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EF3320
Security Analysis and Portfolio Management
Semester B 2009 - 2010
Dr. Anson C. K. Au Yeung
City University of Hong Kong
2
Staff Information Instructor: Dr. Anson C. K. Au Yeung Office: P7315 Phone: 2194-2163 Email: [email protected] Office Hours: By Appointment Class Schedule CA1: Wednesday 13:30 – 16:20; Venue LT-16 CA2: Friday 13:30 – 16:20; Venue P2633 CA3: Monday 15:30 – 18:20; Venue P4802 Course Objectives This course is aimed to provide basic investment theories and applications. After the course is completed, students are expected to apply fundamental principles in portfolio investments. More precisely, students will get familiar with basic financial instruments for adequate applications of investment strategies and portfolio management. References 1. Course Package 2. Bodie, Kane and Marcus, Investments (8th Edition), McGraw Hill 2009. [BKM] Assessment Coursework + Midterm (30%) Examination (70%)
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Course Outline Topic 1 – Expected Utility and Risk Aversion
Readings & References: BKM Chapter 6 (Appendix A)
Topic 2 – Review of Mathematics and Statistics Topic 3 – Risk and Return
Readings & References: BKM Chapter 5
Topic 4 – Portfolio Theory
Readings & References: BKM Chapter 6, 7
Topic 5 – Capital Asset Pricing Model (CAPM)
Readings & References: BKM Chapter 9, 13
Topic 6 – Factor Models
Readings & References: BKM Chapter 8, 10, 13
Topic 7 – Arbitrage Pricing Theory (APT)
Readings & References: BKM Chapter 10, 13
Topic 8 – Anomalies and Market Efficiency
Readings & References: BKM Chapter 11
Topic 9 – Fixed Income Securities
Readings & References: BKM Chapter 14, 16
Topic 10 – Term Structure of Interest Rates
Readings & References: BKM Chapter 15
4
Contents
1 Topic 1 – Expected Utility and Risk Aversion ............................................................... 7
1.1 How to Price a Security? ......................................................................................................... 7
1.1.1 Expected Payoffs............................................................................................................. 7
1.1.2 St. Petersburg Paradox .................................................................................................... 8
1.2 Expected Utility Theory ........................................................................................................ 10
1.2.1 The Axioms of Preference ............................................................................................ 10
1.2.2 Utility Functions and Indifference Curve ..................................................................... 10
1.3 Risk Aversion ........................................................................................................................ 11
1.3.1 Measuring Risk Aversion .............................................................................................. 13
2 Topic 2 – Review of Mathematics and Statistics .......................................................... 15
2.1 Random Variables ................................................................................................................. 15
2.2 Moments ............................................................................................................................... 15
2.3 Comoments ........................................................................................................................... 18
2.4 Properties of Moments and Comoments ............................................................................... 19
2.5 Linear Regression ................................................................................................................. 19
2.6 Calculus and Optimization .................................................................................................... 20
2.6.1 Functions ....................................................................................................................... 20
2.6.2 Limits ............................................................................................................................ 22
2.6.3 Differentiations ............................................................................................................. 23
2.6.4 Optimizations ................................................................................................................ 25
3 Topic 3 – Risk and Return ............................................................................................. 28
3.1 The Definition of Return ....................................................................................................... 28
3.2 The Definition of Risk .......................................................................................................... 29
3.3 The History of U.S. Return ................................................................................................... 30
3.4 International Evidence .......................................................................................................... 31
3.5 Real and Nominal Rates of Interest ...................................................................................... 33
4 Topic 4 – Portfolio Theory ............................................................................................. 35
4.1 Portfolio Risk and Return ..................................................................................................... 36
4.1.1 Portfolio of Two Assets ................................................................................................ 36
4.1.2 Portfolio of Multiple Assets .......................................................................................... 39
4.2 Diversification ....................................................................................................................... 40
4.3 Optimal Portfolio Selection .................................................................................................. 43
4.3.1 Case 1: One Risky Asset + Risk Free Asset ................................................................. 44
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4.3.2 Case 2: Two Risky Assets + Risk Free Asset ............................................................... 50
4.3.3 Case 3: More than Two Risky Assets + Risk Free Asset .............................................. 63
4.4 Appendix – Portfolio Analysis Using Excel ......................................................................... 67
4.4.1 Mean-Variance Efficient Portfolio ................................................................................ 67
5 Topic 5 – Capital Asset Pricing Model (CAPM) .......................................................... 73
5.1 The Market Portfolio ............................................................................................................. 74
5.2 Derivation of the CAPM ....................................................................................................... 76
5.3 Implications of the CAPM .................................................................................................... 79
5.3.1 Two Important Graphs .................................................................................................. 80
5.3.2 Replicating the Beta ...................................................................................................... 83
5.4 Risk in the CAPM ................................................................................................................. 84
5.4.1 CML and SML: A Synthesis ......................................................................................... 85
5.5 Estimating Beta ..................................................................................................................... 85
5.6 Empirical Tests of the CAPM ............................................................................................... 87
5.6.1 Time-series Tests of the CAPM .................................................................................... 87
5.6.2 Cross-sectional Tests of the CAPM .............................................................................. 89
6 Topic 6 – Factor Models ................................................................................................. 90
6.1 Single Factor Model .............................................................................................................. 90
6.1.1 The Market Model Return Decomposition ................................................................... 91
6.1.2 The Market Model Variance Decomposition ................................................................ 91
6.1.3 The Inputs to Portfolio Analysis ................................................................................... 92
6.1.4 The Market Model and Diversification ......................................................................... 92
6.2 Multifactor Models ............................................................................................................... 94
6.2.1 Factor Models for Portfolios ......................................................................................... 95
6.2.2 Tracking Portfolios ....................................................................................................... 96
6.2.3 Pure Factor Portfolio ..................................................................................................... 98
6.2.4 Risk Premiums of Pure Factor Portfolios ...................................................................... 99
7 Topic 7 – Arbitrage Pricing Theory (APT) ................................................................ 101
7.1 Derivation of the APT ......................................................................................................... 101
7.1.1 Single Factor APT ....................................................................................................... 101
7.1.2 Two-Factor APT ......................................................................................................... 103
7.1.3 Multifactor APT .......................................................................................................... 105
7.2 Comments on APT .............................................................................................................. 105
7.2.1 Strength and Weaknesses of APT ............................................................................... 105
7.2.2 Differences between APT and CAPM ........................................................................ 105
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8 Topic 8 – Anomalies and Market Efficiency .............................................................. 106
8.1 CAPM and the Cross-section of Stock Returns .................................................................. 106
8.2 Size Effect ........................................................................................................................... 107
8.3 Value Effect ........................................................................................................................ 110
8.3.1 The Glamour and Value Strategies ............................................................................. 111
8.4 Momentum Investing Strategies ......................................................................................... 115
8.5 Efficient Market Hypothesis ............................................................................................... 117
8.5.1 Empirical Tests of Efficient Market Hypothesis ......................................................... 118
9 Topic 9 – Fixed Income Securities............................................................................... 120
9.1 Fixed Income Securities and Markets ................................................................................. 120
9.1.1 Types of Fixed Income Securities ............................................................................... 120
9.2 Bond Pricing ....................................................................................................................... 121
9.2.1 Coupon Bond .............................................................................................................. 121
9.2.2 Zero-Coupon Bond ..................................................................................................... 124
9.3 Dirty Price, Clean Price and Accrued Interest .................................................................... 124
9.4 Conventional Yield Measures ............................................................................................. 128
9.4.1 Current Yield .............................................................................................................. 128
9.4.2 Yield-to-Maturity ........................................................................................................ 128
9.5 Default Risk ........................................................................................................................ 131
9.5.1 Traditional Credit Analysis ......................................................................................... 131
9.6 Interest Rate Risk ................................................................................................................ 132
9.6.1 Price Volatility and Bond Characteristics ................................................................... 132
9.6.2 Duration ...................................................................................................................... 134
9.6.3 The Determinants of Duration .................................................................................... 138
9.6.4 Convexity .................................................................................................................... 141
9.6.5 Immunization .............................................................................................................. 143
10 Topic 10 – Term Structure of Interest Rates ............................................................. 145
10.1 The Yield Curve .................................................................................................................. 145
10.1.1 Using the Yield Curve to Price a Bond ....................................................................... 146
10.1.2 Constructing the Theoretical Spot-Rate Curve ........................................................... 146
10.2 Spot and Forward Interest Rates ......................................................................................... 148
10.3 Theories of the Term Structure ........................................................................................... 149
10.3.1 The Expectation Hypothesis ....................................................................................... 150
10.3.2 Liquidity Preference .................................................................................................... 151
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1 Topic 1 – Expected Utility and Risk Aversion
1.1 How to Price a Security? One of the most important questions in Finance is how to price a security. In Economics, we know the price of a good can be determined by its demand and supply. Can we follow this approach to price a security? Unlike a good, however, a security does not provide an immediate consumption benefit to you. Rather, it is a saving instrument in which the future payoff is random. In order to study the demand of these uncertain future payoffs, we need a theory to understand investors’ preferences. Our goal here is to understand how investors choose between different securities that have different risks and returns.
1.1.1 Expected Payoffs One possible measure is to assume that investors value risky investment based on expected payoffs.
Example 1.1
Which investment would you choose? Investment A Investment B
Payoff Probability Payoff Probability 15 1/3 20 1/3 10 1/3 12 1/3 5 1/3 4 1/3
The expected payoff of:
Investment A 1 1 115 10 5 103 3 3
= × + × + × =
Investment B 1 1 120 12 4 123 3 3
= × + × + × =
However, the expected payoff is unlikely to be the only criterion.
8
1.1.2 St. Petersburg Paradox You are invited to play a coin-toss game. To enter the game, you have to pay an entry fee. Thereafter, a coin is tossed until the first head appears. The number of tails that appears until the first head is tossed is used to compute your payoff. The probabilities and payoffs for various outcomes:
No of Tails Probability (p) Payoff (x) Probability × Payoff 0 1/2 $1 $1/2 1 1/4 $2 $1/2 2 1/8 $4 $1/2 3 1/16 $8 $1/2
…
…
…
…
n (1/2)n+1 $2n $1/2 How much would you be willing to pay for this game based on its expected value? The expected payoff:
( )
1
1 1 1 11 2 4 82 4 8 161 1 1 12 4 82 2 4 81 1 1 12
i ii
p x∞
=
= ×
= × + × + × + × +
= × + × + × +
= + + +
= ∞
∑
The St. Petersburg Paradox is that the expected value of this game is infinite, but probably you will only pay a moderate, not infinite, amount to play this game. To resolve this paradox, we borrow the concept of expected utility. The insight is that you do not assign the same value per dollar to all payoffs. Specifically, the extra dollar of wealth should increase utility by progressively smaller amounts.
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Graphically, your utility is increasing at a decreasing rate with your wealth. In words, you exhibit diminishing marginal utility of wealth.
Example 1.2
Suppose your utility function is ( ) ( )U x ln x= . What is your expected utility of the coin-toss
game?
( ) ( )
( )( )
( )( )
( )
01
0
0
12
212
1 22
2 12 2
22 1
20 69
i ii
ii
i
i
i
E U x p ln x
ln
lni
ln
ln.
∞
=
+∞
=
∞
=
= ×
= ×
= ×
=
− =
=
∑
∑
∑
The expected utility value is indeed finite, 0.69. And the dollar amount to yield this utility value is $2, which is your maximum amount that you will pay for this game.
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1.2 Expected Utility Theory Expected utility theory is used to explain choice under uncertainty. To develop a theory of rational decision making under uncertainty, we need some precise assumptions about an individual’s behavior.
1.2.1 The Axioms of Preference Axiom 1 – Comparability A person can state a preference among all outcomes. If the person has a choice of outcome A or B, a preference for A to B or B to A can be stated or indifference between them can be expressed. Axiom 2 – Transitivity If a person prefers A to B and B to C, then A is preferred to C. This is an assumption that people are consistent in their ranking of outcomes. Axiom 3 – Independence If a person were indifferent between having a Canon or a Nikon camera, then that person would be indifferent to buying a lottery ticket for $10 that gave 1 in 500 chance of winning a Canon camera or a lottery ticket for $10 that also gave 1 in 500 chance of winning a Nikon camera. Axiom 4 – Certainty Equivalent For every gamble there is a value such that the investor is indifferent between the gamble and the value. The axioms of preference allow us to map the preference into measurable utility function.
1.2.2 Utility Functions and Indifference Curve The utility function is important for us to understand the choice and tradeoff. If we choose among various goods such as apple and orange, the indifference curve is downward sloping. Within the same utility, if we consume fewer apples, in order to stay on the same indifference curve, we have to consume more oranges. Both apple and orange provide positive utility, and therefore, they can substitute for each other. However, in investment, expected return provides positive utility, while risk provides disutility. They cannot substitute for each other. When an investor is asked to take on more risk, he has to be compensated by a higher expected return. That is why the indifference curve is upward sloping.
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Noted that: U3 > U2 > U1
1.3 Risk Aversion The axioms convert preference into a utility function. We can make use of the utility functions to establish a definition of risk aversion. Let’s consider a simple game with two payoffs. • Option A: $100,000 for certain. • Option B: $150,000 if “Head” and $50,000 if “Tail” is tossed.
The expected payoff: ( ) ( )150,000 1 50,000
0.5 150,000 0.5 50,000100,000
E W p p= × + − ×
= × + ×=
Which options do you prefer? The choice will depend on your risk attitude. If you are: • Risk averse, you will prefer option A. • Risk neutral, you will be indifferent between option A & B. • Risk loving, you will prefer option B.
Head = $150,000
Tail = $50,000
p
1 – p
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Graphically, we can use three simple utility functions to demonstrate the idea of risk aversion.
Example 1.3
Suppose the utility function is ( ) ( )U W ln W= . What is the expected utility from the risky
payoff of the simple game?
( ) ( ) ( ) ( )( ) ( )
1 21
0 5 50 000 0 5 150 00011 37
E U W pU W p U W
. ln , . ln ,.
= + − = +
=
Graphically, the log function is consistent with the risk aversion. The utility value of option A is ln(100,000) = 11.51, which is greater than the expected utility. Hence, the risk averse investor will prefer option A than B.
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1.3.1 Measuring Risk Aversion Having established the concept of risk aversion, we can further examine an individual’s behavior in the face of risk. Intuitively, we may suspect the risk attitude of a person is linked to his wealth. Absolute Risk Aversion (ARA) The definition:
( )( )
U WARA
U W′′
= −′
Condition Property Implication
Increasing ARA ( ) 0d ARA WdW
> As wealth increases, hold fewer dollars in risky assets.
