the nominal interest rate (it ). The Survey of Professional Forecasters (SPF) has compiled data on the quarterly inflation expectations of professional forecasters.43 We use those data as our measure of expected inflation. We use three-month Treasury bill returns as our measure of the (risk-free) nominal interest rate.44 We use quarterly data from the fourth quarter of 1968 to the fourth quarter of 2002 to estimate Equation 9-5. Table 9-6 shows the regression results. TABLE 9-6 Results from Regressing T-Bill Returns on Predicted Inflation Intercept Inflation prediction Residual standard error Multiple R-squared Observations Durbin–Watson statistic Coefficient Standard Error t-Statistic 0.0304 0.8774 0.0040 0.0812 7.6887 10.8096 0.0220 0.4640 137 0.4673 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. 43 For this example, we use the annualized median SPF prediction of current-quarter growth in the GDP deflator (GNP deflator before 1992). 44 Our data on Treasury bill returns are based on three-month T-bill yields in the secondary market. Because those yields are stated on a discount basis, we convert them to a compounded annual rate so they will be measured on the same basis as our data on inflation expectations. These returns are risk-free because they are known at the beginning of the quarter and there is no default risk. 350 Quantitative Investment Analysis To make the statistical decision on whether the data support the Fisher effect, we calculate the following t-statistic, which we then compare to its critical value. t= b̂1 − b1 0.8774 − 1 = −1.5099 = sb̂1 0.0812 With a t-statistic of −1.5099 and 137 − 2 = 135 degrees of freedom, the critical t-value is about 1.98. If we have conducted a valid test, we cannot reject at the 0.05 significance level the hypothesis that the true coefficient in this regression is 1 and that the Fisher effect holds. The t-test assumes that the errors are homoskedastic. Before we accept the validity of the t-test, therefore, we should test whether the errors are conditionally heteroskedastic. If those errors prove to be conditionally heteroskedastic, then the test is invalid. We can perform the Breusch–Pagan test for conditional heteroskedasticity on the squared residuals from the Fisher effect regression. The test regresses the squared residuals on the predicted inflation rate. The R 2 in the squared residuals regression (not shown here) is 0.1651. The test statistic from this regression, nR 2 , is 137 × 0.1651 = 22.619. Under the null hypothesis of no conditional heteroskedasticity, this test statistic is a χ2 random variable with one degree of freedom (because there is only one independent variable). We should be concerned about heteroskedasticity only for large values of the test statistic. Therefore, we should use a one-tailed test to determine whether we can reject the null hypothesis. Appendix C shows that the critical value of the test statistic for a variable from a χ2 distribution with one degree of freedom at the 0.05 significance level is 3.84. The test statistic from the Breusch–Pagan test is 22.619, so we can reject the hypothesis of no conditional heteroskedasticity at the 0.05 level. In fact, we can even reject the hypothesis of no conditional heteroskedasticity at the 0.01 significance level, because the critical value of the test statistic in that case is 6.63. As a result, we conclude that the error term in the Fisher effect regression is conditionally heteroskedastic. The standard errors computed in the original regression are not correct, because they do not account for heteroskedasticity. Therefore, we cannot accept the t-test as valid. In Example 9-8, we concluded that a t-test that we might use to test the Fisher effect was not valid. Does that mean that we cannot use a regression model to investigate the Fisher effect? Fortunately, no. A methodology is available to adjust regression coefficients’ standard error to correct for heteroskedasticity. Using an adjusted standard error for b̂1 , we can reconduct the t-test. As we shall see in the next section, using this valid t-test we still do not reject the null hypothesis in Example 9-8. 4.1.3. Correcting for Heteroskedasticity Financial analysts need to know how to correct for heteroskedasticity, because such a correction may reverse the conclusions about a particular hypothesis test—and thus affect a particular investment decision. (In Example 9-7, for instance, MacKinlay and Richardson reversed their investment conclusions after correcting their model’s significance tests for heteroskedasticity.) We can use two different methods to correct the effects of conditional heteroskedasticity in linear regression models. The first method, computing robust standard errors, corrects 351 Chapter 9 Multiple Regression and Issues in Regression Analysis the standard errors of the linear regression model’s estimated coefficients to account for the conditional heteroskedasticity. The second method, generalized least squares, modifies the original equation in an attempt to eliminate the heteroskedasticity. The new, modified regression equation is then estimated under the assumption that heteroskedasticity is no longer a problem.45 The technical details behind these two methods of correcting for conditional heteroskedasticity are outside the scope of this book.46 Many statistical software packages can easily compute robust standard errors, however, and we recommend using them.47 Returning to the subject of Example 9-8 concerning the Fisher effect, recall that we concluded that the error variance was heteroskedastic. If we correct the regression coefficients’ standard errors for conditional heteroskedasticity, we get the results shown in Table 9-7. In comparing the standard errors in Table 9-7 with those in Table 9-6, we see that the standard error for the intercept changes very little but the standard error for the coefficient on predicted inflation (the slope coefficient) increases by about 25.5 percent (from 0.0812 to 0.1019). Note also that the regression coefficients are the same in both tables, because the results in Table 9-7 correct only the standard errors in Table 9-6. TABLE 9-7 Results from Regressing T-Bill Returns on Predicted Inflation (Standard Errors Corrected for Conditional Heteroskedasticity) Intercept Inflation prediction Residual standard error Multiple R-squared Observations Coefficients Standard Error t-Statistics 0.0304 0.8774 0.0038 0.1019 8.0740 8.6083 0.0220 0.4640 137 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. We can now conduct a valid t-test of the null hypothesis that the slope coefficient has a true value of 1, using the robust standard error for b̂1 . We find that t = (0.8774 − 1)/0.1019 = −1.2031. In absolute value, this number is still much smaller than the critical value of 1.98 needed to reject the null hypothesis that the slope equals 1.48 Thus, in this particular example, even though conditional heteroskedasticity was statistically significant, correcting for it had no effect on the result of the hypothesis test about the slope of the predicted inflation coefficient. In other cases, however, our statistical decision might change based on using robust standard errors in the t-test. Example 9-7 concerning tests of the CAPM is a case in point. 4.2. Serial Correlation A more common—and potentially more serious—problem than violation of the homoskedasticity assumption is the violation of the assumption that regression errors are uncorrelated 45 Generalized least squares requires econometric expertise to implement correctly on financial data. See Greene (2003), Hansen (1982), and Keane and Runkle (1998). 46 For more details on both methods, see Greene (2003). 47 Robust standard errors are also known as heteroskedasticity-consistent standard errors or Whitecorrected standard errors. 48 Remember, this is a two-tailed test. 352 Quantitative Investment Analysis across observations. Trying to explain a particular financial relation over a number of periods is risky, because errors in financial regression models are often correlated through time. When regression errors are correlated across observations, we say that they are serially correlated (or autocorrelated). Serial correlation most typically arises in time-series regressions. In this section, we discuss three aspects of serial correlation: its effect on statistical inference, tests for it, and methods to correct for it. 4.2.1. The Consequences of Serial Correlation As with heteroskedasticity, the principal problem caused by serial correlation in a linear regression is an incorrect estimate of the regression coefficient standard errors computed by statistical software packages. As long as none of the independent variables is a lagged value of the dependent variable (a value of the dependent variable from a previous period), then the estimated parameters themselves will be consistent and need not be adjusted for the effects of serial correlation. If, however, one of the independent variables is a lagged value of the dependent variable—for example, if the T-bill return from the previous month was an independent variable in the Fisher effect regression—then serial correlation in the error term will cause all the parameter estimates from linear regression to be inconsistent and they will not be valid estimates of the true parameters.49 In this chapter, we assume that none of the independent