Portfolio Optimization and Passive Portfolio Management Strategy, business and finance homework help

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reading the guide and write stock analysis, the guide name is "Portfolio Optimization and Passive Portfolio Management Strategy ".

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Chapter 5: Risk and Return: Past and Prologue Topics for this chapter Return - HPR (“Holding period return” over one-period buy and sell) - Average HPR over multiple periods = AM, GM, DM - Annualized return (= APR, EAR)  will not be covered. Covered in BUS 170. Risk - Variance, Standard deviation - Value at risk - Skew, kurtosis Risk premium and Risk aversion  will not be covered. Covered in BUS 170. Measure of performance to rank portfolios - Sharpe ratio ---------------------------------------------------------------------------------------------------- HPR (holding period return) = Percentage return on a single holding period 0 1 |-------------------------------------| S0 S1 Income1  𝐷𝑜𝑙𝑙𝑎𝑟 𝑟𝑒𝑡𝑢𝑟𝑛 = 𝑆1 + 𝐼𝑛𝑐𝑜𝑚𝑒1  𝐻𝑃𝑅 = 𝑆1 +𝐼𝑛𝑐𝑜𝑚𝑒1 −𝑆0 𝑆0 = 𝑆1 −𝑆0 + 𝐼𝑛𝑐𝑜𝑚𝑒1 𝑆0 𝑆0 Capital gain yield + Income yield What are the income and income yield called for the followings?i     Stocks Income = $ Dividends Bonds Income = $ Interests Funds Income = $ Distributions (from interests + dividends + sales proceeds) Gold  Convenience (can be monetary; mostly non-monetary) 1 Measuring Investment Returns over Multiple Periods  HPR is the measure of one single period.  What are the measures for the average return over a long period of time? (1) Arithmetic mean (AM) (2) Geometric mean (GM) (3) Dollar-weighted mean (DM). Example (1) AM = 8.75% (2) GM = 7.19% Dollar-weighted Average Return (3) DM = 3.38% Period 0 1 2 3 4 NCF ($mil) -1 -0.1 -0.5 0.8 1.56 -0.6 = 0.96 Negative cash flows represent contribution to the funds (investment costs) from investors’ standpoint. 2 Caution  The table reflects not only cash flows from the performance of the fund but also external cash in/out flows.  Why 1.2 at end of period 1 is not used in the calculation of DM? This $1.2 million at the end of period 1 is used as the beginning costs of period 2. Hence, (+) and (-) they cancel out each other. Period NCF ($mil) 0 -1 1 -0.1 +1.2 -1.2 2 -0.5 +2.0 -2.0 3 0.8 +0.8 -0.8 4 -0.6 +1.56 Information in AM, GM and DM Previously, AM=8.75%, GM=7.19%, DM=3.38%  they are different. Why?  What makes AM and GM different?ii Why volatility causes lower return? Example 0 1 2 0 |-----------------|-----------------| 𝑟1 = 10% 2 |-----------------|-----------------| 𝑟2 = 10% 𝑟1 = 10% AM = 0%  1 𝑟2 = −10% AM = 0% Why volatility causes lower return given the same AMs? Example 0 1 2 0 |-----------------|-----------------| 𝑟1 = 2% 𝑟2 = 4% 1 𝑟1 = 10% 𝑟2 = −4% AM = 3%, GM=2.76% 2 |-----------------|-----------------| 𝑟1 = −4% 2 |-----------------|-----------------| AM = 3%, GM= almost 3.0% 0 1 𝑟2 = 10% AM = 3%, GM= 2.76% 3  What makes GM and DM different?iii Example without any external cash inflows and outflows Period 0 1 2 3 4 $BGN 1 1.1 1.375 1.1 HPR $End 0.1 1.1 0.25 1.375 -0.2 1.1 0.2 1.32 1 In this case, DM (or, IRR) = GM = 7.19% AM GM “Timeweighted average return” DM  Advantage Reflects the most likely return of the asset in the next period Disadvantage AM is not the actual average ROR realized over past periods Reflects average rate of return of the asset realized since the start of the investment Ignore the variation in the fund (inflows and outflows) Reflect the variation in the fund (inflows and outflows) *DM is the actual realized ROR of the investment account. *Meaningful for the account holders. GM is not the ROR on the investor’s account. DM is not the ROR on the asset. Consider a mutual fund. Which should be mandatory to report as the measure of average return on the overall fund?iv 4 Annualizing HPR (This topic will not be covered in the class since they are covered in BUS 170) Animalization of HPR = {Discrete annualized return, continuously compounded annualized return: Discretely annualized return = {APR, EAR} (1) 𝐴𝑃𝑅 = 𝑚 × 𝐻𝑃𝑅 (2) 𝐸𝐴𝑅𝑚 = (1 + 𝐴𝑃𝑅 𝑚 𝑚 ) −1 𝑚 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑖𝑛 𝑎 𝑦𝑒𝑎𝑟 Continuously compounded annualized rate of return (details below) (3) 𝐸𝐴𝑅𝑐𝑐 = (1 + 𝑟𝑐𝑐 𝑚 𝑚 ) − 1 = 𝑒 𝑟𝑐𝑐 − 1, as 𝑚 → ∞ Example: Annualizing HPR (APR and EAR). Given the following information, which investment alternative is better? Assume that your whole holding period is longer than 1 year for Investment A. Investment A B Unit holding HPR period 6 month 2.0% 14 month 4.667% m APR 5 EAR Continuous Compounding Given 𝐸𝐴𝑅𝑚 = (1 + 𝐴𝑃𝑅 𝑚 𝑚 ) −1 𝑟𝑐𝑐 = Quoted APR with continuous compounding The continuously compounded annualized rate of return is 𝐸𝐴𝑅𝑐𝑐 = (1 +  𝑟𝑐𝑐 𝑚 ) − 1 = 𝑒 𝑟𝑐𝑐 − 1, 𝑚 as 𝑚 → ∞ In practice, APY is also used to mean EAR. Even, sometimes, APR stands for EAR. So you have to look at the context. Example Given APR of 10%, the continuously compounded EAR is 𝑒 0.1 − 1 = 0.10517 = 10.52% The monthly compounded EAR is 0.1 12 𝐸𝐴𝑅12 = (1 + ) − 1 = 0.104713 = 10.47% 12 The daily compounded EAR is 𝐸𝐴𝑅365 = (1 + 0.1 365 ) − 1 = 0.105155 = 10.52% 365 6 5.2. Risk and Risk Premiums  Definition of (stand-alone) risk?  