Constant ARA ( ) 0d ARA WdW
= As wealth increases, hold equal dollars in risky assets.
Decreasing ARA ( ) 0d ARA WdW
< As wealth increases, hold more dollars in risky assets.
Example 1.4
Suppose the utility function of an investor is ( ) CWU W e−= − , identify his type of absolute
risk aversion.
( ) ( )
( ) ( )
( )( )
( )
2
2
0
CW CW
CW CW
CW
CW
dU W e CedWdU W Ce C e
dW
U W C eARA CU W Ce
ARA W
− −
− −
−
−
′ = − =
′′ = = −
′′ −= − = − =
′
′ =
An investor with constant ARA cares about absolute losses. For example, he will always pay $100 to avoid a $1,000 fair bet, regardless of his level of wealth.
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Relative Risk Aversion (RRA) The definition:
( )( )
U WRRA W
U W′′
= −′
Condition Property Implication
Increasing RRA ( ) 0d RRA WdW
> Percentage invested in risky assets declines as wealth increases.
Constant RRA ( ) 0d RRA WdW
= Percentage invested in risky assets is unchanged as wealth increases.
Decreasing RRA ( ) 0d RRA WdW
< Percentage invested in risky assets increases as wealth increases.
Example 1.5
Suppose the utility function of an investor is ( ) 2U W W bW= − , identify his type of relative
risk aversion.
( ) ( )
( ) ( )
( )( )
( )( )
2
2
1 2
1 2 2
2 21 2 1 2
2 01 2
dU W W bW bWdWdU W bW b
dW
U W b bWRRA W WU W bW bW
bRRA WbW
′ = − = −
′′ = − = −
′′ −= − = − =
′ − −
′ = >−
In general, most investors exhibit decreasing absolute risk aversion. There is less agreement concerning relative risk aversion.
15
2 Topic 2 – Review of Mathematics and Statistics
2.1 Random Variables Consider two random variables: x and y State 1 2 … n Probability p1 p2 … pn Value of x x1 x2 … xn Value of y y1 y2 … yn
11
n
ii
where p=
=∑
2.2 Moments Mean: the expected value of a random outcome.
( )1
n
i ii
E x x p x=
= = ×∑
Variance: the dispersion of the squared deviation of the realized outcome from its mean.
( ) ( )22
1
n
x i ii
Var x p x xσ=
= = × −∑
Standard deviation: the volatility of a random outcome.
( ) ( )xStd x Var xσ= =
Example 2.1
State Bull Bear Crisis Probability 0.5 0.3 0.2
1r 25% 10% -25%
2r 1% -5% 35% Mean: ( )( )
1
2
0.5 25% 0.3 10% 0.2 25% 10.5%
0.5 1% 0.3 5% 0.2 35% 6%
E r
E r
= × + × + ×− =
= × + ×− + × =
16
Variance:
[ ] [ ] [ ]
[ ] [ ] [ ]
2 2 221
2 2 222
0.5 25% 10.5% 0.3 10% 10.5% 0.2 25% 10.5%3.57
0.5 1% 6% 0.3 5% 6% 0.2 35% 6%2.17
σ
σ
= − + − + − −
=
= − + − − + −
=
Standard deviation:
1
2
18.9%14.7%
σσ
==
In real world data analysis, the mean and variance of a random variable are almost never known, but rather be estimated from a sample. Sample Mean:
1
1 N
ii
x xN =
= ∑
Sample Variance:
( ) ( )22
1
11
N
ii
s x x xN =
= −− ∑
Employing N – 1, instead of N, as the denominator gave an unbiased estimate of the population variance. To prove that the sample variance estimator is unbiased, we have to show that ( )2 2E s σ= .
Proof: Recall that sample variance,
( )22
1
11
N
ii
s x xN =
= −− ∑
Take expectation,
( ) ( )
( )
22
1
2
1
11
11
N
ii
N
ii
E s E x xN
E x xN
=
=
= − − = − −
∑
∑
17
Expand the squared terms,
( ) ( ) ( )2 2 2
1
2 2
2 2
1 2
2
2
N
i ii
i i
i i
N E s E x x x x
E x E x x E x
E x E x x E x
=
− = − + = − + = − +
∑
∑ ∑ ∑∑ ∑ ∑
Given that ix N x= ⋅∑ ,
( ) ( )
( )
2 2 2
2 2 2
2 2
2 2 2
1 2
2
1
i i
i
i
i
N E s E x E x x E x
E x N E x N E x
N E x N E x
N E s E x E xN
− = − + = − ⋅ + ⋅ = ⋅ − ⋅
− = −
∑ ∑ ∑∑
Apply the property that ( ) ( ) 22( )Var y E y E y= − ,
( ) ( )
( )
( )
22
2
22
22
2 22
2 2
1
1
1
1
1
i
i
i
E x Var x E x
Var x xN
Var x xN
Var x xN
xN
xN
σ
σ
= + = +
= +
= +
= +
= +
∑
∑
∑
∑
Substitute it into previous equation,
( )
( )
( )
2 2 2
2 2 2 2
2
2 2
1
1
1
iN E s E x E x
N
x xN
NN
E s
σ σ
σ
σ
− = −
= + − +
−=
=
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2.3 Comoments Covariance: a measure of how much the two random outcomes varies together.
( ) ( )( )1
,N
xy i i ii
Cov x y p x x y yσ=
= = × − −∑
Correlation: a standardized measure of covariation.
( ), xyxy
x y
Corr x yσ
ρσ σ
= =
• xyρ must lie between –1 to +1.
• If 1xyρ = + , the two random outcomes are perfectly positively correlated.
• If 1xyρ = − , the two random outcomes are perfectly negatively correlated.
• If 0xyρ = , the two random outcomes are uncorrelated.
• If one outcome is certain, then 0xyρ = .
Example 2.2
State Bull Bear Crisis Probability 0.5 0.3 0.2
1r 25% 10% -25%
2r 1% -5% 35% With mean and standard deviation,
1 1
2 2
10.5%, 18.9%6%, 14.7%
rr
σσ
= == =
Covariance:
( )( )( )( )( )( )
1 20.5 25% 10.5% 1% 6%
0.3 10% 10.5% 5% 6%
0.2 25% 10.5% 35% 6%0.02405
r rσ = − −
+ − − −
+ − − −
= −
Correlation:
1 2
0.024050.189 0.1470.8656
r rρ −=
×= −
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2.4 Properties of Moments and Comoments Let a and b be two constants.
( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
2
2 2
,
2 ,
, , ,
, ,
E ax aE x
E ax by aE x bE y
E xy E x E y Cov x y
Var ax a Var x
Var ax by a Var x b Var y ab Cov x y
Cov x y z Cov x z Cov y z
Cov ax by ab Cov x y
=
+ = +
= × +
=
+ = + +
+ = +
=
2.5 Linear Regression An intuitive example of least square: • Consider a special case of linear regression: i i iy xβ ε= + so that the population
regression line ( )|E y x xβ= is a ray passing through the origin ( )0,0 .
• We have two pairs of observations, ( ) ( )1 1, 1,1x y = and ( ) ( )2 2, 1, 2x y = .
How do we fit a regression line by suitably choosing β that is meant to represent the data?
1, 1
1, 2
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2
y
x
y = 1.5xy = 1.75x
20
• The line 1.75x is biased towards the observation ( )1,2 .
• The line 1.5x is representative because it passes through halfway between the two observations.
In fact, we can apply the least square principle by choosing β to minimize the residual sum of squares.
( )
( ) ( ){ }( ) ( )
22
2 2
2
min min
min 1 1 2 1
2 1 2 2 0
1.5
i i ii i
y xβ β
β
ε β
β β
εβ β
ββ
= −
= − ⋅ + − ⋅
∂= − − − − =
∂⇒ =
∑ ∑
∑
In general, the relation between two random variables y and x : y xα β ε= + + ,
( )( )
( )( )
( )1
2
1
,ˆ
ˆˆ
n
i ii
n
ii
x x y yCov y xVar x x x
y x
β
α β
=
=
− −= =
−
= −
∑
∑
Note that by assumption: • ε has zero mean: ( ) 0E ε = .
• ε is uncorrelated with x: ( ), 0Cov x ε = .
2.6 Calculus and Optimization Calculus and Optimization are basic concepts to finance theory. In this brief review, we shall summarize the main concepts including: (A) functions, (B) limits, (C) differentiations and (D) optimizations.
2.6.1 Functions A fundamental notion used in finance is the concept of a function. There are three ways to express functions: as (1) mathematical equations, (2) graphs, and (3) tables.
21
Example 2.3
Suppose a variable Y is related to a variable X by the following mathematical equation: 22 3 6Y X X= − +
A shorthand way of expressing this relationship is to write ( )Y f X= , which is read “Y is a
function of the variable X”.
We can also express the function in a tabular and graphical manner. Thus the equation enables us to construct a range of Y values for a given table of X values. The data in the table can then be plotted in a graph.
X Y -2 20 -1 11 0 6 1 5 2 8 3 15 4 26
Example 2.4 From basic capital budgeting concepts, we know that the net present value (NPV) of an investment project is equal to:
( ) 01
0
1:
cash flow in time period the project's initial cash outlaythe firm's cost of capitalthe number of years in the project
Nt
tt
t
CFNPV Ir
whereCF t
Ir
N
=
= −+
====
∑
22
We can express this relationship functionally as: ( )0, , ,tNPV f CF I r N=
Given values for the right-hand-side independent variables, we can determine the left-hand-side dependent variable, NPV. The functional relationship tells us that for every X that is in the domain of the function a unique of Y can be determined.
2.6.2 Limits In finance, it is so important to study the effect on changes of the independent variables on the dependent variable. For example, what happens with the asset return if the market risk premium increases? Assume there is a simple function:
( )Y f X=
If X changes from X0 to X1 , then we can say:
1 0X X X∆ = − Accordingly, Y will change:
( ) ( )( ) ( )
1 0
0 0
Y f X f X
f X X f X
∆ = −
= + ∆ −
The difference quotient measures the change in Y per unit change in X. It is defined as:
( ) ( )0 0f X X f XYX X
+ ∆ −∆=
∆ ∆
Example 2.5
Given ( ) 23 4Y f X X= = − , the difference quotient is:
( ) ( )
( ) ( )
( )
0 0
2 20 0
20
0
3 4 3 4
6 3
6 3
f X X f XYX X
X X X
XX X X
XX X
+ ∆ −∆=
∆ ∆ + ∆ − − − =
∆
∆ + ∆=
∆= + ∆
23
Let 0 3 and 4X X= ∆ = , then the average rate of change of Y will be 6(3) + 3(4) = 30. This means that, on the average, as X changes from 3 to 7, the change in Y is 30 units per unit change in X.
If we assume an infinitesimally small change in X, what would then happen to the change in Y?
( ) ( )0 0
0 0X X
f X X f XYlim limX X∆ → ∆ →
+ ∆ −∆=
∆ ∆
This limit is identified as the derivative of the function ( )Y f X= .
In the above example:
( )0 00 06 3 6
X X
Ylim lim X X XX∆ → ∆ →
∆= + ∆ =
∆
As X∆ approaches zero (meaning that it gets closer and closer to, but never actually reaches zero), ( )06 3X X+ ∆ will approach the value 06X .
We may define the derivative of a given function ( )Y f X= as follows:
( )0X
dY Yf ' X limdX X∆ →
∆≡ ≡
∆
2.6.3 Differentiations
Rules of Differentiation 1. ( ) ( ) 0 where is a constantf X c, f' X , c= =
2. ( ) ( ) 1 n nf X X , f' X nX −= =
3. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f X g X h X , f' X g' X h X h' X g X= ⋅ = ⋅ + ⋅
4. ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) 2 g X g' X h X h' X g X
f X , f' Xh X h X
−= =
5. ( ) ( ) ( ) ( ) where is a constantf X c g X , f' X c g' X , c= ⋅ = ⋅
6. ( ) ( ) ( ) ( ) ( ) ( ) f X g X h X , f' X g' X h' X= + = +
24
Chain Rule Suppose Y is a function of a variable Z:
( )Y f Z=
But Z is in turn a function of another variable X:
( )Z g X=
Because Y depends on Z, and in turn depends on X, Y is also a function of X! We can express this fact by writing Y as a composite function (i.e., a function of a function) of X:
( )Y f g X=
To determine the change in Y from a change in X, the chain rule says:
( ) ( )dY dY dZ f ' Z g' XdX dZ dX
= ⋅ = ⋅
Example 2.6
Suppose we want to differentiate:
( )1023 6Y X= +
( ) ( )
( )
92
92
10 3 6 12
120 3 6
dY X XdX
X X
= +
= +
Higher-order Derivatives The most important of the higher-order derivatives is the second derivative. Understanding the meaning of the second derivative is crucial. We know that the first derivative of a function, f’(X), is the slope of a function or the rate of change of Y as a result of a change in X. The second derivative, f”(X), is the rate of change of the slope of f(X); that is, it is the rate of change of the rate of change of the original function. f’(X) f”(X) f(X) (a) > 0 > 0 Increasing at an increasing rate (b) > 0 < 0 Increasing at a decreasing rate (c) < 0 < 0 Decreasing at an increasing rate (d) < 0 > 0 Decreasing at a decreasing rate
25
Partial Differentiation So far we have only considered differentiation of functions of one independent variable. In practice, functions of two or more independent variables do arise quite frequently.
Since each independent variable influences the function differently, when we consider the instantaneous rate of change of the function, we have to isolate the effect of each of the independent variables.
Let ( )W f X ,Y ,Z= . When we consider how W changes as X changes, we want to hold the
variables Y and Z constant. This gives rise to the concept of partial differentiation. Note that the rules for partial differentiation and ordinary differentiation are exactly the same except that when we are taking partial derivative of one independent variable, we regard all other independent variables as constants.
Example 2.7
2 3
3
2 3
2 2
2
3
W X YZ
W XYZXW X ZYW X YZZ
=
∂=
∂∂
=∂∂
=∂
2.6.4 Optimizations
In portfolio theory, investment manager wants to minimize the portfolio risk for a given level of return. An individual investor seeks to maximize utility when choosing among investment alternatives. Indeed, we are all engaged in optimization problems every day.
If we have a mathematical objective function, then we can solve our optimization problem using calculus.
26
Theorem If f(X) has a relative maximum or minimum at X = a, then f’(a) = 0.
To locate all relative maxima and minima, we differentiate f(X), set the result to zero, and solve for X. That is, find all the solutions to the equation:
( ) 0f ' X =
The above equation is called the first-order condition. To determine which of these solutions are indeed relative maxima or minima, we need the second-order condition.
For maxima, ( ) ( )0 and 0f ' X f X′′= < .
For minima, ( ) ( )0 and 0f ' X f X′′= > .