variables is a lagged value of the dependent variable. When that is the case, the effect of serial correlation appears in the regression coefficient standard errors. We will examine it here for the positive serial correlation case, because that case is so common. Positive serial correlation is serial correlation in which a positive error for one observation increases the chance of a positive error for another observation. Positive serial correlation also means that a negative error for one observation increases the chance of a negative error for another observation.50 In examining positive serial correlation, we make the common assumption that serial correlation takes the form of first-order serial correlation, or serial correlation between adjacent observations. In a time-series context, that assumption means the sign of the error term tends to persist from one period to the next. Although positive serial correlation does not affect the consistency of the estimated regression coefficients, it does affect our ability to conduct valid statistical tests. First, the F -statistic to test for overall significance of the regression may be inflated because the mean squared error (MSE) will tend to underestimate the population error variance. Second, positive serial correlation typically causes the ordinary least squares (OLS)standard errors for the regression coefficients to underestimate the true standard errors. As a consequence, if positive serial correlation is present in the regression, standard linear regression analysis will typically lead us to compute artificially small standard errors for the regression coefficient. These small standard errors will cause the estimated t-statistics to be inflated, suggesting significance where perhaps there is none. The inflated t-statistics may, in turn, lead us to incorrectly reject null hypotheses about population values of the parameters of the regression model more often than we would if the standard errors were correctly estimated. This Type I error could lead to improper investment recommendations.51 49 We address this issue in the chapter on time-series analysis. contrast, with negative serial correlation, a positive error for one observation increases the chance of a negative error for another observation, and a negative error for one observation increases the chance of a positive error for another. 51 OLS standard errors need not be underestimates of actual standard errors if negative serial correlation is present in the regression. 50 In Chapter 9 Multiple Regression and Issues in Regression Analysis 353 4.2.2. Testing for Serial Correlation We can choose from a variety of tests for serial correlation in a regression model,52 but the most common is based on a statistic developed by Durbin and Watson (1951); in fact, many statistical software packages compute the Durbin–Watson statistic automatically. The equation for the Durbin–Watson test statistic is DW = T $ t=2 (ϵ̂t − ϵ̂t−1 )2 T $ (9-6) ϵ̂2t t=1 where ϵ̂t is the regression residual for period t. We can rewrite this equation as T 1 $ 2 (ϵ̂ − 2ϵ̂t ϵ̂t−1 + ϵ̂2t−1 ) T − 1 t=2 t T 1 $ t ϵ̂ T − 1 t=1 2 ≈ Var(ϵ̂t ) − 2 Cov(ϵ̂t , ϵ̂t−1 ) + Var(ϵ̂t−1 ) Var(ϵ̂t ) If the variance of the error is constant through time, then we expect Var(ϵ̂t ) = σ̂ϵ̂2 for all t, where we use σϵ2 to represent the estimate of the constant error variance. If, in addition, the errors are also not serially correlated, then we expect Cov(ϵ̂t , ϵ̂t−1 ) = 0. In that case, the Durbin–Watson statistic is approximately equal to σ̂ϵ2 − 0 + σ̂ϵ2 =2 σ̂ϵ2 This equation tells us that if the errors are homoskedastic and not serially correlated, then the Durbin–Watson statistic will be close to 2. Therefore, we can test the null hypothesis that the errors are not serially correlated by testing whether the Durbin–Watson statistic differs significantly from 2. If the sample is very large, the Durbin–Watson statistic will be approximately equal to 2(1 − r), where r is the sample correlation between the regression residuals from one period and those from the previous period. This approximation is useful because it shows the value of the Durbin–Watson statistic for differing levels of serial correlation. The Durbin–Watson statistic can take on values ranging from 0 (in the case of serial correlation of +1) to 4 (in the case of serial correlation of −1): If the regression has no serial correlation, then the regression residuals will be uncorrelated through time and the value of the Durbin–Watson statistic will be equal to 2(1 − 0) = 2. • If the regression residuals are positively serially correlated, then the Durbin–Watson statistic will be less than 2. For example, if the serial correlation of the errors is 1, then the value of the Durbin–Watson statistic will be 0. • If the regression residuals are negatively serially correlated, then the Durbin–Watson statistic will be greater than 2. For example, if the serial correlation of the errors is −1, then the value of the Durbin–Watson statistic will be 4. • 52 See Greene (2003) for a detailed discussion of tests of serial correlation. 354 Quantitative Investment Analysis Returning to Example 9-8, which explored the Fisher effect, as shown in Table 9-6 the Durbin–Watson statistic for the OLS regression is 0.4673. This result means that the regression residuals are positively serially correlated: DW = 0.4673 ≈ 2(1 − r) r ≈ 1 − DW/2 = 1 − 0.4673/2 = 0.766 This outcome raises the concern that OLS standard errors may be incorrect because of positive serial correlation. Does the observed Durbin–Watson statistic (0.4673) provide enough evidence to warrant rejecting the null hypothesis of no positive serial correlation? We should reject the null hypothesis of no serial correlation if the Durbin–Watson statistic is below a critical value, d ∗ . Unfortunately, Durbin and Watson also showed that, for a given sample, we cannot know the true critical value, d ∗ . Instead, we can determine only that d ∗ lies either between two values, du (an upper value) and dl (a lower value), or outside those values.53 Figure 9-3 depicts the upper and lower values of d ∗ as they relate to the results of the Durbin–Watson statistic. Inconclusive d1 du Reject hypothesis of no serial correlation Fail to reject null hypothesis FIGURE 9-3 Value of the Durbin–Watson Statistic From Figure 9-3, we learn the following: When the Durbin–Watson (DW) statistic is less than dl , we reject the null hypothesis of no positive serial correlation. • When the DW statistic falls between dl and du , the test results are inconclusive. • When the DW statistic is greater than du , we fail to reject the null hypothesis of no positive serial correlation.54 • Returning to Example 9-8, the Fisher effect regression has one independent variable and 137 observations. The Durbin–Watson statistic is 0.4673. If we look at Appendix E in the column marked k = 1, we see that we can reject the null hypothesis of no correlation in favor of the alternative hypothesis of positive serial correlation at the 0.05 level because the Durbin–Watson statistic is far below dl for k = 1 and n = 100 (1.65). The level of dl would 53 Appendix E tabulates the 0.05 significance levels of du and dl for differing numbers of estimated parameters (k = 1, 2, . . ., 5) and time periods between 15 and 100. 54 Of course, sometimes serial correlation in a regression model is negative rather than positive. For a null hypothesis of no serial correlation, the null hypothesis is rejected if DW < dl (indicating significant positive serial correlation) or if DW > 4 − dl (indicating significant negative serial correlation). 355 Chapter 9 Multiple Regression and Issues in Regression Analysis be even higher for a sample of 137 observations. This finding of significant positive serial correlation suggests that the OLS standard errors in this regression probably significantly underestimate the true standard errors. 4.2.3. Correcting for Serial Correlation We have two alternative remedial steps when a regression has significant serial correlation. First, we can adjust the coefficient standard errors for the linear regression parameter estimates to account for the serial correlation. Second, we can modify the regression equation itself to eliminate the serial correlation. We recommend using the first method for dealing with serial correlation; the second method may result in inconsistent parameter estimates unless implemented with extreme care. The most prevalent method for adjusting standard errors was developed by Hansen (1982) and is a standard feature in many statistical software packages.55 An additional advantage of Hansen’s method is that it simultaneously corrects for conditional heteroskedasticity.56 Table 9-8 shows the results of correcting the standard errors from Table 9-6 for serial correlation and heteroskedasticity using Hansen’s method. Note that the coefficients for both the intercept and the slope are exactly the same as in the original regression. The robust standard errors are now much larger, however—about twice the OLS standard errors. Because of the severe serial correlation in the regression error, OLS greatly underestimates the uncertainty about the estimated parameters in the regression. Note also that the Durbin–Watson statistic has not changed from Table 9-6. The serial correlation has not been eliminated, but the standard error has been corrected to account for the serial correlation. TABLE 9-8 Results from Regressing T-Bill Returns on Predicted Inflation (Standard Errors Corrected for Conditional Heteroskedasticity and Serial Correlation) Intercept Inflation prediction Residual standard error Multiple R-squared Observations Durbin–Watson statistic Coefficient Standard Error 0.0304 0.8774 0.0069 0.1729 t-Statistic 4.4106 5.0730 0.0220 0.4640 137 0.4673 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. Now suppose we want to test our original null hypothesis (the Fisher effect) that the coefficient on the predicted inflation term equals 1 (H0 : b1 = 1) against the alternative that the coefficient on the inflation term is not equal to 1 (Ha : b1 ̸= 1). With the corrected standard errors, the value of the test statistic for this null hypothesis is 55 This correction is known by various names, including serial-correlation consistent standard errors, serial correlation and heteroskedasticity adjusted standard errors, robust standard errors, and Hansen–White standard errors. Analysts may also say that they use the Newey–West method for computing robust standard errors. 