Measures of risk = Volatility, Value at Risk (VaR)  Skew(ness) and Kurtosis Scenario Analysis and Probability Distributions  Each scenario = possible events in the future  Random variable = function of events that yields a specific numerical outcome. - Example: Rate of return on an investment  Probability distribution = List of probabilities on possible numerical outcomes  Information from the probability distribution = (we can calculate) Mean, Variance, Skew, Kurtosis and also Value at risk. 7 Quantification of Risk (Uncertainty) with volatility ⋆Before anything else, below, the “r” = capital gain yield + income yield (in either HPR or annualized). What is the mean of those r’s and its variability? The following variables are used. 𝑷𝟏 , … , 𝑷𝑵 = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 𝑜𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 1, … , 𝑁, 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. Probabilities are the best knowledge about how the outcomes will realize. Hence, they describe the uncertainty in the outcomes. 𝒓𝟏 , … , 𝒓𝑵 = 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 1, … , 𝑁, 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. 𝑬[𝒓] = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠. Ex post (using past data or data without probabilities) 𝑴𝒆𝒂𝒏 𝒓𝒆𝒕𝒖𝒓𝒏 (𝒓̅) = 𝒓𝟏 + 𝒓𝟐 + ⋯ + 𝒓𝑵 𝑵 (𝒓𝟏 − 𝒓̅)𝟐 + ⋯ + (𝒓𝑵 − 𝒓̅)𝟐 𝑽𝒂𝒓𝒊𝒂𝒏𝒄𝒆 (𝝈 ) = 𝑵−𝟏 𝟐 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 (𝝈) = √𝝈𝟐 (In the above for variance, the division by N – 1 represents, you think the dataset is a sample. If you think that the data used is the whole population, then you have to divide by N.) Example: Consider a stock with following past data. Period in the past 1 2 3 Realized return 7% 10% 13% What are the ex-post mean return, standard deviation and variance? 𝐸𝑥-𝑝𝑜𝑠𝑡 𝑚𝑒𝑎𝑛 𝑟𝑒𝑡𝑢𝑟𝑛(𝑟̅ ) = 𝐸𝑥-𝑃𝑜𝑠𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝜎 2 ) = 7% + 10% + 13% = 10% 3 (7 − 10)2 + (10 − 10)2 + (13 − 10)2 = 9.0% 3−1 𝐸𝑥-𝑝𝑜𝑠𝑡 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = √9% = 3.0% 8 Ex ante (using forecasted future data, obviously with probabilities) The following diagram shows that when dealing with anticipated data, inevitably, you need probability distribution on the possible outcomes. 𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒓𝒆𝒕𝒖𝒓𝒏 (𝑬[𝒓]) = 𝑷𝟏 𝒓𝟏 + 𝑷𝟏 𝒓𝟐 + ⋯ + 𝑷𝑵 𝒓𝑵 𝑽𝒂𝒓𝒊𝒂𝒏𝒄𝒆 (𝝈𝟐 ) = 𝑷𝟏 (𝒓𝟏 − 𝑬[𝒓])𝟐 + ⋯ + 𝑷𝑵 (𝒓𝑵 − 𝑬[𝒓])𝟐 𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 (𝝈) = √𝝈𝟐 Example Consider a stock with following expected data.  This example is in the body section. Bottom line: please, know whether you are using past data or expected data. If you use the past data, then you don’t have probability distribution. If you use the expected data, then, you have a probability distribution. Suppose that you are expecting 10% return. Even if you don’t know what return will occur next period, you do know the probabilities on each outcome as shown below. How do you quantify the uncertainty in the future outcome? 9 The uncertainty in the outcome is described with the volatility measure, such as variance (or, standard deviation). “How much the outcome can be different from the expected value?”  𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑃𝑟𝑜𝑏1 (𝑜𝑢𝑡𝑐𝑜𝑚𝑒1 − 𝑚𝑒𝑎𝑛)2 + ⋯ + 𝑃𝑟𝑜𝑏𝑁 (𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑁 − 𝑚𝑒𝑎𝑛)2  𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒  𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 0.25(7% − 10%)2 + 0.5(10% − 10%)2 + 0.25(13% − 10%)2 = 4.5%  𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = √4.5% = 2.12%  Measure of uncertainty in the future outcome. 10 Investment Philosophy As To Risk (#1)  Volatility is a primarily measure of RISK. It needs to be estimated and controlled. Especially, risk-averse investors strongly dislike volatility.  Another aspect of volatility is “OPPORTUNITY.” Recall how Warren Buffett commented regarding market declines. The years ahead will occasionally deliver major market declines – even panics – that will affect virtually all stocks. … During such scary periods, you should never forget two things: First, widespread fear is your friend as an investor, because it serves up bargain purchases. Second, personal fear is your enemy. It will also be unwarranted. Investors who avoid high and unnecessary costs and simply sit for an extended period with a collection of large, conservatively-financed American businesses will almost certainly do well.  