Constrained Optimization Let us consider a consumer who wants to maximize his simple utility function: ( )1 2 1 2 12U x ,x x x x= +
Without any constraint, the consumer should purchase an infinite amount of both goods. However, the consumer must also consider his budget constraint into this optimization problem.
If the consumer intends to spend a given sum, say, $60, on the two goods. If the current prices are P1 = $4 and P2 = $2, the budget constraint can be expressed by the linear equation:
1 24 2 60x x+ =
Formally, we can express the above optimization problem as: ( )
1 21 2 1 2 1,
1 2
max , 2
: 4 2 60x x
U x x x x x
st x x
= +
+ =
An easy way to solve this optimization problem is to remove the constraint by rewriting:
12 1
60 4 30 22
xx x−= = −
Combining the constraint with the objective function, the result is an objective function in one variable only:
( ) ( )1
1 1 1 1max 30 2 2x
U x x x x= − +
27
First-order condition: ( )
( )1
1 1
1
1
' 0
2 30 2 2 032 4 0
8
U x
x xx
x
=
⇒ − + − + =
⇒ − =⇒ =
The solution is 1 8x = and ( )2 30 2 8 14x = − = .
Lagrange-Multiplier Method Alternatively, we can set up a Lagrangian function:
( )1 2 1 1 22 60 4 2L x x x x xλ= + + − −
The first-order conditions:
21
12
1 2
2 4 0
2 0
60 4 2 0
L xxL xxL x x
λ
λ
λ
∂= + − =
∂∂
= − =∂∂
= − − =∂
Solving the three system of equations,
2
1
1 2
4 22
4 2 60
xx
x x
λλ
= −=
+ =
The solution is also 1 8x = and 2 14x = .
28
3 Topic 3 – Risk and Return
3.1 The Definition of Return Holding period return (HPR) is capital gain income plus dividend income.
[ ]
1
1
1
where:stock price at time stock price at time 1dividend during the 1, period
t t tt
t
t
t
t
P P DivHPRP
P tP t
Div t t
−
−
−
− +=
== −
= −
HPRt is a random variable from the point of view at time t – 1. t can be one day, one week, one month, one year or any time span. Example 3.1 Suppose you are considering investing some of your money in a stock market index. The price per share is currently $100, and your time horizon is one year. You expect the dividend yield is 4% and the price one year from now is $110.
110 100 4 14%100
capital gain yield + dividend yield=10% 4% 14%
HPR − += =
=+ =
HPR is a simple measure of investment return over a single period. For return over multiple periods, the cumulative return:
( )( ) ( )1 21 1 1 1T TR r r r+ = + + +
Alternatively, since the log of a product is the sum of the logs, then: ( ) ( ) ( ) ( )
( )
1 2
1
ln 1 ln 1 ln 1 1
ln 1
T T
T
tt
R r r r
r=
+ = + + + + + +
= +∑
29
3.2 The Definition of Risk Risk means uncertainty about future rates of return. We can quantify the uncertainty using probability distributions. Example 3.2 Consider three assets:
Mean % Std %
0r 10 0
1r 10 10
2r 10 20
Investors care about expected return and risk.
30
Assumptions on investor preferences: 1. Higher mean in return is preferred. 2. Lower standard deviation in return is preferred. 3. Investor only cares about the first and second moment.
3.3 The History of U.S. Return Some stylized facts about the history of U.S assets returns: 1. Real interest rate has been slightly positive on average.
2. Return on more risky assets has been higher than less risky assets on average.
31
3. Equities were the best performing asset class and bonds proved a disappointing investment in the 20th century.
3.4 International Evidence 1. Equities outperformed bonds in all countries.
32
2. High and unexpected inflation dampened bond markets return.
3. Interestingly, the four countries – Germany, France, Japan and Italy – which suffered
from high inflation during the first half of 20th century, were amongst the best-performing bond markets in the recent 50 years.
33
3.5 Real and Nominal Rates of Interest
• At t = 0, you loan 100 apples to your friend, and he promises to return 120 apples to you
after one year. • The price of an apple at t = 0 is $10 each. • Due to inflation, the price of an apple at t = 1 is $11 each. After a year, what is your real return (in terms of goods)?
1 0 1
0 0
1
120 1 20%100
Q Q QrQ Q−
= = −
= − =
Alternatively, you can offer him an equivalent loan. At t = 0, you lend him $1,000 so he can convert the loan into 100 apples. Then, at t = 1, he repays you the market price of 120 apples. Your nominal return (in terms of money) will be:
1 1 0 0 1 1
0 0 0 0
1
120 $11 1 32%100 $10
Q P Q P Q PRQ P Q P−
= = −
×= − =
×
The nominal return takes into account changes in quantities as well as changes in price. Inflation is the rate of change in prices:
1 0 1
0 0
1
$11 1 10%$10
P P PiP P−
= = −
= − =
P0 = $10 P1 = $11
Q0 = 100 Q1 = 120
34
In general, we let: • R = Nominal rate (in terms of money) • r = Real rate (in terms of real goods) • i = Inflation rate We can show explicitly the interrelationships between the nominal interest rate, real interest rate, and inflation rate.
( ) ( )
1 1
0 0
1 1
0 0
1
1
1 1 1
Q PRQ PQ PQ P
r i
= −
= × −
= + × + −
By expanding the equation, we obtain the Fisher effect: ( ) ( ) ( )1 1 1
1R r i
r i riR r i ri
r i
+ = + × +
= + + += + +≈ +
The Fisher effect tells us that the real rate of interest is approximately equal to the nominal rate minus the inflation rate. Empirically, inflation and interest rates move closely together.
35
4 Topic 4 – Portfolio Theory Consider three stocks: HSBC, CLP and PCCW HSBC CLP PCCW r 19.5% 22.2% 15.5% σ 26.5% 32.5% 34.3%
Which stock will you prefer to invest? • CLP has the highest expected return. • HSBC has lower expected return than CLP but it is less risky. • PCCW has the lowest expected return but highest risk.
In reality, why investors still hold PCCW? • Instead of holding single asset, investors hold diversified portfolios. • Investors care about portfolio risks.
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
0.00% 10.00% 20.00% 30.00% 40.00%
r
sd
HSBC CLP PCCW
36
4.1 Portfolio Risk and Return
4.1.1 Portfolio of Two Assets
Given two stocks: Stock 1 and Stock 2, we can characterize their behavior by their mean, variance and covariance.
Mean return:
Stock 1 2 Mean return 1r 2r
Variance and covariance matrix:
1r 2r
1r 21σ 12σ
2r 21σ 22σ
Two notes: • Covariance of an asset with itself is its variance, i.e., 2
11 1σ σ= and 222 2σ σ= .
• The covariance matrix is symmetric, i.e., 12 21σ σ= . Now let V1 and V2 be the amount that you invest in Stock 1 and 2 respectively. The total value of your portfolio is:
1 2V V V= + The weight on each Stock:
• 11
VwV
= is the weight on Stock 1.
• 22
VwV
= is the weight on Stock 2.
• 1 2 1w w+ =
37
Example 4.1 You invest $113,400 in HSBC and $96,600 in Cheung Kong, then: • 113,400 / 210,000 54%HSBCw = =
• 96,600 / 210,000 46%CKw = = . Now, you sell $138,600 Cheung Kong (by borrowing $42,000 worth of Cheung Kong from your friend and short sell them), then: • 252,000 / 210,000 120%HSBCw = =
• 42,000 / 210,000 20%CKw = − = −
Expected return of a portfolio with two assets:
( ) 1 1 2 2p pE r r w r w r= = +
Variance of return of a portfolio with two assets:
( ) ( )22
2 2 2 21 1 2 2 1 2 122 2 2 21 1 2 2 1 2 1 2 12
2
2
p p p pVar r E r r
w w w ww w w w
σ
σ σ σ
σ σ σ σ ρ
= = −
= + +
= + +
Recall that 12 1 2 12σ σ σ ρ= . An easy way to remember the portfolio variance is to sum up all entries in the variance-covariance matrix.
1 1w r 2 2w r
1 1w r 2 21 1w σ 1 2 12w w σ
2 2w r 1 2 21w w σ 2 22 2w σ
38
Example 4.2 Consider the monthly returns on HSBC and CLP:
Month HSBC CLP 1 0.1205 0.1409 2 0.1527 0.0296 3 -0.0412 0.0719 4 0.0157 0.2439 5 0.0316 0.0006 6 -0.0279 0.0652 7 -0.0897 -0.0875 8 -0.0118 0.0282 9 0.0107 -0.1397 10 0.1275 -0.0806 11 0.0748 -0.0070 12 -0.0094 0.0880
Investment $150,000 $150,000 From the monthly return table, we can summarize the characteristics of the two stocks. HSBC CLP r 0.0295 0.0295 σ 0.0747 0.1051 weight 0.5 0.5
,HSBC CLPρ 0.05
The portfolio return:
0.5 0.0295 0.5 0.02952.95%
p HSBC HSBC CLP CLPr w r w r= +
= ⋅ + ⋅=
The portfolio variance:
( )( ) ( )( )( )( )( )( )( )
2 2 2 2 2,
2 2 2 2
2
0.5 0.0747 0.5 0.1051
2 0.5 0.5 0.0747 0.1051 0.050.0044
p HSBC HSBC CLP CLP HSBC CLP HSBC CLP HSBC CLPw w w wσ σ σ σ σ ρ= + +
= +
+
=
6.61%pσ =
39
4.1.2 Portfolio of Multiple Assets We can extend our analysis into n stocks. Mean return:
Stock 1 2 … n Mean return 1r 2r … nr
Variance and covariance matrix:
21 12 1
221 2 2
21 2
n
n
n n n
σ σ σσ σ σ
σ σ σ
Let wi be the weight of the asset i invested in the portfolio, then 1i
iw =∑ .
The portfolio returns for n stocks:
1 1 2 21
n
p n n i ii
r w r w r w r w r=
= + + + =∑
The variance of the portfolio is the sum of all entries in the variance and covariance matrix.
2 21 1 1 2 12 1 1
2 22 1 21 2 2 2 2
2 21 1 2 2
n n
n n
n n n n n n
w w w w ww w w w w
w w w w w
σ σ σσ σ σ
σ σ σ
( ) 2
1 1
n n
p p i j iji j
Var r w wσ σ= =
= =∑∑
In order to compute the variance of a portfolio, with n stocks, we need to estimate: • Portfolio weights. • Variance of the individual assets. • All covariance among assets.
40
4.2 Diversification
Let us select 33 HSI constituents from January 1977 to August 1996. We then perform the mean-variance analysis and form portfolios with different number of stocks.
No of Stocks r σ 1 33.8 44.2 2 34.3 43.5 3 31.9 42.2 4 31.8 37.3 5 29.1 33.9 10 30.7 35.2 15 29.8 34.8 20 27.9 33.0 25 27.7 33.1 33 28.2 30.7
HSI 21.2 30.3
1
533
HSI20%
22%
24%
26%
28%
30%
32%
34%
36%
20% 25% 30% 35% 40% 45% 50%
r
sd
41
The portfolio risk drops when we add more stocks into the portfolio. Diversification reduces risk!
However, the extent of diversification is up to a limit. Certain risks cannot be diversified away. This is because the individual risk of securities can be diversified away, but the contribution to the total risk caused by the covariance cannot be diversified away.
20%
25%
30%
35%
40%
45%
50%
1 2 3 4 5 10 15 20 25 33
sd
No of Stocks
42
For a well-diversified portfolio: • Variance of each stock contributes little to portfolio risk. • Covariance among stocks determines portfolio risk. To understand this expression, remember the variance of portfolio is the sum of all entries in the variance and covariance matrix.
2 21 1 1 2 12 1 1
2 22 1 21 2 2 2 2
2 21 1 2 2
n n
n n
n n n n n n
w w w w ww w w w w
w w w w w
σ σ σσ σ σ
σ σ σ
2
1 1
2 2
1 1 1
n n
p i j iji j
n n n
i i i j iji i j
i j
w w
w w w
σ σ
σ σ
= =
= = =≠
=
= +
∑∑
∑ ∑∑
Now, we assume equal amounts are invested in each stock. With n stocks, then the proportion invested in each stock is 1/n. We can rewrite the portfolio variance as:
2 2 2
1 1 1
22
1 1 1
22
2 21 1 1
22
2
1 1 1
1 1 1
1
1 average
n n n
p i i i j iji i j
i j
n n n
i iji i j
i j
n n n
i iji i j
i j
i ij
w w w
n n n
n nn n n n n
n nn n
n
σ σ σ
σ σ
σ σ
σ σ
= = =≠
= = =≠
= = =≠
= +
= +
− = + −
− = +
=
∑ ∑∑
∑ ∑∑
∑ ∑∑
( ) ( )2
2 variance average covariancen nn
−+
From this expression, as n becomes very large: • Contribution of variance term goes to zero. • Contribution of covariance term goes to average covariance.
43
4.3 Optimal Portfolio Selection Now, the question is how do we choose a portfolio? Should we: • Minimize risk for a given expected return? • Maximize expected return for a given risk?
Formally, our objective is to maximize utility:
( ) ( )2max : ,U W U r σ=
And subject to investment constraint. We want to construct a portfolio that is feasible. In the meantime, the portfolio can maximize our utility. Consider three cases: 1. Only one risky asset + risk free asset
(e.g. Invest in one of 43 blue chips + borrow or lend at HIBOR.)
2. Two risky assets + risk free asset (e.g. Invest in two of the 43 blue chips + borrow or lend at HIBOR.)
3. More than two risky assets + risk free asset
(e.g. Invest in 43 blue chips + borrow or lend at HIBOR. In reality, we can invest in more than 43 stocks!)
44
4.3.1 Case 1: One Risky Asset + Risk Free Asset • HSBC + HIBOR • 0.195, 0.265HSBC HSBCr σ= = • 0.066f HIBORr r= = • portfolio weight in HSBCy =
Form a complete portfolio of HSBC and HIBOR:
( )0.195 1 0.0660.265
c
c
r y yyσ
= × + − ×
=
y 1-y rc σc
0.0 1.0 6.6% 0.0% 0.1 0.9 7.9% 2.7% 0.2 0.8 9.2% 5.3% 0.3 0.7 10.5% 8.0% 0.4 0.6 11.8% 10.6% 0.5 0.5 13.1% 13.3% 0.6 0.4 14.3% 15.9% 0.7 0.3 15.6% 18.6% 0.8 0.2 16.9% 21.2% 0.9 0.1 18.2% 23.9% 1.0 0.0 19.5% 26.5% 1.1 -0.1 20.8% 29.2% 1.2 -0.2 22.1% 31.8% 1.3 -0.3 23.4% 34.5% 1.4 -0.4 24.7% 37.1% 1.5 -0.5 26.0% 39.8%
45
Capital allocation line (CAL) is the plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky portfolio.