56 We do not always use Hansen’s method to correct for serial correlation and heteroskedasticity because sometimes the errors of a regression are not serially correlated. 356 Quantitative Investment Analysis 0.8774 − 1 b̂1 − b1 = −0.7091 = sb̂1 0.1729 The critical values for both the 0.05 and 0.01 significance level are much larger than 0.7091 (absolute value of the t-test statistic), so we cannot reject the null hypothesis. In this particular case, our conclusion about the Fisher effect was not affected by serial correlation, but the standard error on the slope coefficient after taking into account serial correlation and conditional heteroskedasticity (0.1729) is more than double the OLS standard error (0.0812). Therefore, for some hypotheses, serial correlation and conditional heteroskedasticity could have had a big effect on whether we accepted or rejected those hypotheses.57 4.3. Multicollinearity The second assumption of the multiple linear regression model is that no exact linear relationship exists between two or more of the independent variables. When one of the independent variables is an exact linear combination of other independent variables, it becomes mechanically impossible to estimate the regression. That case, known as perfect collinearity, is much less of a practical concern than multicollinearity.58 Multicollinearity occurs when two or more independent variables (or combinations of independent variables) are highly (but not perfectly) correlated with each other. With multicollinearity we can estimate the regression, but the interpretation of the regression output becomes problematic. Multicollinearity is a serious practical concern because approximate linear relationships among financial variables are common. 4.3.1. The Consequences of Multicollinearity Although the presence of multicollinearity does not affect the consistency of the OLS estimates of the regression coefficients, the estimates become extremely imprecise and unreliable. Furthermore, it becomes practically impossible to distinguish the individual impacts of the independent variables on the dependent variable. These consequences are reflected in inflated OLS standard errors for the regression coefficients. With inflated standard errors, t-tests on the coefficients have little power (ability to reject the null hypothesis). 4.3.2. Detecting Multicollinearity In contrast to the cases of heteroskedasticity and serial correlation, we shall not provide a formal statistical test for multicollinearity. In practice, multicollinearity is often a matter of degree rather than of absence or presence.59 The analyst should be aware that using the magnitude of pairwise correlations among the independent variables to assess multicollinearity, as has occasionally been suggested, is generally not adequate. Although very high pairwise correlations among independent variables can indicate multicollinearity, it is not necessary for such pairwise correlations to be high for 57 Serial correlation can also affect forecast accuracy. We discuss this issue in the chapter on time series. give an example of perfect collinearity, suppose we tried to explain a company’s credit ratings with a regression that included net sales, cost of goods sold, and gross profit as independent variables. Because Gross profit = Net sales − Cost of goods sold by definition, there is an exact linear relationship between these variables. This type of blunder is relatively obvious (and easy to avoid). 59 See Kmenta (1986). 58 To 357 Chapter 9 Multiple Regression and Issues in Regression Analysis there to be a problem of multicollinearity.60 Stated another way, high pairwise correlations among the independent variables are not a necessary condition for multicollinearity, and low pairwise correlations do not mean that multicollinearity is not a problem. The only case in which correlation between independent variables may be a reasonable indicator of multicollinearity occurs in a regression with exactly two independent variables. The classic symptom of multicollinearity is a high R 2 (and significant F -statistic) even though the t-statistics on the estimated slope coefficients are not significant. The insignificant t-statistics reflect inflated standard errors. Although the coefficients might be estimated with great imprecision, as reflected in low t-statistics, the independent variables as a group may do a good job of explaining the dependent variable, and a high R 2 would reflect this effectiveness. Example 9-9 illustrates this diagnostic. EXAMPLE 9-9 Multicollinearity in Explaining Returns to the Fidelity Select Technology Fund In Example 9-3 we regressed returns to the Fidelity Select Technology Fund (FSPTX) on returns to the S&P 500/BARRA Growth Index and the S&P 500/BARRA Value Index. Table 9-9 shows the results of our regression, which uses data from January 1998 through December 2002. The t-statistic of 9.7034 on the growth index return is greater than 2.0, indicating that the coefficient on the growth index differs significantly from 0 at standard significance levels. On the other hand, the t-statistic on the value index return is −1.5953 and thus is not statistically significant. This result suggests that the returns to the FSPTX are linked to the returns to the growth index and not closely associated with the returns to the value index. The coefficient on the growth index, however, is 2.23. This result implies that returns on the FSPTX are more volatile than are returns on the growth index. TABLE 9-9 Results from Regressing the FSPTX Returns on the S&P 500/BARRA Growth and Value Indexes Intercept S&P 500/BARRA Growth Index S&P 500/BARRA Value Index Coefficient Standard Error t-Statistic 0.0079 2.2308 −0.4143 0.0091 0.2299 0.2597 0.8635 9.7034 −1.5953 ANOVA df SS MSS F Significance F Regression Residual Total 2 57 59 0.8649 0.2851 1.1500 0.4324 0.0050 86.4483 5.48E-18 Residual standard error Multiple R-squared Observations 0.0707 0.7521 60 Source: Ibbotson Associates. 60 Even if pairs of independent variables have low correlation, there may be linear combinations of the independent variables that are very highly correlated, creating a multicollinearity problem. 358 Quantitative Investment Analysis Note also that this regression explains a significant amount of the variation in the returns to the FSPTX. Specifically, the R 2 from this regression is 0.7521. Thus approximately 75 percent of the variation in the returns to the FSPTX is explained by returns to the S&P 500/BARRA growth and value indexes. Now suppose we run another linear regression that adds returns to the S&P 500 itself to the returns to the S&P 500/BARRA Growth and Value indexes. The S&P 500 includes the component stocks of these two style indexes, so we are introducing a severe multicollinearity problem. Table 9-10 shows the results of that regression. Note that the R 2 in this regression has changed almost imperceptibly from the R 2 in the previous regression (increasing from 0.7521 to 0.7539), but now the standard errors of the coefficients are much larger. Adding the return to the S&P 500 to the previous regression does not explain any more of the variance in the returns to the FSPTX than the previous regression did, but now none of the coefficients is statistically significant. This is the classic case of multicollinearity mentioned in the text. TABLE 9-10 Results from Regressing the FSPTX Returns on Returns to the S&P 500/BARRA Growth and Value Indexes and the S&P 500 Index Coefficient Intercept S&P 500/BARRA Growth Index S&P 500/BARRA Value Index S&P 500 Index 0.0072 −1.1324 −3.4912 6.4436 Standard Error t-Statistic 0.0092 5.2443 4.8004 10.0380 0.7761 −0.2159 −0.7273 0.6419 ANOVA df SS MSS F Significance F Regression Residual Total 3 56 59 0.8670 0.2830 1.1500 0.2890 0.0051 57.1751 4.73E-17 Residual standard error Multiple R-squared Observations 0.0711 0.7539 60 Source: Ibbotson Associates. Multicollinearity may be a problem even when we do not observe the classic symptom of insignificant t-statistics but a highly significant F -test. Advanced textbooks provide further tools to help diagnose multicollinearity.61 4.3.3. Correcting for Multicollinearity The most direct solution to multicollinearity is excluding one or more of the regression variables. In the example above, we can see that the S&P 500 total returns should not be included if both the S&P 500/BARRA Growth and Value Indexes are included, because the returns to the entire S&P 500 Index are a weighted average of the return to growth stocks and value stocks. In many cases, however, no easy solution is 61 See Greene (2003). 