Buffett also strongly recommends long-term investment without panicking during market declines.  Recall the “Return-risk tradeoff.” A higher average return requires a higher risk under the long-term investment. Professor Jeremy Siegel (in his book, Stocks For The Long Run) puts it as follows: Although most investors express a strong distaste for market fluctuations, volatility must be accepted to reap the superior returns offered by stocks. Accepting risk is required for aboveaverage returns: investors cannot make any more than the risk-free rate unless there is some possibility that they can make less. [An Excerpt from “Stocks For The Long Run”] When stocks are collapsing, worst-case scenarios loom large in investors’ minds. On May 6, 1932, after stocks had plummeted 85 percent from their 1929 high, Dean Witter issued the following memo to its clients: There are only two premises which are tenable as to the future. Either we are going to have chaos or else recovery. The former theory is foolish. If chaos ensues nothing will maintain value; neither bonds nor stocks nor bank deposits nor gold will remain valuable. Real estate will be a worthless asset because titles will be insecure. No policy can be based upon this impossible contingency [chaos]. Policy must therefore be predicated upon the theory of recovery. The present is not the first depression; it may be the worst, but just as surely as conditions have righted themselves in the past and have gradually readjusted to normal, so this will again occur. The only uncertainty is when it will occur…. I wish to say emphatically that in a few years present prices will appear as ridiculously low as 1929 values appear fantastically high. Two months later the stock market hit its all-time low and rallied strongly. In retrospect, these words reflected great wisdom and sound judgment about the temporary dislocations of stock prices. Yet at the time they were uttered, investors were so disenchanted with stocks and so filled with doom and gloom that the message fell on deaf ears. 11 Volatility Index (VIX) As a Forward-Looking Market volatility  The VIX is a measure the volatility that investors expect in the market.  The expected volatility shows the level of anxiety in the market. [See the first figure.]  The periods of high anxiety have often marked turning points for stocks. [See the second figure.] The Relation between VIX and Market Index (S&P500) 12 Another measure of stand-alone risk: Value at Risk VaR = Maximum loss under the worst scenario with a certain confidence interval, either at the 95% or 99%. Example: the current monthly value of fund is $20 million. Its value can be any of values from 14 through 29 with corresponding probabilities as shown below. What is VaR under 95% and 99% confidence interval in return and dollar term, respectively?v 13 Example The following is the 1-day return distribution of an equity portfolio.  What are the VaRs under 95% and 99% confidence intervals, respectively?vi 14 ⋆The practical matter in estimating the VaR is the probability distribution over the next period to assess the worst outcome. Probability distributions for the VaR (1) Normal distribution (it is assumed probability.) (2) Empirical probability distribution (using the past realized data) Since the Normal distribution can be described only by mean and standard deviation (STD). 𝐻𝑃𝑅−𝑀𝑒𝑎𝑛 𝑟𝑒𝑡𝑢𝑟𝑛 Use standard normal variable (sr) = 𝑆𝑇𝐷 expression of the Standard Normal distribution. 15 , then we get a more general It is an assumption that the ROR in the next period are selected from the Normal distribution. Benefit of the assumption of normal distribution Analysis on the distribution becomes very simple in that: (1) The distribution of RORs is completely analyzed only by mean and standard deviation. Example Stock A has expected (mean) return of 15% and standard deviation of 8%. Stock B has expected (mean) return of 7% and standard deviation of 5%. (What about the measures of symmetry and fat tailness of the bell shape?) (2) For the Normal distribution, the skewness (asymmetry) and kurtosis (sharpness and fatness of tails) are the same values (0 and 3) for all the RORs on different securities. Hence, what matter as to the variability (uncertainty) of the return is only the standard deviation. (3) Analysis of the distribution of portfolio returns becomes simple. It is because if component stock returns follow the Normal distribution, then the return on the portfolio also follows the Normal distribution. 16 < Value at risk using the Normal distribution > Consider a stock with the expected return of 15% and the STD of 12%. What is the worst return in the next period that may happen only 5% out of the total frequency (=under 95% confidence interval)? 