The interpretation of this plot is straight forward. The larger the proportion y we put in HSBC, the higher the expected return of our portfolio. The standard deviation of our portfolio is proportional to HSBC’s standard deviation. In other words, each y gives us a pair of ( ) ,c cE r σ that is feasible. The collection of these feasible pairs is the CAL.
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0.0% 10.0% 20.0% 30.0% 40.0% 50.0%
r
sd
CAL - HSBC & HIBOR
46
To derive the exact equation for the CAL, we generalize the previous equation and take expectation: ( ) ( ) ( )
( )
( )
1c HSBC f
f HSBC f
cf HSBC f
HSBC
E r y E r y r
r y E r r
r E r rσσ
= × + − ×
= + −
= + −
The generalized equation describes the expected return and standard deviation tradeoff.
The slope of the CAL, denoted S, equals the increase in the expected return of the portfolio per unit of additional standard deviation – incremental return per incremental risk. Therefore, the slope is called the reward-to-variability ratio.
( )HSBC f
HSBC
E r rS
σ−
=
In our example, the reward-to-variability ratio of holding HSBC is:
( )
0.195 0.0660.265
0.4868
HSBC f
HSBC
E r rS
σ−
=
−=
=
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0.0% 10.0% 20.0% 30.0% 40.0% 50.0%
r
sd
CAL - HSBC & HIBOR
lend
borrow
risk
premium
47
The CAL describes all feasible risk-return combinations available from different asset allocation choices. The question is: how the investor chooses one optimal portfolio?
The investor makes investment decision based on: • Utility function (preference) • CAL (investment opportunity)
In our example, the investor chooses y to maximize his utility:
( )
( ) ( )
2
2 2 2
1max 2
:
c cy
c f HSBC f
c HSBC
U E r A
where
E r r y E r r
y
σ
σ σ
= −
= + − =
Substitute ( ) 2,c cE r σ into the utility function:
( ) 2 21max 2f HSBC f HSBCy
U r y E r r Ay σ = + − −
First order condition with respect to y:
( ) 2 0HSBC f HSBCU E r r Ayy
σ∂= − − =
∂
Therefore, the optimal y is:
( )*2
HSBC f
HSBC
E r ry
Aσ−
=
Note that y* increases when: • risk premium increases, ( )HSBC fE r r − ↑
• less risk averse, A↓ • variance of the stock decreases, 2
HSBCσ ↓
48
Example 4.3 Suppose the investor has a risk aversion parameter, A = 4, and recall that
0.195, 0.265HSBC HSBCr σ= = and 0.066f HIBORr r= = . What is the proportion of money that
he puts in HSBC? ( )*
2 2
0.195 0.066 0.464 0.265
HSBC f
HSBC
E r ry
Aσ− −
= = =×
So 46% of money will be invested in HSBC and 54% will be deposited in the money market. The expected return and standard deviation of the optimal portfolio are:
0.46 0.195 0.54 0.0660.125
0.265 0.460.122
c
c
r
σ
= × + ×=
= ×=
Graphically,
The CAL provided by 1-month T-bills and a board of index of common stocks is called the CML. In Hong Kong, the CML is a straight line that goes through the HIBOR and the Hang Seng Index.
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0.0% 10.0% 20.0% 30.0% 40.0% 50.0%
r
sd
CAL - HSBC & HIBOR
[12.5, 12.2]
y = 0.46
49
Example 4.4 The risk aversion parameter A is difficult to measure. But we can have a rough estimate. For example, in 1987, the total market value of the S&P500 stocks was about 4 times as large as the market value of all outstanding T-bills of less than 6-month maturity. Therefore,
( )
4 0.81 4
Suppose:0.085
0.214m f
m
y
E r rσ
≈ =+
− =
=
Substitute this into y*:
( )*2
2
0.0850.80.214
m f
m
E r ry
A
A
σ−
=
=×
Solving for A: A = 2.32
In reality, individual investors cannot borrow at the risk free rate, simply because we do not have the credit as the government. Therefore individual investors have to borrow at a higher interest rate than the risk free rate.
Suppose we can borrow at 0.09Br = , then the CAL becomes kinked.
50
For the borrowing part of the CAL, the slope now becomes: ( )
0.195 0.090.265
0.3962
HSBC B
HSBC
E r rS
σ−
=
−=
=
4.3.2 Case 2: Two Risky Assets + Risk Free Asset Consider two risky assets:
HSBC Cathy r 0.195 0.127 σ 0.265 0.304
,HSBC Cathyρ 0.68 We can form a portfolio with HSBC and Cathy Pacific:
w(HSBC) w(Cathy) rp σp 0.0 1.0 12.7% 30.4% 0.1 0.9 13.4% 29.2% 0.2 0.8 14.1% 28.2% 0.3 0.7 14.7% 27.3% 0.4 0.6 15.4% 26.6% 0.5 0.5 16.1% 26.1% 0.6 0.4 16.8% 25.8% 0.7 0.3 17.5% 25.6% 0.8 0.2 18.1% 25.7% 0.9 0.1 18.8% 26.0% 1.0 0.0 19.5% 26.5%
51
When forming a portfolio with two risky assets, each pair of portfolio weights [w1 w2] will give a corresponding pair of portfolio mean and standard deviation ( ) ,p pE r σ . The
collection of these feasible portfolio mean and standard deviation is the investment opportunity set of the two risky assets.
Recall that the portfolio standard deviation is: 1
2 2 2 2 21 1 2 2 1 2 1 2 122p w w w wσ σ σ σ σ ρ = + +
Since the investor has to fully invest, the fraction that an investor invests in asset 1 plus the fraction in asset 2 must equal one.
1 2
2 1
11
w ww w+ =
⇒ = −
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
18.0%
19.0%
20.0%
20.0% 22.0% 24.0% 26.0% 28.0% 30.0% 32.0%
r
sd
Investment Opportunity from HSBC & Cathy Pacific
52
Some special cases: 1. Perfect Positive Correlation ( )1ρ = +
( ) ( )( )
122 2 2 2
1 1 1 2 1 1 1 2
1 1 1 2
1 2 1
1
p w w w w
w w
σ σ σ σ σ
σ σ
= + − + − = + −
While the expected return on the portfolio is:
( )1 1 1 21pr w r w r= + −
Thus, with the correlation coefficient equal to +1, both risk and return of the portfolio are simply linear combinations of the risk and return of each stock.
( )1 1 1 2
2 21 2
1 2 1 2
1
1
p
p p
r w r w r
r rσ σ σ σσ σ σ σ
= + −
− − = + − − −
In the case of perfectly correlated assets, the return and risk on the portfolio of the two assets is a weighted average of the return and risk on the individual assets. There is no reduction in risk from purchasing both assets. Nothing has been gained by diversifying rather than purchasing the individual assets.
10.0%11.0%12.0%13.0%14.0%15.0%16.0%17.0%18.0%19.0%20.0%
20.0% 22.0% 24.0% 26.0% 28.0% 30.0% 32.0%
r
sd
Investment Opportunity from HSBC & Cathy Pacific
53
2. Perfect Negative Correlation ( )1ρ = −
( ) ( )( )
122 2 2 2
1 1 1 2 1 1 1 2
1 1 1 2
1 2 1
1
p w w w w
w w
σ σ σ σ σ
σ σ
= + − − − = − −
If two assets are perfectly negatively correlated, it should always be possible to find some combination of these two assets that has zero risk. By setting the above equation equal to zero:
( )1 1 1 2
21
1 2
0 1w w
w
σ σσ
σ σ
= − −
=+
Employing the formula developed, if we set w1 equal to 0.304 / (0.265 + 0.304) = 0.5343, we can form a zero-risk portfolio:
w(HSBC) w(Cathy) rp σp 0 1 12.7% 30.4%
0.1 0.9 13.4% 24.7% 0.2 0.8 14.1% 19.0% 0.3 0.7 14.7% 13.3% 0.4 0.6 15.4% 7.6% 0.5 0.5 16.1% 2.0%
0.5343 0.4657 16.3% 0.0% 0.6 0.4 16.8% 3.7% 0.7 0.3 17.5% 9.4% 0.8 0.2 18.1% 15.1% 0.9 0.1 18.8% 20.8% 1 0 19.5% 26.5%
10.0%11.0%12.0%13.0%14.0%15.0%16.0%17.0%18.0%19.0%20.0%
0.0% 10.0% 20.0% 30.0% 40.0%
r
sd
Investment Opportunity from HSBC & Cathy Pacific
54
3. Zero Correlation ( )0ρ =
( )1
22 2 2 21 1 1 21p w wσ σ σ = + −
The covariance term drops out when there is no relationship between returns on the assets.
10.0%11.0%12.0%13.0%14.0%15.0%16.0%17.0%18.0%19.0%20.0%
0.0% 10.0% 20.0% 30.0% 40.0%
r
sd
Investment Opportunity from HSBC & Cathy Pacific
55
4. Various Correlation Coefficients Portfolio standard deviation for a given correlation: σp w(HSBC) w(Cathy) rp ρ = -1 ρ = -0.7 ρ = -0.3 ρ = 0 ρ = 0.3 ρ = 0.7 ρ = 1
0 1 12.7% 30.4% 30.4% 30.4% 30.4% 30.4% 30.4% 30.4% 0.1 0.9 13.4% 24.7% 25.6% 26.7% 27.5% 28.3% 29.3% 30.0% 0.2 0.8 14.1% 19.0% 21.0% 23.3% 24.9% 26.4% 28.3% 29.6% 0.3 0.7 14.7% 13.3% 16.7% 20.4% 22.7% 24.9% 27.4% 29.2% 0.4 0.6 15.4% 7.6% 13.2% 18.1% 21.1% 23.7% 26.8% 28.8% 0.5 0.5 16.1% 2.0% 11.2% 16.9% 20.2% 23.0% 26.2% 28.5%
0.5343 0.4657 16.3% 0.0% 11.0% 16.8% 20.0% 22.8% 26.1% 28.3% 0.6 0.4 16.8% 3.7% 11.4% 16.9% 20.0% 22.7% 25.9% 28.1% 0.7 0.3 17.5% 9.4% 13.8% 18.0% 20.7% 23.0% 25.8% 27.7% 0.8 0.2 18.1% 15.1% 17.5% 20.2% 22.1% 23.7% 25.8% 27.3% 0.9 0.1 18.8% 20.8% 21.8% 23.1% 24.0% 24.9% 26.1% 26.9% 1 0 19.5% 26.5% 26.5% 26.5% 26.5% 26.5% 26.5% 26.5%
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
18.0%
19.0%
20.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0%
r
sd
p = -1 p = -0.7 p = -0.3 p = 0 p = 0.3 p = 0.7 p = 1
56
There is one point that is worth special attention: the portfolio that has minimum risk. This portfolio can be found in general by looking at the equation for risk:
( ) ( )1
22 2 2 21 1 1 2 1 1 1 2 121 2 1p w w w wσ σ σ σ σ ρ = + − + −
To find the value of w1 that minimizes the equation, we take the derivative of it with respect with w1, set the derivative equal to zero, and solve for w1:
( ) ( )
2 2 21 1 2 1 2 1 2 12 1 1 2 12
121 2 2 2 2
1 1 1 2 1 1 1 2 12
2 2 2 2 412
1 2 1
p w w ww
w w w w
σ σ σ σ σ ρ σ σ ρσ
σ σ σ σ ρ
− + + −∂ = ∂ + − + −
Setting this equal to zero and solving for w1 yields:
22 1 2 12
1 2 21 2 1 2 122
w σ σ σ ρσ σ σ σ ρ
−=
+ −
A summary: 1. In general (when 1ρ ≠ ), the investment opportunity set is a curve that goes through the
two stocks. The portfolio standard deviation is smaller than the standard deviation of either stock. That is, forming portfolio can always enjoy the benefit of diversification.
2. When two stocks are perfectly positively correlated ( 1ρ = ), the investment opportunity
set becomes a straight line. There is no benefit of diversification.
3. When two stocks are perfectly negatively correlated ( 1ρ = − ), it is always possible to reduce the portfolio standard deviation to zero.
4. The closer the correlation is to negative one, the standard deviation for a given expected
return will be lower. That is, the more you can reduce the standard deviation of your portfolio.
57
Now, we add the risk free asset (two risky assets + risk free asset). Then, how an investor chooses his portfolio? In the case of one risky asset plus risk free asset, it is assumed that the investor has decided the composition of the risky portfolio. His only concern is how to allocate his money between the risky asset and the risk free asset.
Now, since the investor has two risky assets, he has to decide: • The weights between the two risky assets within the risky portfolio. • The weights between the risky portfolio and the risk free asset.
Suppose the investor has constructed the investment opportunity set from two risky assets:
CAL(P)E(r)
σp
E(rp)
CAL(B)
CAL(A)
σ
P
B
A
rf
Two possible CALs are drawn from the risk free asset to two feasible portfolios. Portfolio B is better than portfolio A because the reward-to-variability ratio is higher for portfolio B: ( ) ( )B f A f
B A
E r r E r rσ σ
− −>
58
For any level of risk (standard deviation) that the investor is willing to bear, the expected return from portfolio B is higher.