359 Chapter 9 Multiple Regression and Issues in Regression Analysis available to the problem of multicollinearity, and you will need to experiment with including or excluding different independent variables to determine the source of multicollinearity. 4.4. Heteroskedasticity, Serial Correlation, Multicollinearity: Summarizing the Issues We have discussed some of the problems that heteroskedasticity, serial correlation, and multicollinearity may cause in interpreting regression results. These violations of regression assumptions, we have noted, all lead to problems in making valid inferences. The analyst should check that model assumptions are fulfilled before interpreting statistical tests. Table 9-11 gives a summary of these problems, the effect they have on the linear regression results (an analyst can see these effects using regression software), and the solutions to these problems. TABLE 9-11 Problems in Linear Regression and Their Solutions Problem Effect Heteroskedasticity Incorrect standard errors Serial Correlation Incorrect standard errors (additional problems if a lagged value of the dependent variable is used as an independent variable) High R 2 and low t-statistics Multicollinearity Solution Use robust standard errors (corrected for conditional heteroskedasticity) Use robust standard errors (corrected for serial correlation) Remove one or more independent variables; often no solution based in theory 5. MODEL SPECIFICATION AND ERRORS IN SPECIFICATION Until now, we have assumed that whatever regression model we estimate is correctly specified. Model specification refers to the set of variables included in the regression and the regression equation’s functional form. In the following, we first give some broad guidelines for correctly specifying a regression. Then we turn to three types of model misspecification: misspecified functional form, regressors that are correlated with the error term, and additional time-series misspecification. Each of these types of misspecification invalidates statistical inference using OLS; most of these misspecifications will cause the estimated regression coefficients to be inconsistent. 5.1. Principles of Model Specification In discussing the principles of model specification, we need to acknowledge that there are competing philosophies about how to approach model specification. Furthermore, our purpose for using regression analysis may affect the specification we choose. The following principles have fairly broad application, however. 360 • • • • • Quantitative Investment Analysis The model should be grounded in cogent economic reasoning. We should be able to supply the economic reasoning behind the choice of variables, and the reasoning should make sense. When this condition is fulfilled, we increase the chance that the model will have predictive value with new data. This approach contrasts to thevariable-selection process known as data mining, discussed in the chapter on sampling. With data mining, the investigator essentially develops a model that maximally exploits the characteristics of a specific dataset. The functional form chosen for the variables in the regression should be appropriate given the nature of the variables. As one illustration, consider studying mutual fund market timing based on fund and market returns alone. One might reason that for a successful timer, a plot of mutual fund returns against market returns would show curvature, because a successful timer would tend to increase (decrease) beta when market returns were high (low). The model specification should reflect the expected nonlinear relationship.62 In other cases, we may transform the data such that a regression assumption is better satisfied. The model should be parsimonious. In this context, ‘‘parsimonious’’ means accomplishing a lot with a little. We should expect each variable included in a regression to play an essential role. The model should be examined for violations of regression assumptions before being accepted. We have already discussed detecting the presence of heteroskedasticity, serial correlation, and multicollinearity. As a result of such diagnostics, we may conclude that we need to revise the set of included variables and/or their functional form. The model should be tested and be found useful out of sample before being accepted. The term ‘‘out of sample’’ refers to observations outside the dataset on which the model was estimated. A plausible model may not perform well out of sample because economic relationships have changed since the sample period. That possibility is itself useful to know. A second explanation, however, may be that relationships have not changed but that the model explains only a specific dataset. Having given some broad guidance on model specification, we turn to a discussion of specific model specification errors. Understanding these errors will help an analyst develop better models and be a more informed consumer of investment research. 5.2. Misspecified Functional Form Whenever we estimate a regression, we must assume that the regression has the correct functional form. This assumption can fail in several ways: One or more important variables could be omitted from regression. One or more of the regression variables may need to be transformed (for example, by taking the natural logarithm of the variable) before estimating the regression. • The regression model pools data from different samples that should not be pooled. • • First, consider the effects of omitting an important independent variable from a regression (omitted variable bias). If the true regression model was 62 This example is based on Treynor and Mazuy (1966), an early regression study of mutual fund timing. To capture curvature, they included a term in the squared market excess return, which does not violate the assumption of the multiple linear regression model that relationship between the dependent and independent variables is linear in the coefficients. 361 Chapter 9 Multiple Regression and Issues in Regression Analysis Yi = b0 + b1 X1i + b2 X2i + ϵi (9-7) but we estimate the model63 Yi = a0 + a1 X1i + ϵi then our regression model would be misspecified. What is wrong with the model? If the omitted variable (X2 ) is correlated with the remaining variable (X1 ), then the error term in the model will be correlated with (X1 ), and the estimated values of the regression coefficients a0 and a1 would be biased and inconsistent. In addition, the estimates of the standard errors of those coefficients will also be inconsistent, so we can use neither the coefficients estimates nor the estimated standard errors to make statistical tests. EXAMPLE 9-10 Omitted Variable Bias and the Bid–Ask Spread In this example, we extend our examination of the bid–ask spread to show the effect of omitting an important variable from a regression. In Example 9-1, we showed that the natural logarithm of the ratio [(Bid–ask spread)/Price] was significantly related to both the natural logarithm of the number of market makers and the natural logarithm of the market capitalization of the company. We repeat Table 9-1 from Example 9-1 below. TABLE 9-1 (repeated) Results from Regressing ln(Bid–Ask Spread/Price) on ln(Number of Market Makers) and ln(Market Cap) Intercept ln(Number of Nasdaq market makers) ln(Company’s market cap) ANOVA Coefficients Standard Error −0.7586 −0.2790 −0.6635 0.1369 0.0673 0.0246 t-Statistic −5.5416 −4.1427 −27.0087 df SS MSS F Significance F 2 1,816 1,818 2,681.6482 2,236.2820 4,917.9302 1,340.8241 1,2314 1,088.8325 0.00 Residual standard error Multiple R-squared Observations 1.1097 0.5453 1,819 Regression Residual Total Source: FactSet, Nasdaq. If we did not include the natural log of market capitalization as an independent variable in the regression, and we regressed the natural logarithm of the ratio [(Bid–ask spread)/Price] only on the natural logarithm of the number of market makers for the stock, the results would be as shown in Table 9-12. 63 We use a different regression coefficient notation when X2i is omitted, because the intercept term and slope coefficient on X1i will generally not be the same as when X2i is included. 