𝑉𝑎𝑅 = 𝐸(𝑟) − 1.645𝜎 = 15% − 1.645 × 12% = −4.74% Excel function: “=NORMSINV(0.05)”  -1.645 “=NORMSINV(0.01)”  -2.326 < Value at risk using empirical distribution > obs 1 2 3 4 5 … 72 HPR -89% -73% -42% -37% -22% 125% %obs 1/72 2/72 3/72 4/72 5/72 … 72/72 %Cumulative 0.014 0.028 0.042 0.056 0.069 1 VaR =? 17 Example: The Goldman Sachs’ VaR 18 Relation between volatility and VaR 19 Deviation from Normality and VaR The Skew is the measure of asymmetry. If skew > 0, then there is the right tail is fatter than the left tail (“skew to right”) If skew < 0, then there is the left tail is fatter than the right tail (“skew to left”) If skew = 0? Under which distribution is the VaR worse? 20 The Kurtosis is the measure of fatness of the tails of distribution.  If kurtosis = 3?  If kurtosis > 3, then the distribution has fatter tails.  If kurtosis < 3, then the distribution has thin tails. Concept check Which shape of distribution has a worse VaR among the following four combination? (ignored symmetric distribution) 21 Actual return distribution vs. Normal distribution  Differences between the Normal distribution and actual distributionvii Normal Actual stock Distribution Return distribution Kurtosis 3 ? Excess kurtosis 0 ? Skewness 0 ? (Caution: actual distributions of other asset classes have different values) Implications to investment  Normal distribution underestimates realistic VaR. - Hence, it results in unnecessary optimistic views on the futures returns. - The realistic VaR can be determined by using the actual distribution using historical returns.  The normal distribution assumption can be valid only for peaceful time. Why?viii 22 Example with Goldman Sachs – What is the relation between volatility and VaR? The VaR rises with the increase in volatility. Put differently, the VaR grow larger during economic turmoil period. The following chart shows that the VaR of Goldman Sachs increased significantly during the financial crisis. 23 The Historical Record: Risk-Return relation Things to analyze (page 126)   Check the return-risk relation (use Excess return stats). Which between small and large stocks earn high return and why? 24 The Historical Record (I): Normality in distribution?  How information does the data show about the shape of distribution? o Kurtosis? o Skew?  What are the implications on risk in investments? Non-normality and Actual VaR  Discuss the relation between non-normality and the measure of VaR.  What is the downside of assuming the Normal distribution of returns? 25 “Why Risk Management Is Important” (A Lesson from Lehman Brother’s Mismanagement of Risk) Reference: “Uncontrolled Risk” by Williams, Page 113~, For 2002 and 2003 daily VaR exposure increased to about $22 million, and in 2004 daily VaR increased to $30 million. By 2005 daily VaR shot up to $38 million. In 2006 VaR had increased to $55 million. … By 2007 daily VaR doubled, increasing to $124 million. Lehman was able to see past VaR and believed that the excessive credit risk in mortgage- and asset-backed securities needed serious hedging. Reports that have surfaced from the inner workings at Lehman so far indicate that top-level management refused to heed such warnings and continued to bet that the real estate market. At a 2007 risk management committee meeting, Antoncic, the most senior risk officer, was reportedly marginalized by Fuld. At this particular meeting she was asked to leave the room while firm risk was being discussed.” Reference: “A Colossal Failure of Common Sense” Page 267~, I have been told by two close friends who were in attendance at one of those meeting that Dick Fuld, irritated beyond reasonable endurance by Madelyn Antoncic’s warnings, resolutely told her to “shut up.” Dick Fuld decided to make Madelyn’s absence from those [executive committee deal] meetings even more permanent and got rid of her altogether. Reference: “Uncontrolled Risk” by Williams, Page 115, Her replacement was CFO Chris O’Meara. O’Meara met two important requirements: he was Fuldfriendly and had no formal training in risk management—a dangerous combination and hardly an adequate counterbalance against oversized risk taking. Reference: “Lehman Fault-Finding Points to Last Man Fuld as Shares Languish” by Bloomberg, July 22, 2008 Madelyn Antoncic, 55, head of risk, was moved to a government relations job in September 2007 [from her former risk manager position]. Two months later, at a risk management conference in New York, she said that hedging mortgage positions had curtailed Lehman's profit, which was difficult for top management to accept. 