FIN201
FINANCIAL MATHEMATICS
STUDY GUIDE (5CU)
Course Development Team
Head of Programme
:
Dr Ding Ding
Course Developer(s)
:
Dr Tan Chong Hui
Production
:
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CONTENTS
COURSE GUIDE
1. Welcome .............................................................................................................1
2. Course Description and Aims .........................................................................1
3. Learning Outcomes .......................................................................................... 3
4. Learning Material ............................................................................................. 4
5. Assessment Overview ...................................................................................... 5
6. Course Schedule ................................................................................................ 6
7. Learning Mode ..................................................................................................7
STUDY UNIT 1
TIME VALUE OF MONEY
Learning Outcomes ......................................................................................... SU11
Chapter 1 Concept: Interest Rates ................................................................. SU12
Chapter 2 Concept: Time Value of Money ................................................... SU18
Chapter 3 Calculation .................................................................................... SU113
Quiz .................................................................................................................. SU120
Solutions or Suggested Answers ................................................................. SU122
STUDY UNIT 2
STATISTICS AND NUMERICAL METHODS WITH EXCEL
AND PYTHON
Learning Outcomes ......................................................................................... SU21
Chapter 1 Basics of Statistics .......................................................................... SU22
Chapter 2 Basics of Numerical Methods .................................................... SU220
Chapter 3 Using Excel ................................................................................... SU223
Chapter 4 Using Python ................................................................................ SU232
Quiz .................................................................................................................. SU245
Solutions or Suggested Answers ................................................................. SU247
STUDY UNIT 3
EQUITIES
Learning Outcomes ......................................................................................... SU31
Chapter 1 Market ............................................................................................. SU32
Chapter 2 Signals and Quotes ........................................................................ SU34
Chapter 3 Instruments..................................................................................... SU39
Chapter 4 Calculations .................................................................................. SU312
Chapter 5 Concept: The Capital Asset Pricing Model .............................. SU316
Chapter 6 Analyses and Strategies .............................................................. SU319
Quiz .................................................................................................................. SU321
Solutions or Suggested Answers ................................................................. SU323
STUDY UNIT 4
FIXED INCOME
Learning Outcomes ......................................................................................... SU41
Chapter 1 Market ............................................................................................. SU42
Chapter 2 Signals and Quotes ........................................................................ SU44
Chapter 3 Instruments................................................................................... SU410
Chapter 4 Calculations .................................................................................. SU417
Quiz .................................................................................................................. SU451
Solutions or Suggested Answers ................................................................. SU453
STUDY UNIT 5
FOREIGN EXCHANGE
Learning Outcomes ......................................................................................... SU51
Chapter 1 Market ............................................................................................. SU52
Chapter 2 Signals and Quotes ........................................................................ SU54
Chapter 3 Interest Rate Parity ...................................................................... SU512
Chapter 4 Instruments................................................................................... SU515
Chapter 5 Calculations .................................................................................. SU521
Chapter 6 Analyses and Strategies .............................................................. SU526
Quiz .................................................................................................................. SU527
Solutions or Suggested Answers ................................................................. SU529
STUDY UNIT 6
OPTIONS
Learning Outcomes ......................................................................................... SU61
Chapter 1 Market ............................................................................................. SU62
Chapter 2 Signals and Quotes ........................................................................ SU65
Chapter 3 Instruments..................................................................................... SU67
Chapter 4 Concept: Market Price vs Theoretical Price ............................. SU610
Chapter 5 Calculations .................................................................................. SU614
Chapter 6 Analyses and Strategies .............................................................. SU622
Quiz .................................................................................................................. SU626
Solutions or Suggested Answers ................................................................. SU628
COURSE GUIDE
FIN201 COURSE GUIDE
1. Welcome
(Access video via iStudyGuide)
Welcome to the course FIN201 Financial Mathematics, a 5 credit unit (CU) course.
This Study Guide will be your personal learning resource to take you through the
course learning journey. The guide is divided into two main sections – the Course
Guide and Study Units.
The Course Guide describes the structure for the entire course and provides you with
an overview of the Study Units. It serves as a roadmap of the different learning
components within the course. This Course Guide contains important information
regarding the course learning outcomes, learning materials and resources, assessment
breakdown and additional course information.
2. Course Description and Aims
The study of finance requires the understanding of fundamental financial concepts as
well as the proficiency in applying basic mathematics to these concepts with the aid
of robust computational tools. FIN201 Financial Mathematics is designed to equip
students with the ability to understand the realworld issues and the proficiency to
calculate with a full range of products in the world of finance. Software tools used for
the course include a computational platform for statistics (e.g. Python) and
spreadsheet software (e.g. Excel).
Course Structure
This course is a 5credit unit course presented over 6 weeks.
There are six Study Units in this course. The following provides an overview of each
Study Unit.
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FIN201 COURSE GUIDE
Study Unit 1 – The Time Value of Money
This unit helps you understand the concept of the time value of money.
Study Unit 2 – Statistics and Computing with Python and Excel
This unit helps you to recall some statistical concepts essential to finance and
introduces Python and Excel as computing platform for the course.
Study Unit 3  Equities
The unit covers the basic mathematics and computations that you will encounter in
equities.
Study Unit 4 – Fixed Income
The unit covers the basic mathematics and computations that you will encounter in
fixed income.
Study Unit 5 – Foreign Exchange
The unit covers the basic mathematics and computations that you will encounter in
foreign exchange.
Study Unit 6  Options
The unit covers the basic mathematics and computations that you will encounter in
options.
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FIN201 COURSE GUIDE
3. Learning Outcomes
Knowledge & Understanding (Theory Component)
By the end of this course, you should be able to:
Compute present value (PV), future value (FV), compounding, discounting or
other associated notions
Apply the methods of statistical inference in reasoning about data
Compare between the various fixed income instruments and their quantitative
representations
Calculate prices or other quantitative information related to fixed income market
instruments
Calculate various types of interest rates (e.g. spot, forward) from the term
structure
Explain the various foreign exchange instruments and their quantitative
representations
Calculate prices or other quantitative information related to foreign exchange
market instruments
Discuss the various equity instruments and their quantitative representations
Calculate prices or other quantitative information related to equity market
instruments
Key Skills (Practical Component)
By the end of this course, you should be able to:
Use a computing tool (e.g. Excel/Google Spreadsheets or Python) for financial
calculations.
Use a financial information system (e.g. Reuters Eikon, or the Internet) for
obtaining market data and information as well as harnessing welldocumented
API/library/models to make inferencing more expedient
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FIN201 COURSE GUIDE
4. Learning Material
The following is a list of the required learning materials to complete this course.
Required Textbook(s)
Steiner, B. (2007). Mastering financial calculations: A stepbystep guide to the
mathematics of financial market instruments. Harlow, England: Financial Times
Prentice Hall.
Other recommended study material (Optional)
The following learning materials may be required to complete the learning activities:
Special Requirement (Optional)
Any other requirement(s) needed for the course such as the use of lab equipment.
Windows or Mac OS with Excel installed
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FIN201 COURSE GUIDE
5. Assessment Overview
The overall assessment weighting for this course for the Evening Cohort is as follows:
Assessment
Description
Weight
Allocation
Assignment 1
TutorMarked Assignment / TMA 1
25%
Assignment 2
TutorMarked Assignment / TMA 2
25%
Examination
Written examination
50 %
TOTAL
100%
The overall assessment weighting for this course for the Daytime Cohort is as follows:
Assessment
Description
Weight
Allocation
PreCourse Quiz 1
1%
PreCourse Quiz 2
1%
PreCourse Quiz 3
1%
PreCourse Quiz 4
1%
PreCourse Quiz 5
1%
PreCourse Quiz 6
1%
Assignment 2
TutorMarked Assignment / TMA
17%
Assignment 3
Class Test
17%
Class Participation
Class Participation
10%
Examination
Written Examination
50%
Assignment 1
TOTAL
100%
UniSIM’s assessment strategy consists of two components, Overall Continuous
Assessment (OCAS) and Overall Examinable Component (OES) that make up the
overall course assessment score.
(a) OCAS: The subcomponents are reflected in the tables above and are different for
the daytime and evening cohort. The continuous assignments are compulsory and
are nonsubstitutable.
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FIN201 COURSE GUIDE
(b) OES: The Examination is 100% of this component.
To be sure of a pass result you need to achieve scores of 40% in each component. Your
overall rank score is the weighted average of both components.
Nongraded Learning Activities:
Activities for the purpose of selflearning are present in each study unit. These
learning activities are meant to enable you to assess your understanding and
achievement of the learning outcomes. The type of activities can be in the form of Quiz,
Review Questions, ApplicationBased Questions or similar. You are expected to
complete the suggested activities either independently and/or in groups.
6. Course Schedule
To help monitor your study progress, you should pay special attention to your Course
Schedule. It contains study unit related activities including Assignments, Selfassessments, and Examinations. Please refer to the Course Timetable in the Student
Portal for the updated Course Schedule.
Note: You should always make it a point to check the Student Portal for any
announcements and latest updates.
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FIN201 COURSE GUIDE
7. Learning Mode
The learning process for this course is structured along the following lines of learning:
(a) Selfstudy guided by the study guide units. Independent study will require at
least 3 hours per week.
(b) Working on assignments, either individually or in groups.
(c) Classroom Seminar sessions (3 hours each session, 6 sessions in total).
iStudyGuide
You may be viewing the iStudyGuide version, which is the mobile version of the
Study Guide. The iStudyGuide is developed to enhance your learning experience with
interactive learning activities and engaging multimedia. Depending on the reader you
are using to view the iStudyGuide, you will be able to personalise your learning with
digital bookmarks, notetaking and highlight sections of the guide.
Interaction with Instructor and Fellow Students
Although flexible learning – learning at your own pace, space and time – is a hallmark
at SUSS, you are encouraged to engage your instructor and fellow students in online
discussion forums. Sharing of ideas through meaningful debates will help broaden
your learning and crystallise your thinking.
Academic Integrity
As a student of SUSS it is expected that you adhere to the academic standards
stipulated in The Student Handbook, which contains important information
regarding academic policies, academic integrity and course administration. It is
necessary that you read and understand the information stipulated in the Student
Handbook, prior to embarking on the course.
7
STUDY UNIT 1
TIME VALUE OF MONEY
FIN201 STUDY UNIT 1
Learning Outcomes
By the end of this unit, you should be able to:
1. Describe the significance of the time value of money.
2. Express present values and future values through the use of interest rates.
3. Give the value of the NPV from the IRR or vice versa.
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FIN201 STUDY UNIT 1
Chapter 1 Concept: Interest Rates
In this unit, we will discuss one of the most fundamental facts in finance  that an
amount of money now will be valued differently in the future. This is known as the
time value of money.
The reason for the fact can be traced to the existence of demand for idle cash in the
financial environment. This means that if you are able to lend to the demand, you will
be rewarded by means of interest when the loan is repaid.
In a nutshell, the time value of money is a relationship between the present value and
the future value of a cash flow expressed in terms of an interest rate. This and related
notions will be explored here.
1.1 Simple and Compound Interest
Our concern is with the value of a cash flow at different points in time.
Its current value is called present value or PV. Its value in the future is called future
value or FV.
The interpretation is that an amount PV is loaned to someone who has immediate
demand for it. To pay for immediacy, the lender receives an amount FV when the loan
expires which is greater than PV.
PV and FV are related by an equation:
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟),
where 𝑟 is called the interest rate. Written this way, we understand that FV is obtained
from PV by compounding forward at rate 𝑟.
The diagrammatic representation of this relationship is a good way to visualise it:
Figure 1.1
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FIN201 STUDY UNIT 1
Alternately, we may write
𝑃𝑉 =
𝐹𝑉
,
1+𝑟
which says that 𝑃𝑉 may be obtained from FV by discounting backward at rate 𝑟.
The diagrammatic representation of the relationship looks like this:
Figure 1.2
Note that we have not distinguished the future by the actual amount of time that
elapses.
If we do, we will have to specify:
1. the current time, 𝑡 = 0
2. the future time, 𝑡 = 𝑇
Consequently, the duration of the period is 𝑇 − 0 = 𝑇 , and the corresponding
relationships for PV and FV are:
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝑇)
𝑃𝑉 =
𝐹𝑉
1 + 𝑟𝑇
It is customary in finance to regard a duration of 1 year to be 1 unit of time. In this
𝐹𝑉
regard, the formulae 𝑃𝑉 = 1+𝑟 and 𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟) concern PV and FV that are 1year apart.
It is also customary to publish interest rate as annualised. What that means is that if I
say the interest rate is 5%, then by default, it refers to the rate of interest growth for a
period of 1 year. Sometimes, this fact may be emphasised by writing 5% p.a., where
p.a. = per annum.
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FIN201 STUDY UNIT 1
When the interest rate 𝑟 is used this way to relate PV and FV, i.e. in order to obtain PV
from FV or vice versa, one only needs to multiply or divide by a simple factor (1 +
𝑟𝑇), we say that 𝑟 is a simple rate.
If there are more than 1 period over which interest accumulates, the notion of
compound interest rate is applied. For instance, in the following diagram:
Figure 1.3
PV is loaned at time 𝑡 = 0 ("now") until 𝑡 = 𝑇1. At 𝑇1 , interest is paid out. Then right
away, both principal and interest are reinvested until 𝑡 = 𝑇2 . Thus, interest is
compounded twice. Suppose this is done at the rate 𝑟. What that means is that the FV
at time 𝑇2 is given by
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝑇1 ) × (1 + 𝑟(𝑇2 − 𝑇1 ))
where the expression 𝑇2 − 𝑇1 is the length of the period between times 𝑇1 and 𝑇2 .
If the compounding periods have equal lengths (say, all equal to 𝛿𝑡), i.e. 𝑇1 = 𝑇2 − 𝑇1 =
𝛿𝑡, then the above formula may be written as
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝛿𝑡)2 .
Often in practice, the compounding occurs over a period of 1 year, so that twiceayear
1
means "compound every 6 months". In this case, 𝛿𝑡 = 2 and
𝑟
𝐹𝑉 = 𝑃𝑉 × (1 + )2 .
2
Generally, if interest on a loan that is given for a year is to be compounded 𝑛 times a
year at a (compound) interest rate of 𝑟, then
𝑟
𝐹𝑉 = 𝑃𝑉 × (1 + )𝑛 .
𝑛
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FIN201 STUDY UNIT 1
In all these examples, 𝑟 is known as a compound interest rate for the way it is used to
derive FV from PV.
Example
If $100 is deposited at 7% over 35 days, how much is obtained at the end of the period?
We will use the formula
𝐹𝑉 = 𝑃𝑉(1 + 𝑟𝑇)
to compute the answer. The interest rate 𝑟 = 7/100 is regarded to be annualised. The
35
35
length of time is the factor 𝑇. We will taken 𝑇 = 365, i.e. 35 days to be 365 of 1 year.
Note that this is not always true, since 1 year can have 366 days. The way this fraction
is computed is known as the day count convention. Unless it is specified otherwise,
we will always select a simple day count convention, such as the one that is used here,
called ACT/365.
Thus,
𝐹𝑉 = 100 × (1 + 0.07 ×
35
) ≈ 100.67,
365
showing that the amount obtained at the end of the period is $100.67.
1.2 Nominal and Effective Rates
If an interest rate is given to be 𝑟 (assumed annualised) and it is to be applied to a
deposit over a period 𝑡 , the sum of principle plus interest obtained at the end of
the period is given by
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝑡).
The effective rate (aka equivalent annual rate) 𝑟𝑒𝑓𝑓 is defined by the following
equation:
(1 + 𝑟𝑒𝑓𝑓 )𝑡 = 1 + 𝑟𝑡.
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FIN201 STUDY UNIT 1
1
For example, if 𝑡 = 4 , i.e. the period is a quarter, then the effective rate is given by
𝑟
1 + 𝑟𝑒𝑓𝑓 = (1 + )4 .
4
Note that the RHS represents compounding 4 times, while the LHS represents
compoundingforward by the simple rate 𝑟𝑒𝑓𝑓 .
Relative to the effective rate, the rate 𝑟 is known as the nominal rate.
Another way to describe this relationship is: a nominal rate undergoes compounding
in the calculation of interest over a year while the effective rate is always used as a
simple rate in the calculation of interest over a year.
Example
If interest is to be accumulated for $1000 at the rate of 5%, quarterly per year for 3 years, what
is the effective rate?
The stated rate of 5% is the nominal rate.
We may compute
𝐹𝑉 = 1000 × (1 +
0.05 4×3
) .
4
The effective rate is obtained like this:
1000(1 + 𝑟𝑒𝑓𝑓 )3 = 1000(1 +
0.05 4×3
) ,
4
which implies that
𝑟𝑒𝑓𝑓 = (1 +
0.05 4
) − 1 ≈ 0.0509.
4
Example
If 3% is the nominal interest rate, what is the quarterly equivalent?
The question must be interpreted appropriately. Nominal rate is defined relative to
the effective rate or annual equivalent rate. The phrase "quarterly equivalent" is not
consistent with "annual equivalent" but it is used to suggest that the nominal rate is
compounded 4 times in a year and the effective rate is desired to be found.
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FIN201 STUDY UNIT 1
We find its value like this:
1 + 𝑟𝑒𝑓𝑓 = (1 +
0.03 4
)
4
and hence
𝑟𝑒𝑓𝑓 = (1 +
0.03 4
) − 1 ≈ 0.0303.
4
These sections from the Textbook (and do the associated exercises):
Chapter 1 Simple and compound interest
Chapter 1 Nominal and effective rates
Have you ever wondered where the notions of interest and interest rate come from?
Who on earth came up with these ideas?
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FIN201 STUDY UNIT 1
Chapter 2 Concept: Time Value of Money
The time value of money is a fundamental fact of finance. There are many ways to
express it, one of which is this: an amount of money at a certain time is not worth the
same as the same amount at another time. For example, $100 at present is not equal in
value as $100 in a year's time.
Another way is to visualise a cash flow at present and another that is equally valued
at some time 𝑇 later:
Figure 1.4
Each arrow represents a cash flow and the length indicates the amount of cash in the
flow.
The cash flow at present is usually called the present value (abbrev. PV) and that in
the future against which the PV is compared is called the future value (abbrev. FV).
Valuation is always done with respect to a point in time, usually at present. Thus, the
value of the cash flow at present is PV, while the value of FV, also at present, is not
exactly FV  the amount needs to be discounted to the present.
In the simplest case, the PV and the FV are related by an equation:
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟),
Where 𝑟 is the interest rate for the period. The interpretation is: in order to obtain the
FV from the PV, we need to compound the PV into the future at a certain interest rate.
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FIN201 STUDY UNIT 1
Thus, finding FV from PV is called compounding. The factor (1 + 𝑟) is called the
compounding factor.
Conversely, finding PV from FV is called discounting and it is expressed by the
following:
𝑃𝑉 =
𝐹𝑉
.
1+𝑟
1
The factor 1+𝑟 is called the discounting factor.
The above picture relating PV and FV is made simple as the relationship involves only
the interest rate 𝑟 and nothing else. In general, the relationship between PV and FV
involves several more items:
1. interest rate, 𝑟
2. interest accumulation/computation period (usually fraction of a year), 𝛼
3. number of accumulation/computation periods in the entire loan period, 𝑛
A concrete example is this: a loan is made for 3 years at an interest rate of 5% per
annum with interest computed at halfyearly intervals. This may be visualised as
follows:
Figure 1.5
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FIN201 STUDY UNIT 1
Interest that is accumulated over each computation period is commonly assumed to
be reinvested (as is true in the case of a fixed deposit over the entire period). In this
case, the relationship between PV and FV is given by
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝛼)𝑛 .
Here, 𝛼 is the fraction of the year that constitutes the interest
accumulation/computation period, and 𝑛 is the total number of such periods in the
entire loan period.
1
In the current example, 𝛼 = 2 and 𝑛 = 3 × 2 = 6 , and so the formula reads:
𝑟
𝐹𝑉 = 𝑃𝑉 × (1 + )2×3 .
2
Some points to note:
1 1
1
𝛼 is assumed to be 2 , 4 , 12 in the cases of semiannually, quarterly and monthly
compounding respectively. In realworld practice, the actual form that this
fraction takes depends on the day count convention that is assumed. If the actual
number of days is taken into account, we may express 𝛼 as
No.Days in Period
No.Days in Year
. This is
known as ACT/ACT.
Interest rates are commonly assumed to be annualised. For example, a 3%
interest rate on a principal of $100 will garner an interest of $3% × 100 = $3 after
1 year. Sometimes, in order to emphasise the fact, we may write "3% p.a." where
p.a. stands for per annum.
Please check: the product 𝛼𝑛 must always be equal to 𝑇, the length of the entire
loan period.
There are 3 scenarios in which the basic timevalueofmoney equation may be useful
and the equation is expressed slightly differently:
Finding FV from PV and 𝒓
The equation
𝐹𝑉 = 𝑃𝑉 × (1 + 𝑟𝛼)𝑛
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FIN201 STUDY UNIT 1
tells us how to find FV from PV. The process is called compounding, or forwardvaluing. The factor (1 + 𝑟𝛼)𝑛 is known as the compound factor.
Finding PV from FV and 𝒓
In order to find the PV from the FV, we may use the equivalent formula
𝑃𝑉 =
𝐹𝑉
.
(1 + 𝑟𝛼)𝑛
Here, the process is called discounting or presentvaluing and the factor
1
(1+𝑟𝛼)𝑛
is
known as the discount factor.
Finding 𝒓 from FV and PV
If we need to find the rate 𝑟 that is required to compound PV into FV over a total loan
period of length 𝑛𝛼 and with interest computation period of length 𝛼, then we may
use the equivalent formula
𝑟 = ((
𝐹𝑉 1
1
)𝑛 − 1) × .
𝑃𝑉
𝛼
These sections from the Textbook (and do the associated exercises):
Chapter 1 Future value / present value; time value of money
Chapter 1 Discount factors
When I write down "PV" or "FV", with the intention to calculate with them, either
numerically or in formulae, I am applying a very useful methodology of mathematics,
called representation. Crudely put, it's just "writing things down clearly".
Representation is useful for the following reasons:
it allows us to reason about real world entities in mathematical terms
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FIN201 STUDY UNIT 1
it allows us to use computers to help us in our reasoning because symbolic
representations such as PV or FV are readily understood by machines
Take a look at Apple's stock price history.
Represent (i.e. write down) the stock price in a suitable fashion so that you may be
able to discuss the following points with your friends or calculate the values with a
computer:
1.
2.
3.
4.
5.
today's stock price
1day return
annual return
price change over the last week
the price peaked in May last year
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FIN201 STUDY UNIT 1
Chapter 3 Calculations
Investments often involve a series of cash flows into the future, not just a single cash
flow. In order to assess the worth of the investment, it is necessary to find the present
value of the series of cash flows. This is then regarded as the value of the investment.
3.1 Net Present Value (NPV)
The NPV of a series of cash flows is the sum of the PV of each of the constituent cash
flow.
Suppose that the series of cash flow is represented by the following diagram:
Figure 1.6
The amount for each cash flow is indicated by 𝐶𝑖 .
The NPV is given by the formula:
𝑁𝑃𝑉 = 𝑃𝑉(𝐶1 ) + 𝑃𝑉(𝐶2 ) + 𝑃𝑉(𝐶3 ) + 𝑃𝑉(𝐶4 ) + 𝑃𝑉(𝐶5 ).
Note that the PV at each time 𝑡𝑖 depends on the discount factor corresponding to 𝑡𝑖 .
Suppose the discount factor at 𝑡𝑖 is 𝑑𝑖 , then the NPV formula may be expressed as:
𝑁𝑃𝑉 = 𝐶1 𝑑1 + 𝐶2 𝑑2 + 𝐶3 𝑑3 + 𝐶4 𝑑4 + 𝐶5 𝑑5 .
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FIN201 STUDY UNIT 1
Depending on the kind of interest rate information provided, the discount factors may
be calculated differently.
For instance, if it is assumed that there is a constant effective annual yield 𝑦, then
𝑑𝑖 =
1
(1 + 𝑦)𝑡𝑖
and the NPV works out to be
𝑁𝑃𝑉 =
𝐶1
𝐶2
𝐶3
𝐶4
𝐶5
+
+
+
+
.
𝑡
𝑡
𝑡
𝑡
1
2
3
4
(1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)𝑡5
Example
Find the NPV of the following cash flows, discounting at a rate of 5%.
Year Cash Flow
1
100
2
20
3
+180
The NPV is found by:
100
−20
180
+
+
≈ 232.59.
1 + 5% (1 + 5%)2 (1 + 5%)3
3.2 Internal Rate of Return (IRR)
The IRR is the interest rate or yield that needs to be applied to a series of cash flows
such that the PV of the series is equal to a specified amount. Usually, this specified
amount is the upfront payment one is willing to put down into the investment that is
promising the series of cash flows.
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FIN201 STUDY UNIT 1
From the interpretation, the IRR converts information in terms of cash amounts (the
upfront payment) into information in terms of rates or yields (the IRR). This allows us
to compare the attractiveness of the investment with other investments in the market
whose attractiveness is similarly expressed in terms of rates and yields.
To illustrate how IRR is defined and calculated, let's suppose that we are
contemplating an investment that comprises 5 cash flows 𝐶1 , 𝐶2 , … , 𝐶5 as illustrated
from above. Suppose also that we will decide to invest if the PV of the investment is
at least equal to a prespecified amount 𝑋.
Then the IRR is given by the value of 𝑟 that satisfies the following equation:
𝑋=
𝐶1
𝐶2
𝐶3
𝐶4
𝐶5
+
+
+
+
.
𝑡
𝑡
𝑡
𝑡
1
2
3
4
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)𝑡5
Notice that it is not a straightforward matter to compute 𝑟 when all the other
parameters are given. It is not possible to "just plug in" like how 𝑋 may be computed
from all other parameters. It is necessary to "solve for 𝑟".
In Excel, this may be done with the Solver addin. With Python, this may be done
using a root finding method in the scipy.optimize module.
Example
Find the IRR of the following cash flows, given that an upfront of $200 is desired.
Year Cash Flow
1
100
2
20
3
+180
The IRR is found by solving for 𝑟 in
200 =
100
−20
180
+
+
.
1 + 𝑟 (1 + 𝑟)2 (1 + 𝑟)3
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FIN201 STUDY UNIT 1
Using Python's
scipy.optimize.bisect
method (which will be explained in detail in the next unit), its value is equal to 0.12.
Can you find this solution manually or with a calculator?
3.3 Many Measurements of Return
A return is conceptually this: an initial amount of cash 𝐶0 becomes another amount
of cash 𝐶1 some time later. The absolute return is 𝐶1 − 𝐶0 while the relative return is
𝐶1
− 1. "Relative" here refers to "per unit of initial amount".
𝐶
0
But there is another angle to the word "relative" or "rate". It is this: per unit of time.
The latter cash flow 𝐶1 occurs some time later but no mention is made as to exactly
when that is.