362 Quantitative Investment Analysis TABLE 9-12 Results from Regressing ln(Bid–Ask Spread/Price) on ln (Number of Market Makers) Intercept ln(Number of Nasdaq market makers) ANOVA Coefficients Standard Error −0.1229 −1.6629 0.1596 0.0517 t-Statistic −0.7698 −32.1519 df SS MSS F Significance F 1 1,817 1,818 1,783.3549 3,134.5753 4,917.9302 1,783.3549 1.7251 1,033.7464 0.00 Residual standard error Multiple R-squared Observations 1.3134 0.3626 1,819 Regression Residual Total Source: FactSet, Nasdaq. Note that the coefficient on ln(Number of Nasdaq market makers) fell from −0.2790 in the original (correctly specified) regression to −1.6629 in the misspecified regression. Also, the intercept rose from −0.7586 in the correctly specified regression to −0.1229 in the misspecified regression. These results illustrate that omitting an independent variable that should be in the regression can cause the remaining regression coefficients to be inconsistent. A second common cause of misspecification in regression models is the use of the wrong form of the data in a regression, when a transformed version of the data is appropriate. For example, sometimes analysts fail to account for curvature or nonlinearity in the relationship between the dependent variable and one or more of the independent variables, instead specifying a linear relation among variables. When we are specifying a regression model, we should consider whether economic theory suggests a nonlinear relation. We can often confirm the nonlinearity by plotting the data, as we will illustrate in Example 9-11 below. If the relationship between the variables becomes linear when one or more of the variables is represented as a proportional change in the variable, we may be able to correct the misspecification by taking the natural logarithm of the variable(s) we want to represent as a proportional change. Other times, analysts use unscaled data in regressions, when scaled data (such as dividing net income or cash flow by sales) are more appropriate. In Example 9-1, we scaled the bid–ask spread by stock price because what a given bid–ask spread means in terms of transactions costs for a given size investment depends on the price of the stock; if we had not scaled the bid–ask spread, the regression would have been misspecified. EXAMPLE 9-11 Nonlinearity and the Bid–Ask Spread In Example 9-1, we showed that the natural logarithm of the ratio [(Bid–ask spread)/ Price] was significantly related to both the natural logarithm of the number of market makers and the natural logarithm of the company’s market capitalization. But why did Chapter 9 Multiple Regression and Issues in Regression Analysis 363 we take the natural logarithm of each of the variables in the regression? We began a discussion of this question in Example 9-1, which we continue now. What does theory suggest about the nature of the relationship between the ratio (Bid–ask spread)/Price, or the percentage bid–ask spread, and its determinants (the independent variables)? Stoll (1978) builds a theoretical model of the determinants of percentage bid–ask spread in a dealer market. In his model, the determinants enter multiplicatively in a particular fashion. In terms of the independent variables introduced in Example 9-1, the functional form assumed is [(Bid-ask spread)/Price]i = c(Number of market makers)ib1 × (Market capitalization)bi 2 where c is a constant. The relationship of the percentage bid–ask spread with the number of market makers and market capitalization is not linear in the original variables.64 If we take natural log of both sides of the above model, however, we have a log-log regression that is linear in the transformed variables:65 Yi = b0 + b1 X1i + b2 X2i + ϵi where Yi b0 X1i X2i ϵi = the natural logarithm of the ratio (Bid-ask spread)/Price for stock i = a constant that equals ln(c) = the natural logarithm of the number of market makers for stock i = the natural logarithm of the market capitalization of company i = the error term As mentioned in Example 9-1, a slope coefficient in the log-log model is interpreted as an elasticity, precisely, the partial elasticity of the dependent variable with respect to the independent variable (‘‘partial’’ means holding the other independent variables constant). We can plot the data to assess whether the variables are linearly related after the logarithmic transformation. For example, Figure 9-4 shows a scatterplot of the natural logarithm of the number of market makers for a stock (on the X axis) and the natural logarithm of (Bid–ask spread)/Price (on the Y axis), as well as a regression line showing the linear relation between the two transformed variables. The relation between the two transformed variables is clearly linear. If we do not take log of the ratio (Bid–ask spread)/Price, the plot is not linear. Figure 9-5 shows a plot of the natural logarithm of the number of market makers for a stock (on the X axis) and the ratio (Bid–ask spread)/Price (on the Y axis), as well as a regression line that attempts to show a linear relation between the two variables. We see that the relation between the two variables is very nonlinear.66 Consequently, we should not estimate a regression with (Bid–ask spread)/Price as the dependent 64 The form of the model is analogous to the Cobb–Douglas production function in economics. We have added an error term to the model. 66 The relation between (Bid–ask spread)/Price and ln(Market cap) is also nonlinear, while the relation between ln(Bid–ask spread)/Price and ln(Market cap) is linear. We omit these scatterplots to save space. 65 364 Quantitative Investment Analysis variable. Consideration of the need to assure that predicted bid–ask spreads are positive would also lead us to not use (Bid–ask spread)/Price as the dependent variable. If we use the nontransformed ratio (Bid–ask spread)/Price as the dependent variable, the estimated model could predict negative values of the bid–ask spread. This result would be nonsensical; in reality, no bid–ask spread is negative (it is hard to motivate traders to simultaneously buy high and sell low), so a model that predicts negative bid–ask spreads is certainly misspecified.67 We illustrate the problem of negative values of the predicted bid–ask spreads now. Table 9-13 shows the results of a regression with (Bid–ask spread)/Price as the dependent variable and the natural logarithm of the number of market makers and the natural logarithm of the company’s market capitalization as the independent variables. Ln ((Bid–Ask)/Price) 0 !1 !2 !3 !4 !5 !6 !7 !8 !9 !10 0 0.5 1.0 1.5 2.0 2.5 3 3.5 4.0 4.5 5.0 Ln (# of Market Makers) FIGURE 9-4 Linear Regression When Two Variables Have a Linear Relation Suppose that for a particular Nasdaq-listed stock, the number of market makers is 20 and the market capitalization is $5 billion. Therefore, the natural log of the number of market makers equals ln 20 = 2.9957 and the natural log of the stock’s market capitalization (in millions) is ln 5, 000 = 8.5172. In this case, the predicted ratio of bid–ask spread to price is 0.0658 + (2.9957 × −0.0045) + (−0.0068 × 8.5172) = −0.0056. Therefore, the model predicts that the ratio of bid–ask spread to stock price is −0.0056 or −0.56 percent of the stock price. Thus the predicted bid–ask spread is negative, which does not make economic sense. This problem could be avoided by using log of (Bid–ask spread)/Price as the dependent variable.68 67 In our data sample, the bid–ask spread for each of the 1,819 companies is positive. Whether the natural log of the percentage bid–ask spread, Y , is positive or negative, the percentage bid–ask spread found as eY is positive, because a positive number raised to any power is positive. The constant e is positive (e ≈ 2.7183). 68 365 Chapter 9 Multiple Regression and Issues in Regression Analysis (Bid–Ask)/Price 40.0 35.0 30.0 25.0 20.0 15.0 10.0 05.0 0.00 !0.05 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ln (# of Market Makers) FIGURE 9-5 Linear Regression When Two Variables Have a Nonlinear Relation TABLE 9-13 Results from Regressing Bid–Ask Spread/Price on ln(Number of Market Makers) and ln(Market Cap) Intercept ln(Number of Nasdaq market makers) ln(Company’s market cap) ANOVA Regression Residual Total Coefficient Standard Error t-Statistic 0.0658 −0.0045 −0.0068 0.0024 0.0012 0.0004 27.6430 −3.8714 −15.8679 df SS MSS F Significance F 2 1816 1818 0.3185 0.6760 0.9944 0.1592 0.0004 427.8174 0.00 Residual standard error Multiple R-squared Observations 0.0193 0.3203 1,819 Source: FactSet, Nasdaq. Often, analysts must decide whether to scale variables before they compare data across companies. For example, in financial statement analysis, analysts often compare companies using common size statements. In a common size income statement, all the line items 366 Quantitative Investment Analysis in a company’s income statement are divided by the company’s revenues.69 Common size statements make comparability across companies much easier. An analyst can use common size statements to quickly compare trends in gross margins (or other income statement variables) for a group of companies. Issues of comparability also appear for analysts who want to use regression analysis to compare the performance of a group of companies. Example 9-12 illustrates this issue. EXAMPLE 9-12 Scaling and the Relation between Cash Flow from Operations and Free Cash Flow Suppose an analyst wants to explain free cash flow to the firm as a function of cash flow from operations in 2001 for 11 family clothing stores in the United States with market capitalizations of more than $100 million as of the end of 2001. To investigate this issue, the analyst might use free cash flow as the dependent variable and cash flow from operations as the independent variable in single-independentvariable linear regression. Table 9-14 shows the results of that regression. Note that the t-statistic for the slope coefficient for cash flow from operations is quite high (6.5288), the significance level for the F -statistic for the regression is very low (0.0001), and the R-squared is quite high. We might be tempted to believe that this regression is a success and that for a family clothing store, if cash flow from operations increased by $1.00, we could confidently predict that free cash flow to the firm would increase by $0.3579. TABLE 9-14 Results from Regressing the Free Cash Flow on Cash Flow from Operations for Family Clothing Stores Intercept Cash flow from operations Coefficient Standard Error t-Statistic 0.7295 0.3579 27.7302 0.0548 0.0263 6.5288 ANOVA df SS MSS F Significance F Regression Residual Total 1 9 10 245,093.7836 51,750.3139 296,844.0975 245,093.7836 5,750.0349 42.6247 0.0001 Residual standard error Multiple R-squared Observations 75.8290 0.8257 11 Source: Compustat. But is this specification correct? The regression does not account for size differences among the companies in the sample. 