26 Risk Premiums and Risk Aversion (This topic will not be covered in the class since they are covered in BUS 170) Definition of risk aversion Your investment is $100,000. Do you accept the following investment? Event (Scenario) Outcome Probability Good 10% 0.5 Bad - 10% 0.5 Excess return, Risk premium Your investment is $100,000. For how much return (%) for good outcome, do you accept the following investment? Find out excess returns and risk premium. Assume the risk free rate is 3%. Event (Scenario) Outcome Probability Good How much 0.5 Bad -10% (- $10,000) 0.5 Suppose you invest in the asset if the good outcome is 20%. Risk premium = expected return – risk free rate = 5% - 3% = 2% Event (Scenario) Outcome Probability Expected return Good 20% 0.5 0.5*20%+0.5*(-10%) = 5% Bad -10% 0.5 Suppose that you are more risk averse and so you invest in the risky asset only if the good outcome is 40%. Risk premium = expected return – risk free rate = 15% - 3% = 12% Event (Scenario) Outcome Probability Expected return Good 40% 0.5 0.5*40%+0.5*(-10%) = 15% Bad -10% 0.5 Finding: as you become more risk averse, you required more risk premium. The positive relation between risk aversion and risk premium. Bottom line: Risk-return relation and risk aversion?  Positive relation 27 Investment Philosophy As To Risk (#2) Maximizing expected return – is this the goal of investment? As you have seen with the failure of Lehman Brother’s risk governance, it is particularly difficult to properly manage risk under an excessive greed. Imagine the following two return-risk tradeoff alternatives A and B. A. If an investor try to boost up expected return, then she needs to take up more risk. B. If an investor wants to hold down risk, then she needs to sacrifice expected rate of return. Which alternative would a greedy CEO choose? And what is the problem with the choice? -The approach A will be chosen. What is the unwanted ramification of the greed? A greater risk. A greedy executive will simply do not read the reports from risk management teams but will look at only reports of expected returns. There will be several courses to controlling risk. But, one simple way is to combine the measures of return and risk into a risk-adjusted return, like Sharpe ratio. In this way, risk is not ignored in decision making process. 28 Sharpe Ratio It is the measure to rank investment portfolios in terms of risk-return trade-off. Also, called the reward-to-volatility ratio. 𝑆= 𝐸(𝑟𝑃 ) − 𝑟𝑓 𝜎𝑃  What is the meaning of higher S?  Portfolio analysis in terms of mean and standard deviation (or variance) is called “mean-variance analysis.” Exampleix (Risk-free rate = 3%) Portfolio A B Expected Return 9% 17% Standard deviation 12% 30% (1) Find out the Sharpe ratios and interpret them. (2) Which portfolio is better choice? 29 The Historical Record (I): Risk-Return relation Things to analyze (page 126)  Find out the measure of Sharpe ratios using data of 1926-2010 for small stocks and large stocks. Are they estimates are meaningfully different? If not, why?  Historical Arithmetic Mean vs. Geometric Mean  Recall what makes AM and GM differ. Theoretical relation under the assumption that returns follow Normal distribution 𝜎2 𝐴𝑀 = 𝐺𝑀 + 2 Is the theoretic relation valid, at least broadly? Small stock has a relatively big discrepancy between actual and theoretical difference. (5.78 vs. 6.84). 30 i Stocks – dividends (dividend yield) Bonds – coupon interests (current yield) Funds – Income distributions (distribution rate) Gold – No specific monetary income for just holding gold. But, if the gold is lent to your friend, your friend may return some money: this type of return is called “convenience yield.” ii Volatility in HPRs iii Cash inflows into and outflows from the funds iv The appropriate measure of return for the overall fund is the geometric mean. It is because the AM inflate the estimates of returns and that the dollar-weighted mean is only meaningful for the account holder. v Under 95% confidence interval, the value of fund under worst scenario is $17 million. Hence, the VaR in dollar term is $3 million = $20 - $17. The VaR in return is (17 – 20)/20 = -0.15 or 15% is the maximum loss under 95% C.I. vi VaR at ther 99% confidence interval = -5% VaR at ther 95% confidence interval = -1% vii Normal distribution Actual stock return distribution Kurtosis 3 Greater than 3 Excess kurtosis 0 Greater than 0 Skewness 0 Negative (Caution: actual distributions of other asset classes have different values) viii During the period of uncertainty, the return distribution exhibits the negative skewness and higher kurtosis. ix 31 9%−3% (1) 𝑆𝐴 = = 0.5; 𝑆𝐵 = 12% deviation. (2) The portfolio B is better. 