But one can keep moving the CAL upward until it reaches the tangency point P. This portfolio yields the highest reward-to-variability ratio. Therefore, portfolio P is the optimal risky portfolio combining with the risk free asset. Formally, the investor has to solve the following problem:
( )1 2,
max p f
w wp
E r rσ
−
Subject to:
( ) ( ) ( )1 1 2 2
12 2 2 2 21 1 2 2 1 2 1 2 12
1 2
2
1
p
p
E r w E r w E r
w w w w
w w
σ σ σ σ σ ρ
= +
= + + + =
We can substitute the constraints into the objective function:
( ) ( ) ( )
( ) ( )1
1 1 1 21
22 2 2 21 1 1 2 1 1 1 2 12
1max
1 2 1
f
w
w E r w E r r
w w w wσ σ σ σ ρ
+ − −
+ − + −
The solution is:
( ) ( )( ) ( ) ( ) ( )
21 2 2 1 2 12*
1 2 21 2 2 1 1 2 1 2 12
f f
f f f f
E r r E r rw
E r r E r r E r r E r r
σ σ σ ρ
σ σ σ σ ρ
− − − = − + − − − + −
59
Consider our example: HSBC Cathy
r 0.195 0.127 σ 0.265 0.304
,HSBC Cathyρ 0.68 rf 0.066 A 4
How to select the optimal portfolio? First, select the optimal risky portfolio of HSBC and Cathy Pacific:
( ) ( )( ) ( ) ( ) ( )
( )( ) ( )( )( )( )
*
2,
2 2,
20.195 0.066 0.304 0.127 0.066 0.265 0.304 0.680.19
HSBC
HSBC f Cathy Cathy f HSBC Cathy HSBC Cathy
HSBC f Cathy Cathy f HSBC HSBC f Cathy f HSBC Cathy HSBC Cathy
w
E r r E r r
E r r E r r E r r E r r
σ σ σ ρ
σ σ σ σ ρ
− − − = − + − − − + −
− − −=( )( ) ( )( ) ( )( )( )( )2 25 0.066 0.304 0.127 0.066 0.265 0.129 0.061 0.265 0.304 0.681.48
− + − − +
=
* *1
1 1.48 0.48Cathy HSBCw w= −
= − = −
The risky portfolio:
( ) ( )
( )( ) ( )( )( )( )( )( )( )
*
12 2 2 2 2
*
1.48 0.195 0.48 0.127
0.228
1.48 0.265 0.48 0.304
2 1.48 0.48 0.265 0.304 0.68
0.312
p
p
E r
σ
= × + − ×
=
+ − = + −
=
Second, select the optimal (complete) allocation between the risky portfolio and the risk free asset, HIBOR:
( )*2
2
0.228 0.0664 0.312
0.42
p f
p
E r ry
Aσ−
=
−=
×=
60
The complete portfolio:
( ) ( )( )
( )( )( )
* * *
* * *
* *2
2
0.066 0.42 0.228 0.0660.133
0.42 0.3120.130
0.5
0.133 0.5 4 0.130
0.0996
c f p f
c p
c c
E r r y E r r
y
U E r A
σ σ
σ
= + − = + −
=
= ×
= ×=
= −
= −
=
Interpretations:
Risky Portfolio HIBOR * 42%y = *1 58%y− =
HSBC Cathy * *
0.42 1.4862%
HSBCy w⋅= ×=
* *
0.42 0.4820%
Cathyy w⋅= ×−= −
Suppose you have $1 million. The optimal decision is to invest $0.42 million in stocks and deposit $0.58 million at HIBOR. To invest in stocks, you should borrow $0.2 million worth of Cathy Pacific from your friend, and sell them for $0.2 million (short sell). Together with your $0.42 million own money, invest $0.62 million in HSBC.
61
Graphically,
CALOptimal risky portfoliow(HSBC) = 1.48, w(Cathy) = -0.48
[22.8, 31.2]
HSBC[19.5, 26.5]
Cathy[12.7, 30.4]
E(r)
σ
rf
Optimal complete portfolioy = 0.42
[13.3, 13.0]
Example 4.5 More examples from Hong Kong stocks: Stocks 1 & 2 1r 1σ 2r 2σ 12ρ HSBC & Cathy Pacific 0.195 0.265 0.127 0.304 0.68 HSBC & Cheung Kong 0.195 0.265 0.346 0.438 0.70 HSBC & PCCW 0.195 0.265 0.118 0.251 0.54 Optimal risky portfolio: Stocks 1 & 2 1w 2w pr pσ HSBC & Cathy Pacific 1.48 -0.48 0.228 0.312 HSBC & Cheung Kong 0.18 0.82 0.319 0.394 HSBC & PCCW 1.19 -0.19 0.209 0.292
Optimal complete portfolio: Stocks 1 & 2 *y cr cσ U HSBC & Cathy Pacific 0.42 0.133 0.130 0.0966 HSBC & Cheung Kong 0.41 0.169 0.160 0.1175 HSBC & PCCW 0.42 0.126 0.123 0.0962
62
Several observations: 1. If we look at the mean-standard deviation plot, HSBC clearly dominates Cathy Pacific
(higher mean and lower standard deviation). That is why the optimal risky portfolio involves short selling Cathy Pacific. You want HSBC to have a higher portfolio weight because it has a higher expected return.
The standard deviation of the optimal risky portfolio is also higher than that of either HSBC or Cathy Pacific. However, the reward-to-variability ratio is maximized.
The case for HSBC & PCCW is similar.
2. Cheung Kong has a higher expected return than HSBC, but its standard deviation is also
higher. In that case, the optimal risky portfolio invests in both stocks.
3. If we compare the optimal utility levels in the two stocks portfolio with the single stock, the utility is always higher in the two stocks case. We are better off by diversifying our portfolio.
Utility Level at Optimal Complete Portfolio
Stocks 1 & 2 Both stocks Stock 1 Stock 2 HSBC & Cathy Pacific 0.0996 0.0956 0.0710 HSBC & Cheung Kong 0.1175 0.0956 0.1171 HSBC & PCCW 0.0962 0.0956 0.0714
63
4.3.3 Case 3: More than Two Risky Assets + Risk Free Asset In reality, when there are more than two risky assets, the problem becomes more complicated. Since there is no closed form solution, the problem has to be solved numerically.
The Markowitz Portfolio Selection Model Specifically, the Markowitz portfolio selection procedure contains the following three steps: 1. Find the minimum-variance frontier from the expected returns and the variance-
covariance matrix of all individual risky assets.
Usually, we use the historical records or time-series averages.
The expected returns for each risky asset i is:
( )1
1 T
i i itt
E r r rT =
≈ = ∑
The variances for each asset i:
( )22 2
1
11
T
i i it it
r rT
σ σ=
≈ = −− ∑
The covariance for two stocks i and j:
( ) ( )( )1
1cov ,2
T
i j ij it i jt jt
r r r r r rT
σ=
≈ = − −− ∑
Once the mean and variance-covariance matrix of individual risky assets are known, the mean and variance of the portfolio constructed from these individual risky assets are given by:
( ) ( )1
2 2 2
1 1 1
N
p i ii
N N N
p i i i j iji i j
i j
E r w E r
w w wσ σ σ
=
= = =≠
=
= +
∑
∑ ∑∑
For any level of expected return, the investor is interested in the one that yields the lowest risk (standard deviation).
64
Suppose the expected return is 10%, the investor needs to solve the following problem:
( )
1
2
, ,
1
1 2
min
subject to:
10%
1
Npw w
N
i ii
N
w E r
w w w
σ
=
=
+ + + =
∑
This process tries different values of expected return repeatedly until the minimum variance frontier has been plotted.
When more assets are included, the portfolio frontier improves. That is, moves toward upper left – higher mean return and lower risk.
65
2. Find the CAL with the highest reward-to-variability ratio and the optimal risky portfolio. ( )
1 , ,max
N
p f
w wp
E r rσ
−
3. Find the optimal complete portfolio. ( )*
2p f
p
E r ry
Aσ−
=
The most striking feature is that a portfolio manager will offer the same risky portfolio to all clients regardless of their degree of risk aversion. The more risk averse investor will invest more in the risk free asset and less in the optimal risky portfolio.
Separation Theorem The portfolio choice can be separated into two independent tasks: 1. The technical part is to determine the optimal risky portfolio. The best risky portfolio is
the same for all investors.
2. The preference part is to determine the complete portfolio. The allocation between risky and risk free assets depends on the investor’s preference.
66
Some comments: 1. The separation theory is the theoretical basis for the mutual fund industry. Fund
managers focus on the composition of the funds while investors decide how much money to put into mutual funds. The composition of the funds is the same for all investors.
2. In reality, not every individual risky asset is included in constructing the minimum
variance frontier. With modern computer technology, the amount of computation is of less concern. The difficulty is to identify good stocks for the portfolio.
67
4.4 Appendix – Portfolio Analysis Using Excel This appendix provides us an example on how to model the problem of portfolio optimization using Excel. To begin with, recall that the Markowitz portfolio selection contains three steps: 1. Find the mean-variance efficient portfolio. 2. Combine the risk-free asset with the mean-variance efficient portfolio, and find the
optimal risky portfolio. 3. Incorporate the degree of risk aversion of the investors, and solve for the optimal
complete portfolio.
4.4.1 Mean-Variance Efficient Portfolio The inputs for the efficient portfolio are based on the mean and variance of returns for the portfolio. In general,
The portfolio return: 1 1 2 21
n
p n n i ii
r w r w r w r w r=
= + + + =∑
The portfolio variance: ( ) 2
1 1
n n
p p i j iji j
Var r w wσ σ= =
= =∑∑
Step 1: Find the time-series mean and variance of returns of all individual stocks. As an illustration, the sheet “Stock Price” in Markowitz.xls contains the monthly historical closing price from 2001 to 2005. We first calculate the monthly return based on the closing price:
1
1tt
t
PrP−
= −
After getting the monthly returns of each individual stock, we can use the AVERAGE function to calculate the time-series average:
1
1 T
i itt
r rT =
= ∑
We can also use the STDEV function to get the time-series standard deviation:
( )2
1
11
T
i it it
r rT
σ=
= −− ∑
68
Step 2: Find the variance-covariance matrix. The easiest way to get the variance-covariance matrix is to use the COVARIANCE and CORRELATION function under the Data Analysis ToolPak. But the Data Analysis ToolPak is not installed with the standard Excel setup. To use it in Excel, we need to enable it first. To load it: 1. On the Tools menu, click Add-Ins. 2. In the Add-Ins available box, select the check box next to Analysis ToolPak, and then
click OK. 3. Click Tools on the menu bar. When we load the Analysis ToolPak, the Data Analysis
command is added to the Tools menu. Once the Analysis TookPak is installed, click Data Analysis from the Tools on the menu bar, and then select COVARIANCE.
69
Highlight the return columns, and then click OK to output the variance-covariance matrix.
The variance-covariance matrix is listed in the sheet “Risk & Return” in Markowitz.xls Step 3: Finding the portfolio return and variance. Once the mean, variance of returns and the variance-covariance matrix are known, we can construct portfolio expected return and variance. An easy approach is based on Excel’s vector and matrix multiplication. We first denote e and w as the vector of returns and portfolio weights, and the variance-covariance terms by matrix V. Then the portfolio return and variance can be written as simple matrix formulas. They can easily be implemented with Excel array functions. Matrix Excel Formula The portfolio return: ′w e =SUMPRODUCT(w,e) The portfolio variance: ′w Vw =MMULT(TRANSPOSE(w),MMULT(V,w)) The matrix expression simplifies the calculation of portfolio mean and variance when the number of stocks is very large.
70
Step 4: Using Solver to find efficient weights. Given the same data set in Markowitz.xls, suppose we want to construct an efficient portfolio producing a target return of 1%, the problem is to find the split across the stocks that achieve the target return whilst minimizing the variance of return.
( )
1
2
, ,
1
1 2
min
subject to:
1%
1
Npw w
N
i ii
N
w E r
w w w
σ
=
=
+ + + =
∑
We can use Excel Solver to solve for this optimization problem. The steps with Solver are: 1. Invoke Solver by choosing Tools then Options then Solver. 2. Specify in the Solver Parameter Dialog Box:
a. The Target Cell (Portfolio SD) to be optimized. b. The Changing Cell (Portfolio weights).
3. Choose Add to specify the constraints then OK. 4. Click on Options and ensure that Assume Linear Model is not checked. 5. Solve and get the results in the spreadsheet.
71
The Solver Dialog Box to minimize variance:
72
The results:
73
5 Topic 5 – Capital Asset Pricing Model (CAPM) The portfolio theory has been concerned with how an individual, acting upon a set of estimates, could select an optimum portfolio. If investors act as we have prescribed, then we should be able to draw on the analysis to determine how the aggregate of investors will behave, and how prices and returns at which markets will clear are set. CAPM gives us an equilibrium model predicting the relationship between the risk of an asset and its expected return. The assumptions underlying the CAPM: 1. Many investors who are all price takers.
2. All investors plan to invest over the same horizon.
3. There are no taxes or transaction costs.
4. Investors can borrow and lend at the same risk-free rate over the planned investment horizon.
5. Investors only care about expected return and variance.
6. All investors have the same information and beliefs about the distribution of returns.
7. The market portfolio consists of all publicly traded assets.
74
The implications from these assumptions: 1. All investors use the Markowitz portfolio selection model to determine the same set of
efficient portfolios. That is, the efficient portfolios are combinations of the risk-free asset and the tangency portfolio. Therefore, each investor has the same tangency portfolio.
2. Risk averse investors put a majority of wealth in the risk-free asset whereas risk tolerant
investors borrow at the risk-free rate and leverage their holdings. In equilibrium, total borrowing and lending must equalize so that the risk-free asset is in zero net supply when we aggregate across all investors.
3. Given each investor holds the same tangency portfolio and the risk-free asset is in zero net supply, when we aggregate over all investors, the aggregate demand for assets is simply the tangency portfolio. The supply of all assets is simply the market portfolio, in which the weight of an asset in the market portfolio is just the market value of the asset divided by the total market value of all assets. In equilibrium, supply is equal to demand. Therefore, the tangency portfolio is the market portfolio.
4. Since the market portfolio is the tangency portfolio and the tangency portfolio is mean-variance efficient, the market portfolio is also mean-variance efficient.
5. The security market line (SML) relationship holds for all assets and portfolios:
( ) ( )i f i m fE r r E r rβ = + −
Where: ( )( )2
cov ,i mi
m
r rr
βσ
=
5.1 The Market Portfolio The market portfolio is the portfolio of all risky assets traded in the market. Suppose there are a total of n risky assets, the market capitalization of asset i is its total market value:
price per share # shares outstandingi i iMV = ×
Total market capitalization of all risky assets is:
1
n
m ii
MV MV=
=∑
75
The market portfolio is the portfolio with weights in each risky asset i being:
1
i ii n
mi
i
MV MVwMVMV
=
= =
∑
Why the tangency portfolio is the market portfolio? Suppose there are only three risky assets, A, B and C, and the tangent portfolio is:
( ) ( )* * *, , 0.25,0.5,0.25a b cw w w =
There are only three investors in the economy, 1, 2 and 3, with total wealth of 500, 1000 and 1500 million dollars. Their asset holdings are:
Investor Riskless A B C 1 100 100 200 100 2 200 200 400 200 3 -300 450 900 450
Total 0 750 1500 750 In equilibrium, the total dollar holding of each asset must equal its market value: • Market capitalization of A = $750 million • Market capitalization of B = $1500 million • Market capitalization of C = $750 million
The total market capitalization is: 750 + 1500 + 750 = 3000 million
The market portfolio:
750 0.2530001500 0.53000750 0.253000
a
b
c
w
w
w
= =
= =
= =
Since each investor holds the same risky portfolio or the tangency portfolio, the tangency portfolio must be the market portfolio.
76
5.2 Derivation of the CAPM Recall that each investor faces an efficient frontier. When we introduce the risk-free asset, investor will hold the optimal risky portfolio at the tangency point.