If it is specified that 𝐶1 is received 6 months later, then a notion of "rate of return" may
𝐶
encompass "per unit of initial amount" and "per unit of time". The expression 𝐶1 − 1
0
may be called the absolute rate of return (for the 6month period) to distinguish it from
𝐶
2 × (𝐶1 − 1) which is the annualised rate of return, which is more commonly assumed.
0
But note that this usage of the term "absolute" conflicts with the notion of "absolute"
mentioned above.
The point is, there are many ways to measure return. In most realistic scenarios,
various factors may arise to complicate the definition. Some of these factors are:
the length of time over which the return is measured
more than 1 cash flow is involved
the existence of both cash inflows and outflows
The IRR is a measurement of return that takes into account all cash flows, both inflows
and outflows. Other measurements are timeweighted rate of return, equivalent
annual rate of return, daily return, etc.
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3.4 Annuities
An annuity is a regular stream of future cash flows which can be purchased as an
investment. If the stream of cash flows extends indefinitely (i.e. there is no enddate),
then the annuity is known as a perpetuity.
The cash flows in an annuity may be a fixed amount or it may grow over time. We
may for simplicity assume that the cash flows occur annually. The cash flows in an
annuity may be given either at the start of the year or at the end of the year.
In the following example, we will work out the relationship between yield (i.e.
equivalent annual rate) and the NPV of the annuity. We will use the following setup:
the NPV of a series of cash flows 𝐶1 , 𝐶2 , 𝐶3 , … is given by
𝑁𝑃𝑉 =
𝐶1
𝐶2
𝐶3
+
+
+⋯
2
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)3
In the case of an annuity, there are 𝑛 cash flows altogether. In the case of a pertuity,
there is an infinite number of cash flows.
3.4.1 Annuity Pays a Constant Amount at the End of Each Year
As the cash flow is constant, say 𝐶𝑖 = 𝐶 , we may write
𝑁𝑃𝑉 =
𝐶
𝐶
𝐶
𝐶
+
+
+⋯+
.
2
3
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)𝑛
The expression on the RHS is a geometric progression (recall from highschool
mathematics). It has a formula:
𝑁𝑃𝑉
=
1 𝑛
𝐶 1 − (1 + 𝑦 )
=
1+𝑦 1− 1
1+𝑦
𝐶
1 𝑛
(1 − (
) )
𝑦
1+𝑦
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FIN201 STUDY UNIT 1
3.4.2 Annuity Pays a Constant Amount at the Beginning of Each Year
In this case, the NPV is
𝑁𝑃𝑉 = 𝐶 +
𝐶
𝐶
𝐶
𝐶
+
+
+ ⋯+
.
2
3
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)𝑛−1
Using the formula for the geometric progression once again, we obtain
𝑁𝑃𝑉
=
=𝐶×
1
1 − (1 + 𝑦 )𝑛
1
1−1+𝑦
𝐶
1 𝑛−1
(1 + 𝑦 − (
) )
𝑦
1+𝑦
3.4.3 Perpetuity Pays a Constant Amount at the End of Each Year
This is equivalent to allowing 𝑛 to be exceedingly large in the case of an annuity that
pays a constant amount at the end of each year.
Thus, the NPV of the perpetuity is
𝑁𝑃𝑉
=
𝐶1
𝐶2
𝐶3
+
+
+⋯
2
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)3
𝐶
1 𝑛
(1 − (
) )
1+𝑦
𝑛→∞ 𝑦
= lim
=
𝐶
𝑦
3.4.4 Annuity Pays an Amount Which Grows Each Year at the End of
the Year
Suppose that the growth is expressed by a growth rate 𝑔, so that
𝐶𝑖 = (1 + 𝑔)𝑖−1 𝐶.
In other words, the first cash flow is 𝐶, the second is (1 + 𝑔)𝐶 , the third is (1 + 𝑔)2 𝐶 ,
and so on.
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FIN201 STUDY UNIT 1
The NPV is then given by
𝑛
𝑁𝑃𝑉 = ∑
𝑖=1
(1 + 𝑔)𝑖−1 𝐶
.
(1 + 𝑦)𝑖
This is a geometric progression whose first term is
𝐶
1+𝑦
and common ratio is
1+𝑔
1+𝑦
.
Hence the formula tells us that
𝑁𝑃𝑉
=
1+𝑔
1 − ( 1 + 𝑦 )𝑛
𝐶
=
×
1+𝑔
1+𝑦
1−1+𝑦
𝐶
1+𝑔 𝑛
× (1 − (
) )
𝑦−𝑔
1+𝑦
These sections from the Textbook (and do the associated exercises):
Chapter 1 Cashflow analysis, NPV, IRR and timeweighted rate of return
Chapter 1 Annuities
For a further oral elaboration of the concept of the Time Value of Money, watch:
Time Value of Money
(Access video via iStudyGuide)
Practical calculations of interest rates require the concept of Day Count Convention.
For an explanation of the concept, watch:
Day Count Convention
(Access video via iStudyGuide)
The formulae that are derived in this section depend on your knowledge of geometric
progressions and geometric series from high school mathematics.
Review these concepts by following the course tracks on Sequences and Introduction
to Geometric Series at Khan Academy.
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Quiz
1) If 3% is the quarterly equivalent rate, what is the nominal interest rate?
a.
b.
c.
d.
2.95%
2.96%
2.97%
2.98%
2) Time value of money rules out
a.
b.
c.
d.
the possibility that interest rates can be positive
the possibility that interest rates can be negative
the possibility that interest rates can fluctuate
the possibility that interest rates can stagnate
3) The _________ expresses the relationship between the present value and the
future value of a cash flow.
a.
b.
c.
d.
interest
interest rate
principal
term
4) The nominal interest rate undergoes compounding in the calculation of interest
over a year.
a. True
b. False
5) The future value is bigger than the present value
a.
b.
c.
d.
in all cases
if the interest rate is positive
if the interest rate is zero
if the interest rate is negative
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FIN201 STUDY UNIT 1
6) The tool in Excel that is useful for solving IRR problems is
a.
b.
c.
d.
the Analysis ToolPak
the Solver addin
the Ribbon
the worksheet
7) A root finding function can be found in the Python module scipy.stats.
a. True
b. False
8) If interest rate is assumed to be constant over terms, the discount factor for Year
1
a.
b.
c.
d.
is the square root of the discount factor for Year 2
is the square of the discount factor for Year 2
is equal to the discount factor for Year 2
is twice the discount factor for Year 2
9) The return of an investment over half a year must be _________ in order to obtain
the annualised figure.
a.
b.
c.
d.
unchanged
compounded
discounted
doubled
10) Finding the present value of future cash flows through discounting with a
constant interest rate makes the crucial assumption that the interest rate is
_________ over the term of the cash flows.
a.
b.
c.
d.
changing
unchanging
increasing
decreasing
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FIN201 STUDY UNIT 1
Solutions or Suggested Answers
Quiz
Question 1: c
Question 2: b
Question 3: b
Question 4: a
Question 5: b
Question 6: b
Question 7: b
Question 8: a
Question 9: d
Question 10: b
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STUDY UNIT 2
STATISTICS AND NUMERICAL
METHODS WITH EXCEL AND
PYTHON
FIN201 STUDY UNIT 2
Learning Outcomes
By the end of this unit, you should be able to:
1. Review statistical concepts and numerical methods.
2. Summarise financial calculations from Excel.
3. Identify the appropriate Python modules for financial computations.
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FIN201 STUDY UNIT 2
Chapter 1 Basics of Statistics
The purpose of this unit is to collect together some materials and methods that will be
very useful in the measuring, mapping and modelling of things in finance.
These materials and methods are gathered from 4 areas:
Statistics
Numerical Methods
Excel
Python
Statistics is useful whenever we need to deal with lots of numbers. Financial data is a
lot of numbers! Statistical notions are indispensable for us.
Numerical methods are a particular branch of applied mathematics that enables us to
interpolate, extrapolate and solve. For example, the Internal Rate of Return is hard to
work out by hand and needs to be solved using some numerical method, such as the
bisection method.
The theories behind statistics and numerical methods have been implemented in
various software packages. We will explore Excel and Python and use them to aid us
in calculations in finance.
The object of our focus in statistics is data. We will be concerned with data that is a
sequence of numbers:
𝑥1 , 𝑥2 , … , 𝑥𝑛 .
For example, the sequence may be the sequence of S&P 500 index from inception until
now.
Occasionally, we will have to compare between two sequences. For instance, we may
need to measure the relationship between S&P 500 index and Apple's stock price to
see how closely the latter follows the market. In such cases, we will be concerned with
2 sequences of numbers, or 2 datasets:
𝑥1 , 𝑥2 , … , 𝑥𝑛 .
𝑦1 , 𝑦2 , … , 𝑦𝑛 .
Statistics allows us to summarise these datasets in various ways.
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FIN201 STUDY UNIT 2
1.1 Averages
"Average" commonly refers to what is technically called the arithmetic mean in
statistics:
𝑛
1
1
𝜇 = 𝑥 = (𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛 ) = ∑ 𝑥𝑖 .
𝑛
𝑛
𝑖=1
The various expressions shown above are the various abbreviations that are
commonly used for this mean.
Sometimes, in addition to the dataset 𝑥1 , 𝑥2 , … , 𝑥𝑛 , weights are also given: 𝑤1 , 𝑤2 , … , 𝑤𝑛 ,
each weight 𝑤𝑖 corresponding to each 𝑥𝑖 , 𝑖 for 𝑖 respectively. For instance, one can
imagine a scenario whereby the expected return of asset 𝑖 is 𝑥𝑖 and an amount 𝑤𝑖 of
the asset 𝑖 is purchased. Then the overall return of the portfolio thus assembled is
𝑛
𝑤1 𝑥1 + 𝑤2 𝑥2 + ⋯ + 𝑤𝑛 𝑥𝑛 = ∑ 𝑤𝑖 𝑥𝑖 .
𝑖=1
This is called the weighted average of the dataset 𝑥1 , 𝑥2 , … , 𝑥𝑛 with respect to the
weights 𝑤1 , 𝑤2 , … , 𝑤𝑛 .
The geometric mean is the expression
1
(𝑥1 𝑥2 ⋯ 𝑥𝑛 )𝑛 .
It may be applied to find a sort of average between quantities that are naturally
multiplied together, rather than added together. For instance, discount factors.
When there is no confusion as to which mean is being referred to, we'll just say "mean".
All these various types of means are centrality measures, i.e. each measure picks out
a special number that is recognised as a central element of the dataset. In this regard,
two other centrality measures are the median and the mode.
The mode is the data value that occurs the most frequently in the entire dataset. It is
possible to have more than 1 mode in a single dataset.
The median is the middle data value when all the data values in the dataset are ranked
in an increasing order.
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FIN201 STUDY UNIT 2
Example
For the dataset
1,5,8,4,1,8,4,10,
find the
1.
2.
3.
4.
5.
Arithmetic mean
Weighted average according to the weighting rule: "half for odd, quarter for even"
Geometric mean
Mode
Median
For (1), the arithmetic mean is:
1 + 5 + 8 + 4 + 1 + 8 + 4 + 10
= 5.125.
8
For (2), the weighted average is
1
1
1
1
1
1
1
1
1 × 2 + 5 × 2 + 8 × 4 + 4 × 4 + 1 × 2 + 8 × 4 + 4 × 4 + 10 × 4
2.75
= 4.3636.
For (3), the geometric mean
1
(1 × 5 × 8 × 4 × 1 × 8 × 4 × 10)8 ≈ 3.878.
For (4), there are 3 modes: 1, 4 and 8.
For (5), in order to find the median, firstly, we have to rank the data in increasing order:
1,1,4,4,5,8,8,10.
There are 8 elements altogether. Thus, there are 2 central values: 4 and 5. The median
4+5
is therefore 2 = 4.5.
1.2 Expectation of a Random Variable
A random variable 𝑋 is a representation of a dataset of values. In other words, it can
assume any value from the dataset with a certain chance factor attached to each value.
Think of a biased coin. The chance that it will be heads is some probability 𝑝 and the
chance that it will be tails is 𝑞 = 1 − 𝑝.
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FIN201 STUDY UNIT 2
In the same way, a random variable 𝑋 may be visualised as a table:
Value Probability
𝑥1
𝑝1
𝑥2
𝑝2
𝑥3
𝑝3
The expectation of 𝑋, written 𝐸(𝑋), is the average value of 𝑋, given by the weighted
average
𝐸(𝑋) = 𝑝1 𝑥1 + 𝑝2 𝑥2 + 𝑝3 𝑥3 .
The expectation has the following properties:
1.
If 𝑋 and 𝑌 are two random variables, then
𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌).
2.
If 𝑋 is a random variable and 𝑎 is a number, then
𝐸(𝑎𝑋) = 𝑎𝐸(𝑋).
1.3 Spreads
A spread measure is a formula that computes how spreadout a dataset is.
The most commonly used spread measures in statistics are the variance:
(𝑥1 − 𝑥)2 + (𝑥2 − 𝑥)2 + ⋯ + (𝑥𝑛 − 𝑥)2 ∑𝑛𝑖=1( 𝑥𝑖 − 𝑥)2
𝜎 =
=
𝑛
𝑛
2
and the standard deviation:
𝜎.
Yes, the standard deviation is the square root of the variance. Essentially, they
measure the same thing. One being the square root of the other, they differ in terms of
units. The standard deviation is of the same unit as the data values in the dataset and
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FIN201 STUDY UNIT 2
consequently we may use the standard deviation to measure distances between data
values. Thus, if 2 data values 𝑥𝑖 and 𝑥𝑗 satisfy 𝑥𝑖 − 𝑥𝑗 = 2𝜎, then we may say that they
are 2 standard deviations apart.
Take a look at the formula and try to see what it's trying to compute: the variance is
an average of distances between each data point 𝑥𝑖 and the mean 𝑥 (distance is here
defined to the square of the difference).
Example
For the dataset
1,5,8,4,1,8,4,10,
find the standard deviation and the variance. How many standard deviations is 10 away from
the mean?
The mean is 5.125 (see above).
The variance is 9.609.
The standard deviation is 𝜎 = √9.609 = 3.100.
Since
10−5.125
3.100
= 1.573, which may be rewritten as 10 − 5.125 = 1.573 × 𝜎, the value 10
is 1.573 standard deviations away from the mean.
1.4 Biased versus Unbiased
In statistics, the variance and standard deviation that are defined above are called
biased variance and biased standard deviation respectively. The word biased means
that in a theoretical sense, these are not the most accurate estimates of spreads.
The unbiased variance is given by the formula
𝜎^2 =
(𝑥1 − 𝑥)2 + (𝑥2 − 𝑥)2 + ⋯ + (𝑥𝑛 − 𝑥)2 ∑𝑛𝑖=1( 𝑥𝑖 − 𝑥)2
=
𝑛−1
𝑛−1
and the unbiased standard deviation is
𝜎^.
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FIN201 STUDY UNIT 2
The distinction is due to the denominator "𝑛 − 1" instead of "𝑛".
So when do we use the unbiased version and when do we use the biased version?
The answer is this: if we are computing the spread of actual data values, we will use
the biased version. If we intend to fit the standard deviation to a probability
distribution, then we will use the unbiased version.
1.5 Variance of a Random Variable
If 𝑋 is a random variable, then the variance of 𝑋 is defined in the following way:
𝑉𝑎𝑟(𝑋) = 𝐸((𝑋 − 𝐸(𝑋))2 ).
In other words, first we compute the number
𝜇 = 𝐸(𝑋).
Then we plug 𝜇 into the above formula to compute the variance:
𝑉𝑎𝑟(𝑋) = 𝐸((𝑋 − 𝜇)2 ).
From this expression, we see that the variance of a random variable may be interpreted
as the average (squared) distance between a value of 𝑋 and the mean value 𝜇. This is
exactly the definition of spread as we have stated above.
An alternative way to write the variance is this:
𝑉𝑎𝑟(𝑋) = 𝐸(𝑋 2 ) − 𝐸(𝑋)2 .
The two formulae are exactly equivalent.
Note that according to this definition, if 𝑎 is a number, then
𝑉𝑎𝑟(𝑎𝑋) = 𝑎2 𝑉𝑎𝑟(𝑋).
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FIN201 STUDY UNIT 2
1.6 Correlation and Covariance: Relationships between Two
Datasets
The most common measures of relationships between 2 datasets is the covariance and
the Pearson correlation coefficient (or correlation coefficient or just correlation).
The covariance of the datasets 𝑥1 , 𝑥2 , … , 𝑥𝑛 and 𝑦1 , 𝑦2 , … , 𝑦𝑛 is
𝐶𝑜𝑣(𝑥, 𝑦) =
∑𝑛𝑖=1 𝑥𝑖 𝑦𝑖 − (∑𝑛𝑗=1 𝑥𝑗 )(∑𝑛𝑘=1 𝑦𝑘 )
𝑛
.
The correlation is
𝜌=
𝐶𝑜𝑣(𝑥, 𝑦)
,
𝜎𝑥 𝜎𝑦
where 𝜎𝑥 and 𝜎𝑦 are the standard deviations of the dataset 𝑥 and 𝑦 respectively.
The correlation coefficient is a number that lies between −1 and 1. If it turns out to be
1, then the two datasets are related linearly, i.e. the pairs (𝑥𝑖 , 𝑦𝑖 ) all lie on a single
straight line with a positive slope. If it turns out to be −1, then the pairs lie on a single
straight line with a negative slope. If it turns out to be 0, then we interpret that the
relationship between the two datasets is random. Any value inbetween is interpreted
between being linear related and random.
The following scatter plots make the intuition for correlation coefficient clear:
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FIN201 STUDY UNIT 2
Figure 2.1
Incidentally, scatter plots are very powerful in rendering relationships that may exist
within bivariate data (i.e. a dataset of the form (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), … , (𝑥𝑛 , 𝑦𝑛 ) ).
Notice how the covariance and the correlation are related. The correlation is obtained
from the covariance by normalising (i.e. dividing) by the standard deviations of the
two datasets. Hence, the covariance contains 3 pieces of information:
•
•
•
the relationship that is represented by the correlation coefficient
the standard deviation of the dataset 𝑥
the standard deviation of the dataset 𝑦
As it involves 3 concepts together, it is generally hard to interpret exactly what the
covariance tells us apart from whether the relationship is a positive or negative one.
1.7 Covariance of 2 Random Variables
The notion of covariance has an analogue for random variables. If 𝑋 and 𝑌 are 2
random variables, their covariance is
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐸(𝑋𝑌) − 𝐸(𝑋)𝐸(𝑌).
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FIN201 STUDY UNIT 2
Note that
𝐶𝑜𝑣(𝑋, 𝑋) = 𝐸(𝑋 2 ) − 𝐸(𝑋)2
by this formula. This shows that the covariance of 𝑋 and itself is its variance.
The covariance has the following properties: if 𝑋, 𝑌, 𝑍 are 3 random variables, then
1.
2.
3.
𝐶𝑜𝑣(𝑋, 𝑌 + 𝑍) = 𝐶𝑜𝑣(𝑋, 𝑌) + 𝐶𝑜𝑣(𝑋, 𝑍)
𝐶𝑜𝑣(𝑋 + 𝑌, 𝑍) = 𝐶𝑜𝑣(𝑋, 𝑍) + 𝐶𝑜𝑣(𝑌, 𝑍)
𝐶𝑜𝑣(𝑎𝑋, 𝑌) = 𝐶𝑜𝑣(𝑋, 𝑎𝑌) = 𝑎𝐶𝑜𝑣(𝑋, 𝑌)
1.8 Variance of a Sum of Random Variables
Suppose 𝑋 and 𝑌 are 2 independent random variables, then it is known that
𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) = 𝜎𝑋2 + 𝜎𝑌2 ,
where 𝜎𝑋 and 𝜎𝑌 are the standard deviations of 𝑋 and 𝑌 respectively.
In general, without assuming independence, it is known that
𝑉𝑎𝑟(𝑋 + 𝑌) = 𝜎𝑋2 + 𝜎𝑌2 + 2𝜎𝑋 𝜎𝑌 𝜌𝑋𝑌 ,
where 𝜌𝑋𝑌 is the correlation coefficient of 𝑋 and 𝑌.
If there are 3 random variables 𝑋, 𝑌, 𝑍, then
𝑉𝑎𝑟(𝑋 + 𝑌 + 𝑍) = 𝜎𝑋2 + 𝜎𝑌2 + 𝜎𝑍2 + 2𝜎𝑋 𝜎𝑌 𝜌𝑋𝑌 + 2𝜎𝑌 𝜎𝑍 𝜌𝑌𝑍 + 2𝜎𝑍 𝜎𝑋 𝜌𝑍𝑋 .
If there are 4 random variables 𝑊, 𝑋, 𝑌, 𝑍, then
𝑉𝑎𝑟(𝑊 + 𝑋 + 𝑌 + 𝑍)
2
= 𝜎𝑊
+ 𝜎𝑋2 + 𝜎𝑌2 + 𝜎𝑍2 + 2𝜎𝑋 𝜎𝑌 𝜌𝑋𝑌 + 2𝜎𝑌 𝜎𝑍 𝜌𝑌𝑍
+2𝜎𝑍 𝜎𝑋 𝜌𝑍𝑋 + 2𝜎𝑊 𝜎𝑋 𝜌𝑊𝑋 + 2𝜎𝑊 𝜎𝑌 𝜌𝑊𝑌 + 2𝜎𝑊 𝜎𝑍 𝜌𝑊𝑍 .
The same pattern holds for any number of random variables.
This formula may be applied to calculate the volatility of a portfolio of assets.
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1.9 Histograms
When we compute a measure (mean or spread), we are computing a single number
from a dataset which may comprise many data values. Hence, while statistical
measures are useful summaries, they lose information.
Plotting the dataset as a histogram retains more information about the set than just
the mean or standard deviation.
The following histogram describes the daily returns of Yahoo! stock between 2010 and
2015.
Figure 2.2
A histogram categorises the data into a small number of bins and displays the count
of each bin, thus simplifying the dataset and enables visualisation at the same time.
A histogram tells us something about the dataset. It gives us information about the
source. From the first histogram above, the daily returns of Yahoo! stock are mostly
concentrated between 5% and 5%. There are some extremes near 10% and 10%. This
gives us a sense of how the stock price fluctuates on a daily basis.
1.9.1 Bin Size
One choice that needs to be made in plotting histograms is the bin size. If the bin size
is chosen too large, we will obtain a histogram that looks like this:
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Figure 2.3
The result is a histogram with only 4 categories (bin width = 0.5). This is too rough and
it hides the intricacies of how the data is distributed.
If the bin size is chosen too small, we obtain:
Figure 2.4
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The result is a histogram that is quite complex as it reveals too much details from
within the dataset.
The goals of simplifying the dataset and revealing its distribution are opposite to each
other and requires a bit of balancing.
1.9.2 Types of Histograms: Count, Relative Frequency, Normalised
In the histograms above, each vertical bar represents the number of data values in the
dataset that belongs to the category beneath the bar. For example, in the second
histogram, the bar that extends over the interval 0 to 5 has a height of about 630. This
means that roughly 630 days of Yahoo! returns are positive and less than 5%.
Thus, the histogram shows the count for each category.
There are two other kinds of histograms: relative frequency and normalised.
A bar in a relative frequency histogram shows the fraction of the dataset that belongs
to it. The following histogram is a relative frequency version of the 4bar histogram
from above:
Figure 2.5
A normalised histogram has the property that the total area in the vertical bars is 1. It
allows the histogram to be interpreted as a probability distribution since total
probability in a distribution is always 1.
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The normalised version of the above histogram is this:
Figure 2.6
The count histogram, relative frequency histogram and normalised histogram have
equal shapes. They differ from each other only by a scale factor in the vertical direction.
1.10 Probability Distributions
Another simplification of the dataset is through interpretation by a mathematical
probability distribution. Such a probability distribution is visualised by means of its
density function 𝑦 = 𝑓(𝑥). When this function is plotted, it gives a description of how
the data is spread out in the same way a histogram displays the spread.
The distributions encountered in basic statistics are:
the normal distribution
the Student's t distribution
the chisquared distribution
the F distribution
The normal distribution is commonly used in applications. It is used to give a
mathematical interpretation of datasets that clearly display a concentration of data
values and a spread of data values away from the central position.
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For instance, consider a dataset of IQ scores whose histogram looks like this:
Figure 2.7
The histogram is informative about its underlying dataset, but it's not so easy to deal
with in writings.
Instead, we overlay the density function of a normal distribution, making sure that
the mean and the standard deviation of the distribution agree with the mean and the
standard deviation of the dataset:
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Figure 2.8
The overlay allows the underlying dataset to be interpreted in terms of the normal
distribution.
For example, to the question: what proportion of the dataset is greater than 100?, we
may answer half since the normal distribution that has been calibrated to the dataset
has mean equal to 100, and thus half of its area is to the right of 100 and the other half
is to the left. Notice that we have not sought out the dataset for this answer. Instead,
we use the normal distribution that has been fitted to the dataset.
1.10.1 Continuous and Discrete Distributions
A continuous distribution is used to summarise a huge dataset. The distributions
mentioned above  normal, Student's t, chisquared, F  are all continuous distributions.
A continuous probability distribution is specified by its probability density function
(PDF) which is intended to be a close approximation of the normalised histogram of
the dataset. For instance, the PDF of the normal distribution with mean 𝜇 and
standard deviation 𝜎 is
𝑓(𝑥) =
1
𝜎√2𝜋
1
2
𝑒 −2𝜎(𝑥−𝜇) .
This is a very concise description. It is for this conciseness that datasets are often
interpreted in terms of PDFs.
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Another key feature of continuous distributions is that areas under the graph of the
density function are interpretable as probabilities.
For instance, the probability that IQ is greater than 120 is given by the shaded area
below:
Figure 2.9
On the other hand, if there are lots of data and they are required to be grouped in
several discrete categories, then a discrete probability distribution is more useful.
A discrete probability distribution is described by a table:
Value Probability
𝑥1
𝑝1
𝑥2
𝑝2
⋯
⋯
𝑥𝑛
𝑝𝑛
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Here, the distribution has 𝑛 categories and the probability that the category 𝑥𝑖 occurs
is 𝑝𝑖 . Due to the fact that the 𝑝𝑖 's are probabilities, they must be nonnegative and
satisfy
𝑝1 + 𝑝2 + ⋯ + 𝑝𝑛 = 1.
For example, a fair dice is described by the following probability distribution:
Value Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Probability distributions are very important in finance as we have to deal with lots of
data as well as uncertainty concerning the future.
As probability distributions are approximations of histograms, they allow us to work
with lots of data in terms of a concise representation. As the language of probability
distributions revolves around probabilities, they allow us to talk about reason and
compute with chances concerning uncertainty which is inherent in financial
transactions and investments.