17%−3% 14% = 1.0. Portfolio B earns 1% for every 1% of standard 32 Chapter 6: Efficient Diversification The topic “single index model” which is about the determination of alpha and beta coefficients will be covered in chapter 7. TOPICS 1) Mean-Variance analysis 2) Efficient diversification 3) Passive Strategy --6.1 Diversification and Portfolio Risk Firm-specific risk (diversifiable risk, nonsystematic risk) - Risk that is unique to a firm. - Example: Default risk, lawsuits, strikes, unsuccessful investment in projects, etc. - It can be eliminated by portfolio diversification. Market risk (non-diversifiable risk, systematic risk) - Risk that comes from the market (Macroeconomic uncertainty). Hence, this risk is common to all the assets. - Examples: war, inflation, recessions, exchange rate risk, market liquidity, sudden economic policy change (such as interest rate), global market concerns, etc. - Can conventional portfolio diversification reduce market risk in a portfolio?i Combination of the two stock into a new portfolio results in a reduced volatility. Why?  There are firm-specific unique risk factors which cancel each other. Illustration 1 As the above example shows, the risk of a portfolio decreases as it becomes more diversified. Caution that the (conventional) diversification has the uncontrollable part of portfolio risk which is not reduced. Concept Checkii • • What risk do well-diversified portfolio have? What is the measure of (sensitivity to) market risk? 2 6.2. Asset allocation with two risky assets There are two risky assets. (Assumption: no mispriced assets)iii Period ROR (Apple) 1 2 3 Mean STDP 7.50% -3.49% -1.20% 0.94% 4.73% ROR (Exxon) 1.06% -4.21% 5.49% 0.78% 3.97% Based on the historical monthly data you got mean return and standard deviations. (The following is hypothetical example.) Mean return Standard deviation Apple 0.94% 4.73% Exxon Mobile 0.78% 3.97% Total investment = $10,000. Asset allocation  How much would you spend on Apple and Exxon Mobile, respectively? Suppose you spend $2,000 on Apple out of total $10,000 investment. 𝑤= $2,000 $10,000 = 0.2. You would achieve: 𝑀𝑒𝑎𝑛 = 0.81%, 𝑆𝑡𝑑 𝑑𝑒𝑣. = 3.53%. However, with 𝑤 = 0.5, you would achieve: 𝑀𝑒𝑎𝑛 = 0.86%, 𝑆𝑡𝑑 𝑑𝑒𝑣. = 3.44%. The allocation with w=0.5 is better. Why? Weight on Apple Std dev. Mean 0 … … 0.1 … … 0.2 3.53% 0.81% 0.3 … … 0.4 … … 0.5 3.44% 0.86% 0.6 … … 0.7 … … 0.8 … … 0.9 … … 1 … … 3 To determine which allocation is most preferred by you, you need to find (1) Individual portfolio return  “Rule 1” (2) Expected portfolio return  “Rule 2” (3) Portfolio standard deviation  “Rule 3” Rule 1: 𝑤1 𝑟1 + 𝑤2 𝑟2 = 𝑟𝑃 (Individual portfolio return) W1 0.2 R1 7.5% W2 0.8 R2 1.06% Rp 0.2 × 7.5% + 0.8 × (1.06%) = 2.78% Rule 2: 𝑤1 𝐸[𝑟1 ] + 𝑤2 𝐸[𝑟2 ] = 𝐸[𝑟𝑃 ] (Expected portfolio return) W1 0.2 E[R1] 0.94% W2 0.8 E[R2] 0.78% E[Rp] 0.2 × 0.94% + 0.8 × 0.78% = 0.81% Rule 3: σP = √w12 σ12 + w22 σ22 + 2w1 w2 ρσ1 σ2 (Portfolio standard deviation) σP = √(0.2)2 (4.73%)2 + (0.8)2 (3.97%)2 + 2(0.2)(0.8)ρ(4.73%)(3.97%) = 3.53% Where 𝜌 = 0.2464  Correlation coefficient. 4 Covariance and correlation 𝑪𝒐𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 = (𝑟𝐴,1 − 𝐸[𝑟𝐴 ])(𝑟𝐵,1 − 𝐸[𝑟𝐵 ]) + ⋯ + (𝑟𝐴,𝑁 − 𝐸[𝑟𝐴 ])(𝑟𝐵,𝑁 − 𝐸[𝑟𝐵 ]) 𝑁 Period ROR (Apple) 1 2 3 Mean STDP ROR (Exxon) 7.50% -3.49% -1.20% 0.94% 4.73% 1.06% -4.21% 5.49% 0.78% 3.97% 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (7.5 − 0.94)(1.06 − 0.78) + (−3.49 − 0.94)(−4.21 − 0.78) + (−1.2 − 0.94)(5.49 − 0.78) = 3 = 4.627% 𝑪𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏(𝝆) = 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝜎1 𝜎2 = 4.627% 4.73%×3.97% = 0.2464 Going back to the previous example, Rule 3: σP = √w12 σ12 + w22 σ22 + 2w1 w2 ρσ1 σ2 σP = √(0.2)2 (4.73%)2 + (0.8)2 (3.97%)2 + 2(0.2)(0.8)(0.2464)(4.73%)(3.97%) = 3.53% When variance is estimated from a sample of n observed returns, it is common to divide the squared deviations by n – 1 rather than by n. In Excel, the function STDEVP computes standard deviation dividing by n, while the function STDEV uses n – 1. Excel’s covariance and correlation functions both use n. We ignored this fine point, and divided by n throughout. In any event, the correction for the lost degree of freedom is negligible when there are plentiful observations. 5 Concept checkiv Calculate the covariance and correlation coefficient. (Page, 152) Scenario Severe recession Mild recession Normal growth Boom ROR on stock Probability fund 0.05 0.25 0.4 0.3 -37 -11 14 30 ROR on Bond Fund -9 15 8 -5 The Risk-Return Trade-off with Two Risky Assets Portfolios Mean return Standard deviation Correlation to Apple Apple 0.94% 4.73%. Exxon Mobile 0.78% 3.97% 0.2464 Investment opportunity set Weight on Apple Port STDP (%) Port ROR (%) -0.2 4.62% 0.75% -0.1 4.27% 0.77% 0 3.97% 0.78% 0.1 3.72% 0.80% 0.2 3.53% 0.81% 0.3 3.42% 0.83% 0.4 3.39% 0.85% 0.5 3.44% 0.86% 0.6 3.58% 0.88% 0.7 3.78% 0.89% 0.8 4.05% 0.91% 0.9 4.37% 0.92% 1 4.73% 0.94% 1.1 5.12% 0.96% 1.2 5.53% 0.97% Investment opportunity set shows the set of feasible portfolios depending on weights on them. 6 Going back to the original question, how much would you spend on Apple and Exxon Mobile, respectively? Without any risk-free asset, your choice should be “efficient.” Given any particular standard deviation, the expected return should be larger. Or, equivalently, Given any particular expected return, the standard deviation should be smaller. Such investment opportunity set is called “Efficient frontier”. This weight that achieve the least standard deviation is called “Minimum variance portfolio.” Concept checkv (1) Which portfolio in the efficient frontier exactly do you choose? (2)What is the relation between return and risk as you observe from the efficient frontier? 