If all investors have homogeneous expectations (A6) and they all face the same lending and borrowing rate (A4), then they will each face a diagram above, and furthermore, all of the diagrams will be identical. Therefore, all investors will end up with portfolios somewhere along the capital market line (CML) and all efficient portfolios would lie along the CML.
77
We know that if market equilibrium is to exist, the prices of all assets must adjust until all are held by investors. There can be no excess demand. In other words, prices must be established so that the supply of all assets equals the demand for holding them. Suppose a portfolio consisting of a% invested in risky asset i and (1 – a)% in the market portfolio will have the following mean and standard deviation:
( ) ( ) ( ) ( )
( ) ( )1
22 2 2 2
1
1 2 1
p i m
p i m im
E r aE r a E r
a a a aσ σ σ σ
= + −
= + − + −
The change in the mean and standard deviation with respect to a is determined as follows:
( ) ( ) ( )
( ) ( )
2 2 2
122 2 2 2
2 2 2 2 412
1 2 1
pi m
p i m m im im
i m im
E rE r E r
aa a a
aa a a a
σ σ σ σ σ σ
σ σ σ
∂= −
∂∂ − + + −
= ⋅∂ + − + −
In equilibrium, the market portfolio already has the value weight wi invested in the risky asset i. Therefore, the percentage a in the above equation is the excess demand for an individual risky asset. But we know the excess demand in equilibrium must be zero. Therefore, we can evaluate the partial derivatives where a = 0.
( ) ( ) ( )0
2
0
pi m
a
p im m
ma
E rE r E r
a
aσ σ σ
σ
=
=
∂= −
∂
∂ −=
∂
The slope of the risk-return tradeoff evaluated at point M, the market equilibrium, is:
( ) ( ) ( )2
0
p i m
p im m ma
E r a E r E raσ σ σ σ
=
∂ ∂ −=
∂ ∂ −
Note that ( )
0
p
p a
E r aaσ
=
∂ ∂
∂ ∂
is equal to the slope of the CML. Therefore,
( ) ( ) ( )2
m fi m
im m m m
E r rE r E rσ σ σ σ
−−=
−
78
This relationship can be arranged to solve for ( )iE r :
( ) ( )
( )
2im
i f m fm
f i m f
E r r E r r
r E r r
σσ
β
= + −
= + −
This equation is known as capital asset pricing model (CAPM). Graphically, it is also called the security market line (SML).
79
5.3 Implications of the CAPM What does beta really mean? Beta measures the extent to which individual risky asset moves with the market. Suppose: ( ) 112%, 6%, 0.5m fE r r β= = =
Then:
( ) ( )( )
1 1
6% 0.5 12% 6%9%
f m fE r r E r rβ = + − = + −
=
If the market goes up to ( ) 18%mE r = , then:
( ) ( )1 6% 0.5 18% 6%12%
E r = + −
=
In other words, the stock return should go up by: 3% = [ ( ) 10.5 18% 12% change in market returnβ× − = × ]
What if another stock has a 2 2β = , then the stock should up by ( )2 18% 12% 12%× − = .
In general,
( )stock return market return
stock return market returnf fr beta r
beta
− = × −
∆ = ×∆
Example 5.1 Suppose that CAPM holds. The expected market return is 14% and T-bill rate is 5%. 1. What should be the expected return on a stock with β = 0?
2. What should be the expected return on a stock with β = 1? 3. What should be the expected return on a portfolio made up of 50% T-bills and 50%
market portfolio? 4. Can beta be negative? What should be the expected return on stock with β = −0.6?
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5.3.1 Two Important Graphs Capital Market Line (CML) Recall that:
( ) ( )m fp f p
m
E r rE r r σ
σ−
= +
The slope ( )m f mE r r σ− is the reward-to-variability ratio. CML describes an investment
opportunity, feasible mean and standard deviation pairs from combining the risk-free asset and the market portfolio.
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Security Market Line (SML) Mathematically:
( ) ( )i f i m fE r r E r rβ = + −
The slope ( )m fE r r− is the premium on the market portfolio. SML describes an equilibrium
result – a relation between expected return and risk as measured by beta.
82
Notice ( ) ( )i f i m fE r r E r rβ = + − for any stock i, i = 1, ... , N is equivalent to every stock
lies on SML.
What if some stocks are off the SML? For example, stocks XYZ and ABC are off the SML, what will happen then? Which stock would you like to buy?
For the same beta risk, return on XYZ is higher than the market. Therefore, every investor wants to buy XYZ. So the price of XYZ will up and the expected return will drop. In equilibrium, XYZ should lie on the SML.
Likewise, investor will sell ABC, so the price of ABC will go down until the expected return lie on the SML.
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5.3.2 Replicating the Beta We can use market portfolio plus risk-free asset to replicate the beta of any stock. Let x be the proportion of your money invested in the market portfolio, then 1 – x is the proportion in the risk-free asset. The return on this portfolio is:
( )1p m fr xr x r= + −
The beta of the portfolio is:
( ) ( )( )
( ) ( )( )
( )
2 2
2
2
1
1
0
p m m f mp
m m
m m f m
m
m m
m
cov r ,r cov xr x r ,r
cov xr ,r cov x r ,r
x cov r ,r
x
βσ σ
σ
σ
+ −= =
+ −=
+=
=
In words, if you want to construct a portfolio that has the same beta as an individual stock, say 1β , you only need to invest 1β proportion in the market portfolio and 11 β− proportion in the risk-free asset. Example 5.2 Suppose you want to construct a portfolio with a beta of 0.5, what should you do? You should invest 0.5 in the market portfolio and 0.5 in the risk-free asset.
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5.4 Risk in the CAPM Remember, investors can always diversify away all risk except the covariance of an asset with the market portfolio. The only risk that investors will pay a premium to avoid is covariance risk.
This figure shows there are two ways to receive an expected return of ( )HSBCE r – simply buy
shares in HSBC, or buy portfolio A. For a risk averse investor, portfolio A is preferred to an investment solely in HSBC since it produces the same return with less risk. The total risk of HSBC can therefore be decomposed into systematic risk (the minimum risk required to earn that expected return) and unsystematic risk (portion of the risk that can be eliminated without sacrificing any expected return by diversifying). Investors are rewarded for bearing systematic risk, but they are not rewarded for bearing unsystematic risk, because it can easily be diversified at no cost!
85
5.4.1 CML and SML: A Synthesis
5.5 Estimating Beta In practice, we estimate the beta of an individual stock from an OLS regression:
it i i mt itr rα β ε= + +
For example, to obtain the beta of China Mobile, we run a regression of the return of China Mobile on HSI return from January 2003 to December 2008. The regression results show that:
941 0 02 1 19 HSIr . . r= +
86
The slope 1.19 is the beta coefficient of China Mobile within the recent 5 years. The beta 1.19 > 1 suggests that the return of China Mobile is more sensitive to the variability from Hang Seng Index return. Now, consider we construct a portfolio of three stocks with the following beta and portfolio weight.
Beta Weight China Mobile 1.19 40% HSBC 0.85 30% PCCW 0.92 30%
What will be the portfolio beta?
( )
( )( )
( )
2
2
2
p mp
m
i i m
m
i i m
m
i i
cov r ,r
cov w r ,r
w cov r ,r
w
βσ
σ
σ
β
=
=
=
=
∑
∑
∑
In the example, the portfolio beta is:
0 4 1 19 0 3 0 85 0 3 0 92
1 007p . . . . . .
.β = × + × + ×
=
Ri = 1.1867Rm + 0.0154R² = 0.7173
-0.3000
-0.2000
-0.1000
0.0000
0.1000
0.2000
0.3000
-0.3000 -0.2000 -0.1000 0.0000 0.1000 0.2000
Ri
Rm
87
5.6 Empirical Tests of the CAPM
5.6.1 Time-series Tests of the CAPM If CAPM holds, we should expect that the individual stock return follows this relationship:
( ) ( )i f i m fE r r E r rβ = + −
Alternatively, we can rewrite:
( ) ( )i f i m fE r r E r rβ − = −
Graphically, if we plot the average risk premiums from portfolios with different betas, we should expect the portfolios are fitting on the SML.
However, Black (1993) finds that: • High beta portfolios fall below SML. • Low beta portfolios land above SML.
88
Separating into different sub periods, CAPM does not seem to work well since the late 1960s.
89
5.6.2 Cross-sectional Tests of the CAPM Two Step Approach First, betas were estimated with a set of time-series regressions, one for each security. For example, the returns on China Mobile and the HSI might be the respective left-hand-side and right-hand-side values for a single observation. Each of these regressions, one for each security i can be represented by:
it i i mt itr rα β ε= + + The second step obtains estimates of the intercept and slope coefficient of a single cross-sectional regression.
0 1 2i i i iˆr Xγ γ β γ δ= + + +
If the CAPM is true, the second step regression should have the following features: 1. The intercept, 0γ , should be the risk free return.
2. The slope, 1γ , should be the market portfolio’s risk premium.
3. 2γ should be zero since variables other than beta should not explain the mean returns once beta is accounted for.
Both the time-series and cross-sectional tests find evidence that is not supportive of the CAPM.
90
6 Topic 6 – Factor Models
6.1 Single Factor Model The simplest factor model is the market model. Casual observation of stock prices reveals that when the market index goes up, most stocks tend to increase in price; and when the market index goes down, most stocks tend to decrease in price.
This suggests that stock return may be correlated to market changes. We can relate the return on a stock to the return on a stock market index as:
i i i m ir rα β ε= + +
( )( )
( )
:0
, 0
0
i
i m
i j
whereE
Cov r
E
ε
ε
ε ε
=
=
=
To understand the properties of the market model, we can examine the return and risk decomposition.
91
6.1.1 The Market Model Return Decomposition The market model suggests we can decompose a stock’s return into three pieces:
iα • The stock’s expected return if the market is neutral, that is, if the market’s excess return is zero.
i mrβ • The component of return due to movements in the overall market. • iβ is the stock’s responsiveness to market movements.
iε • The unexpected component due to an unexpected event that is relevant only to this stock – firm specific.
By construction, ( ) 0iE ε = , the expected return of a stock only has two components: a unique
part iα and a market related part i mrβ . Mathematically,
( ) ( )i i i mE r E rα β= +
6.1.2 The Market Model Variance Decomposition Each security has two sources of risk: systematic risk, attributable to its sensitivity to macroeconomic factors as reflected in mR , and unsystematic risk, as reflected in iε . We can decompose the risk on each stock by taking variance on both sides:
( ) ( )( ) ( )2
2 2 2 2
i i i m i
i m i
i i m
Var r Var r
Var r Var
ε
α β ε
β ε
σ β σ σ
= + +
= +
= +
Note: iα is a constant term therefore has no variance. The covariance between mr and iε is
zero ( ), 0i mCov rε = because iε is defined as firm specific, that is, independent of
movements in the market.
What about the covariance between the rates of return on two stocks?
( ) ( )( )
2
i j i i m i j j m j
i m j m
i j m
Cov r ,r Cov r , r
Cov r , r
α β ε α β ε
β β
β β σ
= + + + +
=
=
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6.1.3 The Inputs to Portfolio Analysis To define the efficient frontier, we have to determine the expected return and standard deviation on a portfolio.
Recall that the expected return and the standard deviation on any portfolio are:
1 1 2 21
n
p n n i ii
r w r w r w r w r=
= + + + =∑
12
2 2
1 1 1
n n n
p i i i j iji i j
j i
w w wσ σ σ= = =
≠
= + ∑ ∑∑
Suppose we are going to analyze 50 stocks. This means that we have to estimate: • n = 50 estimates of expected returns • n = 50 estimates of variances • (n2 – n)/2 = 1,225 estimates of covariances • A total of 1,325 estimates What if the number of stocks increases to 3,000, roughly the number of NYSE stocks? We need more than 4.5 million estimates.
Comparing the set of estimates needed with the Markowitz model, the market model only needs: • n estimates of the extra-market expected excess return, iα
• n estimates of the sensitivity coefficients, iβ
• n estimates of the firm-specific variances, 2εσ
• 1 estimate for the expected market return, ( )mE R
• 1 estimate for the variance of the common macroeconomic factor, 2mσ
Then, the 3n + 2 estimates will enable us to prepare the entire input list for this single-index universe. For a 50 stocks portfolio, we will need 152 estimates rather than 1,325!
6.1.4 The Market Model and Diversification Suppose that we choose an equally weighted portfolio of n stocks. The portfolio return can be written as:
p p p m pr rα β ε= + +
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Since this is an equally weighted portfolio, each portfolio weight 1iw n= .
( )
1 1
1
1 1 1
1
1
1 1 1
n n
p i i ii i
n
i i m ii
n n n
i i m ii i i
r w r rn
rn
rn n n
α β ε
α β ε
= =
=
= = =
= =
= + +
= + +
∑ ∑
∑
∑ ∑ ∑
And the portfolio variance is:
2 2 2 2p p M εσ β σ σ= +
2 2p Mβ σ is the systematic risk component of the portfolio variance. This part of the risk
depends on portfolio beta and 2Mσ , and will persist regardless of the extent of portfolio
diversification.
In contrast, the unsystematic component 2εσ is attributable to firm-specific components iε .
Because these iε are independent and all have zero expected value, when more stocks are added to the portfolio, the firm-specific components tend to cancel out.
22 2 2
1
1 1n
ii n nε εσ σ σ=
= =
∑
Graphically,
94
6.2 Multifactor Models The single factor provides a simple description of stock returns, but it is not realistic. For example, a security can be sensitive to interest rate risk other than market risk alone. In real life, there exists more than one common factor that generates stock returns. The multifactor model: 1 1 2 2i i i i iK K ir F F Fα β β β ε= + + + + +
The F can be thought of as proxies for new information about macroeconomic variables. Some common factors proposed by Chen, Roll and Ross (1986):
i i i ,MP i ,DEI i ,UI i ,UPR i ,UTS ir MP DEI UI UPR UTSα β β β β β ε= + + + + + + 1. Changes in the monthly growth rate of the GDP (MP).
• This alters investor expectations about future industrial production and corporate earnings.
2. Changes in expected inflation (DEI).
• Measured by changes in the short-term T-bill yield. • Changes in expected inflation affect government policy, consumer confidence, and
interest rate levels.
3. Unexpected changes in the price level (UI). • Measured by the difference between actual and expected inflation. • Unexpectedly high or low inflation alters the values of most contracts. These include
contracts with suppliers and distributors, and financial contracts such as a firm’s debt instruments.
4. Changes in the default risk premium (UPR).
• Measured by the spread between the yields of AAA and Baa bonds of similar maturity. • As the spread widens, investors become more concerned about default.