These sections from the Textbook (and do the associated exercises):
Chapter 2 Averages  arithmetic and geometric means, weighted averages,
median and mode
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Chapter 2 Variance, standard deviation and volatility
Chapter 2 Correlation and covariance
Chapter 2 Histograms, probability density and the normal probability
function
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Chapter 2 Basics of Numerical Methods
Two sets of numerical methods are the most important for practical finance:
interpolation and extrapolation methods
o help to find approximate values between data points or outside of data
range
root finding methods
o help to find solutions of equations
We will discuss them here so that you will be able to confidently use them in Excel or
Python.
2.1 Interpolation and Extrapolation
"Interpolation" means finding a number between some given ones. For example, given
𝑥+𝑦
𝑥 and 𝑦 , interpolating fairly between them, we obtain the midpoint 2 .
"Extrapolation" means finding a number that lies beyond some given ones.
Interpolation and extrapolation are often applied where data is missing. For instance,
if it is known what the 4year and 5year interest rates are, it's possible to deduce a 4.5year interest by taking the average of the 2 known rates. More precisely, if the 4year
rate is 5% and the 5year rate is 6%, then we may estimate that the 4.5year rate is
5+6
%, i.e. 5.5%.
2
How to find the 4yearand3month rate (i.e. 4.25 since 3month = 0.25)?
Let's put the interest rates of concern sidebyside:
𝑟(4), 𝑟(4.25), 𝑟(5)
The simplest assumption one can make is that the points
(4, 𝑟(4)), (4.25, 𝑟(4.25)), (5, 𝑟(5))
lie on a straight line.
Obtaining 𝑟(4.25) this way is known as linear interpolation.
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Solving for 𝑟(4.25), we have:
𝑟(4.25) − 𝑟(4) 𝑟(5) − 𝑟(4)
=
,
4.25 − 4
5−4
i.e.
𝑟(4.25) = (𝑟(5) − 𝑟(4)) × 0.25 + 𝑟(4) = (5 − 4) × 0.25 + 4 = 4.25.
2.2 Root Finding
Root finding is the problem that arises when we need to find the value of 𝑥 for which
a given equation
𝑓(𝑥) = 0
holds.
For example, to find the IRR means to solve for 𝑟 in
𝑃𝑉 =
𝐶1
𝐶2
𝐶3
𝐶4
𝐶5
+
+
+
+
.
(1 + 𝑟)𝑡1 (1 + 𝑟)𝑡2 (1 + 𝑟)𝑡3 (1 + 𝑟)𝑡4 (1 + 𝑟)𝑡5
In other words, if we write
𝑓(𝑟) =
𝐶1
𝐶2
𝐶3
𝐶4
𝐶5
+
+
+
+
− 𝑃𝑉,
(1 + 𝑟)𝑡1 (1 + 𝑟)𝑡2 (1 + 𝑟)𝑡3 (1 + 𝑟)𝑡4 (1 + 𝑟)𝑡5
then the IRR is the solution to the equation
𝑓(𝑟) = 0.
Many methods are available to find roots to equations. Two commonest ones are:
•
•
the bisection method
the NewtonRaphson method
Do not worry about how these methods work  you will not be tested on this. Just be
aware of their names, where they can be found in Excel or Python and how they may
be applied.
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These sections from the Textbook (and do the associated exercises):
Chapter 1 Interpolation and extrapolation
The following YouTube videos explain the mechanics behind the bisection method
and the NewtonRaphson method:
How to locate a root : Bisection Method : ExamSolutions
Newton's Method
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Chapter 3 Using Excel
We will discuss the following aspects of Excel in order to get you uptospeed quickly:
Basic objects of Excel
The several categories of "things" that we manipulate all the time in Excel  if
we are aware of them, we'll be able to work with them more efficiently
Excel financial formulae
There are several categories of worksheet functions in Excel. Apart from certain
breadandbutter type functions (e.g. AVERAGE), the category of financial
functions is the most useful for a finance student. We'll make use of a few
financial functions in this course.
The Data Analysis addin
This addin helps to deal with basic statistics.
The Solver addin
This addin helps in rootfinding. It is required for IRR or YTM problems.
3.1 Basic Objects of Excel
Excel is made up of parts. As the software is large and has undergone development
for many years, it is quite complex. For this reason, we may at times think of its
structure as being hierarchical.
The most important objects making up Excel are the following:
Application
Workbooks
Worksheets
Cells
Ranges
Formulae
Addins
The term "objects" is not imaginary. At its core, Excel is created in terms of these objects.
You may refer to Excel Object Model for a glimpse of this core:
https://msdn.microsoft.com/enus/library/office/ff194068.aspx
Let me show you what these objects are in pictures.
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When Excel is installed onto a computer, there is a single Excel application residing
on the computer.
Figure 2.10
Each file that one creates in order to use Excel is known as a workbook.
Figure 2.11
Opening a workbook, 3 worksheets are precreated, ready to be used.
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Figure 2.12
The cell A1 above is selected. There are altogether 1,048,576 rows and 16,384 columns
in Excel 2007, 2010 and 2013. That makes a total of 17,179,869,184 (17 billion) cells.
A rectangular block is known as a range.
Figure 2.13
Values of various types may be entered into the cells. What makes Excel very useful
is that in addition to values, formulae can also be written into cells. For example,
typing "=NOW()" into cell A1 gives the current date and time.
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Figure 2.14
These functions are prepackaged functionalities that are very often used. They greatly
facilitate calculations with data.
In order to use them efficiently, it is necessary to know what functions there are. Excel
functions can be grouped into the following categories:
Compatibility functions
Cube functions
Database functions
Date and time functions
Engineering functions
Financial functions
Information functions
Logical functions
Lookup and reference functions
Math and trigonometry functions
Statistical functions
Text functions
User defined functions that are installed with addins
Web functions
For more details, you may refer to: https://support.office.com/enus/article/Excelfunctionsbycategory5f91f4e97b4246d29bd163f26a86c0eb
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3.2 Excel Financial Formulae
The most important Excel functions for the finance student are the financial functions.
A listing of them can be found here: https://support.office.com/enus/article/Financialfunctionsreference5658d81e60354f2489c1fbf124c2b1d8
Some of the more commonly used ones, also important for us, are:
ACCRINT
DISC
DURATION
EFFECT
FV
INTRATE
IRR
MDURATION
NPV
NPER
PMT
PRICE
PV
RATE
YIELD
3.3 The Data Analysis Addin
The Data Analysis addin contains a library of analysis tools for basic statistics.
Figure 2.15
For example, the Descriptive Statistics tool computes the basic descriptive statistics for
a dataset that is set within a range.
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Figure 2.16
The output of the above setting is this:
Figure 2.17
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3.4 The Solver Addin
The solver addin implements some rootfinding algorithms. It basically enables us
to solve for the roots of equations, such as the following:
𝑓(𝑦) =
5
5
105
+
+
− 100.
2
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)3
The solution 𝑦0 to 𝑓(𝑦) = 0 is the yieldtomaturity of a par bond which matures in 3
years and has an annual coupon rate of 5% p.a.
This may be solved with the solver as follows.
First, set up the worksheet this way:
Figure 2.18
We don't know what the correct YTM should be at first, so we set it at some arbitrary
value: 2%. As the PV (i.e. the sum of all the discounted cash flows) is not 0, we know
that 2% is not the correct YTM.
Next, we open the solver window and type in the necessary parameters:
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Figure 2.19
Running it, we obtain:
Figure 2.20
We see that 5% is the correct answer.
The solver is not the only way to solve this particular problem.
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The Excel function IRR is designed with this purpose in mind. We may invoke:
in any cell on the worksheet and we'll obtain the answer 5%.
Another solution is to observe that
100𝑦
100𝑦
100(1 + 𝑦)
+
+
= 100
1 + 𝑦 (1 + 𝑦)2
(1 + 𝑦)3
is always true regardless of the value of 𝑦. Hence, if we compare this with the equation
5
5
105
+
+
− 100 = 0
2
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)3
which we need to solve, immediately we see that 𝑦 = 0.05 is an answer.
One point to remember is this: there can be many approaches to finding an answer.
Having many tools around  mathematical formulae, Excel functions, Excel addins makes the task easier.
1.
Browse the Excel functions (by category) webpage. Try to commit to memory
the most important things on the webpage pertaining to how the functions
are classified.
2.
Browse the Financial functions (reference) webpage. Locate the key financial
functions mentioned above and read what these functions do.
3.
Browse the PV function webpage. Read to become familiar with the sort of
information that you can find on the page.
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Chapter 4 Using Python
We will discuss the following aspects of Python in order to get you uptospeed
quickly:
Installing and Starting Python
Using Python as a Calculator
Basic Objects of Python
Useful Modules
4.1 Installing and Starting Python
Python is a very popular programming language. Some of its features make it very
useful for beginners:
It has a command line interpreter, allowing it to be used like a calculator.
It has a rich set of builtin modules. This means that, once installed, it can be
used to perform many different kinds of commonly encountered calculations
without having to install any further modules.
It has a strong community of developers and contributors. This means that
apart from its builtin libraries, there are a lot of very useful, more specialised
and wellmaintained libraries residing on the internet, ready to be installed and
used.
It is largely opensource. This means that using it for calculations is free.
It has a clean syntax. This means that codes written in Python are quite humanreadable and easy to keep and maintain.
Do go ahead and install Python on your machine. We'll use it freely in this course.
Python is currently in its version 3. However, version 2 has existed for a long time and
there is rich support for it on the internet. Thus, we will use the latest version of Python
2 (i.e. 2.7) here.
Python may be downloaded from its website. However, we will not install Python
directly from its source site. Instead, we will install Python from here: http://pythonxy.github.io/
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Here, Python is packaged with many libraries apart from its builtin suite to facilitate
scientific computing. A part of scientific computing is data analysis. Python data
analysis libraries are very useful in finance.
Go ahead and install the latest version of Pythonxy (as of 30 Dec 2015, the latest
version is Python 2.7.10.0).
Once installed, run the Spyder IDE (i.e. integrated development environment). You
will find yourself in this environment:
Figure 2.21
The pane on the left is the Editor pane. It is used for writing scripts to be saved away
in files.
The pane on the right is the Console pane. It allows Python to be used like a calculator.
What that means is that the mode of interaction between Python and the user goes
like this:
User issues commands
Python computes and replies
This interactivity makes Python friendlier to use than some older languages such as
C or C++.
There are quite a few other panes. To see them, check out View>Panes.
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4.2 Using Python as a Calculator
Go ahead and type something into the righthandside console.
Figure 2.22
Every line typed into the console and ended with an "Enter" is interpreted by Python.
If it doesn't make sense, Python will let you know. Otherwise, it will report the value
of the statement that you have just typed.
You may follow what I have typed as shown above to get started. Don't worry about
typing the wrong things. It's not easy to get Python stuck.
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4.3 Basic Objects of Python
A computer does a few things very well and very efficiently. We may understand
these things in this manner:
computers manipulate values and data structures
data structures are like containers, and values are like contents in the containers
values and data structures can be stored in variables, which are named, for us
to manipulate
The first step in using Python is to be aware of the kinds of values and data structures
that exist.
The basic value types are:
integer (e.g. 1, 0, 100)
float (e.g. 4.5, 0.0, 34.23)
string (e.g. 'abcd', "superman")
The basic data structures are:
list (e.g. [1,2,3], ['a', 'b', 4])
tuple (e.g. (1,2,3), ('a', 'b', 4))
dictionary (e.g. {'name': 'John', 'age': 15, 'school': 'XYZ Sec. Sch.'})
Type the following examples into the console and have a look:
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Figure 2.23
In order to use these values and data structures, they must be stored away, kept in
variables which are named:
Figure 2.24
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Now we may manipulate the values and data structures by manipulating the names
(which refer to the variables or containers of these values):
Figure 2.25
These basic facts about values, data structures and names/variables play a role in the
writing of a Python program (aka script) which is similar to bricks and mortar that go
into the construction of a building.
To see the overall structure of a building, it's very useful to examine examples of
scripts that perform intuitive tasks.
4.4 Useful Modules
Python comes with a rich set of builtin modules. This practically means that any basic
calculation or computational task is no further than a few commands away provided
you know where to look for the relevant ones.
The internet Python community has also built and maintained a rich ecology of
modules that extend the builtin ones that come installed with a minimal installation
of Python.
The following modules are useful for handling data in finance:
pandas_datareader
numpy, scipy, matplotlib and pandas
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4.4.1 pandasdatareader
The pandasdatareader module allows data from the internet (various sources) to be
downloaded with Python.
Check to see if the module has been installed by typing on the command line.
If Python complains (with some error message), that means that the module is not on
your system. Thus, you must install it before proceeding.
To do so, open the Command Prompt window and execute the command
pip install pandas_datareader
which should install the module onto your system. Make sure that you are connected
to the internet when you're doing this.
Figure 2.26
Now, back in Spyder, type the following statements into the console.
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Figure 2.27
The first statements
import pandas_datareader as web
import datetime as dt
import pandas as pd
loads the required modules into the program. datetime is a standard Python
module manipulating dates and times. pandas is a robust Python module for
handling time series. Our focus here is on the pandas_datareader module which
provides the functionality to access data found on the internet. In particular, the
web
object is loaded into memory.
After the module importation segment, parameters that will be needed for the body
of the program are defined:
startdate = dt.date(2017, 8, 1)
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enddate = dt.date.today()
engine = 'google'
s = ['GOOGL','AAPL']
x = pd.DataFrame()
Here, the start and end dates of data download are specified to be between 1/8/2017
and the current date (i.e. today). Data will be downloaded from Google’s servers –
hence the engine variable is specified with the string ‘google’. We want to download
Google and Apple stock price data. Hence their ticker symbols are stored in the list s.
The data downloaded will be stored in a Pandas DataFrame – hence the variable x is
defined.
The loop
for ticker in s:
y = web.DataReader(ticker, engine, startdate, enddate)
y = y.loc[:, ['Close']]
y = y.rename(columns={'Close': ticker})
x = pd.concat([x,y], axis = 1, join='outer')
goes into the list of tickers, and for each ticker symbol in the list, downloads the data
from Google, extracts only the ‘Close’ data column into the variable y, rename the
dataframe object y corresponding to the column that has just been extracted with
‘Close’ (otherwise, some default name will be given in y), and finally, this y is
concatenated onto x. The result is that x is a dataframe that contains the close columns
of the price data of Google and Apple over the given period.
The pandas_datareader module is a boon for the finance student who wishes to
analyze financial data. With Python, this makes efficient analysis possible. Previously,
for many years, the module that takes this place was yahoofinance. This was
discontinued after Yahoo! took down its historical data service.
4.4.2 numpy, scipy, matplotlib and pandas
The trio  numpy, scipy and matplotlib  are very important and useful Python
modules for data analysis. You may think of numpy as the primitive package that
provides the nuts and bolts for scipy to latch on, scipy contains functions much like
the Excel functions, only much more, and matplotlib is the chart and graph plotting
library. pandas is a more recent module that facilitates the handling of time series. It
is built on top of the trio.
We will use pandas to visualise the returns of Yahoo! stock price.
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Type the following into the Spyder console:
# import statements
import pandas_datareader.data as web
import datetime as dt
import pandas as pd
import matplotlib.pyplot as plt
# set parameters
startdate = dt.date(2016, 1, 1)
enddate = dt.date.today()
engine = 'google'
ticker = 'YHOO'
# download data
y = web.DataReader(ticker, engine, startdate, enddate)
# calculate and plot returns
prices = list(y['Close'])
s = pd.Series(prices)
ret = (s  s.shift(1)) / s * 100
ret.hist()
# need this to actually see the graph
plt.show()
In addition to what has been entered before, there are these additional lines:
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FIN201 STUDY UNIT 2
s = pd.Series(prices)
ret = (s  s.shift(1)) / s * 100
ret.hist()
The
import
statement imports the pandas library into Python.
The statement
s = pd.Series(prices)
stores the list of prices that was obtained earlier as a pandas timeseries object. This
facilitates manipulation as the next line shows:
ret = (s  s.shift(1)) / s * 100
What this does is to compute the daily returns from the prices.
Given the sequence of daily prices:
… , 𝑝𝑖−1 , 𝑝𝑖 , 𝑝𝑖+1 , …,
the return on the 𝑖th day is
𝑟𝑖 =
𝑝𝑖 − 𝑝𝑖−1
.
𝑝𝑖−1
Expressed as a percentage, it is 100 × 𝑟𝑖 %.
In the expression above,
s
stands for 𝑝𝑖 ,
s.shift(1)
stands for 𝑝𝑖−1 , and so
(s  s.shift(1)) / s.shift(1)
stands for
𝑝𝑖 −𝑝𝑖−1
𝑝𝑖−1
.
The mulitplicationby100 is to convert the return into a percentage.
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FIN201 STUDY UNIT 2
Finally, the statement
ret.hist()
instructs Python to draw this picture:
Figure 2.29
Sections 1 to 4 in The Python Tutorial that's found at Python HQ site.
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FIN201 STUDY UNIT 2
Watch this video on Python 100 in which I talk about the nature of the programming
language, how it may be useful to you and the steps to install it on your computer:
Starting to Use Python
(Access video via iStudyGuide)
Watch this video in which I explain the components behind Excel that you may not be
aware of as a user of its worksheets interface:
Excel Nuts and Bolts
(Access video via iStudyGuide)
As a supplement to your reading assignment for Python, you may view the video
explanations of Python by Khan Academy at YouTube.
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Quiz
1) The version of Python that is recommended for usage in this course is
a.
b.
c.
d.
2.5
2.7
3.0
3.5
2) Google's version of Excel is called
a.
b.
c.
d.
Giselle
gExcel
Google Spreadsheets
Sheets@Google
3) If you arrange some numbers in an increasing order, the number in the middle is
a.
b.
c.
d.
the mean
the median
the mode
the standard deviation
4) If X is a random variable, then Var(3X) is equal to
a.
b.
c.
d.
0
Var(X)
3Var(X)
9Var(X)
5) If the correlation coefficient of a set of data points on the xyplane is 1, then
a.
b.
c.
d.
the data points are randomly distributed
the data points lie on a straight line with positive slope
the data points lie on a straight line with negative slope
the data points lie on two lines, one with positive slope and the other with
negative slope
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6) Which of the following does not affect the value of the covariance of X and Y?
a.
b.
c.
d.
The correlation coefficient of X and Y
The variance of X
The variance of Y
The mean of X
7) The last version of Python in the 2.x series is
a.
b.
c.
d.
2.1
2.3
2.5
2.7
8) Which of the following is not a Python module for numerical analysis and
visualisation?
a.
b.
c.
d.
numpy
scipy
matplotlib
yahoofinance
9) As a finance student, which of the following categories of Excel worksheet
functions are you most unlikely to use?
a.
b.
c.
d.
Financial functions
Statistical functions
Engineering functions
Date and time functions
10) The Excel formula for calculating interest rate is PV.
a. True
b. False
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Solutions or Suggested Answers
Quiz
Question 1: b
Question 2: c
Question 3: b
Question 4: d
Question 5: b
Question 6: d
Question 7: d
Question 8: d
Question 9: c
Question 10: b
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STUDY UNIT 3
EQUITIES
FIN201 STUDY UNIT 3
Learning Outcomes
By the end of this unit, you should be able to:
1. Outline significant features of the equity market.
2. Recognise basic equity instruments.
3. Report on company characteristics through ratios and valuation models.
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Chapter 1 Market
The stock market refers collectively to the activity of stock trading. It is mainly
organised through centralised exchanges. For example, in Singapore, the Singapore
Stock Exchange (SGX) is the centralised exchange for stock trading. In the United
States, the New York Stock Exchange and the Nasdaq are the major stock exchanges.
How stocks are traded has been continually evolving since the first stock exchange
was established in Amsterdam in 1602. With internet on the scene to facilitate
communication and interaction, various nontraditional means of trade congregation
have arisen, such as electronic exchanges, dark pools, and so on. This trend can be
understood from the following perspective. Centralised and organised exchanges
possess some important features that facilitate trading. Being centralised aids in the
location of trade counterparties and helps keep prices liquid and current. Modern
stock exchanges are highly complex and organised to handle the huge volume of
trades, process the transactions and help to ameliorate credit risk. These features are
not immediately present in the OTC market. So instead of expecting new technology
to make traditional exchanges obsolete, it is more reasonable to foresee coevolution
between the various market forms on the road ahead.
Statistics on the exchanges around the world are collected by the World Federation of
Exchanges. These numbers tell us the shape of stock exchanges and hence the state of
the stock market internationally. For instance, there were 2868 listed companies on
Nasdaq, 2441 listed companies on NYSE and 771 listed companies on SGX in
November 2015. The values of share trading (measured in terms of No. Shares Traded
x Stock Price) in the same month are respectively USD 893,981.3 million, USD
1,272,753.2 million and USD 13,728.6 million for Nasdaq, NYSE and SGX.
Each stock exchange actually holds two markets  the primary market and the
secondary market. The primary market for stocks is also known as the market for
Initial Public Offerings (IPOs). After its IPO, a stock proceeds to be traded in the
secondary market.
As a publicly listed company is accountable to the public for its finances, its life in the
stock market goes through a regular cycle within a year. Four times annually, a
company has to announce its earnings. An earnings season is a period of time when
many companies release their earnings announcements. In the US, these seasons are
from early to midJanuary, April, July and October, during which a flurry of stock
market activities is always expected.
Other regular activities in the life of stocks are stock splits and rights issues.
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A split is an adjustment to the total number of shares in issue. It is an attempt to
improve the liquidity for trading since a cheaper share price allows for finer groupings
of shares in a portfolio and avails itself to be traded by more market participants. A
stock split naturally diminishes the price of each share. Hence, stock splits always
compensate existing shareholders with more shares on a prorata basis. Conversely, a
reverse split replaces old shares by fewer new ones. This may be viewed as an attempt
to increase the price of each share which is psychologically advantageous to do so if
the share price has fallen too low.
On the other hand, a rights issue occurs when a company wishes to raise new capital
by issuing new shares to existing shareholders. Related to a rights issue is a share (or
private) placement. Here, new shares are issued to external shareholders, usually
small in number (i.e. not public) and institutional in scale. In any case, the purpose is
to raise funds for the company and invariably results in the dilution of existing shares.
Thus, the move needs to be justified to shareholders.
1. These sections from the Textbook (and do the associated exercises):
Chapter 12 Introduction
Chapter 12 Stock splits and rights issues
2. The Amsterdam stock exchange is the first stock exchange in the world. Read
about it at Wikipedia.
How do people come up with the idea of a stock?
1. Visit the websites of major stock exchanges in your cognitive vicinity (e.g. SGX,
NYSE, Nasdaq). What are the types of information that you can find there?
What are the top level sections at the websites?
2. By using the statistics provided by the World Federation Exchange, plot
suitable charts to compare between the stock exchanges around the world in
terms of (a) the number of listed companies, (b) the values of share trading, and
(c) the domestic market capitalisation.
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Chapter 2 Signals and Quotes
The stock prices at which trades occur are a result of the activity of trading.
Stock prices provide traders and investors an important signal that relates to the wellbeing of a company. If a company creates a brilliant product that has a good potential
with the mass market, then the stock price rises in anticipation. If it then turns out that
the mass market does not appreciate the product even though it is brilliant, then there
is a selloff and the stock price falls.
Notice that there is separation between the state of the company in terms of its stock
price, and its state with regard to what it does fundamentally as a business. This
results in two points:
the performance of the stock price and the fundamental performance of the
company in the business may lag each other in time;
the performance or potential performance of the company in the business
world is perceived by the traders and investors before it is reflected in the stock
price. This means that there can be underreaction or overreaction in the stock
price relative to its performance in the business world.
Thus, stock price trajectories never proceed in a straight line. There is always some
volatility intrinsic to it.
The following is the trajectory of Apple's stock price (Src: Yahoo! Finance):
Figure 3.1
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FIN201 STUDY UNIT 3
Notice that it traces a path through time. This path appears pretty uneven. In finance,
this unevenness is referred to as volatility. Several things are pertinent on the
trajectory. Rises are times when investors who have invested in the stock are happy.
Falls are when they are unhappy.
But notice that a fall may then lead to a further rise. Hence, some investors regard
themselves as longterm investors as they believe that the fundamental characteristics
of the stock will eventually drive it to a "correct" high level that is commensurate to
the fundamental characteristics.
If one takes a step back and observes what goes on in the stock market, it is fair to say
that no one can absolutely predict every move that the stock price makes. There is a
high level of unpredictability. However, there is also some evidence that some
investors possess the knack of getting their investments substantially correct. Warren
Buffet is the most prominent example.
The opportunity to make money from stocks creates a lot of interest in price
movements. There is a general interest in assessing the price trajectory, attempting to
grasp what it is doing and how it is going to move next as well as communicating
about it, as journalist and commentators do.
Allow me to point out an important fact: price trajectories such as the one that you
visualise above, can be represented as a sequence of numbers, called the price data:
… , 𝑝𝑖−2 , 𝑝𝑖−1 , 𝑝𝑖 , 𝑝𝑖+1 , 𝑝𝑖+2 , ….
The sequence of price points is obtained by noting the price that is recorded at regular
time intervals.
In actuality, price is recorded whenever a trade occurs. For liquid stocks, this may
mean many price data points within a minute. The most easily available price data on
the internet are recorded at the daily frequency. For example, Yahoo! Finance data for
Apple (recorded daily) is freely available for downloading.
Price data, being a sequence of numbers, allows us to measure various aspects of it.
We list some common measures here:
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FIN201 STUDY UNIT 3
Return
The return at time 𝑖 is given by
𝑟𝑖 =
𝑝𝑖 − 𝑝𝑖−1
.
𝑝𝑖−1
If the frequency is daily, then the return is called the daily return. If it is annually, then
the return is called an annual return. Annual returns are commonly used by fund
managers to report the performance of their funds.
Continuous Return
The continuous return at time 𝑖 is given by
𝑟𝑖 = ln(
𝑝𝑖
).
𝑝𝑖−1
The continuous return is approximately equal to the return. It arises theoretically and
is used often in textbooks and academic papers.
Price Change
The price change at time 𝑖 is given by
𝛿𝑖 = 𝑝𝑖 − 𝑝𝑖−1 .
The price change is less often used than the return as a measure because price changes
occur at different scales for different stocks while price returns may be compared
uniformly across all stocks.
Volatility
The volatility of a stock is the standard deviation of the returns of the stock price taken
over some appropriate period.
Suppose I would like to assess how volatile the stock is over the past month. Then I
would collect daily price data from the past month (say of 31 days):
𝑝𝑖−30 , 𝑝𝑖−29 , … , 𝑝𝑖−1 , 𝑝𝑖 ,
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FIN201 STUDY UNIT 3
compute the daily returns:
𝑝𝑖−29 − 𝑝𝑖−30
𝑝𝑖−1 − 𝑝𝑖−2 𝑝𝑖 − 𝑝𝑖−1
,…,
,
,
𝑝𝑖−30
𝑝𝑖−2
𝑝𝑖−1
or
𝑟𝑖−29 , … , 𝑟𝑖−1 , 𝑟𝑖
for brevity, and then the standard deviation:
(𝑟
− 𝑟)2 + ⋯ + (𝑟𝑖−1 − 𝑟)2 + (𝑟𝑖 − 𝑟)2
√ 𝑖−29
.