69 For more on common size statements, see White, Sondhi, and Fried (2003). Free cash flow and cash flow from operations are discussed in Stowe, Robinson, Pinto, and McLeavey (2002). 367 Chapter 9 Multiple Regression and Issues in Regression Analysis We can account for size differences by using common size cash flow results across companies. We scale the variables by dividing cash flow from operations and free cash flow to the firm by the company’s sales before using regression analysis. We will use (Free cash flow to the firm/Sales) as the dependent variable and (Cash flow from operations/Sales) as the independent variable. Table 9-15 shows the results of this regression. Note that the t-statistic for the slope coefficient on (Cash flow from operations/Sales) is 1.6262, so it is not significant at the 0.05 level. Note also that the significance level of the F -statistic is 0.1383, so we cannot reject at the 0.05 level the hypothesis that the regression does not explain variation in (Free cash flow/Sales) among family clothing stores. Finally, note that the R-squared in this regression is much lower than that of the previous regression. TABLE 9-15 Results from Regressing the Free Cash Flow/Sales on Cash Flow from Operations/Sales for Family Clothing Stores Intercept Cash flow from operations/Sales Coefficient Standard Error −0.0121 0.4749 0.0221 0.2920 t-Statistic −0.5497 1.6262 ANOVA df SS MSS F Significance F Regression Residual Total 1 9 10 0.0030 0.0102 0.0131 0.0030 0.0011 2.6447 0.1383 Residual standard error Multiple R-squared Observations 0.0336 0.2271 11 Source: Compustat. Which regression makes more sense? Usually, the scaled regression makes more sense. We want to know what happens to free cash flow (as a fraction of sales) if a change occurs in cash flow from operations (as a fraction of sales). Without scaling, the results of the regression can be based solely on scale differences across companies, rather than based on the companies’ underlying economics. A third common form of misspecification in regression models is pooling data from different samples that should not be pooled. This type of misspecification can best be illustrated graphically. Figure 9-6 shows two clusters of data on variables X and Y , with a fitted regression line. The data could represent the relationship between two financial variables at two stages of a company’s growth, for example. In each cluster of data on X and Y , the correlation between the two variables is virtually 0. Because the means of both X and Y are different for the two clusters of data in the combined sample, X and Y are highly correlated. The correlation is spurious (misleading), however, because it reflects scale differences across companies. 368 Quantitative Investment Analysis Series Y 10 8 6 4 2 0 !2 !2 !1 0 1 2 3 4 5 6 7 8 9 Series X FIGURE 9-6 Plot of Two Series with Changing Means 5.3. Time-Series Misspecification (Independent Variables Correlated with Errors) In the previous section, we discussed the misspecification that arises when a relevant independent variable is omitted from a regression. In this section, we discuss problems that arise from the kinds of variables included in the regression, particularly in a time-series context. In models that use time-series data to explain the relations among different variables, it is particularly easy to violate Regression Assumption 3, that the error term has mean 0, conditioned on the independent variables. If this assumption is violated, the estimated regression coefficients will be biased and inconsistent. Three common problems that create this type of time-series misspecification are including lagged dependent variables as independent variables in regressions with serially correlated errors, • including a function of a dependent variable as an independent variable, sometimes as a result of the incorrect dating of variables, and • independent variables that are measured with error. • The next examples demonstrate these problems. Suppose that an analyst has estimated a linear regression with significant serial correlation in the errors. That serial correlation could be corrected by the methods discussed previously in this chapter. Nevertheless, suppose that the analyst includes as an additional independent variable the first lagged value of the dependent variable. For example, the analyst might use the regression equation Yt = b0 + b1 X1t + b2 Yt−1 + ϵt (9-8) 369 Chapter 9 Multiple Regression and Issues in Regression Analysis Because we assume that the error term is serially correlated, by definition the error term is correlated with the dependent variable. Consequently, the lagged dependent variable, Yt−1 , will be correlated with the error term, violating the assumption that the independent variables are uncorrelated with the error term. As a result, the estimates of the regression coefficients will be biased and inconsistent. EXAMPLE 9-13 Fisher Effect with a Lagged Dependent Variable In our discussion of serial correlation, we concluded from a test using the Durbin– Watson test that the error term in the Fisher effect equation (Equation 9-5) showedpositive (first-order) serial correlation, using three-month T-bill returns as the dependent variable and inflation expectations of professional forecasters as the independent variable. Observations on the dependent and independent variables were quarterly. Table 9-16 modifies that regression by including the previous quarter’s three-month T-bill returns as an additional independent variable. TABLE 9-16 Results from Regressing T-Bill Returns on Predicted Inflation and Lagged T-Bill Returns Intercept Inflation prediction Lagged T-bill return Residual standard error Multiple R-squared Observations Coefficient Standard Error t-Statistic 0.0046 0.2753 0.7553 0.0040 0.0631 0.0495 1.5718 4.3610 15.2510 0.0134 0.8041 137 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. At first glance, these regression results look very interesting—the coefficient on the lagged T-bill return appears to be highly significant. But on closer consideration, we must ignore these regression results, because the regression is fundamentally misspecified. As long as the error term is serially correlated, including lagged T-bill returns as an independent variable in the regression will cause all the coefficient estimates to be biased and inconsistent. Therefore, this regression is not useable for either testing a hypothesis or for forecasting. A second common time-series misspecification in investment analysis is to forecast the past. What does that mean? If we forecast the future (say we predict at time t the value of variable Y in period t + 1), we must base our predictions on information we knew at time t. We could use a regression to make that forecast using the equation Yt+1 = b0 + b1 X1t + ϵt+1 (9-9) 370 Quantitative Investment Analysis In this equation, we predict the value of Y in time t + 1 using the value of X in time t. The error term, ϵt+1 , is unknown at time t and thus should be uncorrelated with X1t . Unfortunately, analysts sometimes use regressions that try to forecast the value of a dependent variable at time t + 1 based on independent variable(s) that are functions of the value of the dependent variable at time t + 1. In such a model, the independent variable(s) would be correlated with the error term, so the equation would be misspecified. As an example, an analyst may try to explain the cross-sectional returns for a group of companies during a particular year using the market-to-book ratio and the market capitalization for those companies at the end of the year.70 If the analyst believes that such a regression predicts whether companies with high market-to-book ratios or high market capitalizations will have high returns, the analyst is mistaken. For any given period, the higher the return during the period, the higher the market capitalization at the end of the period. It is also true that the higher the return during the period, the higher the market-to-book ratio at the end of the period. So in this case, if all the cross-sectional data come from period t + 1, a high value of the dependent variable (returns) actually causes a high value of the independent variables (market cap and market-to-book), rather than the other way around. In this type of misspecification, the regression model effectively includes the dependent variable on both the right- and left-hand sides of the regression equation. The third common time-series misspecification arises when an independent variable is measured with error. Suppose a financial theory tells us that a particular variable Xt , such as expected inflation, should be included in the regression model. We do not observe Xt ; instead, we observe actual inflation, Zt = Xt + ut , where ut is an error term that is uncorrelated with Xt . Even in this best of circumstances, using Zt in the regression instead of Xt will cause the regression coefficient estimates to be biased and inconsistent. Let’s see why. If we want to estimate the regression Yt = b0 + b1 Xt + ϵt but we observe Zt not Xt , then we would estimate Yt = b0 + b1 Zt + (−b1 ut + ϵt ) But because Zt = Xt + ut , Zt is correlated with the error term (−b1 ut + ϵt ). Therefore, our estimated model violates the assumption that the error term is uncorrelated with the independent variable. Consequently, the estimated regression coefficients will be biased and inconsistent. EXAMPLE 9-14 The Fisher Effect with Measurement Error Recall from Example 9-8 on the Fisher effect that we could not reject the hypothesis that three-month T-bill returns moved one-for-one with expected inflation. 