5. Changes in the spread between the yields of long-term and short-term government bonds
(UTS). • The average slope of the term structure of interest rates as measured by the yields on
U.S. Treasury notes and bonds. • This would affect the discount rates for obtaining present values of future cash flows.
95
The factor betas β describe how sensitive the stock’s return is to changes in the common factors.
Source: Table 4, Chen, Ross & Ross (1986)
6.2.1 Factor Models for Portfolios Given the K-factor model for each stock i, a portfolio of N securities with weights wi on stock i has a factor equation of:
1 1 2 2p p p p pK K pr F F Fα β β β ε= + + + + +
Where:
1 1 2 2
1 1 11 2 21 1
2 1 12 2 22 2
1 1 2 2
1 1 2 2
p N N
p N N
p N N
pK K K N NK
p N N
w w ww w ww w w
w w ww w w
α α α α
β β β β
β β β β
β β β β
ε ε ε ε
= + + +
= + + +
= + + +
= + + +
= + + +
96
Example 6.1 Consider the following two-factor model for the returns of three securities: Cheung Kong, Cathy Pacific and HSBC.
1 2
1 2
1 2
0.03 4
0.05 3 2
0.10 1.5 0
CK CK
CP CP
HSBC HSBC
r F F
r F F
r F F
ε
ε
ε
= + − +
= + + +
= + + +
Write out the factor equation for a portfolio that equally weights all three securities.
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1
2
1 1 10 03 0 05 0 10 0 063 3 31 1 11 3 1 5 1 833 3 31 1 14 2 0 0 673 3 3
p
p
p
. . . .
. .
.
α
β
β
= + + =
= + + =
= − + + = −
Thus, the equation:
1 20.06 1.83 0.67p pr F F ε= + − +
6.2.2 Tracking Portfolios One of the most important applications of the multifactor model is that we can design a portfolio that targets a specific factor beta in order to track the risk of a security. Suppose you invest in ICBC. ICBC’s stock price will drop by 10% for every 1% decline in the growth of China’s GDP and 5% drop when the RMB depreciates 1%. Hence, investing in ICBC has two sources of risk: currency risk and a slowing of China’s economy. You can hedge these sources of risk by short selling a portfolio that tracks the sensitivity of ICBC’s stock to these two sources of risk.
97
Example 6.2 Suppose ICBC has a factor beta of 2 on the first factor and a factor beta of 1 on the second factor. Design a portfolio of stocks in Example 6.1 that tracks the ICBC return. Recall that from Example 6.1,
1 2
1 2
1 2
0.03 4
0.05 3 2
0.10 1.5 0
CK CK
CP CP
HSBC HSBC
r F F
r F F
r F F
ε
ε
ε
= + − +
= + + +
= + + +
To design a portfolio with these characteristics, it is necessary to find portfolio weights, wCK, wCP and wHSBC, that make the portfolio weighted averages of the betas equal to the target betas. To make the weights sum to one,
1CK CP HSBCw w w+ + = To have a factor beta of 2 on the first factor, 1 3 1 5 2CK CP HSBCw w . w+ + = To have a factor beta of 1 on the second factor,
4 2 0 1CK CP HSBCw w w− + + = With three equations and three unknown, the solution is:
0 10 30 8
CK
CP
HSBC
w .w .
w .
= − = =
98
6.2.3 Pure Factor Portfolio Pure factor portfolios are portfolios with a sensitivity of one to one of the factors and zero to the remaining factors. Example 6.3 What are the weights of the two pure factor portfolios constructed from the three stocks in Example 6.1?
1 2
1 2
1 2
0.03 4
0.05 3 2
0.10 1.5 0
CK CK
CP CP
HSBC HSBC
r F F
r F F
r F F
ε
ε
ε
= + − +
= + + +
= + + +
To construct the pure factor portfolio for the first factor, find portfolio weights that result in a portfolio with a target factor beta of one on the first beta and zero on the second beta.
1 3 1 5 14 2 0 0
1
CK CP HSBC
CK CP HSBC
CK CP HSBC
w w . ww w w
w w w
+ + =− + + = + + =
Thus,
0 20 4
1 6
CK
CP
HSBC
w .w .
w .
= − = − =
To find pure factor portfolio for the second factor, solve the following system of equations:
1 3 1 5 04 2 0 1
1
CK CP HSBC
CK CP HSBC
CK CP HSBC
w w . ww w w
w w w
+ + =− + + = + + =
The weights become,
0 91 3
3 2
CK
CP
HSBC
w .w .
w .
= − = − =
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6.2.4 Risk Premiums of Pure Factor Portfolios The respective risk premiums of the K-factor model are denoted by 1 2 K, , ,λ λ λ . By definition,
( )
( )where:
expected return of pure factor portfolio
risk premium of pure factor portfolio risk free rate
FPi f i
FPi
i
f
E r r
E r i
ir
λ
λ
= +
=
==
Example 6.4 Suppose we have three stocks: A, B and C, and their hypothetical factor equations (with factor means of zero) are:
1 2
1 2
1 2
0 08 2 3
0 10 3 2
0 10 3 5
A
B
C
r . F F
r . F F
r . F F
= + +
= + +
= + +
Write out the pure factor equations for the two factor portfolios and determine their risk premiums if the risk free rate is 4%. First, find the weights that satisfy pure factor portfolio 1 and 2.
For pure factor portfolio 1:
213
43
A
B
C
w
w
w
= = = −
For pure factor portfolio 2:
32343
A
B
C
w
w
w
= = − = −
Second, the α for pure factor portfolios 1 and 2:
( ) ( ) ( )
( ) ( ) ( )
1
2
1 42 0 08 0 1 0 1 0 063 32 43 0 08 0 1 0 1 0 043 3
FP
FP
. . . .
. . . .
α
α
= + − =
= − − =
100
The pure factor equations:
1 1 2
2 1 2
0 06 1 0
0 04 0 1FP
FP
r . F F
r . F F
= + +
= + +
By assumption, the factor means are zero. The risk premiums become:
1
2
0 06 0 04 0 020 04 0 04 0. . .. .
λλ= − == − =
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7 Topic 7 – Arbitrage Pricing Theory (APT) Arbitrage pricing theory (APT) is a different approach to determining asset prices. It is based on the law of one price: two items that are the same cannot sell at different prices. The advantage of APT is that we do not need strong assumptions made in CAPM. The APT requires only four assumptions: 1. The returns can be described by a factor model.
2. There are no arbitrage opportunities. 3. There are a large number of securities, so that it is possible to form portfolios that
diversify the firm specific risk of individual stocks.
4. The financial markets are frictionless.
7.1 Derivation of the APT
7.1.1 Single Factor APT Suppose the return generating process for an individual stock follows a single factor model:
i i ir Fα β ε= + + Now consider an arbitrage portfolio consisting of w in Stock A, and (1 – w) in Stock B. The two stocks share a common risk factor F . The risk free rate is rf. Here, by arbitrage portfolio, we mean zero risk portfolio. The portfolio return is therefore:
(1 )p A Br wr w r= + −
If an investor holds a well-diversified portfolio, residual risk will go to zero and only factor risk will matter. So that:
i i i ir Fα β= +
Therefore,
( ) ( )( )( ) ( )
1p A A B B
B A B B A B
r w F w F
w w F
α β α β
α α α β β β
= + + − +
= + − + + −
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By definition, this portfolio is to be risk free. Thus we need to choose a w so that:
( ) 0B A Bw Fβ β β+ − =
This implies this portfolio has a zero factor risk.
( ) 0B A B
B
A B
w
w
β β β
ββ β
+ − =
⇒ = −−
We can replace the w into the portfolio return equation:
( ) ( )
( )
p B A B B A B
BB A B
A B
r w w Fα α α β β β
βα α αβ β
= + − + + −
= − −−
Since this portfolio has zero risk, it must earn a risk free return.
( )Bp B A B f
A B
r rβα α αβ β
= − − =−
Rearrange the terms,
( )
( )
( ) ( )
Bf B A B
A B
A B A B Bf B A B
A B A B A B
f A f B B A B B A B B B
A B A B A B
f A f B B A A B
A B A B A B A B
B A f A B f
A f B f
A B
r
r
r r
r r
r r
r r
βα α αβ β
β β β β βα α αβ β β β β β
β β α β α β α β α ββ β β β β β
β β α β α ββ β β β β β β β
β α β α
α αβ β
= − −−
− −= − −
− − −
− − +− =
− − −
− = −− − − −
− = −
− −=
Since ( )A AE rα = and ( )B BE rα = , we can rewrite:
( ) ( )A f B f
A B
E r r E r rλ
β β− −
= =
The equation tells us the excess return over factor loading is constant across stocks. The constant λ is the risk premium of a stock with unit factor beta.
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The expected return on any stock can be written as: ( )i f iE r r β λ= +
This is the single factor APT.
7.1.2 Two-Factor APT Suppose the return of a security is described by the following two-factor model:
1 1 2 2i i i i ir F Fα β β ε= + + +
The theory does not say what the factors are but you may think of the individual stock has two common factors like unexpected growth in GDP and unexpected inflation. If an investor holds a well-diversified portfolio, residual risk will go to zero and only factor risk will matter. So the investor will concern about the risk and return of the portfolio by looking at the three attributes: ( ) 1 2, ,i i iE r β β .
Consider three diversified portfolios:
Portfolio ( )iE r 1iβ 2iβ A 15% 1.0 0.6 B 14% 0.5 1.0 C 10% 0.3 0.2
From the concepts of geometry, the equation of the plane defined by the three portfolios: ( ) 0 1 1 2 2i i iE r λ λ β λ β= + +
By substituting in the values of ( ) 1 2, ,i i iE r β β for portfolios A, B and C, we obtain three
equations with three unknown.
0 1 2
0 1 2
0 1 2
0 15 1 0 60 14 0 5 10 10 0 3 0 2
. .
. .
. . .
λ λ λλ λ λλ λ λ
= + + = + + = + +
Solving the system of equations and we can get the equation of the plane: ( ) 1 27 75 5 3 75i i iE r . .β β= + +
Let’s create two additional portfolios here. Portfolio E has the following factor risks and expected return: ( ) 1 215%, 0.6, 0.6E i iE r β β= = = . We can compare this portfolio with a
portfolio D constructed by placing one-third of the funds in each portfolio A, B and C.
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The risk and return for the two portfolios: Portfolio D Portfolio E ( )iE r ( ) ( ) ( )1 1 1
3 3 315 14 10 13% % % %+ + = 15%
1iβ ( ) ( ) ( )1 1 13 3 31 0 0 5 0 3 0 6. . . .+ + = 0.6
2iβ ( ) ( ) ( )1 1 13 3 30 6 1 0 0 2 0 6. . . .+ + = 0.6
By the law of one price, two portfolios that have the same risk cannot sell at a different expected return. In this situation, arbitrageurs will step in and buy portfolio E while selling an equal amount of portfolio D short. Assume an arbitrageur short sells $100 worth of portfolio D to finance the purchase of portfolio E, the payoff will be: Beginning
Cash Flow Ending
Cash Flow 1iβ 2iβ
Portfolio D +$100 –$113 –0.6 –0.6 Portfolio E –$100 +$115 0.6 0.6 Arbitrage Portfolio $0 +$2 0 0 The arbitrage portfolio involves zero investment, has no systematic factor risk, and earns $2. Arbitrage will continue until the expected return of portfolio E becomes 13%. In general, all investments and portfolios must be on a plane in the ( ) 1 2, ,i i iE r β β space. If an
investment were to lie above or below the plane, an opportunity would exist for riskless arbitrage. The arbitrage would continue until all investment converged to a plane. Recall that the equation of a plane in our illustration: ( ) 1 27 75 5 3 75i i iE r . .β β= + +
We can confirm portfolio D is on the plane and no arbitrage opportunity exists. ( ) ( ) ( )13 7 75 5 0 6 3 75 0 6 13DE r % . . . . %= ≡ + + =
However, portfolio E is not on the plane. ( ) ( ) ( )15 7 75 5 0 6 3 75 0 6 13EE r % . . . . %= ≠ + + =
In equilibrium, all portfolios must obey the two-factor APT model.
( )
( )
0 1 1 2 2
0
where:
risk premium of pure factor portfolio
i i i
f
i i f
E r
r
E r r i
λ β λ β λ
λ
λ
= + +
=
= − =
105
7.1.3 Multifactor APT The analysis can be generalized into the K-factor case.
1 1 2 2i i i i iK K ir F F Fα β β β ε= + + + + +
By analogous arguments it can be shown that all securities and portfolios have expected returns described by the K-dimensional hyper plane: ( )
( )
0 1 1 2 2
0
where:
risk premium of pure factor portfolio
i i i iK K
f
i i f
E r
r
E r r i
λ β λ β λ β λ
λ
λ
= + + + +
=
= − =
7.2 Comments on APT
7.2.1 Strength and Weaknesses of APT Strength • The model gives a reasonable description of return and risk. • Factors seem plausible. • No need to measure market portfolio correctly. Weaknesses • Model itself does not say what the right factors are. • Factors can change over time. • Estimating multi-factor models requires more data.
7.2.2 Differences between APT and CAPM
• APT is based on the factor model of returns and the no arbitrage argument. • CAPM is based on investors’ portfolio demand and equilibrium argument.
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8 Topic 8 – Anomalies and Market Efficiency
8.1 CAPM and the Cross-section of Stock Returns The CAPM states that:
( ) ( )i f m fE r r E r rβ = + −
The expected return on a security is positively and linearly related to the security’s beta. In other words, if one stock has a high beta, then the realized return should also be high. Example 8.1 Suppose: • China Mobile has a beta of 1.5. • CLP has a beta of 0.7. • The risk free rate is assumed to be 3%, and the market risk premium is assumed to be 8%.
The expected return for China Mobile: 15% = 3% + 1.5 * 8%
The expected return for CLP: 8.6% = 3% + 0.7 * 8%
Here, the beta of China Mobile is higher than the beta of CLP. Therefore, CAPM says that the return of China Mobile will be higher than CLP (15% > 8.6%) in this example. In reality, this relationship is NOT true. Fama & French (1992) test the CAPM simply by looking at the realized return of 10 beta portfolios. In each year, they rank stocks according to its beta. Their Low-β portfolio, on average, has a beta of 0.87, while the High-β portfolio has a beta of 1.72.
Source: Table 1, Fama & French (1992).
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If CAPM holds, then we expect the High-β portfolio should have the highest return. But, let’s take a look at the evidence:
Source: Table 1, Fama & French (1992).
There is NO relationship between beta and return. We do not observe any monotonic-increasing return from the Low-β portfolio to High-β portfolio.
The evidence suggests that high risk (high beta) does not truly mean high return.
The market beta fails to explain the cross-section of expected returns.
We call the empirical contradictions to the benchmark asset pricing models (e.g. CAPM) as anomalies.