30
As the data is daily in frequency, this expresses the daily volatility. To obtain an
annualised volatility (which is what is standardly assumed when one says "volatility"
in finance), we multiply by a factor of √252 (252 is the number of business days in a
year):
(𝑟𝑖−29 − 𝑟)2 + ⋯ + (𝑟𝑖−1 − 𝑟)2 + (𝑟𝑖 − 𝑟)2
𝜎31 = √252 × √
.
30
I have written the 31day (annualised) volatility as 𝜎31 . We may also compute an 𝑛day (annualised) volatility 𝜎𝑛 for any 𝑛, depending on the period of time that we're
interested in.
Stock prices at present are quoted up to cents. Thus we see that Apple's stock price is
quoted at 99.52 or 97.39. The broker will quote the price of the stock in the form of a
bid and ask, such as 99.5199.53. The lower price 99.51 is the bid price or the price at
which the broker will buy from the investor. The higher price 99.53 is the ask price or
the sell price at which the broker will sell to the investor. The price 99.52, being right
in between, is called the midmarket price. The difference of 99.53  99.51 = 0.02 or 2
cents, is called the spread. It represents the profit that the broker makes per share for
every buysell round trip.
In order to maintain an orderly market, stock exchanges have trading halts on extreme
price moves and volatility. For example, Rule 80B of NYSE stipulates in detail the
conditions under which trading will be halted if the market becomes too volatile. This
implies that within a day, the stock market index is guaranteed not to drop beyond a
certain level. This puts investors psychologically at ease.
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FIN201 STUDY UNIT 3
Rule 80B of NYSE on trading halts due to extraordinary market volatility.
1. Find out what the shape of the distribution of Apple's daily return look like.
2. Find the date(s) on which Apple's 31day volatility is the highest in its history.
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Chapter 3 Instruments
3.1 Stocks and Indices
The basic instruments in the stock market are the stocks themselves.
Stocks can be managed in a collection or portfolio. Mutual funds do this as a business.
A stock index is a weighted average of the prices of a basket of stocks. For example,
the Dow Jones Industrial Average (DJIA) is computed as the average of 30 stock prices.
The S&P 500 is a weighted average of the stock prices of 500 stocks in the US market
and it is the most prominent stock market signal from the US. The weighting for DJIA
is the simple one (all weights equal 1) while the weighting for S&P 500 is the number
of tradable shares of a stock in the market. Thus, the S&P 500 index is said to be
weighted by market capitalisation.
A formula for an index weighted by market capitalisation may be written this way:
Index =
∑(number of shares issued x price of share)
.
divisor
3.2 Single Stock Futures
Single stock futures are futures contract whose underlying asset is a single stock.
When two parties enter into a single stock futures contract, the following happens:
the contract is entered into on a start date, call it time 0
party A agrees to buy a single stock from party B at a price 𝐾 that is determined
at time 0
at time 𝑇, called the maturity, party A buys the stock from party B at price 𝐾
In the above scenario, A is said to be in the long position while B is said to be in the
short position. The price 𝐾 is called the strike price.
For example, suppose A and B enter into a futures contract to trade AAPL stock 1
month from now at $100. Suppose that at maturity, the stock price of AAPL is $110.
Then A will buy the stock from B at $100. This is seen as a gain for A, since he can
immediately sell the stock in the market to make $10. Conversely, if the stock price of
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FIN201 STUDY UNIT 3
AAPL is $90 at maturity, then it is seen as a loss for A, since he can otherwise obtain
the stock from the market at $90. Thus, he makes a loss of $10.
You would enter into a short position in a single stock futures for one of two reasons:
because you think that the stock price will fall by maturity (speculation)
because you want to protect the value of the stock that you have already
owned (hedging)
The reasons for the long position are similar (i.e. speculation or hedging).
If a futures contract is traded at an exchange, the listed price is the strike price.
There is however, another notion of price associated to a futures contract  its
theoretical price, or fair price, or theoretical stock futures price. This must be
distinguished from its strike price.
At the exchange, a futures contract is entered into without any initial charge. Its fair
price is 0. However, its strike price is never 0.
The strike price is empirically determined by trading activities in the market.
The theoretical price is derived mathematically from assumptions. It is derived from
the perspective of one who is trying to hedge the futures contract by trading the
underlying stock.
The theoretical stock futures price is given by the following formula:
(current share price − PV of any dividends) × (1 + 𝑖 × 𝛼),
where 𝛼 is the maturity of the contract expressed as a fraction of the year (1 year = 1
unit by convention in finance), 𝑖 is the shortterm interest rate and PV stands for
"PresentValue".
3.3 Stock Index Futures
A stock index futures is a futures contract whose underlying is a stock index. It is
similar in concept and purpose to a single stock futures, except that you will use it to
speculate or hedge the marketwide movements instead of a single stock.
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One difference is that while the short party in a single stock futures contract may
deliver the single stock asset at maturity, the short party in an index futures contract
must deliver cash (i.e. the futures contract is cashsettled) as it is impossible to deliver
all the stocks in an index.
There is similarly a notion of theoretical stock index futures price or the fair price. And
again, this must be distinguished from the contract's strike price.
These sections from the Textbook (and do the associated exercises):
Chapter 12 Stock indices
Chapter 12 Single stock futures
Chapter 12 Stock index futures
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Chapter 4 Calculations
4.1 Ratios
Ratios are fractions computed from the characteristics of a company and its stock. It is
commonly used in fundamental analysis to compare between stocks so as to select a
more attractive one for investment.
Ratios do not have an absolute unit, unlike, for example, the metric system  everyone
knows the meaning of 1 cm. Ratios only make sense in comparative terms  when they
are compared against themselves over time, or when they are compared between
different companies.
Some common ratios are:
Return on Equity (ROE)
ROE =
earnings
book value
This indicates the return that the management is achieving with the capital that is
available in the company.
Dividend Yield
Dividend yield =
dividend per share
share price
This indicates the dividend return that is received on the capital that is invested in the
share.
Earnings Yield and Price/Earnings Ratio (PE ratio)
Earnings yield =
PE ratio =
earnings per share
share price
share price
earnings per share
These two ratios are reciprocal to each other.
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The earnings yield is the return that is achieved for the capital that is invested in the
share.
The PE ratio is the opposite of this: it is the price that is paid in order to achieve a given
earnings by the company.
Price/Book Ratio (PB ratio)
PB ratio =
share price
book value per share
This ratio indicates the value of the share as given by its share price over its book value.
Earnings Per Share (EPS)
EPS =
earnings
number of shares issued
This ratio indicates literally what its name suggests.
Dividend Cover
Dividend cover =
earnings
total dividend payout
This measures how able the company is in paying its dividends.
Projected Earnings Growth (PEG ratio)
PEG =
PE ratio
forecast earnings growth rate x 100
This ratio is used in the following manner: if 2 companies have the same PE ratio but
one has a lower PEG ratio, it means that its forecast earnings growth rate is higher.
Hence it's a better buy.
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4.2 Valuation
In stock valuation, the value of a stock is derived theoretically, and not from the price
that is traded in the market. Thus, it is important when trades in shares of private
(nonlisted) companies occur.
Two theoretical models are considered here. They both derive a relationship between
the value (i.e. theoretical price) of the stock and its yield (from future cash flows).
4.2.1 The Gordon Model
The Gordon model assumes that owning a stock provides an infinite series of cash
flows due to dividends which grow at a constant rate 𝑔 every year. It then relates the
value of the stock with the yield from the cash flow series.
Suppose currently it is time 0.
At time 1, a dividend of amount 𝑑(1 + 𝑔) is received.
At time 2, a dividend of amount 𝑑(1 + 𝑔)2 is received.
At time 𝑖, a dividend of amount 𝑑(1 + 𝑔)𝑖 is received.
And so on.
Suppose a rate of return or yield is expected on the cash flows. This may be found by
applying the Capital Asset Pricing Model (refer to the Concept section).
The theoretical price of the stock is given by (using the formula for geometric
progressions)
𝑃=
𝑑(1 + 𝑔) 𝑑(1 + 𝑔)2
𝑑(1 + 𝑔)𝑖
+
+
⋯
+
+⋯
1+𝑦
(1 + 𝑦)2
(1 + 𝑦)𝑖
=
𝑑(1 + 𝑔)
1
×
1+𝑔
1+𝑦
1−1+𝑦
=
𝑑(1 + 𝑔)
𝑦−𝑔
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The formula can also be inverted to express the yield in terms of the price:
𝑦=
𝑑(1 + 𝑔)
+ 𝑔.
𝑃
4.2.2 The T Model
The T Model posits that the yield of a stock is given by the following formula:
yield =
dividend + excess cash + price increase
.
share price
The T model's yield is computed based on cash flows that occur over a year. This is in
contrast with the case of the Gordon Model, whose yield is derived from cash flows
that occur indefinitely.
These sections from the Textbook (and do the associated exercises):
Chapter 12 Ratios
Chapter 12 Valuation
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Chapter 5 Concept: The Capital Asset Pricing Model
In the academic finance or theoretical finance, there is an effort to explain the
significance of a stock price. This effort consists of a search for underlying factors that
drive price changes.
Such a wellknown model is the Capital Asset Pricing Model (or CAPM). It posits that
𝑟 − 𝑟𝑓 = 𝛽(𝑟𝑚 − 𝑟𝑓 ),
where
• 𝑟 is the expected return of the stock
• 𝑟𝑓 is the riskfree rate in the market
• 𝑟𝑚 is the expected return of the market
• 𝛽 is a sensitivity measure of the stock return with respect to the market return
A theoretical argument is used to establish the model. The model expresses a
relationship between the market return and the return of an individual stock in the
form of a linear equation.
In more details, the meaning of the model is as follows. The lefthandside is the excess
return of a company stock over the riskfree rate. The righthand side is the product
of beta, the sensitivity measure, and the excess return of the market over the riskfree
rate. The equation tells us that, in order to find the excess return of a company stock
over the riskfree rate, one only needs to multiply beta to the excess return of the
market over the riskfree rate. In other words, information about the expected return
of the stock is encoded in the factor beta.
To apply the model to real world data, the riskfree rate may be read from the money
market, while the expected return of the market is commonly measured as the
historical return on a selected market index. The market index is taken as a proxy for
the entire market.
Theoreticians devise sophisticated methods to measure beta from market data. We'll
not be concerned with such technical issues here. Instead, let's see how the model is
used in some theoretical calculations.
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Example
Assume that the expected return on the market is 14%, risk free rate is 6% and beta of a security
is 1.2. Calculate the expected return on the security.
Since
𝐸(𝑅) = 𝑟𝑓 + 𝛽(𝐸(𝑅𝑚 ) − 𝑟𝑓 ) = 6% + 1.2(14% − 6%),
where 𝐸(𝑅) denotes the expected return of the security return, and 𝐸(𝑅𝑚 ) denotes the
market return. Thus,
𝐸(𝑅𝑖 ) = 15.6%.
Example
Consider the following project that requires an investment of $3 million. The cash flows are
estimated as follows:
Year Cash Flow
1
700,000
2
900,000
3
1,500,000
4
1,800,000
If 𝑟𝑓 = 6%, 𝛽 = 1.3, 𝑟𝑚 = 14%, should the project be undertaken? (Apply the Capital
Asset Pricing Model to answer the question.)
The project will be undertaken if the NPV of the project is positive,
i.e. NPV = PV of future cash inflows – Initial investment > 0.
The PV of future cash flows are calculated as follows.
First, we estimate the opportunity cost or the cost of capital from the CAPM. This is
none other than the term 𝑟 in the CAPM equation.
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(Digression: When interpreted in terms of stocks, it is the expected stock return. This
interpretation and the one that is required for this question are consistent in this
manner: the expected return of a stock is the rate at which money invested grows.
Thus, the term 𝑟 in the CAPM model is the rate of growth of cash that is associated to
an asset or a project.)
We have
Opportunity cost = 𝑟𝑓 + 𝛽(𝑟𝑚 − 𝑟𝑓 ) = 0.06 + 1.3(0.14 − 0.06) = 16.4%.
Thus,
𝑁𝑃𝑉 =
700,000 900,000 1,500,000 1,800,000
+
+
+
− 3,000,000 = 197,275.
1.164
1.1642
1.1643
1.1644
Since NPV > 0, the project should be undertaken.
Example
Assume that a company has recently paid a dividend of $2. The dividend is expected to grow
at a constant rate of 6%. The riskfree rate is 4%, beta is 1.2 and the expected market return is
10%.
What is the value of the stock?
The plan is this: first we'll derive the yield or expected rate of return, from the CAPM.
Then we will apply the Gordon Model to price the stock.
By the CAPM,
𝑟 = 0.04 + 1.2 × (0.10– 0.04) = 11.2%.
By the Gordon Model,
𝑃=
2(1 + 0.06)
= $40.77.
0.112– 0.06
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FIN201 STUDY UNIT 3
Chapter 6 Analyses and Strategies
The workflow in trading or investment proceeds like this:
1. Analysis
2. Make a choice and enter into a trading/investment position
3. Exit the position
It is necessary to analyse carefully the choices that one may have before entering into
a trading or investment position as it involves a real chance of losing money.
As so many things are going on in the market, different styles of analysis have
developed to focus on different aspects. Analysis of stocks is commonly classified as
either fundamental or technical.
In fundamental analysis, a collection of companies are pooled together and compared
against each other with respect to certain distinguished features. Some of these
features are quantitative in nature, such as the stock price itself, its returns, the PE
ratio or the EPS. Other features may be qualitative in nature, such as the perceived
quality of the executives, the business the company undertakes or the economy at
large. The purpose of the analysis, is to make a choice for the most attractive stock in
its potential to gain in price in the future.
In technical analysis, what's analysed is price trajectories themselves. Certain patterns
that occur along the price trajectories are deemed as signals that are indicative of the
impending price movement. The purpose of the analysis, is to make suitable choice of
an entry point, i.e. a time at which the trader presses the buy or the sell button and
takes on a long or a short position in the stock.
Apart from fundamental analysis and technical analysis, there is also something
known as quantitative analysis. This makes use of statistics as a basis for decision
making. For example, one may think that within a year, there are regularly
distinguished periods of time, such as the 4 seasons, Christmas buying period,
earnings seasons, and so on. And during each season, a certain category of stocks
tends to rise appreciably in price. So an attempt may be made to formulate a strategy
that selects stocks based on category per season on an annual basis. The law of large
numbers in statistics helps to ensure that the decisions are correct most of the time.
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It's illuminative to put the strategies sidebyside for comparison:
Analysis
Features
Choices
Fundamental
Characteristics
Best company
of companies
from a collection
Patterns in price
Time of entry
Technical
trajectories
Quantitative
Statistical
Categories of
regularities
stocks, periods of
times, associated
with the
regularities
Watch this video on the basic steps of how to use Thomson Reuter’s Eikon system to
carry out fundamental analysis research:
Fundamental Analysis on Eikon
(Access video via iStudyGuide)
Watch this video in which I explain the distinction between stock price and stock value:
Stock Price vs Valuation
(Access video via iStudyGuide)
This is a marketing video for an automatic calendarbased trading system called
“Absolute Profits”. View the video (6:00 – 27:00) and pay attention to the nature of the
trading strategy that is purportedly behind the system. In other words, what
information is the strategy picking out from the market and how does it use this
information to trade?
(As a finance student, you should be as neutral as possible towards the marketing
stance of the video. You should try to appreciate what is being articulated in terms of
information and decisions.)
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Quiz
1)
The Amsterdam Stock Exchange was started in
a.
b.
c.
d.
2)
Opportunity cost is also known as
a.
b.
c.
d.
3)
NYSE
Nasdaq
Shanghai Stock Exchange
Korea Exchange
There are about _________ listed companies on the SGX.
a.
b.
c.
d.
5)
chance cost
cost of capital
lucky price
cost of opportunity
The largest stock exchange in the world (2016) is
a.
b.
c.
d.
4)
1602
1702
1802
1902
250
750
1250
1750
Which of the following does not represent a bet on the economy?
a.
b.
c.
d.
A position in an index option
A position in an index futures
Holding a share of an S&P 500 ETF
Holding a share of Apple stock
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6)
To verify whether or not the executives of a company are doing their best with
available capital or not, an analyst will look at
a.
b.
c.
d.
7)
ROE
dividend yield
PE ratio
PB ratio
The application of the Gordon model in the valuation of a publicly listed company
will produce a price that is equal to the stock price.
a. True
b. False
8)
If you are interested to speculate on Apple, the most expensive option is
a.
b.
c.
d.
9)
to buy Apple option
to long Apple futures contract
to buy Apple stock
to buy an iPhone
The index of Singapore's stock market is the
a. FTSE Straits Times Index
b. MSCI Straits Times Index
10) Stock price generally drops right after dividend payout day.
a. True
b. False
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Solutions or Suggested Answers
Quiz
Question 1: a
Question 2: b
Question 3: a
Question 4: b
Question 5: d
Question 6: a
Question 7: b
Question 8: c
Question 9: a
Question 10: a
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STUDY UNIT 4
FIXED INCOME
FIN201 STUDY UNIT 4
Learning Outcomes
By the end of this unit, you should be able to:
1. Describe significant features of the fixed income market.
2. Calculate with quotes of various fixed income instruments.
3. Describe significant types of price information.
4. Apply discounting methods to the pricing of basic fixed income instruments.
5. Compute with prices and rates.
6. Describe significant features of the term structure.
7. Apply bootstrapping to construct the yield curve.
8. Compute with yield curves.
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FIN201 STUDY UNIT 4
Chapter 1 Market
The fixed income market is conventionally categorised into:
money market (trade in shortterm instruments, i.e. 1 year or less in maturity)
bond market (trade in longterm instruments, i.e. more than 1 year in maturity)
In the money market, forward rate agreements (FRAs) and interest rate futures (IRFs)
are the major instruments. FRAs are found in the OTC market while IRFs are traded
on exchanges. Bonds and their derivatives are the major instruments in the longterm
interest rate market.
Interest rates signals arise from the fixed income market. When we piece together an
interest rate for each term, across all terms, we obtain a yield curve, which is a state of
the fixed income market at a given time.
The fixed income market is the market for borrowing and lending. In a borrowerlender relationship, we say that the borrower obtains credit from the lender. Hence
the market is also called the market for credit. Another name is the debt market. This
highlights the role of debt as one of two major financing means for companies. "Fixed
income" refers to the series of cash flows associated with such investments.
Buying a fixed income instrument is equivalent to lending money, while selling it is
equivalent to borrowing money. The consequence of crystallising the borrowerlender
relationship into an instrument allows it to be traded in what is called the secondary
market. The primary market precedes the secondary market. It is where a fixed income
instrument is first issued. All subsequent trades occur in the secondary market. The
primary market is generally structured differently from the secondary market. For
instance, the first issuance of the US Treasury debt instruments occurred in an auction
market that involves only big market players like the major banks.
The auction schedule for US Treasury securities is published regularly at its website
(https://www.treasurydirect.gov/instit/annceresult/annceresult.htm). It gives a sense
of the types of instruments auctioned, the meaning of settlement date and the great
number of debts that is issued by the government.
A crucial measure of the secondary market is liquidity. It is the single reason for the
existence of the market  the ability for buyers and sellers to locate each other easily.
Prior to its maturity, an instrument can change hands any number of times. The ease
of trading facilitates the growth of the secondary market and the existence of the
secondary market in turn improves the ease of trading.
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Borrowing and lending has a very long history in human civilisation. It preceded all
other forms of financial transactions. Clay tablets from Babylonia dated back to 3000
BCE have been excavated and found to be inscribed with loan contractual obligations.
In contrast, the first stock traded was that of the Dutch East Indies Company which
was founded in 1602, King Alyattes of Lydia (located in modern day Turkey) first
minted coins from electrum around 600 BCE, and the earliest paper currency has been
traced back to the Tang dynasty in China around 800 CE. These snapshots put into
perspective how the major features of today's financial landscape have evolved into
its present form.
These sections from the Textbook (and do the associated exercises):
Chapter 3 Overview
Chapter 6 Overview of capital market instruments
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FIN201 STUDY UNIT 4
Chapter 2 Signals and Quotes
There are several standard ways that interest rate signals are commonly classified into:
Long term vs short term
–
shortterm rates come from the money market
–
longterm rates are computed from yields in the capital market
Benchmark vs nonbenchmark
–
benchmark shortterm rates are averages of rates bidded/offered by top
banks
–
benchmark longterm rates are yields of riskfree government bonds
Geographical
–
due to global linkages, there is a mix of dependence and independence
among international rates
Instrument type
–
e.g. shortterm loans, bonds, mortgages and other collateralised loans
Prices in the interest rate and bond derivative markets
Benchmark rates (both short and long terms) give us the term structure  a barometer
of the fixed income market. Let's take a look at them.
2.1 ShortTerm Rates
"LIBOR" stands for "London Interbank Offered Rate". It is an average of the interest
rates offered (i.e. ask rate or selling rate) by top banks to one another in London for
shortterm loans (o/n, 1w, 2w, 1m, 2m, 3m, 4m, 5m, 6m, 7m, 8m, 9m, 10m, 11m, 12m).
It is calculated by the British Bankers' Association (BBA) daily.
The LIBOR is a benchmark shortterm interest rate for the money market. Other
interest rates are often expressed in its terms (e.g. a rate is LIBOR + 3 b.p.) and
derivative contracts make reference to it as underlying.
One typically finds a LIBOR for each international currency, e.g. an oftmentioned
LIBOR is the USD LIBOR.
In the US, the Fed Funds rate is keenly watched as it is a key instrument used by the
central bank to intervene in the money market.
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FIN201 STUDY UNIT 4
Figure 4.1
Significant market events are reflected on the trajectory of the Fed Funds rate. The
peak was a deliberate policy by the thenchairman Paul Volcker as he raised rates to
put an end to stagflation (stagnation plus inflation) that developed in the US over the
1970s. The interest rate environment is the lowest for decades currently (2015) as
reflected by the rightend of the curve. The Fed has been lowering rates over the years
to help boost the economy.
2.2 LongTerm Rates
Longterm benchmark rates are found from the yields of government bonds. In the
US, these are referred to as treasury instruments and are important for several reasons:
they are generally regarded as of the lowest risk, hence yields derived from them
are taken as benchmark yields, relative to which other bonds are priced (e.g.
quotes of bonds are often expressed in terms of a credit spread over a base yield)
government borrowing may be regarded as bedrock and driver of the debt
market in general
government borrowing impacts the economy at large
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FIN201 STUDY UNIT 4
Investors of these debts are generally large institutions, such as banks and pension
funds. Pension funds, which have longterm liabilities, understandably prefer to lock
rates over the long term, and hence are investors of bonds.
2.3 The Term Structure or Yield Curve
When longterm yields and shortterm rates at a given instant are pieced together
across terms (i.e. terms of maturity of the associated loans), the yield curve (aka term
structure) of the fixed income market is obtained
The yield curve allows us to assess the cost of borrowing in the market at the
benchmark level. Actual borrowing is derived by adding an estimated credit spread
to the benchmark rate (e.g. benchmark + X basis points).
The term structure (aka yield curve) is commonly displayed as a curve, with the
horizontal axis representing time into the future, and levels on the curve representing
general borrowing cost (in terms of interest rate) for a loan that lasts the length of time:
Figure 4.2
The common representation depicts the term structure as an upward sloping curve,
which means that the longer the term, the higher the borrowing costs. The economists
have provided reasons for this "selfevident truth" (as it seems intuitively clear that
the lender would naturally charge more for a longer loan period), enshrining them
into 3 "theories"  liquidity preference theory, market segmentation theory, pure
expectation theory. Theory aside, it is not wholly appropriate to think that the market
would behave in a certain way because a rational lender would behave in the same
way, for the market is not made up of a single lender.
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FIN201 STUDY UNIT 4
The following chart shows the term structure of GBP on 9/2/2005 (ref: Wikipedia
https://en.wikipedia.org/wiki/Yield_curve):
Figure 4.3
It is quite unlike the upward sloping model that is depicted above.
Thus, remember this: the common way to think about a term structure is that it's an
upward sloping curve which indicates that the longer the term the higher the
borrowing cost; however, real term structure displays more complexities than that.
The term structure may be divided into 2 halves. The lefthalf (short end), to the left
of the 1year mark, is made up of data points from the money market. The righthalf
(long end) consists of yields that are extracted from bonds from the capital market.
The money market thus accounts for the short end of the yield curve.
The shape of the term structure changes over time as the market evolves and the prices
and rates of fixed income instruments change. It is important to be aware that there
are 2 dimensions of time variations in the study of interest rates: 1) from the present
until time 𝑡 into the future which is called the "term"; 2) the change of a certain
quantity as it evolves for time. Thus, one may talk about the 3month interest rate as
it is now, or as it was a while ago, or as it may be some time into the future. And
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FIN201 STUDY UNIT 4
similarly, we may think about the 5year yield as it may evolve in the future as we
plan our longterm investments.
There is not one standard term structure. Anybody can construct one for himself using
data collected from the market. However, when this is done, it is usually based on
benchmark interest rates and yields such as those from the LIBOR and the US treasury
instruments. Other rates and yields in the market lie close to these rates as competitive
pricing action in the market keeps rates reasonably together.
Attempting to construct the yield curve by hand would give us data points at a
discrete set of terms: 1 day, 1wk, 1mth, 2mths, 3mths, 6mths, 12mths; 2y, 3y, 5y, 7y,
10y, 20y, 30y. These data points are then stitched together by linear interpolation to
produce a yield curve. The curve provides for an easier reading than the data points.
It also provides interpolation and extrapolation of rates and yields to terms that are
not found in the benchmark instruments used to plot it.
Currently (Jan 2016), the term structure (for AAA rated bonds and all bonds) in the
Eurozone looks like this (ref: ECB
https://www.ecb.europa.eu/stats/money/yc/html/index.en.html):
Figure 4.4
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FIN201 STUDY UNIT 4
Notice that the short end has dipped below the zero level, while the entire curve
exhibits the standard upward sloping model behaviour. The gap between the two
curves represents credit spread  there are nonAAA rated bonds in the collection of
instruments that are used to plot the yield curve, and these give higher yields than the
AAArated one.
We will go through how a term structure is put together starting from rates and yields
that are obtainable from instruments that are traded in the market in the next chapter.
Plot the yield curve for the US fixed income market using data that is found at the
Federal Reserve website.
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FIN201 STUDY UNIT 4
Chapter 3 Instruments
We will be mainly concerned with these instruments:
loans
forwardforwards
forward rate agreements (FRAs)
bonds
floating rate notes (FRNs)
3.1 Loans
We will consider simple loans for illustrating calculations in simple settings.
Real world loans can be complicated, with the possible additions of various clauses
(e.g. change of terms, collaterals, etc.) in order to make the loan attractive or to secure
the loan for the creditor in case the debtor defaults.