70 ‘‘Market-to-book ratio’’ is the ratio of price per share divided by book value per share. 371 Chapter 9 Multiple Regression and Issues in Regression Analysis TABLE 9-16 (repeated) Results from Regressing T-Bill Returns on Predicted Inflation Intercept Inflation prediction Coefficient Standard Error t-Statistic 0.0304 0.8774 0.0040 0.0812 7.6887 10.8096 Residual standard error Multiple R-squared Observations Durbin–Watson statistic 0.0220 0.4640 137 0.4673 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. What if we used actual inflation instead of expected inflation as the independent variable? Note first that π = πe + ν where π = actual rate of inflation πe = expected rate of inflation ν = the difference between actual and expected inflation Because actual inflation measures expected inflation with error, the estimators of the regression coefficients using T-bill yields as the dependent variable and actual inflation as the dependent variable will not be consistent.71 Table 9-17 shows the results of using actual inflation as the independent variable. The estimates in this table are quite different from those presented in the previous table. Note that the slope coefficient on actual inflation is much lower than the slope coefficient on predicted inflation in the previous regression. This result is an illustration of a general proposition: In a single-independent-variable regression, if we select a version of that independent variable that is measured with error, the estimated slope coefficient on that variable will be biased toward 0.72 TABLE 9-17 Results from Regressing T-Bill Returns on Actual Inflation Intercept Actual inflation Residual standard error Multiple R-squared Observations Coefficient Standard Error t-Statistic 0.0432 0.5066 0.0034 0.0556 12.7340 9.1151 0.0237 0.3810 137 Source: Federal Reserve Bank of Philadelphia, U.S. Department of Commerce. 71 A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increases. 72 This proposition does not generalize to regressions with more than one independent variable. Of course, we ignore serially-correlated errors in this example, but because the regression coefficients are inconsistent (due to measurement error), testing or correcting for serial correlation is not worthwhile. 372 Quantitative Investment Analysis 5.4. Other Types of Time-Series Misspecification By far the most frequent source of misspecification in linear regressions that use time series from two or more different variables is nonstationarity. Very roughly, nonstationarity means that a variable’s properties, such as mean and variance, are not constant through time. We will postpone our discussion about stationarity until Chapter 10, but we can list some examples in which we need to use stationarity tests before we use regression statistical inference.73 Relations among time series with trends (for example, the relation between consumption and GDP). • Relations among time series that may be random walks (time series for which the best predictor of next period’s value is this period’s value). Exchange rates are often random walks. • The time-series examples in this chapter were carefully chosen such that nonstationarity was unlikely to be an issue for any of them. But nonstationarity can be a very severe problem for analyzing the relations among two or more time series in practice. Analysts must understand these issues before they apply linear regression to analyzing the relations among time series. Otherwise, they may rely on invalid statistical inference. 6. MODELS WITH QUALITATIVE DEPENDENT VARIABLES Financial analysts often need to be able to explain the outcomes of a qualitative dependent variable. Qualitative dependent variables are dummy variables used as dependent variables instead of as independent variables. For example, to predict whether or not a company will go bankrupt, we need to use a qualitative dependent variable (bankrupt or not) as the dependent variable and use data on the company’s financial performance (e.g., return on equity, debt-to-equity ratio, or debt rating) as independent variables. Unfortunately, linear regression is not the best statistical method to use for estimating such a model. If we use the qualitative dependent variable bankrupt (1) or not bankrupt (0) as the dependent variable in a regression with financial variables as the independent variables, the predicted value of the dependent variable could be much greater than 1 or much lower than 0. Of course, these results would be invalid. The probability of bankruptcy (or of anything, for that matter) cannot be greater than 100 percent or less than 0 percent. Instead of a linear regression model, we should use probit, logit, or discriminant analysis for this kind of estimation. Probit and logit models estimate the probability of a discrete outcome given the values of the independent variables used to explain that outcome. The probit model, which is based on the normal distribution, estimates the probability that Y = 1 (a condition is fulfilled) given the value of the independent variable X . The logit model is identical, except that it is based on the logistic distribution rather than the normal distribution.74 Both models must be estimated using maximum likelihood methods.75 73 We include both unit root tests and tests for cointegration in the term ‘‘stationarity tests.’’ These tests will be discussed in Chapter 10. 74 The logistic distribution e (b0 +b1 X ) /[1 + e (b0 +b1 X ) ] is easier to compute than the cumulative normal distribution. Consequently, logit models gained popularity when computing power was expensive. 75 For more on probit and logit models, see Greene (2003). Chapter 9 Multiple Regression and Issues in Regression Analysis 373 Another technique to handle qualitative dependent variables is discriminant analysis. In his Z-score and Zeta analysis, Altman (1968, 1977) reported on the results of discriminant analysis. Altman uses financial ratios to predict the qualitative dependent variable bankruptcy. Discriminant analysis yields a linear function, similar to a regression equation, which can then be used to create an overall score. Based on the score, an observation can be classified into the bankrupt or not bankrupt category. Qualitative dependent variable models can be useful not only for portfolio management but also for business management. For example, we might want to predict whether a client is likely to continue investing in a company or to withdraw assets from the company. We might also want to explain how particular demographic characteristics might affect the probability that a potential investor will sign on as a new client, or evaluate the effectiveness of a particular direct-mail advertising campaign based on the demographic characteristics of the target audience. These issues can be analyzed with either probit or logit models. EXAMPLE 9-15 Explaining Analyst Coverage Suppose we want to investigate what factors determine whether at least one analyst covers a company. We can employ a probit model to address the question. The sample consists of 2,047 observations on public companies in 1999. All data comefrom Disclosure, Inc. The analyst coverage data on Disclosure come from I/B/E/S. The variables in the probit model are as follows: ANALYSTS = the discrete dependent variable, which takes on a value of 1 if at least one analyst covers the company and a value of 0 if no analysts cover the company LNVOLUME = the natural log of trading volume in the most recent week LNMV = the natural log of market value ESTABLISHED = a dummy independent variable that takes on a value of 1 if the company’s financial data has been audited for at least five years LNTA = the natural log of total assets(book value) LNSALES = the natural log of net sales In this attempt to explain analyst coverage, the market (volume and value) and the book (value and sales) variables might be expected to explain coverage through various dimensions of size and, hence, importance.76 The audit history variable reflects a possible comfort level that analysts could be expected to have with audited statements. The model includes three variables (LNMV, LNTA, and LNSALES) that we may expect 76 For more information on tests of multicollinearity, see Greene (2003). 