7 Consider another stock (PG&E) as shown below. It has the same mean return and standard deviation. The difference is the correlation coefficient. What would be the resultant investment opportunity set? Mean return Standard deviation Correlation to Apple Apple 0.94% 4.73%. Exxon Mobile 0.78% 3.97% 0.2464 Which is better between Exxon Mobile and PG&E for a portfolio? 8 PG&E 0.78% 3.97% -0.3 6.3 The Optimal Risky Portfolio With A Risk-Free Asset You have two risky assets (AAPLE and Exxon Mobile) and a risk-free asset (T-bills). Mean return Standard deviation Correlation to Apple Apple 0.94% 4.73%. Exxon Mobile 0.78% 3.97% 0.2464 The problem remains the same: Find out the “efficient frontier.” 9 T-bills 0.7% 0% We can utilize the investment opportunity set with the two risky assets. Suppose that you use the minimum variance portfolio (MVP) to form a new portfolio including the risk-free asset. The portfolio with 50% on T-bills and 50% on the MVP The portfolio with 30% on T-bills and 70% on the MVP 10 Imagine all portfolios with T-bills and the MVP Then, we actually form a line. The line represents all possible portfolios between the risk-free asset. This line has a name, “Capital Allocation Line (CAL).” 11 But, the portfolios that can be formed using the T-bills and MVP are not efficient. Imagine another CAL that connects the risk-free rate and the risky portfolio on which the line is tangent. The risky portfolio on which a CAL is tangent is called “Optimal risky portfolio.” Why? Go back to the “efficiency” of a portfolio. A portfolio is efficient if it achieves the greatest expected return given a certain standard deviation. 12 In short, the efficient asset allocation with two risk assets and a risk-free asset is achieved when the CAL becomes tangent on the optimal risky portfolio. Practically, how much do you have to spend on Apple and Exxon Mobile to determine the optimal risky portfolio? Determination of the weights on the optimal risky portfolio 𝐸[𝑟1 ] = 0.94%, 𝐸[𝑟2 ] = 0.78%, 𝜎1 = 4.73%, 𝜎2 = 3.97%, 𝜌 = 0.2464, 𝑟𝑓 = 0.7% 𝑤1 = (𝐸 [𝑟1 ] − 𝑟𝑓 )𝜎22 − (𝐸 [𝑟2 ] − 𝑟𝑓 )𝜌𝜎1 𝜎2 (𝐸 [𝑟1 ] − 𝑟𝑓 )𝜎22 + (𝐸 [𝑟2 ] − 𝑟𝑓 )𝜎12 − (𝐸 [𝑟1 ] − 𝑟𝑓 + 𝐸 [𝑟2 ] − 𝑟𝑓 )𝜌𝜎1 𝜎2 𝑤1 = 0.83 ≈ 0.8. 𝑤2 = 1 − 𝑤1 ≈ 0.2. 13 Concept checkvi (1) Suppose the total investment is $10,000, and the optimal risky portfolio weights are 0.8 and 0.2 on two risky assets for Apple and Exxon Mobile, respectively. If you want hold $3000 on T-bills, then how much to you spend on Apple stock? (2) At which portfolio the risk is minimized? a. Minimum variance portfolio b. Optimal risky portfolio (3) What is the meaning of Sharpe ratio? (4) What is meaning of capital allocation line (CAL)? (5) What is the relation between Sharpe ratio and the CAL? (6) At which portfolio the Sharpe ratio is maximized? a. Minimum variance portfolio b. Optimal risky portfolio Concept check In the following mean-variance analysis, identify the efficient frontier investment opportunity set.vii 14 6.4 Efficient Diversification With Many Risky Assets We can imagine the construction of a portfolio with more than two risky assets as: We can imagine now the investment opportunity sets using stock 1 and 2 and the one using stock 2 and 3. Consider the portfolios of “a” and “b” from the investment opportunity sets. (say, the portfolio “a” is composed of 0.4 and 0.6 on stocks 1 and 2. The portfolio “b” is composed of 0.3 and 0.7 on stocks 2 and 3.) We can also form a portfolio “c” using the combination of “a” and “b”. Suppose that the “c” is composed of 0.2 and 0.8 on “a” and “b”. Then, what are the ultimate weights on stocks 1, 2 and 3 to achieve the portfolio “c”? 0.08, 0.36 and 0.56 on stocks 1, 2 and 3, respectively. We can repeat forming all possible investment opportunity set in this way. 15 When is an asset shorted? Is shorting lead to a higher Sharpe ratio? There are three stocks A, B and C. Imagine “kinda” Sharpe ratio for these individual stocks. Then, the stock A is most inferior in that regard. (Recall, the Sharpe ratio is used only for a large portfolio.) (Risk-free rate=1%) A B C Sharpe MEAN Ratio = Std dev RETURN (mu-rf)/std 0.047 0.011 0.025
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Explanation & Answer