Loans are repaid with interests whose levels are mediated by the prevailing interest
rates. The current prevailing interest rates are known as spot rates.
3.2 ForwardForwards
A forward interest rate is an interest rate that is posted at present but pertains to a loan
that will be carried out for a period of time in the future.
This is illustrated by the following diagram:
Figure 4.5
A forward interest rate allows a borrower to hedge risk by fixing an interest rate which
he finds reasonable at present for a loan which he intends to take out some time in the
future.
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FIN201 STUDY UNIT 4
Such a loan is known as a forward loan or a forwardforward (because the start and
end dates occur in the future).
3.3 Forward Rate Agreements (FRAs)
A forward rate agreement (FRA) is a contract that binds two parties over the future
outcome of a reference interest rate. The reference interest rate is usually some money
market benchmark rate such as the LIBOR. The contract is found in the OTC market
(i.e. not traded at centralised exchanges like stocks are).
A brief description of how it works is this. There are two parties A and B who are
engaged in an FRA contract. The FRA contract specifies the following items:
a floating rate 𝑓
a fixed rate 𝑟
a notional amount 𝑁
maturity of the contract 𝑠
length of loan period referred to by the contract 𝑡
At the end of the loan period 𝑠 + 𝑡, one party, say A, pays
𝑓×𝑡×𝑁
to his counterparty, and his counterparty, B, pays
𝑟×𝑡×𝑁
to A.
Note that the floating rate is unknown at the start of the contract. It is only known at
time 𝑠, at which the floating rate is said to be fixed.
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Diagrammatically, the exchange may be visualised this way:
Figure 4.6
Let's try to understand this.
A and B are in opposite positions with respect to their views about the floating interest
rate which will apply to the future loan period between the times 𝑠 and 𝑠 + 𝑡.
Figure 4.7
At the fixed rate 𝑟, the loan interest over the period is 𝑟𝑡𝑁. At the floating rate 𝑓 , the
loan interest will be 𝑓𝑡𝑁. However, the value of 𝑓 is unknown initially (at time 0); only
the fixed rate is known.
If 𝑓 turns out higher than 𝑟, then the floating rated loan will generate a bigger interest
amount than the fixed rated interest, and vice versa.
Hence, an FRA is a bet of which of the two loans  fixed rated or floating rated  is
higher priced at time 𝑠.
B is said to be in the long position of the contract (pays fixed and receives floating)
and A is said to be in the short position. The long party stands to gain from an
increased floating rate, while the short party stands to gain from a decreased floating
rate.
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3.3.1 Alternative Definition
We may combine the two cash flows in the description into one this way: at loanend
time 𝑠 + 𝑡, the following amount needs to be paid by the long party to the short party:
(𝑓 − 𝑟) × 𝑡 × 𝑁.
Thus, if 𝑓 > 𝑟 at maturity, B needs to pay this amount to A. And conversely, if 𝑓 < 𝑟 ,
A needs to pay the negative of this amount to B.
3.4 Bonds
Bonds are the key instruments of capital markets which deal with longterm interest
rate risk.
In order to understand bonds, we must appreciate the following key dimensions:
Issuers
Primary and secondary markets
Credit quality
Types
Pricing
Issuers of bonds may be classified into several large groups:
governments (sovereign bonds)
municipalities (states, cities, towns, etc.)
corporates (corporate bonds)
Bonds are commonly issued in auctions. Then they are processed, stripped or resold
in secondary markets.
The nature of the issuer affects the bond in the sense of credit quality. Credit is a
measure of the likelihood of default. If credit is good, lenders are willing to lend to the
borrower without further surcharge. Conversely speaking, if the issuer has a good
credit quality, he may expect to borrow relatively cheaply from the market.
Three major agencies rate the credit quality of firms and institutions: Standard &
Poor's, Moody's and Fitch.
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S&P's credit rating scale is ranged from AAA to D and NR (notrated) (Ref:
https://www.standardandpoors.com/en_US/web/guest/article//view/sourceId/504352).
As of Jan 2016, the S&P's rating of longterm debt for some entities are:  Singapore
and the United States are countries with the AAA rating  Only a small handful of US
companies are with the AAA rating (Microsoft, ExxonMobil, Johnson & Johnson) Japan is rated A+ for its weak economic outlook  Banks in general do not have the
strongest of credit ratings due to the nature of their business. DBS is rated AA.
Goldman Sachs Group Inc. is rated BBB+.
A bond basically promises a series of cash flow payments in the form of coupons and
a final principal payout at maturity.
Many variations exist because these cash flows can vary and be made dependent on
market factors.
Some important classes of bonds or bondlike instruments are:
Treasury bonds
Corporate bonds
Zerocoupon bonds
Assetbacked securities
Floating rate notes
We will focus on treasury bonds and zero coupon bonds.
3.4.1 Treasury Bonds
The US Treasury issues 3 kinds of debts:
Tbills (4w, 3m, 6m, 1y)
Tnotes (2y, 3y, 5y, 7y, 10y)
Tbonds (20y, 30y)
For bonds and notes (longterm instruments), coupons are issued at semiannual
frequency. The day count convention for them is ACT/ACT. For bills, the day count
convention is ACT/360.
3.4.2 Zero Coupon Bonds (ZCBs)
Zerocoupon bonds (ZCBs) are the simplest bonds conceptually, for there is only 1
cash flow at its maturity.
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FIN201 STUDY UNIT 4
Figure 4.8
Suppose a ZCB has a maturity of 𝑇 and a face value of $100, i.e. the holder of such a
bond will expect to receive $100 at maturity.
Suppose the market price of this bond is 𝑃.
Then we may hypothesise a yield 𝑦 such that
𝑃=
100
.
(1 + 𝑦)𝑇
The point to note here is that, in the absence of coupons, the relationship between price
and yield is very simple.
Given a spectrum of ZCBs of varying maturities, we would be able to compute yield
numbers 𝑦𝑇1 , 𝑦𝑇2 , … , 𝑦𝑇𝑛 . These numbers together form a yield curve.
The yield curve gives a snapshot of the interest rate environment in terms of cost of
borrowing for various period lengths.
Thus, the usefulness of ZCBs is that a collection of them of various maturities provides
us with a hint of what the interest rate environment is like in terms of the cost of debt.
3.5 Floating Rate Notes
In contrast to treasury bonds whose coupons are fixed rated, floating rate notes
promise a series of cash flows whose amounts are dependent on some benchmark
market rate.
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In fact, a shortterm deposit that is rolled over periodically can be interpreted as a
floating rate note. As it rolls over to the next period of deposit, the interest rate is
marked to the latest market level. Effectively, the rolling deposit receives interest
payouts that depend on the market interest rate level as well as the final principal
repayment.
Floating rate notes may be diagrammatically represented by the following series of
cash flows, where the squiggly payouts refer to amounts that are not determined until
some time in the future:
Figure 4.9
These sections from the Textbook (and do the associated exercises):
Chapter 3 Day / year conventions
Chapter 3 Money market instruments
Appendix 2 A summary of market day / year conventions and government
bond markets
Chapter 4 Forwardforwards, FRAs and futures
Chapter 6 Features and variations
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Chapter 4 Calculations
4.1 Day / Year Conventions
"Day / Year" here refers to a fraction, namely the day count fraction (DCF) that appears
in the computation of interest:
Interest = Interest Rate x Day Count Fraction
Interest is paid on a loan that is held over a fraction of the year  this is what the
formula tells us. The simple concept is made complicated by the actual form that the
fraction takes.
How the fraction is computed depends on the convention that is assumed in the
calculation. This convention is called the day count convention.
Some common conventions are listed here:
ACT/ACT
ACT/365
ACT/360
30/360
"ACT" means "actual number of days" and it implies that the actual number of days
that are found in the loan period is to be counted.
ACT/365 is said to be on a 365day basis, while ACT/360 is said to be on a 360basis.
Most money markets assume the ACT/360 convention which is thus commonly called
the money market basis. Exceptions exist, the common of which is ACT/365.
Consider the following example to see how these different conventions work out in
calculations:
Example
Compute the interest on $100 based on a rate of 5% over a period of 62 days that belong to a
single year, assuming each of the following conventions:
1.
2.
3.
4.
ACT/ACT
ACT/365
ACT/360
30/360
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For ACT/ACT, we need to know whether or not the year is a leap year.
Suppose it is a leap year, then the DCF is 62/366 and the interest is
62
100 × 0.05 × 366 ≈ 0.847.
For ACT/365, the interest is 100 × 0.05 × 365 ≈ 0.849.
For ACT/360, the interest is 100 × 0.05 × 360 ≈ 0.861.
62
62
For 60/360, the DCF is 60/360 as the 62period spans 2 months which is
60
rounded to 2 × 30 for this convention. The interest is 100 × 0.05 × 360 ≈ 0.833.
4.2 Simple and Effective (or Equivalent) Rates on Different
Bases
Suppose it is given that the principal is $100, the interest is $3 and the period of loan
is 180 days. What is the interest rate that is implied?
The answer to this question depends on assumption made on the interest rate and the
DCF.
If 𝑟 is regarded as a simple rate on the 365basis (which by default, means ACT/365),
then we will have
(1 + 𝑟 ×
180
) × 100 = 103.
365
If 𝑟 is regarded as a simple rate on the 360basis (which by default, means ACT/360),
then we will have
(1 + 𝑟 ×
180
) × 100 = 103.
360
If 𝑟 is regarded as an effective or equivalent (or annual equivalent) rate on the 365basis (which by default, means ACT/365), then we will have
180
(1 + 𝑟)365 × 100 = 103.
Thus, simple rates involve multiplying DCF to the rate while effective (or equivalent
or annual equivalent rates) involve exponentiating the factor (1 + 𝑟) by the DCF.
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4.3 Certificate of Deposit (CD)
A certificate of deposit is like a time deposit only that it is
transferrable/tradeable/negotiable. When a depositor deposits at a bank by means of
a CD, he in fact purchases the CD from the bank. There is a liquid CD market for him
to then trade away (i.e. liquidate) his investment if he so desires to not hold it till
maturity.
A CD generally pays interest in the form of coupons.
As it is not a discount instrument, when a CD is first issued, it is usually issued at par,
i.e. at a price that is equal to its face value.
The net amount received at maturity (coupons being compounded forward in time) is
called the maturity proceeds.
After issuance, its fair price may be computed as the NPV of the cash flows that come
from owning it.
Let's consider the simplest case of a CD that issues a coupon and matures at time 𝑇.
The current time is assumed to be 0.
Figure 4.10
Let the face value be denoted by 𝐹 and the coupon rate be denoted by 𝑟.
Then the maturity proceeds is
𝐹 × (1 + 𝑇 × 𝑟).
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The NPV of the CD is obtained by discounting the maturity proceeds using a suitable
yield. If the price is given, then this yield is implied. If the price is unknown and needs
to be calculated, then this yield must be read from the market.
In any case, the important thing to remember is that yield of an instrument is as
informative as the price as one determines another.
Let 𝑦 be the yield of the CD and 𝑃 be its price. Then
𝑃=
𝐹(1 + 𝑟𝑇)
.
(1 + 𝑦𝑇)
More generally, let 𝑃(𝑡) be the price of the CD at time 𝑡. Then the price of the CD in
terms of its yield is
𝑃(𝑡) =
𝐹(1 + 𝑟𝑇)
.
(1 + 𝑦(𝑇 − 𝑡))
Figure 4.11`
From this formula, we may compute the return from holding the CD.
Suppose the CD is bought at time 𝑡1 and sold at time 𝑡2 , where 𝑡1 < 𝑡2.
Let's try and compute the simple return on this investment 𝑟′:
(1 + 𝑟′(𝑡2 − 𝑡1 ))𝑃(𝑡1 ) = 𝑃(𝑡2 ).
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FIN201 STUDY UNIT 4
This implies that the simple return on investment on the CD is
𝑟′ =
1
𝑃(𝑡2 )
(
− 1).
𝑡2 − 𝑡1 𝑃(𝑡1 )
4.4 Discount Instruments
The pricing methods of money market instruments are generally classified into 2 types:
discount and yield.
4.4.1 Present Value by Yield
In the yield method, the instrument promises to pay a sequence of cash flows in the
future, which when suitably discounted to the present with a suitable yield, gives the
market price of the instrument.
Examples of yield instruments are: CDs, Eurocurrency CPs.
The relevant equation is explained as follows.
Suppose:
the market price of a discount instrument is 𝑃
its face value is 𝐹
the yield on it is 𝑦
the time to maturity is 𝑇
Then these quantities are related by the equation:
𝑃=
𝐹
.
1 + 𝑦𝑇
Example
A sterling commercial paper with a face value £10 million is issued for 91 days at a yield of
10%. What is its fair price given this information?
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FIN201 STUDY UNIT 4
The fair price is
10 × 106
𝑃=
= 9,975,130.49.
91
1 + 0.10 ×
365
Note that the 365basis is used because the UK money market convention is ACT/365.
4.4.2 Present Value by Discount Rate
In the discount method, the instrument is priced at a discount to its face value.
Ownership of the instrument receives a single cash flow corresponding to the face
value at maturity. The market price and the face value are related by means of a
suitable discount rate. Different market conventions mean that different relationships
and calculation methods govern discount instruments. We'll consider some of them
here.
Examples of discount instruments are: Tbills and commercial papers that are
domestically denominated.
Suppose:
the market price of a discount instrument is 𝑃
its face value is 𝐹
the discount rate on it is 𝑟
the time to maturity is 𝑇
Then these quantities are related by the equation:
𝑃 = 𝐹(1 − 𝑟𝑇).
Example
A US Tbill of $1 million is issued for 91 days at a discount rate of 6%. Find the discount and
the price of the instrument.
By the formula,
𝑃 = 𝐹 − 𝐹𝑟𝑇 = 106 − 106 × 0.06 ×
91
The discount is 106 × 0.06 × 360 = 15,166.67.
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91
= 984,833.33.
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FIN201 STUDY UNIT 4
4.5 ForwardForwards
A forwardforward loan can be synthetically created in the following manner:
Borrow $ X for time units of 𝑇1 at interest rate of 𝑟1
Lend $ X for time units of 𝑇2 at interest rate of 𝑟2
Diagrammatically, these are represented like this:
Figure 4.12
In terms of cash flows is this:
Time 0: effectively no cash flow
Time 𝑇1 : Cash outflow of $𝑋(1 + 𝑟1 𝑇1 )
Time 𝑇2 : Cash inflow of $𝑋(1 + 𝑟2 𝑇2 )
Let us work out the interest rate 𝑓 for the forwardforward contract (or just forward
rate):
𝑋(1 + 𝑟1 𝑇1 ) × (1 + 𝑓(𝑇2 − 𝑇1 )) = 𝑋(1 + 𝑟2 𝑇2 ).
This implies that
(1 + 𝑟1 𝑇1 ) × (1 + 𝑓(𝑇2 − 𝑇1 )) = (1 + 𝑟2 𝑇2 )
and the interest rate for the forwardforward loan is given by:
𝑓=(
1 + 𝑟2 𝑇2
1
− 1) ×
.
1 + 𝑟1 𝑇1
𝑇2 − 𝑇1
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Notation
The 𝑎 × 𝑏 forward interest rate is the forward interest rate that refers to a loan over a
future period from time 𝑎 to time 𝑏. Hence, a 3 × 6 forward rate is the forward interest
rate for the loan period which begins in 3 months and which ends in 6 months.
4.5.1 Significance of the Forward Rate
Consider 3 snapshots in time: 0 (now), 3 months' time ( 𝑡3 ), 6 months' times (𝑡6 ).
Thus, we do not know what the 3month interest rate will be in 3 months' time. But (1)
we can expect what it may be, or (2) we may calculate the 3x6 forward interest rate.
These two quantities are conceptually different.
The forward rate is determinable from current information. However, conditions in
the market may change and the 3month interest rate that is actually realised in 3
months' time may be different from the forward rate.
Market expectations of the 3month interest rate in 3months' time are different from
the forward rate as well as the actual rate that is realised in 3 months' time. But such
expectations have an impact on traders' decisions that ultimately shapes the reality.
Example
If the 3month spot rate is equal to the 6month spot rate, does it mean that the 3x6 forward
rate is equal to either of these rates?
In other words, if the market rate for loan up to 3 months is equal to the market rate for loan
up to 6 months, does it mean that the 3x6 forward rate is equal to either rate?
And in this case, what is the market's expectation of the 3month forward rate in 3 months'
time?
Suppose for simplicity that 3month (resp. 6month) is 0.25 (resp. 0.5) in time length.
Let the common 3month and 6month rate be 𝑟 and let the 3x6 forward rate be 𝑓.
Then
(1 + 0.25𝑟) × (1 + 0.25𝑓) = (1 + 0.5𝑟).
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FIN201 STUDY UNIT 4
This implies that
𝑓=(
1 + 0.5𝑟
− 1) × 4.
1 + 0.25𝑟
If 𝑟 = 0.1, then
𝑓=(
1 + 0.5 × 0.1
− 1) × 4 = 0.09756.
1 + 0.25 × 0.1
Thus, the forward rate may not be equal to the common spot rates that bracket it.
The market's expectation is a psychological fact that can be partially sampled through
surveying money market participants. From the given information here, we do not
know anything about the market's expectation.
4.6 Pricing an FRA
The price of an FRA refers to the fixed rate. Hence, in an FRA contract, the long party
buys the floating rate at the price of the fixed rate.
The price of an FRA, i.e. the fixed rate, needs to be theoretically determined.
This is done by setting it to be the forward interest rate. For example, if the FRA is
based on LIBOR, then the fixed rate is a LIBOR forward rate.
This is justified because the forward interest rate is the theoretically fair interest rate
to charge for the future loan period.
4.6.1 Quotation
An FRA is more specifically referred to as a 5x8 FRA or a 3x6 FRA.
In a description like this: 𝑎 × 𝑏 FRA, 𝑎 is the start month (from now) and 𝑏 is the end
month (from now). The length of the future loan period is 𝑏 − 𝑎.
Thus a 5x8 FRA started 5 months from now and ends 8 months from now.
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4.6.2 Settlement
Suppose that a 3x6 FRA refers to the 3month LIBOR which is realised as 11.5% in 3
months' time. Suppose also that the fixed rate of the FRA is 12.88%, the notional is
USD 10,000 and that the loan period is 92 days in length.
In 6 months' time, at expiry, the long party will pay
(0.1288 − 0.1150) ×
92
× 10000 = 35.27
360
to the short party.
Note that the floating interest will have been known in 3 months' time. Hence, it is
possible to discount the cash flow from 6month point to 3month point and for the
long party to pay this amount at the 3month point to the short party:
(0.1288 − 0.1150) ×
92
1
1
× 10000 ×
= 35.27 ×
92
92
360
1 + 0.1288 × 360
1 + 0.1288 × 360
= 34.14.
These are 2 possible ways to settle an FRA. The former method, in which payment
occurs at the end of the loan period, is called paymentinarrear. The latter method, in
which payment occurs at the start of the loan period, is called paymentinadvance.
Figure 4.13
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Example
The current market rates are as follows for SEK:
Term
Rate
3 months (91 days)
9.87/10.00%
6 months (182 days) 10.12/10.25%
9 months (273 days) 10.00/10.12%
Find the theoretical FRA 3x9 for SEK now.
Note that SEK is the ISO currency symbol for Swedish Kronor. SEK uses ACT/360 in
the money market.
The theoretical FRA refers to the fixed rate of the FRA that is computed using spot
data.
3x9 refers to a loan period that spans the 3month and the 9month marks.
Let the FRA fixed rate be 𝑓 𝑏 /𝑓 𝑎 .
Then
(1 + 0.0987 ×
91
273 − 91
273
)(1 + 𝑓 𝑏 ×
) = (1 + 0.1000 ×
),
360
360
360
(1 + 0.1000 ×
91
273 − 91
273
)(1 + 𝑓 𝑎 ×
) = (1 + 0.1012 ×
).
360
360
360
and
This implies that
𝑓 𝑏 = 9.82% and 𝑓 𝑎 = 9.92%.
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FIN201 STUDY UNIT 4
4.7 Bond Pricing and Yield
A bond is a financial instrument for borrowing money. The notion of a bond does 2
interesting things to the plain notion of borrowing. First, "borrowing money" becomes
"selling a bond" and "lending money" becomes "buying a bond". Second, the bond
makes an entity out of borrowinglending so that buyers of the bond can transfer the
debt to other buyers in the secondary market.
4.7.1 Basic Notions
The owner of a bond can expect a cash flow sequence that looks like this:
Figure 4.14
The arrows represent cash flows. These are inflows, from the perspective of the buyer.
The length of the arrows represents the size of the cash flows. Thus the long arrow is
the principal repayment, called the face value. The short arrows are the coupons, all
of equal magnitude a certain percentage of the face value.
The frequency of the bond is the number of coupon payments within a year. Thus an
annual bond pays coupons on a yearly basis. US Treasury bonds are semiannual
bonds  they pay coupons every half a year. In that case, the coupon payment is given
by
1
× 𝑐 × 𝐹,
2
where 𝑐 is the coupon rate (e.g. 0.10 for a 10% bond) and 𝐹 is the face value.
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FIN201 STUDY UNIT 4
In general, if the frequency of the bond is 𝑓 times a year, then the coupon is
1
𝑓
× 𝑐 × 𝐹.
The entire length of the bond is called its maturity or expiry  we denote it by 𝑇.
Bonds are first issued in a primary market. Thereafter, it changes hands in the
secondary market at market price 𝑃𝑡 (at time 𝑡).
In the diagram above, the cash flow sequence is to be expected by the buyer of the
bond in the primary market at time 0. For a buyer in the secondary market, the
coupons would not be the complete set as some may have already been issued. Trade
may occur inbetween coupon dates.
The phrase "face value" needs to be explained. The word "face" is used because in the
olden days, bonds are issued as paper certificates with appendages that represent
coupons to be cut out and redeemed. The body of the certificate is printed with a large
picture representing the issuing party, and that has been referred to as the face.
The face value affects several features:
the price
the coupons
the principal repayment
If the price of the bond is equal to its face value, then the bond is said to be
issued/selling at par. If the price is lower than face, then it is said to be issued/selling
at a discount. If the price is higher than face, then it is said to be issued/selling at a
premium.
The coupons are priced at a percentage off the face value. The principal repayment is
often just the face value. Sometimes, the repayment is referred to as redemption. Then
the notion of a face value is similar to that of a notional  it does not directly correspond
to any physical cash flow; rather, cash flows are computed relative to it. For example,
if you look at the "PRICE" function in Excel, the redemption is explained this way (ref:
https://support.office.com/enus/article/PRICEfunction3ea9deac8dfa436fa7c817ea02c21b0a).
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FIN201 STUDY UNIT 4
4.8 Price and Yield
Price and yield of a bond are 2 facets of the same thing. Price is the more familiar
notion. The yield of a bond is the constant discount rate such that if all the cash flows
are discounted to the present using this rate, the NPV is exactly equal to the price. This
yield is also known as the yieldtomaturity (YTM).
Thus, we have the fundamental bond pricing relationship:
Price of Bond = NPV of Cash Flows using YieldtoMaturity
Suppose 𝑃 is the price, 𝑦 is the yield, coupon rate is 𝑐, face value is 𝐹, and the bond is
annual and has a maturity of 𝑛 years. Then the relationship between price and yield
is given by
𝑃=
𝑐𝐹
𝑐𝐹
𝑐𝐹
𝐹
+
+⋯+
+
.
2
𝑛
1 + 𝑦 (1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)𝑛
If the frequency of the bond is twice a year, then
𝑃=
0.5𝑐𝐹
0.5𝑐𝐹
0.5𝑐𝐹
0.5𝑐𝐹
𝐹
+
+
+ ⋯+
+
.
2
3
2𝑛
1 + 0.5𝑦 (1 + 0.5𝑦)
(1 + 0.5𝑦)
(1 + 0.5𝑦)
(1 + 0.5𝑦)2𝑛
If the frequency is 𝑓 times a year, then
𝑃=
𝑐𝐹/𝑓
𝑐𝐹/𝑓
𝑐𝐹/𝑓
𝑐𝐹/𝑓
𝐹
+
+
+⋯+
+
.
2
3
𝑓𝑛
1 + 𝑦/𝑓 (1 + 𝑦/𝑓)
(1 + 𝑦/𝑓)
(1 + 𝑦/𝑓)
(1 + 𝑦/𝑓) 𝑓𝑛
In practice, the day convention needs to be respected in the discounting process. In
that case, the pricing formula reads:
𝑘
𝐶𝑘
𝑃=∑
(1 + 𝑦/𝑓)
𝑑𝑘 𝑓 ,
𝑌
where 𝐶𝑘 is the 𝑘th cash flow, 𝑑𝑘 is the number of days until 𝐶𝑘 , 𝑌 is the number of
days in the conventional year.
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FIN201 STUDY UNIT 4
Note that coupons are taken to be
𝑐𝐹
𝑓
regardless of the day count convention.
We may conclude from the relationship that price and yield are inverse to one another.
If the price goes up, the yield comes down. Conversely, if the price comes down, the
yield goes up.
Note that when "yield" is mentioned in the news, it can refer to different things. First
of all, there are many yields due to the great abundance of bonds trading at varying
prices. Second of all, there is a distinction between the yield of a specific bond and the
yield of a market.
The yield of a specific instrument is wholly obtained from its market price as
explained here.
The yield of a market is usually referenced to some benchmark instruments, such as
Treasury bonds. Thus, the suite of Treasury debt instruments provide a spectrum of
yields across the terms, giving us the term structure. This provides us with a snapshot
of the state of the interest rate market in terms of borrowing costs for varying terms.
Though the benchmark yields do not directly apply to specific instruments, they are
comparable. The prices of nonTreasury bonds are often quoted in terms of the market
yield plus a credit spread. For example, I may say that the yield of a corporate bond is
"3% + 100 bp" (bp = basis point), where "3%" is understood to be a market yield (i.e.
derived from a benchmark, usually riskfree, bond of the same term) and "100 bp" is
the credit spread that is added to account for the credit quality of the issue. Notice that
a higher credit spread implies a lower price. From the perspective of the issuer, its
lower credit quality implies that its funding cost is higher when compared to other
issuers. From the perspective of the investor, the bond of an issuer with a lower credit
quality ought to be priced lower. Hence, the whole concept pans out nicely.
Example
Find the price of a bond with these conditions:
maturity: 5 years
coupon rate: 5%
face value: $1000
frequency: annual
yield: 3%
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FIN201 STUDY UNIT 4
The price is given by
𝑃=
50
50
50
50
50
1000
+
+
+
+
+
.