374 Quantitative Investment Analysis to be correlated. Based on analysis not shown here, our probit regression did not exhibit the classic symptom of multicollinearity. Table 9-18 shows the results of the probit estimation. TABLE 9-18 Explaining Analyst Coverage Using a Probit Model Coefficient Standard Error Intercept LNVOLUME LNMV ESTABLISHED LNTA LNSALES Percent correctly predicted −7.9738 0.1574 0.4442 0.3168 0.0548 0.0507 0.4362 0.0158 0.0369 0.1045 0.0296 0.0266 t-Statistic −18.2815 9.9482 12.0268 3.0320 1.8494 1.9059 73.67 Source: Disclosure, Inc. As Table 9-18 shows, three coefficients (besides the intercept) have t-statistics with an absolute value greater than 2.0. The coefficient on LNVOLUME has a t-statistic of 9.95. That value is far above the critical value at the 0.05 level for the t-statistic (1.96), so we can reject at the 0.05 level of significance the null hypothesis that the coefficient on LNVOLUME equals 0, in favor of the alternative hypothesis that the coefficient is not equal to 0. The second coefficient with an absolute value greater than 2 is LNMV, which has a t-statistic of 12.03. We can also reject at the 0.05 level of significance the null hypothesis that the coefficient on LNMV is equal to 0, in favor of the alternative hypothesis that the coefficient is not equalto 0. Finally, the coefficient on ESTABLISHED has a t-statistic of 3.03. We can reject at the 0.05 level of significance the null hypothesis that the coefficient on ESTABLISHED is equal to 0. Neither of the two remaining independent variables is statistically significant at the 0.05 level in this probit analysis. Neither of the t-statistics on these two variables is larger in absolute value than 1.91, so neither one reaches the critical value of 1.96 needed to reject the null hypothesis (that the associated coefficient is significantly different from 0). This result shows that once we take into account a company’s market value, trading volume, and the existence of a five-year audit history, the other factors—book value of assets and value of sales—have no power to explain whether at least one analyst will cover the company. CHAPTER 10 TIME-SERIES ANALYSIS 1. INTRODUCTION As financial analysts, we often use time-series data to make investment decisions. A time series is a set of observations on a variable’s outcomes in different time periods: the quarterly sales for a particular company during the past five years, for example, or the daily returns on a traded security. In this chapter, we explore the two chief uses of time-series models: to explain the past and to predict the future of a time series. We also discuss how to estimate time-series models, and we examine how a model describing a particular time series can change over time. The following two examples illustrate the kinds of questions we might want to ask about time series. Suppose it is the beginning of 2003 and we are managing a U.S.-based investment portfolio that includes Swiss stocks. Because the value of this portfolio would decrease if the Swiss franc depreciates with respect to the dollar, and vice-versa, holding all else constant, we are considering whether to hedge the portfolio’s exposure to changes in the value of the franc. To help us in making this decision, we decide to model the time series of the franc/dollar exchange rate. Figure 10-1 shows monthly data on the franc/dollar exchange rate. (The data are monthly averages of daily exchange rates.) Has the exchange rate been more stable since 1987 than it was in previous years? Has the exchange rate shown a long-term trend? How can we best use past exchange rates to predict future exchange rates? As another example, suppose it is the beginning of 2001. We cover retail stores for a sell-side firm and want to predict retail sales for the coming year. Figure 10-2 shows monthly data on U.S. real retail sales. The data are inflation adjusted but not seasonally adjusted, hence the spikes around the holiday season at the turn of each year. Because the reported sales in the stores financial statements are not seasonally adjusted, we model seasonally unadjusted retail sales. How can we model the trend in retail sales? How can we adjust for the extreme seasonality reflected in the peaks and troughs occurring at regular intervals? How can we best use past retail sales to predict future retail sales? Some fundamental questions arise in time-series analysis: How do we model trends? How do we predict the future value of a time series based on its past values? How do we model seasonality? How do we choose among time-series models? And how do we model changes in the variance of time series over time? We address each of these issues in this chapter. 2. CHALLENGES OF WORKING WITH TIME SERIES Throughout the chapter, our objective will be to apply linear regression to a given time series. Unfortunately, in working with time series we often find that the assumptions of the 375 376 Quantitative Investment Analysis Swiss Franc/U.S. Dollar 3.0 2.5 2.0 1.5 1.0 0.5 0 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 Year FIGURE 10-1 Swiss Franc/U.S. Dollar Exchange Rate, Monthly Average of Daily Data Source: Board of Governors of the Federal Reserve System. $ Millions 200,000 180,000 160,000 140,000 120,000 100,000 80,000 60,000 40,000 20,000 0 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 Year FIGURE 10-2 Monthly U.S. Real Retail Sales Source: U.S. Department of Commerce, Census Bureau. linear regression model are not satisfied. To apply time-series analysis, we need to assure ourselves that the linear regression model assumptions are met. When those assumptions are not satisfied, in many cases we can transform the time series, or specify the regression model differently, so that the assumptions of the linear regression model are met. We can illustrate assumption difficulties in the context of a common time-series model, an autoregressive model. Informally, an autoregressive model is one in which the independent variable is a lagged (that is, past) value of the dependent variable, such as the model xt = b0 + b1 xt−1 + ϵt .1 Specific problems that we often encounter in dealing with time series include the following: 1 We could also write the equation as yt = b0 + b1 yt−1 + ϵt . 377 Chapter 10 Time-Series Analysis The residual errors are correlated instead of being uncorrelated. In the calculated regression, the difference between xt and b0 + b1 xt−1 is called the residual error. The linear regression assumes that this error term is not correlated across observations. The violation of that assumption is frequently more critical in terms of its consequences in the case of time-series models involving past values of the time series as independent variables than for other models (such as cross-sectional) in which the dependent and independent variables are distinct. As we discussed in the chapter on multiple regression, in a regression in which the dependent and independent variables are distinct, serial correlation of the errors in this model does not affect the consistency of our estimates of intercept or slope coefficients. By contrast, in an autoregressive time-series regression such as xt = b0 + b1 xt−1 + ϵt , serial correlation in the error term causes estimates of the intercept (b0 ) and slope coefficient (b1 ) to be inconsistent. • The mean and/or variance of the time series changes over time. Regression results are invalid if we estimate an autoregressive model for a time series with mean and/or variance that changes over time. • Before we try to use time series for forecasting, we may need to transform the time-series model so that it is well specified for linear regression. With this objective in mind, you will observe that time-series analysis is relatively straightforward and logical. 3. TREND MODELS Estimating a trend in a time series and using that trend to predict future values of the time series is the simplest method of forecasting. For example, we saw in Figure 10-2 that monthly U.S. real retail sales show a long-term pattern of upward movement—that is, a trend. In this section, we examine two types of trends, linear trends and log-linear trends, and discuss how to choose between them. 3.1. Linear Trend Models The simplest type of trend is a linear trend, one in which the dependent variable changes at a constant rate with time. If a time series, yt , has a linear trend, then we can model the series using the following regression equation: yt = b0 + b1 t + ϵt , t = 1, 2, . . . , T (10-1) where yt b0 b1 t ϵt = the value of the time series at time t (value of the dependent variable) = the y-intercept term = the slope coefficient = time, the independent or explanatory variable = a random-error term In Equation 10-1, the trend line, b0 + b1 t, predicts the value of the time series at time t (where t takes on a value of 1 in the first period of the sample and increases by 1 in each subsequent period). Because the coefficient b1 is the slope of the trend line, we refer to b1 as the trend coefficient. We can estimate the two coefficients, b0 and b1 , using ordinary least squares, denoting the estimated coefficients as b̂0 and b̂1 .2 2 Recall that ordinary least squares is an estimation method based on the criterion of minimizing the sum of a regression’s squared residuals. 378 Quantitative Investment Analysis Now we demonstrate how to use these estimates to predict the value of the time series in a particular period. Recall that t takes on a value of 1 in Period 1. Therefore, the predicted or fitted value of yt in Period 1 is ŷ1 = b̂0 + b̂1 (1). Similarly, in a subsequent period, say the sixth period, th