Attached.

Date
Price_FB Price_AAPL Price_KO
1-Jun-17
151,53
153,18
45,79
2-Jun-17
153,61
155,45
45,89
5-Jun-17
153,53
153,93
45,99
6-Jun-17
152,81
154,45
45,98
7-Jun-17
153,12
155,37
45,51
8-Jun-17
154,71
154,45
45,13
9-Jun-17
149,6
155,37
45,32
12-Jun-17
148,44
145,42
45,33
13-Jun-17
150,68
146,59
45,03
14-Jun-17
150,25
145,16
45,3
15-Jun-17
149,8
144,29
45,25
16-Jun-17
150,64
142,27
45,31
19-Jun-17
152,07
146,34
45,38
20-Jun-17
152,25
145,01
45,61
21-Jun-17
153,91
145,87
45,22
22-Jun-17
153,4
145,63
45,07
23-Jun-17
155,07
146,28
45,25

Mean

152,0835 149,1211765 45,43294

R_Mean

R_FB

R_AAPL R_KO

0,013633
-0,000521
-0,004701
0,002027
0,01033
-0,033587
-0,007784
0,014978
-0,002858
-0,003
0,005592
0,009448
0,001183
0,010844
-0,003319
0,010828

0,01471
-0,009826
0,003372
0,005939
-0,005939
0,005939
-0,066183
0,008013
-0,009803
-0,006011
-0,014099
0,028206
-0,00913
0,005913
-0,001647
0,004453

0,002182
0,002177
-0,000217
-0,010274
-0,008385
0,004201
0,000221
-0,00664
0,005978
-0,001104
0,001325
0,001544
0,005056
-0,008588
-0,003323
0,003986

0,001443 -0,002881 -0,000741

ER_FB
0,012190298
-0,001963935
-0,006143668
0,000583608
0,008887469
-0,035030331
-0,009227229
0,013534547
-0,004300809
-0,004442502
0,004148813
0,008005057
-0,000260035
0,009401109
-0,004762127
0,009384739

ER_AAPL ER_KO
0,01759
-0,006946
0,006252
0,008819
-0,003059
0,008819
-0,063303
0,010893
-0,006923
-0,003131
-0,011219
0,031086
-0,00625
0,001233
0,001233
0,007333

0,002922
0,002917
0,000523
-0,009534
-0,007645
0,004941
0,000961
-0,0059
0,006718
-0,000364
0,002065
0,002284
0,005796
-0,007848
-0,002583
0,004726

ERt=Rt-Rmean

FB
AAPL
KO

STD DEV
0,011786
0,019889
0,005201

MEAN RETURN
0,001443
-0,00288
-0,00074

Calculating standard deviations
VARIANCE-CO-VARIANCE MATRIX
R_FB
R_AAPL
R_KO
R_FB
0,000138914 6,08267E-05 -2,18469E-05
R_AAPL
6,08267E-05 0,000395564 -7,98543E-06
R_KO
-2,18469E-05 -7,98543E-06 2,70491E-05

VAR (AAPL) = 0.000395564, Standard Deviation for AAPL = Sqrt (0.000395564) = 0.01

VAR (FB) = 0.000138914, Standard Deviation for FB = Sqrt (0.000138914) = 0.01178

VAR (KO) = 2.70491E-05, Standard Deviation for KO =Sqrt(2.70491E-05) = 0.005

Constant
Asset 1
Asset 2
Asset 3

W on FB
W on AAPL
W on KO
Total

Concept of Investment Opportunity Set
Take 0.00133 as first constant. Monthly free-risk = 0.133%
0,00133
0,1
Portfolio 1
Portfolio 2
0,333
0,333
Taking equal weights for t
0,333
0,333
0,333
0,333
0,999
0,999
Max
Min

0.000395564) = 0.019888879

00138914) = 0.011786183

.70491E-05) = 0.005201

.133%

g equal weights for the initial weights of the stock

Monthly
FB
AAPL
KO

Annual
Std Dev
0,011786
0,019889
0,005201

MEAN RETURN
0,001443
-0,00288
-0,00074

FB
AAPL
KO

Std Dev

MEAN RETURN

Constant
Asset 1
Asset 2
Asset 3

0,00133
0,1
Portfolio 1
Portfolio 2
W on FB
0,3333
0,3333
W on AAPL
0,3333
0,3333
W on KO
0,3333
0,3333
Total
1
1
Max
Min

Portfolio Mean
Portfolio Variance
Portfolio Standard Deviation
Covariance

Correlation
Sharpe Ratio

Monthly
FB
AAPL
KO

-0,000725594 -0,000725594
7,11635E-05 7,11635E-05
0,008435849 0,008435849
7,11635E-05
1
-0,243673657 -11,94018503

Using the Solver Function

Constant
Asset 1
Asset 2
Asset 3

0,00133
0,01
Portfolio 1
Portfolio 2
W on FB
1
-1,2452
W on AAPL
-0,3333
1,5919
W on KO
0,3333
0,6533
Total
1
1
Max
Min

Portfolio Mean
Portfolio Variance
Portfolio Standard Deviation
Covariance

Correlation
Sharpe Ratio

0,002156262 -0,006864938
0,000130552 0,001022001
0,011425922 0,031968744
-0,000252704
-0,69182225
1
-1,2452

W1
-1
-0,9
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0,3
0,4
0,5
0,6

Portfolio mean =(0.1 x0.002156262) + (0.9x-0.006864938) =
Portfolio Variance =

{(0.1x0.1143)^2 +(0.9x0.03197)^2 + 2(0.1 x0.9)x
(-0.6918x0.1143x0.03197) = 0.000503368

Portfolio standard deviation = SQRT(0.000503368) =

Investing in on risky

W2
0,8
...


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Great study resource, helped me a lot.

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