2
3
4
5
1 + 0.03 (1 + 0.03)
(1 + 0.03)
(1 + 0.03)
(1 + 0.03)
(1 + 0.03)5
Using Excel, we may compute this in one of the following ways:
Worksheet Calculations
Figure 4.15
Figure 4.16
Excel PRICE Function
Figure 4.17
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FIN201 STUDY UNIT 4
Figure 4.18
Note how the price is computed using the Excel PRICE function. As the
documentation
(ref:
https://support.office.com/enus/article/PRICEfunction3ea9deac8dfa436fa7c817ea02c21b0a) says that the face is assumed to be 100, we
multiply by 10 the value that is computed from the PRICE function at the end.
Example
Find the YTM of a bond with these conditions:
maturity: 3 years
coupon rate: 3%
face value: $1000
frequency: semiannual
Price: $999
The priceyield relationship is given by:
999
15
15
15
15
15
15
+
+
+
+
+
1
2
3
4
5
(1 + 𝑦/2)
(1 + 𝑦/2)
(1 + 𝑦/2)
(1 + 𝑦/2)
(1 + 𝑦/2)
(1 + 𝑦/2)6
1000
+
.
(1 + 𝑦/2)6
=
It is not straightforward to solve for 𝑦.
Let's see how this may be done using Excel.
Excel and its Solver Addin
Set up the worksheet like this, where the yield is arbitrarily inserted for now:
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FIN201 STUDY UNIT 4
Figure 4.19
Figure 4.20
Then we open the Solver window and set it up this way:
Figure 4.21
Clicking on "Solve" gives us the answer:
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Figure 4.22
Example
Show that the annual par bond has a yield that is equal to the coupon rate.
The bond pricing formula states:
𝑃=
𝑐𝑃
𝑐𝑃
𝑐𝑃
𝑃
+
+⋯+
+
,
1
2
𝑛
(1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)
(1 + 𝑦)𝑛
where 𝑃 is the common face value and price, 𝑐 is the coupon rate, 𝑦 is the yield, and 𝑛
is the maturity.
This expression may be simplified using the formula for geometric progressions:
Hence,
𝑐 = 𝑦.
In other words, the coupon rate is equal to the yield for the par bond.
4.9 InBetween Coupon Dates
Suppose a bond has been trading in the market for a while and at present, the time
to the next coupon is not a whole coupon period:
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FIN201 STUDY UNIT 4
Figure 4.23
How may we price the remaining cash flows? In other words, what is the bond pricing
equation in this case?
Let 𝑃 be the bond price, 𝑦 be the yield, 𝐹 be the face value, 𝑐 be the coupon rate, 𝑓 be
the frequency. Let the remaining times be 𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 and 𝑇 = 𝑡5 .
Then the price is given by the NPV equation:
𝐹𝑐
𝐹𝑐
𝐹𝑐
𝐹𝑐
𝑓
𝑓
𝑓
𝑓
𝑃=
+
+
+
𝑓(𝑡
−𝑡)
𝑓(𝑡
−𝑡)
𝑓(𝑡
−𝑡)
1
2
3
(1 + 𝑦/𝑓)
(1 + 𝑦/𝑓)
(1 + 𝑦/𝑓)
(1 + 𝑦/𝑓) 𝑓(𝑡4 −𝑡)
+
𝐹𝑐/𝑓
𝐹𝑐/𝑓
+
(1 + 𝑦/𝑓) 𝑓(𝑡5 −𝑡) (1 + 𝑦/𝑓) 𝑓(𝑡5 −𝑡)
Note that the discounting is done by raising to the power of the length of time to the
cash flow. For example, the first cash flow is discounted by the factor (1 + 𝑦/𝑓) 𝑓(𝑡1 −𝑡) .
The exponent is 𝑓 × (𝑡1 − 𝑡). The factor 𝑓 is due to the frequency. The factor 𝑡1 − 𝑡 is
the remaining time to the first upcoming coupon, expressed using the day count
fraction that is assumed in the calculation.
Example
The diagram below represents the remaining cash flows of a US 6% Treasury bond.
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FIN201 STUDY UNIT 4
Figure 4.24
The dates 𝑡, 𝑡1 , 𝑡2 and 𝑡3 are, respectively, 1/12/2013, 1/1/2014, 1/6/2014 and 1/1/2015.
Assume that the bond is trading at par. Find its yield.
Computation of yield does not require the actual face value. So let's assume that the
face value is 1; hence its price is also 1.
The bond pricing equation is
1=
0.03
0.03
1.03
+
+
.
2(𝑡
−𝑡)
2(𝑡
−𝑡)
(1 + 𝑦/2) 1
(1 + 𝑦/2) 2
(1 + 𝑦/2)2(𝑡3 −𝑡)
No leap year is involved and the day count convention of the US Treasury bond is
ACT/ACT. Let's calculate the number of days between the dates with Excel:
Figure 4.25
31
182
396
Hence, 𝑡1 − 𝑡 = 365, 𝑡2 − 𝑡 = 365 and 𝑡3 − 𝑡 = 365.
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4.9.1 Accrued Interest
Another issue that arises from being inbetween coupon dates is the notion of accrued
interest.
Let's refer to this diagram again. We'll use it for illustration only, as the general case
can involve more coupons upstream.
Figure 4.26
The context is that there is a bond holder and he wishes to sell his bond to a buyer.
It is currently inbetween coupon dates. Hence, part of the upcoming coupon is due
to the bond holder as he has been holding the bond for the initial coupon period.
However, once the trade settles, the ownership of the bond falls to the bond buyer and
he will redeem the upcoming coupon in whole.
Hence, a portion of the coupon corresponding to the length of time that the bond
holder has been holding on to it during the current coupon period must be added to
the price. This is called the accrued interest. It is given by the formula
Accrued Interest = Fraction of Hold × 𝑐𝐹,
i.e. it is the fraction of the upcoming coupon amount according to the fraction of the
coupon period it is held on to.
The bond price that is computed as the NPV of upcoming cash flows is actually the
net price of the bond.
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The difference between the dirty price and the accrued interest is called the clean price.
We have the following relationship:
Dirty Price = Clean Price + Accrued Interest.
In the market, the price of a bond that is quoted usually as the clean price, not the dirty
price. Hence, in order to find the price of transaction, a calculation using this equation
or the NPV equation is necessary.
One attractive feature of the clean price in contrast to the dirty price shows up when
we compare how the two prices behave across coupon periods.
If we monitor the dirty price of a bond, we will notice that price jumps occur at coupon
periods because coupons are suddenly removed from their price as they are issued.
On the other hand, according to the definition of the clean price, as we approach a
coupon date, the contribution to the pricing of the bond by the upcoming coupon is
gradually diminished to zero thanks to the removal of the accrued interest. Thus, if
we monitor the clean price of a bond, it will not exhibit a price jump across coupon
dates.
Example
A 9% semiannual bond matures on 14 Aug 2018 and its yield is 8% on 11 Jun 2013. Find its
price and accrued interest on 11 Jun 2013, assuming a basis of ACT/360 and a face value of
100.
The upcoming coupon dates are 14 Aug 2013, 14 Feb 2014, 14 Aug 2014, ..., 14 Aug
2018  11 dates altogether. The last coupon date is 14 Feb 2013.
The number of days between 14 Feb 2013 and 11 Jun 2013 is 117. The accrued interest
is
117
× 4.5 = 1.4625.
360
The dirty price is given by
4.5
4.5
4.5
+
+ ⋯+
2×(𝑡
−𝑡)
2×(𝑡
−𝑡)
1
2
(1 + 0.08/2)
(1 + 0.08/2)
(1 + 0.08/2)2×(𝑡𝑛 −𝑡)
100
+
,
(1 + 0.08/2)2×(𝑡𝑛−𝑡)
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FIN201 STUDY UNIT 4
where the times 𝑡1 , 𝑡2 , … are the coupon dates, and the durations 𝑡1 − 𝑡, 𝑡2 − 𝑡, … are
computed on a ACT/360 basis.
4.10 Term Structure Construction
We will show how a term structure is constructed in the following sections.
4.10.1 Zero Rates
We assume that there is a series of zero coupon bonds (ZCBs) of varying terms. Note
that in reality, ZCBs are found in the form of instruments such as the US treasury bills
(1m, 3m, 6m, 1y maturity). Longertermed instruments are usually not zerocoupon.
We will disregard this fact of reality and assume that ZCBs, all with face value 1, are
traded at prices
𝑃1 , 𝑃2 , … , 𝑃𝑛
and have maturities
𝑇1 , 𝑇2 , … , 𝑇𝑛 .
Let's consider the first ZCB with price 𝑃1 and maturity 𝑇1 . Write its YTM as 𝑦1 . By the
bond pricing equation, we have
𝑃1 =
1
.
(1 + 𝑦1 )𝑇1
𝑃𝑖 =
1
(1 + 𝑦𝑖 )𝑇𝑖
Similarly, we have
for each 𝑖 = 2,3, ….
These yields 𝑦1 , 𝑦2 , … , 𝑦𝑛 that are implied by the ZCBs, are called zero/spot yields/rates
(any combination is a proper name).
When the points (𝑡1 , 𝑦1 ), (𝑡2 , 𝑦2 ), … , (𝑡𝑛 , 𝑦𝑛 ) are plotted, we obtain a chart like this:
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FIN201 STUDY UNIT 4
Figure 4.27
These points constitute an outline of the term structure. The yield curve is obtained by
connecting the points together by some method of interpolation.
For example, we may connect successive points together by straight lines (called linear
interpolation), giving us the following chart:
Figure 4.28
Or we may try to connect the points together into a smooth curve, like this:
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FIN201 STUDY UNIT 4
Figure 4.29
In any case, we end up having a yield curve, which may be thought of as a function
𝑡 ↦ 𝑌(𝑡).
This means that, whenever we specify a term 𝑡, we are able to read off a yield value,
𝑌(𝑡), corresponding to that term. This yield allows us to presentvalue future cash
flows.
For instance, a cash flow of 100 at time 𝑇 in the future has a PV of
100
.
(1 + 𝑌(𝑇))𝑇
Example
Assume the following zerocoupon yield structure:
Term
YTM
1year 10.000%
2year 10.526%
3year 11.076%
4year 11.655%
What are the zerocoupon discount factors? What is price and yield of a 4year, 5% coupon
bond?
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The discount factors are:
Term
YTM
Discount Factor
1year 10.000%
1
≈ 0.90909
(1 + 10%)1
2year 10.526%
1
≈ 0.81860
(1 + 10.526%)2
3year 11.076%
1
≈ 0.72969
(1 + 11.076%)3
4year 11.655%
1
≈ 0.64341
(1 + 11.655%)4
The 4year 5% coupon bond comprises 4 cash flows of sizes 5, 5, 5 and 105 respectively.
From the bond pricing equation,
𝑃=
5
5
5
105
+
+
+
2
3
(1 + 10%) (1 + 10.526%)
(1 + 11.076%)
(1 + 11.655%)4
= 5 × 0.90909 + 5 × 0.818602 + 5 × 0.729693 + 105 × 0.643414
= 79.84495
4.11 Bootstrapping
We may not be able to find a good set of ZCBs that span a wide spectrum of terms to
paint a good picture of the term structure. Hence we will derive zero rates from
coupon bonds in a process called bootstrapping.
First, we will describe the principle of how this is done. Then we will illustrate using
some examples.
Suppose we have annual coupon bonds 𝐵1 , 𝐵2 , … , 𝐵𝑛 over 𝑛 years, where the maturity
of 𝐵𝑖 is 𝑖year. Let's assume that the face value of each of these bonds to be 1, their
coupon rates are all equal to 𝑐, and they have prices equal to 𝑃1 , 𝑃2 , … , 𝑃𝑛 respectively.
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We will find the zero rates 𝑌(1), 𝑌(2), … , 𝑌(𝑛) from this information in an iterative
manner. Note the meanings of these zero rates: they give us the discount factors across
terms to presentvalue future cash flows.
The first bond does not have any interim coupons, hence:
𝑃1 =
1+𝑐
,
1 + 𝑌(1)
which implies that
1+𝑐
− 1.
𝑃1
𝑌(1) =
From the bond pricing equation for the second bond, we have
𝑃2 =
𝑐
1+𝑐
+
,
1 + 𝑌(1) (1 + 𝑌(2))2
which implies that
𝑌(2) = (
1
1+𝑐
2 − 1.
)
𝑐
𝑃2 −
1 + 𝑌(1)
Note: the formula looks complicated, but that's not the key point. The key point is that
𝑌(2) can be obtained in terms of 𝑌(1), which has already been obtained. This is what
"iterative" refers to.
Let's look at the third bond. Its price is given by
𝑃3 =
𝑐
𝑐
1+𝑐
+
+
,
2
1 + 𝑌(1) (1 + 𝑌(2))
(1 + 𝑌(3))3
which implies that
𝑌(3) = (
1+𝑐
𝑃3 −
𝑐
𝑐
−
1 + 𝑌(1) (1 + 𝑌(2))2
1
)3 − 1.
In general, the 𝑖th zero rate 𝑌(𝑖) is obtained from the pricing equation of the 𝑖th bond
this way:
𝑃𝑖 =
𝑐
1+𝑐
1+𝑐
1+𝑐
+
+ ⋯+
+
2
𝑖−1
1 + 𝑌(1) (1 + 𝑌(2))
(1 + 𝑌(𝑖 − 1))
(1 + 𝑌(𝑖))𝑖
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implies that
𝑌(𝑖) = (
1
1+𝑐
) 𝑖 − 1.
𝑐
𝑐
𝑐
𝑃𝑖 −
−
+ ⋯+
1 + 𝑌(1) (1 + 𝑌(2))2
(1 + 𝑌(𝑖 − 1))𝑖−1
Example
The 1year interest rate is 10% and the following bonds are currently trading:
Maturity
Price
Coupon
1year
97.409
9%
2year
85.256
5%
3year
104.651
13%
Find the term structure from the given information.
The given information allows us to find 𝑌(1), 𝑌(2), 𝑌(3), 𝑌(4).
𝑌(1) is plainly given to be 10%.
The other zeros are found by applying the formulae above:
109
𝑌(2) = (
97.409 −
𝑌(3) = (
9
1 + 𝑌(1)
1
)2 − 1,
1
105
)3 − 1,
5
5
85.256 −
−
1 + 𝑌(1) (1 + 𝑌(2))2
113
𝑌(4) = (
104.651 −
1
13
13
13
−
−
2
1 + 𝑌(1) (1 + 𝑌(2))
(1 + 𝑌(3))3
Plugging the value of 𝑌(1) into
0.105260529055 ≈ 10.5%.
the first
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)4 − 1.
equation, we
obtain 𝑌(2) =
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Plugging the values of 𝑌(1) and 𝑌(2) into the second equation, we obtain 𝑌(3) =
0.110760266184 ≈ 11.1%.
Plugging the values of 𝑌(1) and 𝑌(2) and 𝑌(3) into the second equation, we obtain
𝑌(4) = 0.116550704358 ≈ 11.7%.
The method applies also to bonds whose coupons are not annual.
Example
The 0.5year interest rate is 10% and the following semiannual bonds are currently trading:
Maturity
Price
Coupon
1year
97.409
9%
1.5year
85.256
5%
2year
104.651
13%
Find the term structure from the given information.
The given information allows us to find 𝑌(0.5), 𝑌(1), 𝑌(1.5), 𝑌(2) . We will regard
these yields as annual equivalent yields for the purpose of discounting in the
following.
𝑌(0.5) is plainly given to be 10%.
The other zeros are found by iteration.
From the bond pricing equation,
97.409 =
4.5
104.5
+
,
0.5
(1 + 0.10)
1 + 𝑌(1)
it's implied that
𝑌(1) = (
104.5
4.5
97.409 −
(1 + 0.10)0.5
) − 1 = 0.122226964906 ≈ 12.2%.
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From the bond pricing equation,
85.256 =
2.5
2.5
102.5
+
+
,
(1 + 0.10)0.5 1 + 0.122 (1 + 𝑌(1.5))1.5
it's implied that
𝑌(1.5) = (
1
102.5
)1.5 − 1 = 0.173367059053 ≈ 17.3%.
2.5
2.5
85.256 −
−
(1 + 0.10)0.5 1 + 0.122
From the bond pricing equation,
104.651 =
7.5
7.5
7.5
107.5
+
+
+
,
(1 + 0.10)0.5 1 + 0.122 (1 + 0.173)1.5 (1 + 𝑌(2))2
it's implied that
107.5
𝑌(2) = (
104.651 −
7.5
7.5
7.5
−
−
(1 + 0.10)0.5 1 + 0.122 (1 + 0.173)1.5
1
)2 − 1
= 0.125174188454 ≈ 12.5.
4.12 Forward and Par Yields
Apart from zero yields, we may also create a term structure that is made up of par
yields or forward yields.
We'll start by assuming that we have the zero yields
𝑌(1), 𝑌(2), 𝑌(3), … , 𝑌(𝑛).
For simplicity of illustration, we'll assume frequency of compounding within a year
to be 1 here.
The par yield 𝑃(𝑖) of the 𝑖th year is the coupon rate of the par bond of 𝑖year maturity.
Thus,
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1=
𝑃(𝑖)
𝑃(𝑖)
𝑃(𝑖)
1 + 𝑃(𝑖)
+
+ ⋯+
+
.
2
𝑖−1
1 + 𝑌(1) (1 + 𝑌(2))
(+𝑌(𝑖 − 1))
(1 + 𝑌(𝑖))𝑖
Hence,
1−
𝑃(𝑖) =
1
(1 + 𝑌(𝑖))𝑖
1
1
1
1
+
+ ⋯+
+
𝑖−1
1 + 𝑌(1) (1 + 𝑌(2))2
(+𝑌(𝑖 − 1))
(1 + 𝑌(𝑖))𝑖
.
Note that the first par yield is just the spot yield:
𝑃(1) = 𝑌(1).
Using this formula, we may derive a term structure of par yields
𝑃(1), 𝑃(2), … , 𝑃(𝑛)
from the zero rates.
The term structure of par yields gives a picture of coupon rates of par bonds across
terms as implied by the zero rates.
Example
Find the par yields from the zero yields 𝑌(1) = 0.1 , 𝑌(2) = 0.105 , 𝑌(3) = 0.111 ,
𝑌(4) = 0.117.
The first par yield is 𝑃(1) = 𝑌(1) = 0.1.
The other par yields are obtained using the formulae above:
1
(1 + 𝑌(2))2
𝑃(2) =
= 0.104750056548 = 10.5%,
1
1
+
1 + 𝑌(1) (1 + 𝑌(2))2
1−
1
(1 + 𝑌(3))3
𝑃(3) =
= 0.110194909102 ≈ 11.0%,
1
1
1
+
+
1 + 𝑌(1) (1 + 𝑌(2))2 (+𝑌(3))3
1−
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1
(1 + 𝑌(4))4
𝑃(4) =
= 0.115375937612
1
1
1
1
+
+
+
1 + 𝑌(1) (1 + 𝑌(2))2 (+𝑌(3))3 (1 + 𝑌(4))4
≈ 11.5%.
1−
The 𝑎 × 𝑏 forward yield is the forward rate that is derived from spot yields that
pertain to a forward deposit over the period [𝑎, 𝑏]. It is also called the forwardforward
yield. Let's denote this to be 𝐹(𝑎, 𝑏).
From the spot yields 𝑌(𝑎) and 𝑌(𝑏), we have the relationship
(1 + 𝑌(𝑎))𝑎 (1 + 𝐹(𝑎, 𝑏))𝑏−𝑎 = (1 + 𝑌(𝑏))𝑏 .
Note that the forward yields thus expressed are automatically assumed to be
annualised.
This allows the forward yield to be expressed as
(1 + 𝑌(𝑏))𝑏 1
𝐹(𝑎, 𝑏) = (
)𝑏−𝑎 − 1.
(1 + 𝑌(𝑎))𝑎
Example
Find the forward yields 𝐹(0,1), 𝐹(1,2), 𝐹(2,3), 𝐹(3,4) from the zero yields 𝑌(1) = 0.1 ,
𝑌(2) = 0.105, 𝑌(3) = 0.111, 𝑌(4) = 0.117.
The first forward rate is
𝐹(0,1) = 𝑌(1) = 0.1.
The other forward rates are obtained by the formula above:
𝐹(1,2) = (
(1 + 𝑌(2))2 1
)2−1 − 1 = 0.110022727273 ≈ 11.0%,
(1 + 𝑌(1))1
(1 + 𝑌(3))3 1
𝐹(2,3) = (
)3−2 − 1 = 0.123097914457 ≈ 12.3%,
(1 + 𝑌(2))2
𝐹(3,4) = (
(1 + 𝑌(4))4 1
)4−3 − 1 = 0.135195120367 ≈ 13.5%.
(1 + 𝑌(3))3
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These sections from the Textbook (and do the associated exercises):
Chapter 3 Money market calculations
Chapter 3 Discount instruments
Chapter 4 Applications of FRAs
Chapter 6 Introduction to bond pricing
Chapter 6 Different yield measures and price calculations
Chapter 8 Zerocoupon yields, par yields and bootstrapping
Chapter 8 Forwardforward yields
Chapter 8 Summary
Watch this video in which I discuss some examples and work out some solutions
under the topic of fixed income:
Worked Examples in Fixed Income
(Access video via iStudyGuide)
Watch this video in which I explain the mechanics behind the construction of the yield
curve:
Bootstrapping the Yield Curve
(Access video via iStudyGuide)
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FIN201 STUDY UNIT 4
Quiz
1)
Shortterm interest rates on the yield curve are obtained from the
a.
b.
c.
d.
2)
Longterm interest rates on the yield curve are obtained from the
a.
b.
c.
d.
3)
bond market
cash market
credit market
debt market
The process of creating a yield curve from bond prices is called
a.
b.
c.
d.
5)
money market
capital market
equity market
foreign exchange market
The fixed income market is conventionally categorised into money market and
a.
b.
c.
d.
4)
money market
capital market
equity market
foreign exchange market
bootlicking
bootstrapping
bootlegging
booting
The short end of the yield curve is found using
a.
b.
c.
d.
notes and bonds
equity instruments
money market instruments
FRAs
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6)
The long end of the yield curve is found using
a.
b.
c.
d.
7)
The duration of cash is
a.
b.
c.
d.
8)
0
1
2
3
The duration of a ZCB is
a.
b.
c.
d.
9)
notes and bonds
equity instruments
money market instruments
FRAs
0
1
its midterm
its maturity
A 4x7 FRA pertains to a loan period
a.
b.
c.
d.
between April and July
that begins 4 months from now and lasts for 3 months
that begins 4 months from now and lasts for 7 months
that is 28 months in length
10) The notion of yieldtomaturity assumes that coupons are
a.
b.
c.
d.
reinvested
recomputed
reissued
none of the above
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Solutions or Suggested Answers
Quiz
Question 1: a
Question 2: b
Question 3: a
Question 4: b
Question 5: c
Question 6: a
Question 7: a
Question 8: d
Question 9: b
Question 10: a
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FOREIGN EXCHANGE
FIN201 STUDY UNIT 5
Learning Outcomes
By the end of this unit, you should be able to:
1. Describe significant features of the foreign exchange market.
2. Compute with foreign exchange spot quotes.
3. Apply the Interest Rate Parity.
4. Compute with foreign exchange forward quotes.
5. Apply the Principle of No Arbitrage.
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Chapter 1 Market
The foreign exchange market is the major market in finance that's concerned with the
exchange of currencies.
Some indications of its importance are these:  Without foreign exchange, international
trade will not be possible.  It is the largest market in the world by dollar volume of
daily trade.  It is closely related to the money market via the interest rate parity.
In this unit, we will discuss how rates are commonly quoted in the foreign exchange
market. In the next unit, we will delve into calculations concerning these quotes and
FX instruments.
According to the 2013 Triennial Central Bank Survey of Foreign Exchange and OTC
Derivatives Markets Activity conducted by the Bank for International Settlements, the
daily trade in the foreign exchange market is over $5 trillion in Apr 2013. By
comparison, the average daily trading volume at the NYSE is about US$169 billion in
2013.
The market begins trading on Monday mornings in Wellington, New Zealand and
ends on Friday evenings in New York, USA. Apart from weekends and major
international holidays, the market operates nonstop throughout the year.
Unlike stock exchanges which are centrally organised, the foreign exchange market is
distributed in nature. There is no one single geographically centralised place of trade
nor is there a single regulatory body that governs the functioning of the market.
However, that does not make it anarchic either since some of the largest participants
are banks which are highly regulated, particularly after the Global Financial Crisis of
200708. In addition, central banks intermittently intervene in the market in
implementing their monetary policies.
Such a state of the market has not been planned internationally. Evolution is a better
word to describe how it has come to be. The major form of currency at present is fiat
money which comprises pieces of paper which on their own are worth little. By norm,
fiat money is readily exchanged for goods and services which in turn reinforce and
sustain their status of being money.
If we look back in history a hundred years, most countries in the world were on the
gold standard while China and Hong Kong were on the silver standard. This means
that the value of the money was derived directly from the metals to which they are
pegged. Towards the end of World War II, the gold standard was ratified by the
international community at the Bretton Woods Conference in 1944.
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Events entering the 1970s saw gold price rising and consequently the dollar
strengthening which hurt the US economy. President Nixon thus decided to halt
convertibility of dollar to gold. Since then, international fiat currencies have not been
pegged to anything but themselves in what has been called the floating rate regime.
This has been the state of the foreign exchange market till today.
1. These sections from the Textbook (and do the associated exercises):
Chapter 9 Introduction
2. The article The Gold Standard, Bretton Woods and Other Monetary Regimes: A
Historical Appraisal (pp. 160−183) to find out more about the background of the
foreign exchange market as we witness it today.
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Chapter 2 Signals and Quotes
2.1 Spot Exchange Rates and Quotes
Spot exchange rates refer to current rates for exchanges of currencies that are meant
to take place at present.
A typical exchange rate looks like this:
EUR/USD 1.0810,
which may also be written as
EURUSD 1.0810.
Generally, exchange rates are quoted as XY, where X and Y are the national currency
codes as accorded by ISO 4217. X is referred to as base currency and is thought of as
the asset (1 unit of this is to be priced), and Y is sometimes referred to as counter
currency and is thought of as the unit of account that is used to price X. "Buying/selling
X with Y" is sometimes described as "buying/selling X against Y".
Thus, EUR is regarded as an asset while USD is regarded as the currency used to price
it. The fact that EURUSD is quoted at 1.0810 means that EUR 1 costs USD 1.0810. EUR
is also called the base currency and USD is called the counter currency in this pair.
Most major currency pairs are quoted up to the 4th decimal place. The unit in the 4th
decimal place (i.e. 0.0001) is called the pip. The unit in the second decimal place is
called the big figure. Thus in 1.0810, there are 10 pips and the big figure is 8.
The pair USDJPY is a notable exception. It is typically of such a magnitude: USDJPY
117.96. In this case, the 2nd decimal place (i.e. 0.01) is known as the pip and the unit is
the big figure. Thus, there are 96 pips and 17 big figures in USDJPY 117.96.
The distribution by international usage (i.e. volume of transaction) of currency and
pair, according to the Triennial Survey of BIS, shows that, by far, USD, EUR and JPY
are the most traded currencies internationally. The top 3 exchange rates are EURUSD,
USDJPY and GBPUSD.
Note that in the chart above, the pairs are not written in any particular order. For
example, USD / EUR does not refer to the exchange rate USDEUR. In practice, even
though USDEUR and EURUSD refer essentially to the same pair, only one of them is
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conventionally quoted in the market  namely, EURUSD. This is also true for other
pairs. Thus, USDJPY is conventionally quoted but not JPYUSD.
2.2 Bid and Ask
Participants in the foreign market trade for one reason or another and each has his
own strategies and methods. A distinguished group of participants are the market
makers. These are traders who do not typically trade a currency because they think
that it may rise or fall. Instead, money makers publicise two quotes for each currency
pair, one for buying and the other for selling the base currency.
For instance, a bidask quote for EURUSD may look like this:
EURUSD 1.0810  1.0825.
What this means is that the market maker is willing to buy EUR 1 at USD 1.0810 and
to sell EUR 1 and USD 1.0825.
The lower price is called the bid price and the higher price is called the ask price.
The bidask quote may be shorten into
10  25
This makes sense to active market participants who are highly aware of what the big
figure is at any one time.
The difference 25  10 = 15 pips is known as the bidask spread.
The bidask spread represents the profit that a market maker makes. In a single
exchange, the market maker merely trades one of EUR or USD for the other. Suppose
he sells EUR for USD in a first leg, and then he sells USD for EUR in a second leg.
He sold EUR 1 for USD 1.0825. Then he sells USD 1.0825 at EURUSD 1.0810.
Note that, "selling USD" is equivalent to "buying EUR". Thus, he receives
1.0825
= 1.00138760407
1.0810
in euros. In the "round trip", the initial EUR 1 has become EUR 1.001.
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The business proposition of a market maker is thus to avoid the risk of directional
trades and to focus on carrying out as many trades as possible in order to profit from
the bidask spread.
Electronic market makers can afford to keep the spread low, e.g. around 1 pip, as the
high speed of trades assures them of a substantial volume. Nonelectronic market
makers have to make do with a larger spread. For instance, money changers can quote
at spreads of about 200 pips. The spread is a measure of competition  the lower the
spread, the more competitive is the market maker, but it also means that the profit per
trade is tiny and has to be compensated by volume.
Finally, note that with respect to a pair XY, buying Y is equivalent to selling X, and
selling Y is equivalent to buying X.
For the obvious reason, the bid rate must be smaller than the ask rate. If not, the market
maker will bleed his profits away into a mountain of losses. From the opposite
perspective, the low ask and high bid implies that I can arrange for two trades that
will guarantee me profits without any risk.
For example, if a market maker mistakenly quotes:
EURUSD 1.08251.0810.
Then I will sell EUR 1 for USD 1.0825 and buy EUR with USD 1.0825, resulting in EUR
1.0825 / 1.0810 = EUR 1.00138760407.
Such a riskless profitable trade is known as an arbitrage opportunity. The careless
market maker has thus created an arbitrage opportunity for traders.
We will explain this important concept further later.
For now, in order to distinguish "bid" from "ask", just remember that these terms are
from the perspective of the market maker and he needs to "buy low and sell high" to
stay in business.
Example
In one deal, Bank A buys EUR 1 million against CHF at 1.2830. In another deal, Bank A
sells EUR 1 million against CHF 1.2855.
What is the net result?
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FIN201 STUDY UNIT 5
Deal #
Cash inflow
Cash outflow
1
EUR 1,000,000 CHF 1,283,000
2
CHF 1,285,500 EUR 1,000,000
The net result requires evaluating the state of holdings for both EUR and CHF.
For EUR, the net flow is 0.
For CHF, there is a net inflow of 1,285,500  1,283,000 = 2,500.
2.3 Cross Rates
A cross rate is an exchange rate that is deduced from two exchange rates that share a
common currency.
For example, if EURUSD and USDSEK are quoted, then the exchange rate EURSEK
that is deduced is known as a cross rate.
Cross rates exist because not all pairs of currencies have their exchange rates quoted
directly. In this case, it will be necessary to exchange one currency for another
indirectly through an intermediate series of exchanges.
For example, in order to buy SEK from EUR, and if only EURUSD and USDSEK are
quoted, then a possible route would be:
buy USD from EUR
buy SEK from USD
Example
Given that the following spot rates have been quoted:
USDJPY: 83.17  83.21
USDSGD: 1.2882  1.2892
Find a suitable bidask quote for SGDJPY.
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Let's write the bidask quote for SGDJPY to be
SGDJPY: x  y
Let's find the bid SGDJPY x first. The dealer would buy SGD 1 with JPY x.
With respect to the given quotes, the dealer would buy USD 1 for JPY 83.17. This
𝑥
means that he will buy USD 83.17 for JPY x.
And he would sell USD 1 for SGD 1.2892. This means that he will buy SGD 1.2892 ×
𝑥
83.17
𝑥
for USD 83.17.
𝑥
Putting these together, he would buy SGD 1.2892 × 83.17 for JPY x,
i.e. he would buy SGD 1 for JPY 83.17/1.2892 = JPY 64.52,
𝑥 = 64.52.
Now, let's try to find the ask SGDJPY y. The dealer would sell SGD 1 for JPY y.
With respect to the given quotes, the dealer would sell SGD 1 for USD 1 / 1.2882.
And he would sell USD 1 for JPY 83.21, i.e. USD 1 / 1.2882 for JPY 83.21 / 1.2882.
Putting these together, he would sell SGD 1 for JPY 83.21 / 1.2882.
Thus
𝑦=
83.21
= 64.59.
1.2882
Here is an alternative argument that avoids the use of algebra and is somewhat more
direct.
From the quotes:
USDJPY: 83.17  83.21
USDSGD: 1.2882  1.2892
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FIN201 STUDY UNIT 5
suppose that we want to find the bid price of SGDJPY. This means that the dealer is
willing to buy SGD 1 by selling some units of JPY.
Focussing on “buy SGD, sell JPY”, we should focus on the quotes
USDJPY: 83.17 (bid)
USDSGD: 1.2892 (ask)
This says that the dealer is willing to

buy USD 1, sell JPY 83.17
sell USD 1, buy SGD 1.2892
Combining these, he’s willing to sell JPY 83.17 and buy SGD 1.2892, which is
equivalent to saying: buy SGD 1, sell 83.17/1.2892 = 64.52
Thus, the bid price SGDJPY 64.52.
You may try to reconstruct this same argument for the ask price.
We may abstract the general rule for computing cross rates from the above this way.
Suppose we have 3 currencies X, Y, Z.
XY is quoted as:
𝑋𝑌: 𝑏(𝑥, 𝑦)  𝑎(𝑥, 𝑦),
XZ is quoted as:
𝑋𝑍: 𝑏(𝑥, 𝑧)  𝑎(𝑥, 𝑧).
With reference to the above example, we may take X to be USD, Y to be JPY and Z to
be SGD.
Then ZY is given by
𝑍𝑌:
𝑏(𝑥, 𝑦) 𝑎(𝑥, 𝑦)

.
𝑎(𝑥, 𝑧) 𝑏(𝑥, 𝑧)
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FIN201 STUDY UNIT 5
What if instead of XY, YX is quoted instead?
Then we'll begin with the quote for YX:
𝑌𝑋: 𝑏(𝑦, 𝑥)  𝑎(𝑦, 𝑥)
and convert it into the quote for XY:
𝑋𝑌: 𝑏(𝑥, 𝑦)  𝑎(𝑥, 𝑦) =
1
1

.
𝑎(𝑦, 𝑥) 𝑏(𝑦, 𝑥)
We may summarise these like this:
Suppose that for any 2 currencies X and Y, we write their bid and ask rates as 𝑏(𝑥, 𝑦)
and 𝑎(𝑥, 𝑦).
Then the following statements are true:
𝑎(𝑥, 𝑦) =
1
,
𝑏(𝑦, 𝑥)
𝑏(𝑥, 𝑦) =
1
.
𝑎(𝑦, 𝑥)
𝑏(𝑧, 𝑦) =
𝑏(𝑥, 𝑦)
,
𝑎(𝑥, 𝑧)
𝑎(𝑧, 𝑦) =
𝑎(𝑥, 𝑦)
.
𝑏(𝑥, 𝑧)
(1)
(2)
Example
Given the spot rates:
EURUSD: 1.3166  1.3171
NZDUSD: 0.7634  0.7639
Find the spot bidask quotes for EURNZD.
Let Z denote EUR, Y denote NZD, X denote USD.
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FIN201 STUDY UNIT 5
Then the bidask for EURNZD is given by
𝑏(𝑧, 𝑦) =
𝑏(𝑥, 𝑦)
,
𝑎(𝑥, 𝑧)
𝑎(𝑧, 𝑦) =
𝑎(𝑥, 𝑦)
.
𝑏(𝑥, 𝑧)
Let's compute 𝑏(𝑧, 𝑦):
1
𝑏(𝑥, 𝑦) 𝑎(𝑦, 𝑥) 𝑏(𝑧, 𝑥) 1.3166
𝑏(𝑧, 𝑦) =
=
=
=
= 1.7235
1
𝑎(𝑥, 𝑧)
𝑎(𝑦, 𝑥) 0.7639
𝑏(𝑧, 𝑥)
and
1
𝑎(𝑥, 𝑦) 𝑏(𝑦, 𝑥) 𝑎(𝑧, 𝑥) 1.3171
𝑎(𝑧, 𝑦) =
=
=
=
= 1.7253.
1
𝑏(𝑥, 𝑧)
𝑏(𝑦, 𝑥) 0.7634
𝑎(𝑧, 𝑥)
Thus, the quote for EURNZD is
EURNZD: 1.7235  1.7253
2.4 Forward Exchange Rates
The forward exchange rate is an exchange rate at which a currency exchange will
occur in the future but is fixed at present. The concept is analogous to the concept of
a forward price or forward interest rate.
For example, this information may be flashed to the trader on the quotes board:
Instrument
Bid
Spot EURUSD
1.4066
1.4071
1m Forward
EURUSD
1.4181
1.4191
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FIN201 STUDY UNIT 5
It means:
the dealer is willing to buy EUR 1 at USD 1.4066 currently
the dealer is willing to sell EUR 1 at USD 1.4071 currently
the dealer is willing to buy EUR 1 at USD 1.4181 one month from now
the dealer is willing to sell EUR 1 at USD 1.4191 one month from now
The forward instrument in the foreign exchange market is called the forward outright.
This is to distinguish it from another instrument called the forward swap.
We'll explain these instruments later.
These sections from the Textbook (and do the associated exercises):
Chapter 9 Spot exchange rates
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Chapter 3 Concept: Interest Rate Parity
The interest rate parity is a mathematical relationship between the following terms
related to two currencies X and Y:
the interest rate of X
the interest rate of Y
the spot exchange rate between X and Y
the forward exchange rate between X and Y
We assume that the terms of the interest rates and the forward exchange rate are the
same (i.e. all 1month in length, or all 3month in length, etc.), and that there is no bidask spread in the rates.
In the following, we'll see two different ways for investing an amount of money. The
returns from both must be equal otherwise it would be possible to lock in a riskfree
profit.
Step 1 Quotes
Suppose we have 2 currencies X and Y.
The spot exchange rate is XY s, and the forward exchange rate is XY f. The forward
exchange is to occur at 𝑇 in the future.
In other words, 1 unit of X is to trade for s units of Y currently. And 1 unit of X is to
trade for f units of Y at time 𝑇 in the future.
Step 2 Courses of Action
Let's imagine that we have 1 unit of X to begin with.
There are 2 things that we can do to it:
1.
exchange it into Y right away and deposit that in the bank at interest rate 𝑟𝑌 until
time 𝑇
2.
deposit X in the bank at interest rate 𝑟𝑋 until time 𝑇 and exchange the deposit into
𝑌
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Step 3 Reconciling the 2 Actions by the Principle of No Arbitrage
Assuming that the market is efficient in coming up with prices and rates, the two ways
of investing the initial unit of X must lead to the same amount of Y at time 𝑇.
By method (1), the amount of 𝑌 at time 𝑇 is given by
𝑠 × (1 + 𝑟𝑌 × 𝑇).
By method (2), the amount of 𝑌 at time 𝑇 is given by
(1 + 𝑟𝑋 × 𝑇) × 𝑓.
Equating them, we have
𝑠 × (1 + 𝑟𝑌 × 𝑇) = (1 + 𝑟𝑋 × 𝑇) × 𝑓.
Rearranging, so that we obtain a formula for the forward rate:
𝑓 =𝑠×
1 + 𝑟𝑌 𝑇
.
1 + 𝑟𝑋 𝑇
This formula for the forward exchange rate is known as the interest rate parity.
The relationship holds due to the Principle of No Arbitrage. This means that the
violation of the relationship implies the existence of a trading strategy that is riskfree.
In reality, the relationship only roughly holds because fees and commissions will
reduce profits to be worth the effort.
To let the notation remind us about the direction of the exchange, we may write:
𝑓𝑋𝑌 = 𝑠𝑋𝑌 ×
1 + 𝑟𝑌 𝑇
.
1 + 𝑟𝑋 𝑇
The forward rate, as expressed by this formula, is also known as the theoretical
forward outright or simply the theoretical forward.
There are two things to note when we apply the formula in practice:
1. The time 𝑇 needs to take into account the day count fraction and convention of
the currency.
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FIN201 STUDY UNIT 5
2. We may use it to compute the theoretical bid and ask forward rates. Inputting
the bid rate for spot will give us the bid rate for forward. And similarly, for the
ask rate.
Example
Given the following information:
31day USD interest rate: 5%
31day EUR interest rate: 3%
Spot EURUSD rate: 1.4068
What is the forward outright?
By the interest rate parity, the forward outright is
1.4068 ×
31
1 + 0.05 × 360
31
1 + 0.03 × 360
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= 1.4092.
FIN201 STUDY UNIT 5
Chapter 4 Instruments
The major classes of instruments in the foreign exchange market are:
spot instruments (i.e. currencies)
forward instruments (forward outrights, forward swaps)
derivatives (futures and options)
Spot instruments have been considered in the previous section. We will consider
forward outrights and forward swaps in the following. We will not discuss FX
derivatives in this course.
A financial instrument that implements the concept of the forward exchange rate is
called a forward outright. This is a contract that fixes the rate of exchange between
two currencies in the future at a level that is determined at present.
A financial instrument that simultaneously involves a spot exchange and a forward
outright in the opposite direction is known as a forward swap.
We will discuss these two instruments in order.
4.1 Forward Outrights
Forward outrights are instruments that allow traders to trade currencies in the future
at an exchange rate that is fixed today.
The quotes of the instrument are called forward rates (or forward outright rates).
By the interest rate parity, a forward rate is affected by the spot rate level as well as
the interest rates underlying the currency pair.
Naturally, forward outrights fall into various terms according to when the contract
expires and the trade is supposed to occur.
4.2 Forward Swaps
The forward swap is conceptually the difference between a spot exchange and a
forward outright. Its underlying signal is the forward swap rate.
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Let's look at the formula for the forward swap rate to understand what's going on:
Forward Outright Rate = Spot Rate + Forward Swap Rate.
This means that
Forward Swap Rate = Forward Outright Rate − Spot Rate.
The forward swap as an instrument has the same cash flow profile as a long forward
and a short spot. In other words, it may synthetically be put together with one forward
outright in the long position and one spot in the short position.
Figure 5.1
Example
Given the following information:
Spot EURUSD: 1.4066  1.4071
Forward Swap: 0.0115  0.0120
Find the forward outright.
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For the bid side of the forward outright:
1.4066 + 0.0115 = 1.4181.
For the ask side of the forward outright:
1.4071 + 0.0120 = 1.4191.
We may use the interest rate parity to express the forward swap in terms of the interest
rate differential of the two currencies involved in the exchange.
Let the forward swap be denoted by 𝑤𝑋𝑌 . We have
𝑓𝑋𝑌 = 𝑠𝑋𝑌 + 𝑤𝑋𝑌 ,
which follows from the definition of the forward swap.
We also have
𝑓𝑋𝑌 = 𝑠𝑋𝑌 ×
1 + 𝑟𝑌 𝑇
,
1 + 𝑟𝑋 𝑇
which follows from the interest rate parity.
Equating the RHSs:
𝑠𝑋𝑌 + 𝑤𝑋𝑌 = 𝑠𝑋𝑌 ×
1 + 𝑟𝑌 𝑇
.
1 + 𝑟𝑋 𝑇
This implies that
𝑤𝑋𝑌 =
𝑠𝑋𝑌 (𝑟𝑌 𝑇 − 𝑟𝑋 𝑇)
.
1 + 𝑟𝑋 𝑇
Note that in the formula above, the day count convention must be taken into account
in substituting the appropriate fraction for 𝑇 corresponding to the two currencies X
and Y.
For a quick and dirty method to approximate the swap rate, we may assume that
1 + 𝑟𝑋 𝑇 ≈ 1,
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and therefore,
𝑤𝑋𝑌 ≈ 𝑠𝑋𝑌 𝛥𝑌𝑋 𝑇,
where the interest rate differential 𝑟𝑌 − 𝑟𝑋 is denoted by 𝛥𝑌𝑋 .
Vice versa, we may also express the interest rate differential as
𝛥𝑌𝑋 ≈
𝑤𝑋𝑌
.
𝑠𝑋𝑌 𝑇
Example
Find the forward swap and the approximate swap (using the formulae above) from the following
information:
29day USD interest rate: 5%
29day EUR interest rate: 3%
Spot EURUSD: 1.4055
Using the formula that is derived from the interest rate parity, the forward swap is
given by
1.4055 ×
29
29
0.05 × 360 − 0.03 × 360
29
1 + 0.03 × 360
= 0.0023.
Note that both EUR and USD are ACT/360.
The approximate swap rate is
1.4055 × (0.05 − 0.03) ×
29
= 0.0023.
30
Thus, at 4 decimal places, the approximation is as good as the exact formula.
From the swap rate formula derived from the interest rate parity,
𝑤𝑋𝑌 =
𝑠𝑋𝑌 (𝑟𝑌 𝑇 − 𝑟𝑋 𝑇)
,
1 + 𝑟𝑋 𝑇
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FIN201 STUDY UNIT 5
we see that if the interest rate for the counter currency Y is higher than the interest rate
for the base currency X, then the swap rate is positive. In this case, we say that the base
currency is selling at a premium to the counter currency.
Conversely, if the interest rate for the counter currency Y is lower than the interest rate
for the base currency X, then the swap rate is negative and we say that the base
currency is selling at a discount to the counter currency.
When bidask quotes are involved, swap rates are given in a pair. The pair of swap
rates are then used to calculate the forward exchange rates from spot rates in a peculiar
manner as the following examples illustrate.
Example
If the spot EURUSD is 1.406671 and the 1month swap rate is 2022, find the 1month
forward quote.
The swap rate is interpreted in terms of pips. Thus 2022 means 0.00200.0022.
As the pair is increasing, we will add correspondingly to the spot quote to obtain the
forward quote:
1.4066 + 0.0020  1.4071 + 0.022
or
1.4086  1.4093
Example
If the spot EURUSD is 1.406671 and the 1month swap rate is 2220, find the 1month
forward quote.
The swap rate is interpreted in terms of pips. Thus 2220 means 0.00220.0020.
As the pair is decreasing, we will subtract correspondingly to the spot quote to obtain
the forward quote:
1.4066  0.0022  1.4071  0.0020
or
1.4044  1.4051
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Thus the rule to obtain forward (outright) rates from (forward) swap rates is:
if the swap rates are increasing in the bidask quote, add them to the spot bidask quote to obtain the forward bidask quote
if the swap rates are decreasing in the bidask quote, subtract them from the
spot bidask quote to obtain the forward bidask quote
These sections from the Textbook (and do the associated exercises):
Chapter 9 Forward exchange rates
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Chapter 5 Calculations
5.1 ForwardForward Swap
A forwardforward swap (or simply forwardforward) is a swap deal that involves 2
forward dates. Diagrammatically, it looks like this:
Figure 5.2
This may be interpreted as a combination of two ordinary forward swaps in opposite
directions:
Figure 5.3
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FIN201 STUDY UNIT 5
in the long position and
Figure 5.4
Let's consider the following example.
Example
Given the following quotes:
Name
Rate
EURCHF spot 1.2325 / 35
1month swap
65 / 61
3month swap
160 / 155
If the client wishes to sell EUR one month forward and buy EUR three months forward, what
are the rates involved?
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From the perspective of the dealer, this is equivalent to:
sell EUR spot and buy EUR one month forward
buy EUR spot and sell EUR three months forward
Thus the rate to charge the client is 1.2330 − 0.0065 = 1.2265 for the onemonth
forward swap and 1.2330 − 0.0155 = 1.2175 for the threemonth forward swap. The
midmarket of 1.2330 is chosen for convenience.
In other words, the dealer is to buy EUR for CHF at EURCHF 1.2265 at 1month and
to sell EUR for CHF at EURCHF 1.2175 at 3month.
Just as forward swaps are characterised by the difference between the forward and
spot exchange rates, the forwardforward swap may be characterised by the difference
between the two forward exchange rates, in this case, namely,
1.2175 − 1.2265 = (−0.0155) − (−0.0065) = −0.009.
Conversely, if the client wishes to buy EUR one month forward and sell EUR three
months forward, the dealer will do this:
buy EUR spot and sell EUR one month forward
sell EUR spot and buy EUR three months forward
at the respective exchange rates of 1.2330 − 0.0061 = 1.2269 and 1.2330 − 0.0160 =
1.2170. The difference between them is
(−0.0160) − (−0.0061) = −0.0099.
Thus the price of the forwardforward swap is
99/90.
Note that the negative sign is implied by the reversed ordering of the two numbers as
is the case for forward swaps.
The general rule to remember is this:
Suppose 𝑡 and 𝑇 are two times in the future, with 𝑡 < 𝑇.
Suppose the bid and ask swap rates are 𝑤𝑡𝐵  𝑤𝑡𝐴 and 𝑤𝑇𝐵  𝑤𝑇𝐴 .
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Then the forwardforward bidask rates are
𝑤𝑇𝐵 − 𝑤𝑡𝐴  𝑤𝑇𝐴 − 𝑤𝑡𝐵 .
5.2 NonDeliverable Forwards
In a forward outright, two currencies are exchanged on a future date. Some currencies
may not be easily exchanged, perhaps due to capital controls. In this case, a forward
outright wouldn't be implementable.
Instead, a nondeliverable forward (NDF) would be the suitable instrument to trade
in order to bet on the future movement of the exchange rate.
A NDF is an agreement to exchange two currencies on a certain date 𝑇 at a certain
exchange rate and to counterexchange the same pair of currencies in an offsetting
trade at the spot rate 2 days before 𝑇.
Here's an example.
Example
It is currently 15 April.
A company needs to buy TWD 100 million against USD for settlement in 3 months on 17 July.
It does this with a 3month NDF at the rate of USDTWD 30.06 entered into with a dealer.
Two days before maturity on 15 July, the spot rate is USDTWD 29.43.
If actual cash flows were involved, they would look like this, from the perspective of the
company:
15 July:
Currency
Flow
TWD
 100,000,000
USD
+ 100,000,000 / 29.43
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17 July:
Currency
Flow
TWD
+ 100,000,000
USD
 100,000,000 / 30.06
Netting the cash flows and disregarding the 2day interest, we obtain an inflow of
100,000,000 100,000,000
−
= 71213.33
29.43
30.06
in USD for the company.
In an NDF, actual cash flows do not occur. Instead, the company receives a settlement of USD
71,213.33 on 17 July.
The advantages of an NDF are:
it allows the forward exchange of a hardtotrade currency
it avoids the risk involved in exchanging principal amounts
it can be used for both hedging and speculation
These sections from the Textbook (and do the associated exercises):
Chapter 9 Forwardforwards
Chapter 9 Nondeliverable forwards
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Chapter 6 Analyses and Strategies
It is known in the FX market that when the spot exchange rate nears a round figure
(e.g. 1.1000 as opposed to 1.1155), it becomes more volatile.
The article on trading near a round figure that is to be distributed in class prior to
discussion.
Obtain data from a suitable Financial Information System (e.g. Eikon, internet or
metatrader). Find out if it is true that volatility increases near round figures. If so, are
you able to formulate a suitable trading strategy?
Watch this video for a visualoral presentation of the steps that are required to
compute cross rates in foreign exchange:
Computing Cross Rates
(Access video via iStudyGuide)
Watch this video for a presentation of further worked examples under the topic of
forex:
Worked Examples in Forex
(Access video via iStudyGuide)
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Quiz
1)
A pip for EURUSD is
a. 0.1
b. 0.01
c. 0.001
d. 0.0001
2)
A pip for USDJPY is
a.
b.
c.
d.
3)
Foreign exchange trading begins weekly from
a.
b.
c.
d.
4)
0.1
0.01
0.001
0.0001
Singapore, Singapore
Wellington, New Zealand
Tokyo, Japan
Sydney, Australia
The "value" of the currency is largely determined relative to other international
currencies. This state of affairs is known as
a. the silver standard
b. the gold standard
c. the Bretton Woods system
d. the floating rate regime
5)
An exchange rate that is determined from a chain of other exchange rates
involving the pair at the ends of the chain is known as a/an
a.
b.
c.
d.
cross rate
interest rate
chain rate
swap rate
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FIN201 STUDY UNIT 5
6)
Interest rate parity is a relationship between the interest rates of two currencies
that depends on the spot and forward exchange rates.
a.
b.
7)
Chinese capital controls made _________ an attractive instrument to bet on
movements in exchange rates that involve the CNY.
a.
b.
c.
d.
8)
nondeliverable forwards
Chinese stocks
binary options
currency swaps
A forward outright and a forward swap are different because the former involves
a counter exchange of currencies at initiation.
a.
b.
9)
True
False
True
False
The second most traded currency is
a.
b.
c.
d.
USD
EUR
SGD
DEM
10) There was a significant drop in international foreign exchange activity on 1 Jan
1999 because on that day many currencies ceased to exist and the _________ came
in existence.
a.
b.
c.
d.
Deutsche mark
euro
sterling
none of the above
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Solutions or Suggested Answers
Quiz
Question 1: d
Question 2: b
Question 3: b
Question 4: d
Question 5: a
Question 6: a
Question 7: a
Question 8: b
Question 9: b
Question 10: b
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OPTIONS
FIN201 STUDY UNIT 6
Learning Outcomes
By the end of this unit, you should be able to:
1. Describe significant features of the options market.
2. Compute option prices with the BlackScholes model.
3. Compute option prices with the binomial model.
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FIN201 STUDY UNIT 6
Chapter 1 Market
Options and forward/futures contracts are the primary types of financial derivatives
in the financial markets. Futures contracts were introduced at the Dojima Rice
Exchange (Osaka, Japan) in 1710 (Ref: Wikipedia on Dojima Rice Exchange). Both
types of instruments were used in speculation during tulipmania in Holland in the
1600s (Ref: Economist article: Was tulipmania irrational?). The first organised
exchange in futures contracts that was established in the United States in 1848 was the
Chicago Board of Trade (CBOT), while the first organised exchange in options
contracts established was the Chicago Board Options Exchange (CBOE) in the United
States in 1973. This shows that there is a history to the trading of derivatives,
particularly options.
What are new in the 20th century, therefore, are the advent of information technology,
which affects the trading of all financial instruments, and the invention of
mathematical modelling methods that are used in the pricing and risk management
of options.
At CBOE, trading activities in the options produce prices for the instrument. Off the
exchanges, in the OTC market, where options are also traded, the pricing of options
cannot be found simply by watching the market because it is nonexistent. It is in this
context that mathematical modelling methods play the important role of putting a
price on an option so that it may be rationally and accountably traded and used in
hedging.
CBOE is not the only exchange that trades options. The International Securities
Exchange (ISE) is another example. But if we browse the Products page at CBOE, we
may obtain an idea of the sort of options and optionsrelated contracts that are
standardly traded globally at the moment. I'll list the headlines here:
VIX Index & Volatility
o VIX is the foremost volatility index. It is sometimes called dubbed the
"fear index" for the stock market. When there is a lot of anxiety in the
stock market, VIX goes up. On the contrary, when the market is calm,
VIX comes down. While that is the qualitative manner to understand
VIX, it is in fact precisely stated in terms of a mathematical formula that
involves the prices of options on the S&P 500 index. In other words, the
level of VIX commensurates with the general level of option prices in the
market as it is represented by the prices of a prominent market index.
CBOE introduced the VIX index in the 1990s. After the index has been
firmly established in the psyche of market participants, the exchange
introduced futures and options on VIX in the 2000s.
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SPX & Stock Index Options
o SPX refers to the S&P 500 index. These options have market indices as
underlying. Generally, they are cashsettled and are European in style.
Options on Single Stocks, ETFs & ETNs
o This is the major category of options at CBOE. Generally, such options
are physically delivered (i.e. assetsettled) and are American in style.
Mini Options
o Mini options have smaller sizes (i.e. each price change (in tick)
corresponds to a smaller P&L on a single contract). Mini options allow
more investors to participate in their trades.
Weekly Options
o These options have maturities of a week. This is in contrast with the
main category of single stock options, which have maturities of several
months.
EndofMonth & Quarterly Options
o These are S&P500 index options that have maturity at monthends or
quarterends. They facilitate fund managers who want to match their
endofmonth or endofquarter fund performance with the market
index.
FLEX Options/CFLEX
o These are customisable option contracts.
Strategy Benchmark Indices
o These indices keep track of some basic option strategies.
The description above gives you an idea of some of the key dimensions in options and
how they are used by market participants.
Options are financial instruments that are relatively hard to understand for the general
public. The general media either refer to options as boon or bane, depending on the
mood of the times. From the perspective of economists, derivatives are useful because
they can be applied to the hedging of risk, just as insurance contracts protect us from
highly undesirable states of affairs in our lives. As with all financial instruments, there
is more than one side to the story. Options are also used for speculation.
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1. These sections from the Textbook (and do the associated exercises):
Chapter 11 Overview
Chapter 11 OTC options vs exchangetraded options
2. The history of CBOE at this link.
3. What is Tulipmania? Get a glimpse here.
The CBOE 40th anniversary video describes the development of the institution over
40 years of innovation.
Pay attention to how the options exchange developed over the years. This is
interesting because it reflects the development of the options market in general.
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Chapter 2 Signals and Quotes
The following picture shows what an option chain for the S&P 500 index (Ref: CBOE
delayed quotes) looks like:
Figure 6.1
The tables look complicated but organised. The complication is due to the fact that
options are categorised according to the underlying reference price, rate or index.
Here, the underlying is the S&P 500 index. And corresponding to this single index,
there are many options that are currently traded in the market. These options are
organised according to:
Maturity date
Strike price
Call or put
The statistics in the tables are interesting. It is an empirical fact that options are most
actively traded atthemoney. What this means is that those options whose strike
prices are near to the current level of the underlying have the highest trade volumes
and open interests. Can you find where the strike prices  1615, 1622  are printed
above?
It is also an empirical fact that the activity dies down when the options are near
maturity. This is evident above since this picture is taken in January 2016 and the
maturities of the contracts here are in the same month.
As with all financial instruments that are traded through a middleman, there are bid
and ask prices, opening and closing prices.
One can only holistically make sense of options prices by taking into account the
following:
1. The price of an option comes in a trajectory, not just snapshots of price
points.
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2. An option makes a contractual reference to its underlying. Hence what
happens to the underlying price/rate/index affects the option price greatly.
3. An option does not exist in isolation. It exists within the option chain. Thus,
the behaviour of any one option in an option chain is associated with the
behaviours of other options in the chain.
4. The VIX index provides a background level for option prices, since it is
roughly a weighted average of prices of options on the S&P 500 index and
this index is the most prominent index of the US stock market.
Read the contract specification for equity options at CBOE.
By browsing option chains of the S&P 500 index, pick out patterns of how the index
option prices are distributed. (Sources of option chains data may be found at CBOE's
website, marketwatch.com and Reuters Eikon.)
How does movement in the S&P 500 index affect movement in its option chain? For
example, if there is a 60pointmove in the S&P 500 index over the course of a single
day, what magnitudes of price moves would we see empirically in its option chain
that may be associated with the 60point move?
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Chapter 3 Instruments
An option is a contract between two parties. One party is called the long (i.e. he is in
the long position), while the counterparty is called the short (i.e. he is in the short
position).
An option is a derivate contract, in the sense that it makes a reference to some
underlying asset or index.
There are two pertinent times in the contract  initiation and expiry (or maturity).
The simplest types of contracts are classified by the following:
call or put
cashsettled or physically delivered
European or American
These are all regarded as vanilla as they are commonly found in the market.
At initiation, when a call contract is entered into, the short promises to sell the
underlying to the long at a specified strike price on expiry. Thus, if the underlying
asset is Apple stock, the strike price is $100 and the maturity is 1 month, then the short
is promising the long to sell Apple stock to him at $100 per share 1 month from now.
In more legalistic terms, a call option contract is a right of the long party to buy an
underlying asset at a stipulated strike price on a stipulated date in the future from the
short party.
Analogously, a put option contract is a right of the long party to sell an underlying
asset at a stipulated price on a stipulated date in the future from the short party.
From the perspective of the party who is long call, at maturity, the payoff to his option
position looks like this:
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Figure 6.2
The parameter 𝐾 is the strike price, and 𝑆𝑇 is the underlying price at maturity.
If the underlying price ends at 𝐾 exactly or at a value that is less than 𝐾, then the
payoff to the long party is 0. This is because, the market price is less than the strike
price, there is no reason for the long to buy the underlying from the short since he
could get it from the market at a cheaper price.
If the underlying price ends at a value greater than 𝐾, then the payoff is the excess
amount over the strike price. The reason is this: once the long buys the underlying
from the short at the strike price 𝐾, he can right away sell it in the market at the price 𝑆𝑇 ,
making the amount 𝑆𝑇 − 𝐾 in the procedure.
In the same way, the payoff to a put option looks like this:
Figure 6.3
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Since the long party has the right to sell in a put option contract at the stipulated strike
price, he would only do so if the market price is lower than 𝐾, thereby pocket a profit
that is equal to the deficiency below 𝐾. If the market price were higher than 𝐾, there
is no reason for the party in the long position to sell the underlying to his option
counterparty.
In the scenarios described above, it is assumed the short party will deliver physical
underlying (for stocks, this is some shares of the stock; for commodities, this is a
physical good). In some cases, it is practically impossible to deliver physically  such
is the case for index options. Then cashsettlement is the alternative mode that works.
The cashsettlement procedure builds the "buyunderlyingfromcounterparty/sellunderlyingtothemarket" procedure into the contract.
The final dimension is European/American. European options do not allow the option
holder from exercising the contract any time prior to expiry. On the other hand,
American options allow the option holder to exercise at any time in the life of the
option.
The payoff of call and put options may be mathematically expressed this way:
Call option: max{𝑆𝑇 − 𝐾, 0},
Short option: min{𝐾 − 𝑆𝑇 , 0}.
𝑇 here presents time at maturity. For American options, it's more appropriate to write
these this way:
Call option: max{𝑆𝑡 − 𝐾, 0},
Short option: min{𝐾 − 𝑆𝑡 , 0},
where 𝑡 represents the current time.
If the underlying price is equal to the strike price, the option is said to be atthemoney
(ATM). If the options gives a positive payoff on exercise, it is said to be inthemoney
(ITM). Otherwise, it is said to be outofthemoney (OTM).
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Chapter 4 Concept: Market Price vs Theoretical Price
In economic or financial theories, prices are explained, i.e. there are principles, more
fundamental than the current price, that can explain the current price.
In the market, the price is determined by interactions and constraints. A trade is
conducted at a price, thus producing a price point. Many trades over time produce a
price trajectory. Prices are not quoted arbitrarily. For example, the current price level
has a significant impact on where it will be next. If a price is too high or too low as
compared to the current market price (relative), it's considered too high or too low
(absolute).
When you learn about options and their prices, you must keep this distinction in mind.
When you observe the prices at CBOE, these are market prices. They are not produced
by a calculation, by a person or a computer. They are produced by a process  the
activity of trading.
On the other hand, the academic profession produces theoretical prices to reason with.
These prices are useful in the market when the corresponding instrument is not traded
widely and therefore when it is hard to gauge the price of the instrument.
The wellknown BlackScholes model and the binomial model are theoretical prices.
They do not explain market prices. They are used in the market when market prices
are absent.
Two major principles of theoretical pricing are these:
Discounted expected future payoff
The Principle of No Arbitrage
4.1 Expected Present Value of Future Payoff
Suppose there is a financial instrument that needs to be priced.
Its value is dependent on the state of the world. For example, if the economy is not
performing well, its price may be deemed to be low. And so on.
So suppose there are 𝑛 states of the world: 𝜔1 , 𝜔2 , … , 𝜔𝑛 . These states occur at some
time 𝑇 in the future (e.g. 1 month from now). In each state 𝜔𝑖 , the price of the
instrument is 𝑃𝑖 .
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Put in another way: we know affirmatively what the prices of the instrument will be
in the different scenarios in the future, but we do not know its price presently.
Two uncertainties are pertinent here: the existence of several scenarios (i.e. states), so
that we do not know which scenario will actually arise; and the uncertainty about the
future.
The way theoretical finance solves these problems is as follows.
In order to price something in the future at present, we discount the future cash flow
to the present with a riskfree rate. So suppose 𝑟 is a riskfree interest rate.
Second, to handle the probable outcomes 𝜔1 , 𝜔2 , … , 𝜔𝑛 , we posit that there are 𝑛
corresponding probabilities 𝑝1 , 𝑝2 , … , 𝑝𝑛 , so that
𝑝1 + 𝑝2 + ⋯ + 𝑝𝑛 = 1
and the following table gives the probability distribution of the various outcome states:
State Probability Price / Cash Flow
𝜔1
𝑝1
𝑃1
𝜔2
𝑝2
𝑃2
⋯
⋯
⋯
𝜔𝑛
𝑝𝑛
𝑃𝑛
The present value of the instrument is then expected to be given by the following
expression:
𝑒 −𝑟𝑇 (𝑝1 𝑃1 + 𝑝2 𝑃2 + ⋯ + 𝑝𝑛 𝑃𝑛 ),
i.e. the discounted probabilityweighted future prices.
Note that this is a generalisation of the method of PV as it includes a mechanism for
discounting to the present with an interest rate. The generalisation is due to the
uncertainty by the various states that can occur in the future. This uncertainty is
"removed" or "balanced" by the probability weights and the averaging.
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If the discounting is removed from the expression, then the value is the expected
future value.
Here are some applications to the pricing of financial instruments.
Example
Assume that a security is likely to provide returns over the next year as follows:
Return (R) Probability (p)
6%
0.3
7.5%
0.2
8.5%
0.2
10%
0.3
What is its expected return?
The expected return is calculated as:
4
𝐸(𝑅) = ∑ 𝑝𝑖 𝑅𝑖 = 0.3 × 6% + 0.2 × 7.5% + 0.2 × 8.5% + 0.3 × 10% = 0.08 = 8%.
𝑖=1
Example
There are two possible states of the economy 𝜔1 , 𝜔2 one year from now. In 𝜔1, the price of
Apple stock is deemed to be $90, while in 𝜔2 , it is deemed to be $110. Given that the riskfree
interest rate is 5% and the states are equally likely to occur, find the expected present value of
the stock.
The expected present value is given by (in $)
1
1
𝑒 −0.05×1 × ( × 90 + × 110) = 95.12.
2
2
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4.2 The Principle of No Arbitrage
An arbitrage opportunity is a trading strategy that (1) has no chance of losing money
and (2) has a positive chance of making money.
The Principle of No Arbitrage states that in an efficient market, there is no arbitrage
opportunity.
This is the key principle behind the theoretical pricing of options (more generally, any
financial derivatives) in theoretical economics and theoretical finance.
When applied to option pricing, the result is the BlackScholes option pricing model
and its famous pricing formulae (for vanilla European call and put options).
The way these option pricing formulae are derived goes as follows, in principle: when
the option writer sells an option to the long position, he does not regard it as a bet. He
is all aware that he is under the risk of having to deliver the stock (in the case of a call
option), because if the stock price were to rise at maturity, he is compelled to buy the
stock from the market at a relatively higher price to sell to the long at a relatively
cheaper price. In order to protect himself, he buys a certain number of shares of the
stock to hedge. This hedging strategy is in effect an insurance mechanism and
therefore, like any insurance policy, needs to be charged a premium. The theoretical
BlackScholes option prices give the amount that the option buyer needs to pay to the
option writer at the start of the contract.
As a consequence, note that theoretical option prices are not the present value of
expected future cash flows!
These sections from the Textbook (and do the associated exercises):
Chapter 11 The ideas behind option pricing
To what extent can life insurance premiums be interpreted as the present value of
expected future cash outflows by insurance companies?
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Chapter 5 Calculations
The simplest options are the vanilla, European, call and put options. The standard
mathematical models in finance that are used to price such options are the BlackScholes model and the binomial model.
5.1 The BlackScholes Model
The de facto theoretical pricing model is the BlackScholes Model.
This says that the price of a call option is given by the formula:
𝐶(𝑆, 𝑡) = 𝑁(𝑑1 )𝑆 − 𝑁(𝑑2 )𝐾𝑒 −𝑟(𝑇−𝑡) ,
where
𝑑1
=
𝑆
𝜎2
(ln( ) + (𝑟 + )(𝑇 − 𝑡)),
𝐾
2
𝜎√𝑇 − 𝑡
1
𝑑2 = 𝑑2 − 𝜎√𝑇 − 𝑡
and the price of the put option is given by the formula:
𝑃(𝑆, 𝑡) = 𝑁(−𝑑2 )𝐾𝑒 −𝑟(𝑇−𝑡) − 𝑁(−𝑑1 )𝑆.
Some things to note about the formulae are discussed here.
There are 5 parameters: underlying price 𝑆, strike price 𝐾, riskfree rate 𝑟, time to
maturity 𝑇 − 𝑡 and volatility 𝜎 . The time 𝑡 represents the current time when the
options are priced, and 𝑇 is the time at maturity.
The pricing functions 𝐶(𝑆, 𝑡) and 𝑃(𝑆, 𝑡) have 𝑆 and 𝑡 only in their arguments because
the other parameters 𝐾, 𝑟 and 𝑇 are regarded as fixed (i.e. strike price and maturity
are fixed in the contract, while riskfree rate is relatively stable as compared to stocks
and their option prices).
The function 𝑁(−) is the standard normal cumulative distribution function (CDF).
Hence, there are these relationships arising from their interpretations as probabilities:
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𝑁(𝑑1 ) + 𝑁(−𝑑1 ) = 1,
𝑁(𝑑2 ) + 𝑁(−𝑑2 ) = 1.
Associated to the pricing formula, is the concept of the delta. For the call option, the
delta is given by the formula:
𝛥𝐶 (𝑆, 𝑡) = 𝑁(𝑑1 ),
while for the put option, it is given by
𝛥𝑃 (𝑆, 𝑡) = 𝑁(𝑑1 ) − 1.
These formulae were originally derived by Fisher Black and Merton Scholes in 1973
with the interpretation given below:
Step 1: Initiation
An option buyer buys an option from an option writer and pays him an option
premium, i.e. either 𝐶(𝑆0 , 0) or 𝑃(𝑆0 , 0), where 𝑆0 is the stock price at time 0, the
initiation of the contract.
Step 2: Hedging the short option position
The option writer hedges his position by maintaining this portfolio:
𝛥𝑂 (𝑆𝑡 , 𝑡) × 𝑆𝑡 + Cash.
𝑂 here stands for 𝐶 (call) or 𝑃 (put), depending on whether he has sold a call or put
option, and 𝑆𝑡 stands for the stock price at time 𝑡. He holds 𝛥𝑂 (𝑆, 𝑡) shares of the stock.
And he holds a certain amount of cash with the bank.
Delta is computed based on the formula given above. Everyday, as the stock price
changes, the option writer needs to recompute Delta. That will help him determine
the new number of shares that he should hold according to the BlackScholes Theory.
If Delta goes up, he needs to withdraw or borrow cash from the bank to buy shares. If
Delta goes down, he needs to sell shares to reduce his holding, and deposit the
proceeds into the bank.
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Step 3: At expiry
At expiry, the BlackScholes Theory guarantees that the option writer has exactly 1
share of stock to deliver to the option buyer if it's a call option that's inthemoney, or
the cash to pay to the option buyer if it's a put option that's inthemoney.
The key point is this: if the option writer charges the option buyer a premium that is
given by the BlackScholes formula, and if he then hedges his position in accordance
to the BlackScholes dynamic hedging recipe listed down here, he will be fully and
exactly hedged at expiry of the contract. An option writer in a bank will make money
from providing this service by charging a fee that is added on top of the premium.
5.2 The Binomial Model
The binomial model is an approximation of the BlackScholes model. The answer that
it produces becomes closer and closer to the BlackScholes model answer if the tree in
the model becomes more refined (i.e. has more steps).
The input to the binomial model consists of the following parameters:
• initial stock price 𝑆0
• strike price 𝐾
• time to maturity 𝑇
• number of time steps 𝑛
• up factor 𝑢
• down factor 𝑑
• riskfree interest rate 𝑟
• call or put
We will consider only the case of 𝑛 = 1 here, i.e. the future is represented by only 1
time step:
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Figure 6.4
The single node on the left represents the situation at time 0, i.e. now.
The two nodes on the right represent two possible states at time 1, i.e. future.
In deriving the option price, there will be two passes, which are called forward
induction and backward induction.
Step 1: Forward induction
Fill the initial node with the initial stock price 𝑆0 .
As it evolves into the future, time 1, it can either go up or come down.
It goes up by multiplication by the upfactor 𝑢, and it comes down by multiplication
by the down factor 𝑑.
Thus we obtain a diagram like this:
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Figure 6.5
Step 2: Evaluate the option payoff
Given the states at time 𝑇 = 1, the option payoff at maturity would be determined. In
other words, though we may not know that option payoff would be in the future, we
will know what it will be provided we are told what the stock price will be. This is
simply because the contract ties the option payoff to the value of the stock price at
maturity.
And the model posits that there are two possibilities: the stock price can be either 𝑢𝑆0
or 𝑑𝑆0 .
Let's assume that the option is a call, so its payoff formula is given by
max(𝑆 − 𝐾, 0),
and we obtain the following picture after evaluating with the stock price:
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Figure 6.6
We have used the notation 𝐶1 (𝑢) to denote the call option price at time 1 when the
state is "up", and 𝐶1 (𝑑) to denote the call option price at time 1 when the state is
"down".
Step 3: Backward induction
We apply the following formula to the two payoffs above to obtain the option price at
present:
𝑇
𝐶0 = 𝑒 −𝑟𝑛 (𝑝̃𝐶1 (𝑢) + 𝑞̃𝐶1 (𝑑)),
where 𝐶0 denotes the call option price time time 0, i.e. at present, and
𝑇
𝑝̃
𝑒 𝑟𝑛 − 𝑑
=
𝑢−𝑑
𝑞̃ = 1 − 𝑝̃
This formula is known as the riskneutral pricing formula. It is analogous to the
method of the discounted expected present valuation in form, but the probability
weights 𝑝̃ and 𝑞̃ are not real probabilities  they are called risk neutral probabilities. (If
you find the terminology very strange, it's OK, because it is indeed very strange. These
concepts arise from theoretical reasoning in academic finance, in particular, from the
Principle of No Arbitrage. They don't necessarily have to relate to anything that we
may deem real from the financial markets.)
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At this point, we have finally found the present value of the option price.
On diagram, this looks like:
Figure 6.7
The entire argument can be compressed into one step as the following example shows.
Example
Given the following parameters:
initial stock price 𝑆0 = 100
strike price 𝐾 = 100
up factor 𝑢 = 2
down factor 𝑑 = 0.5
riskfree interest rate 𝑟 = 0.05
Find the prices of the 1year expiry call and put options by using the 1step binomial model.
In the forward induction step, the two possibilities for the stock price at time 1 are: 200
and 50.
For the call option, the corresponding payoffs are: 100, 0
For the put option, the corresponding payoffs are: 0, 50
The risk neutral probabilities are:
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1
𝑝̃
𝑒 0.05×1 − 0.5
=
= 0.367514064251,
2 − 0.5
𝑞̃ = 1 − 𝑝̃ = 0.632485935749.
The current price of the call option is therefore:
1
𝐶0 = 𝑒 −0.05×1 (𝑝̃ × 100 + 𝑞̃ × 0) = 34.9590191833.
The current price of the put option is therefore:
1
𝑃0 = 𝑒 −0.05×1 (𝑝̃ × 0 + 𝑞̃ × 50) = 30.0819616334.
These sections from the Textbook (and do the associated exercises):
Chapter 11 Pricing models
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Chapter 6 Analyses and Strategies
There are several ways options may be traded. They are listed/categorised here:
As a standalone
Together with its underlying asset
In a combination with other options
As in any financial position, there are 2 reasons why one may enter into a trading
position: for hedging or for speculation. In addition, as options have expiries, one may
hold an option until maturity or close the position prior to expiry (which may be easily
achieved in a liquid market such as in an exchange setting).
As a standalone, there are 4 possibilities, as one may:
1.
2.
3.
4.
long a call
long a put
short a call
short a put
Let me analyse some strategies here.
When one longs a call, one gains when the price of the call option rises. Since in turn,
when the underlying stock rises in price, the call option price also rises, longing a call
may be viewed as a trading strategy that bets on rising stock price. It is an alternative
to buying a stock that is relatively cheaper (since the price of an option is on the order
of about 10% of the stock price) and at the same time, a protected form of investment
since drastic price fall in the stock has little effect on the option price.
When one longs a put option, one gains when the stock price falls. Thus, this is a
trading position that is entered into to either bet on the fall of a stock price or to hedge
against the fall of the stock price. Shorting a stock directly is generally frowned upon
or disallowed. Taking a long position in put option is an alternative to shorting.
When one longs a stock and shorts a call option, the position is called a covered call.
This position may be interpreted this way. Suppose that we own a stock and its price
is currently stagnant. We may sell a call option to earn the premium in the meanwhile.
If our expectation of stagnancy remains, then we would earn the premium. Otherwise,
if the price of the stock suddenly rises to the extent that it crosses the strike, then the
inthemoney option will require us to deliver the stock at the strike price. Since we
hold the stock for delivery, the position is said to be "covered".
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FIN201 STUDY UNIT 6
Option strategies are thus put together to achieve certain trading, hedging or
investment goals. To guide us in formulating these goals, we have to remember what
the payoff profile of the call and put options looks like:
Figure 6.8
Figure 6.9
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FIN201 STUDY UNIT 6
These payoff diagrams are from the perspective of the long position and they include
the upfront option premium.
Thinking about how strategies work is to consider what happens to the payoff when
the underlying price ends up at option maturity at the various locations relative to the
strike price 𝐾.
Combinations of options allow payoff diagrams of other shapes to be constructed.
These shapes may be used in a bet that stock price will be in certain ranges at option
expiry.
These sections from the Textbook (and do the associated exercises):
Chapter 11 Trading with calls and puts
Chapter 11 Hedging with options
Nick Leeson is wellknown to have brought down the venerable Barings Bank which
was founded
in 1762. He took big bets on the Nikkei 225 index by shorting options.
If Leeson was neutral to the direction of the market, he was likely to have sold equal
numbers of call and put options on the index.
1. What is the resulting position commonly called?
2. Under what condition would he have profited from the position?
3. Use historical data on Nikkei 225 options to analyse the beforeandafter of
4. the option strategy around Jan 17, 1995.
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FIN201 STUDY UNIT 6
Watch this video to appreciate the key concepts behind the theory of option pricing:
The Idea behind Option Pricing
(Access video via iStudyGuide)
Watch this video to see how to solve problems with the binomial model:
The Binomial Model
(Access video via iStudyGuide)
SU625
FIN201 STUDY UNIT 6
Quiz
1)
The CBOE was founded in
a.
b.
c.
d.
2)
Which of the following is not a parameter in the BlackScholes options model?
a.
b.
c.
d.
3)
volatility
interest rate
strike price
option price
_________ options are relatively cheaper.
a.
b.
c.
d.
4)
1970
1971
1972
1973
Inthemoney
Atthemoney
Outofthemoney
Makealotofmoney
If one takes on a long position in a put option, one is _________ on the underlying.
a. bearish
b. bullish
c. ambivalent
5)
An owner of an American option has _________ choices, namely, _________.
a.
b.
c.
d.
6)
1; wait for expiry
1; exercise opportunistically
2; wait for expiry or exercise opportunistically
3; trade it, wait for expiry or exercise opportunistically
The Principle of NoArbitrage states that there is no investment that has both a
positive chance of making profits and a positive chance of making losses.
a. True
b. False
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FIN201 STUDY UNIT 6
7)
The 3 steps of implementation of the binomial model are _________, evaluation
of option payoff and _________.
a.
b.
c.
d.
8)
A futures contract does not require this but an option contract requires this. This
is
a.
b.
c.
d.
9)
reflection, action
digestion, execution
forward induction, backward induction
deduction, induction
premium
foresight
hindsight
cost
When the stock market is full of fear and anticipation, prices in the options market
will
a.
b.
c.
d.
fall
rise
be unaffected
rise and fall in regular succession
10) The implied volatility is the volatility parameter in the BlackScholes model that
is found by substituting market option price into the BlackScholes pricing
formula.
a. True
b. False
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FIN201 STUDY UNIT 6
Solutions or Suggested Answers
Quiz
Question 1: d
Question 2: d
Question 3: c
Question 4: a
Question 5: d
Question 6: a
Question 7: c
Question 8: a
Question 9: b
Question 10: a
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