In practice, some important anomalies include: • Size effect: small stocks earn a higher return than large stocks. • Value effect: value stocks earn a higher return than growth stocks. • Momentum effect: winning stocks keep winning over a short-period of time.
8.2 Size Effect The size effect refers to the negative relationship between returns and the market capitalization (market value) of a firm. Note that market capitalization is defined as the share price times share outstanding.
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
Low β β2 β3 β4 β5 β6 β7 β8 β9 High β
Ave
rage
Mon
thly
Ret
urn
(%).
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A simple way to show the size effect is to form portfolios based on market capitalization.
Source: Table 5, Loughran (1997). We can see the average return on small stocks is quite a bit higher than the average return on large stocks.
Can this size effect be explained by risk differences? Is it possible that small stocks are riskier than large stocks and therefore the return difference is merely compensation for the extra risk?
The risk story cannot fully explain the size effect.
Source: Table 1, Fama & French (1992).
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
18.00%
20.00%
Small 2 3 4 Large
Ave
rage
Ret
urn
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When we control for risk (beta), the stocks that are in the same risk class also have the size anomaly. For example, in the High-β portfolio, the size premium (i.e., the difference between Small-ME and Large-ME) is 0.86%. And this effect applies to other beta portfolios. One explanation of this size anomaly is related to the January effect. The January effect is a calendar effect where stocks, especially small-cap stocks, tend to rise markedly in price during the period starting on the last day of December and ending on the early days of January. This effect is owed to year-end selling to create tax losses, recognize capital gains, effect portfolio window dressing, or raise holiday cash. Because such selling depresses the stocks but has nothing to do with their fundamental worth, bargain hunters quickly buy in, causing the January rally. To test whether January causes this size effect, a simply way is to get rid of the return from January and then observe the return from the same size portfolios.
Source: Table 5, Loughran (1997).
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Panel A includes the return from January. We can see the SMALL has an average return of 18.9% while the LARGE is 11.91%. But once we exclude the return from January in Panel C, the size effect is gone!
How about the evidence from Pacific-Basin markets?
Chiu & Wei (1998) show there is size effect in Hong Kong, Korea, Malaysia and Thailand except Taiwan. The size sorted portfolios from 1984 to 1993 show that the size premium in Hong Kong is ranging from 0.04% to 1.24%.
Source: Table 3, Chiu & Wei (1998).
8.3 Value Effect The value effect refers to the positive relationship between returns and the ratio of value to market price of a security. For example, some ratios include: • B/M (book value of equity/market value of equity) • E/P or C/P (earnings or cash flow/price).
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8.3.1 The Glamour and Value Strategies Value stocks refer to stocks with low prices relative to book equity, earnings or cash flows. Therefore, they have high B/M, E/P or C/P. Glamour stocks refer to stocks with high prices relative to book equity, earnings or cash flows. Therefore, they have low B/M, E/P or C/P.
E/P and C/P are easy to understand:
0.05 20
0.15 6.67
E P glamourP EE P valueP E
= ⇒ = ⇒
= ⇒ = ⇒
B/M is a proxy for growth opportunities. The book value of equity refers to shareholders’ equity in the balance sheet and market value of equity is simply the market price * shares outstanding.
A large extent of book value is based on historical costs. It does not reflect the value of future prospects.
On the other hand, the market value of the stock does reflect these future prospects. When the market perceives a firm with good future prospects, the book value will be small relative to the market value. Think of a company that has recently introduced a new and exciting product. The historical cost of its assets-in-place may be small, but sales and earnings are up. Since the company has great prospects, the market is willing to pay a higher stock price. Therefore, the B/M will be small for growth firms.
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And here is the evidence.
Source: Table 1, Lakonishok, Shleifer & Vishny (1994).
Both ratios show that value stocks outperform glamour stocks subsequently. Also, value stocks outperform the market indices such as S&P500. If you buy value stocks, you can beat the market!
B/M
0%
5%
10%
15%
20%
25%
Glamour 2 3 4 5 6 7 8 9 Value
Ave
rage
Ret
urn
C/P
0%
5%
10%
15%
20%
25%
Glamour 2 3 4 5 6 7 8 9 Value
Ave
rage
Ret
urn
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In practice, fund managers apply the size and value to describe their investing style. For example, Morningstar, one of the largest mutual fund information providers, classifies funds according to their market capitalization and investment style.
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The Morningstar style box is a nine-square grid that classifies securities by size along the vertical axis and by value-and-growth characteristics along the horizontal axis. The size is divided into large, medium and small, whereas the growth characteristic is specified as value, balanced and growth. The chart below illustrates a mutual fund which its investment style is described as "Large-cap Value".
Are the value stocks riskier than the glamour stocks?
Again, this is not the case. The difference in two risk measures – beta and standard deviations are too small to justify the difference in returns.
Source: Table 8, Lakonishok, Shleifer & Vishny (1994).
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Lakonishok, Shleifer & Vishny (1994) propose the extrapolation story can explain the value effect. The idea is that: 1. Value stocks are underpriced because investors have low expectation.
• These stocks performed poorly in the past. • Investors expect them to continue to perform poorly. • But once actual performance improved, stock prices rise and returns become high.
2. Glamour stocks are overvalued because investors have high expectation.
• These stocks performed well in the past. • Investors expect them to perform well in the future. • When future performance does not meet expectation, investors are disappointed, stock
prices drop and returns become low.
8.4 Momentum Investing Strategies Stocks with prices on an upward (downward) trajectory over a prior period of 3 to 12 months have a higher than expected probability of continuing on that upward (downward) trajectory over the subsequent 3 to 12 months. This temporal pattern in prices is referred to as momentum. In layman terms, momentum investing is based on a simple rule: buy stocks that perform the best (winner) and sell stocks that perform the worst (loser) in recent past. This strategy focuses on cross-sectional patterns of return continuation instead of the time-series predictability known as technical analysis. Jegadeesh and Titman (1993) construct a simple J-K momentum strategy – select stocks on the basis of returns over the past J months and hold them for K months. • At the beginning of each month t the stocks are ranked in ascending order on the basis of
their returns in the past J months. • Based on these rankings, form ten equally-weighted deciles portfolios. • In each month t, the strategy buys the winner portfolio and sells the loser portfolio and
holds this position for K months. • The strategy closes out the position initiated in month t – K.
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The result:
Source: Table 1, Jegadeesh and Titman (1993) Buy-sell means a zero dollar investment strategy. The investors short the loser portfolio and use the proceeds to long the winner portfolio. Without putting money, the momentum strategy generates positive return!
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Two explanations: • Risk-based explanations for momentum. • Behavioral explanations for momentum.
The risk story is not consistent because they find that the beta of the winning portfolio (P10) is not significantly higher than the losing portfolio (P1).
Source: Table 2, Jegadeesh and Titman (1993)
The behavioral explanations challenge the often-assumed fully rational behavior of investors. Some psychological factors may explain: • Investors are over confident in their private signals related to the value of a firm. (Daniel
et al., 1998) • Investors are conservative (Barberis et al., 1998). Conservatism relates to slow updating
of beliefs when new evidence is presented.
8.5 Efficient Market Hypothesis In an efficient market, an asset’s price should be the best possible estimate of its economic values. A financial market is informational efficient when market prices reflect all available information about value.
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What does it mean to “reflect all available information”? There are three types of market efficiency.
Weak Form Efficiency • Stock prices reflect all historical information. • No investor can earn abnormal returns by developing trading rules based on historical
price or return information.
Semi-strong Form Efficiency • Stock prices reflect all public information. • Publicly available information includes published accounting statements and information
found in annual reports. Strong Form Efficiency • Stock prices reflect all public and private information. • No investor can earn abnormal returns by developing trading rules based on private
information.
8.5.1 Empirical Tests of Efficient Market Hypothesis In general, the tests of efficient market hypothesis often start with:
( )
( ):
risk adjusted expected return from a pricing model (e.g. CAPM)
= residual term
i i i
i
i
r E r ewhereE r
e
= +
=
The efficient market hypothesis says that the residual term ie must be (1) zero on average; (2) unpredictable based on current information.
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Example 8.2 The annual return and beta of the three assets are as follows:
Average Annual Return
Beta
The Franklin Income Fund 12.9% 1.000 Dow Jones Industrial Average 11.1% 0.683 Salomon’s High Grade Bond Index 9.2% 0.367
Suppose the market return is 13% and the risk free rate is 7%. Using CAPM, the expected returns are:
Expected Return The Franklin Income Fund 13.0% Dow Jones Industrial Average 11.1% Salomon’s High Grade Bond Index 9.2%
Consider this example, the expected return on the Franklin Income Fund was higher than the realized return. The market expected 13% performance the Fund delivered 12.9%. The difference is the abnormal return.
Is the existence of this abnormal return evidence market inefficiency? However, we cannot conclusively state that the market is inefficient because: 1. The model that we used to risk adjusted the returns (CAPM) could be incorrect. 2. Our estimates of beta may be incorrect. 3. Our estimate of the expected return of the market may be incorrect.
As a result, all statements about market efficiency should be conditioned in terms of the model used to test efficiency. That is, any test of efficiency is a joint test of efficiency and the asset pricing model.
Given a particular pricing model, you might find evidence against market efficiency. Another explanation, however, is that the market is efficient and you are using the wrong pricing model. This is a common dilemma in testing joint hypotheses.
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9 Topic 9 – Fixed Income Securities
9.1 Fixed Income Securities and Markets
Fixed income securities are financial claims with promised cash flows of fixed amount paid at fixed dates. A bond is a basic fixed income security. The issuer sells a bond to the bondholder for some amount of cash. This arrangement obligates the issuer to make specified interest payments to the bondholder on specified dates.
A typical coupon bond obligates the issuer to make semiannual interest payments to the bondholder. These payments are called coupon payments. When the bond matures, the issuer repays the debt by paying the bondholder the bond’s par value.
Some bonds do not make any periodic coupon payments. Zero-coupon bond is issued that makes no coupon payments. Instead, the bondholder realizes interest by the difference between the maturity value and the purchase price.
9.1.1 Types of Fixed Income Securities
Treasury securities • Treasury bills
Treasury bills (T-bills) mature in one-year or less. Like zero-coupon bonds, they do not pay interest prior to maturity.
• Treasury notes
Treasury notes have maturities between 1 to 10 years. Like straight bonds, they make semiannual coupon payments.
• Treasury bonds
Treasury bonds have maturities usually ranging from 10 to 30 years. They also make semiannual coupon payments.
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Corporate bonds • Mortgage bonds
The issuer has granted the bondholders a first-mortgage lien on substantially all of its properties. A lien is a legal right to sell mortgaged property to satisfy unpaid obligations to bondholders.
• Debentures
Debentures are not secured by a specific pledge of designated property. They are unsecured debt backed only by the goodwill of the corporation. In the event of liquidation, debenture holders are paid after mortgage bondholders.
• Convertible bonds
Convertible bonds give bondholders an option to exchange each bond for a specified number of shares of common stock of the corporation.
• Callable / Puttable bonds
The call provisions on corporate bonds allow the issuer to repurchase the bond at a specified call price before the maturity date. The puttable bond grants the bondholder the right to sell the issue back to the issuer at par value on designated dates.
9.2 Bond Pricing
The price of a bond is equal to the present value of the expected cash flow.
9.2.1 Coupon Bond
The cash flow for a coupon bond consists of an annuity of fixed coupon interest and the par value at maturity.
1 2 3 T
$C $C $C $C+$F
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In general, the price of a bond is given by:
( ) ( ) ( )
( ) ( ) ( )
( ) ( )( ) ( )
2 1
2
1 1 1 1
1 1 1 11 1 1 1
1 1 111 1
Coupon Annunity Factor , Par PV Factor ,
T T
T T
T T
C C C C FPy y y y
C Fy y y y
C Fy y y
y T y T
−
+= + + + +
+ + + +
= + + + +
+ + + + = × − + × + +
= × + ×
Example 9.1 A 30-year bond with an 8% (4% per 6 months) coupon rate and a par value of $1,000 has the following cash flows: Semiannual coupon = $1,000×4% = $40 Par value at maturity = $1,000
Therefore, there are 60 semiannual cash flows of $40 and a $1,000 cash flow 60 six-month periods from now.
1 2 3 60
$40 $40 $40 $40+$1,000
The price of this 30-year bond:
( ) ( ) ( )2 59 6040 40 40 1040
1 1 1 12 2 2 2:
annual discount rate
P y y y y
wherey
= + + + ++ + + +
=
Suppose the discount rate is 8% annually or 4% per 6-month, the price of the bond is:
( ) ( )
( ) ( )60 60
1 1 111 1
1 1 140 1 10000.04 1.04 1.04
904.94 95.061000
T TP C Fy y y
= × − + × + + = × − + ×
= +=
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What if the discount rate rises to 10% annually, then the bond price will fall by $189.29 to $810.71.
( ) ( )60 601 1 140 1 1000
0.05 1.05 1.05
757.17 53.54810.71
P
= × − + ×
= +=
The central feature of fixed income securities is that there is an inverse relation between bond price and discount rate.
In this example, the price/yield relationship for a 30-year, 8% coupon bond:
Yield 4% 6% 8% 10% 12% Price 1,695.22 1,276.76 1,000.00 810.71 676.77
Note: • When the coupon rate equals the discount rate, the price equals the par value. • When the coupon rate is less than the discount rate, the price is less than the par value. • When the coupon rate is greater than the discount rate, the price is greater than the par
value.
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9.2.2 Zero-Coupon Bond
In the case of a zero-coupon bond, the only cash flow is the par value. Therefore, the price of a zero-coupon bond is simply the present value of the par value.
Example 9.2 A zero-coupon bond that matures in 20 years has a par value of $1,000. If the required yield is 4.3%, then the value is:
20
1000 430 831 043
P ..
= =
9.3 Dirty Price, Clean Price and Accrued Interest
Typically, an investor will purchase a bond between coupon dates. So how to determine the price when the settlement date falls between coupon periods?
Suppose we have a coupon bond that matures in 4 years has a par value of $1,000. Assume the bond pays 10% coupon annually and the market yield is also 10%.
The price of the bond at t = 0 is:
0 2 3 4
100 100 100 11001 1 1 1 1 1 1 11000
tP. . . .= = + + +
=
After half year, if the bondholder wishes to sell the bond, the price of the bond at t = 0.5 becomes:
.5 1 2 4
100100 100 1100
3
0 5 0 5 1 5 2 5 3 5
100 100 100 11001 1 1 1 1 1 1 11048 81
t . . . . .P. . . .
.
= = + + +
=
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In general, the present value formula should be modified because the cash flows will not be received one full period from now. For a bond with T coupon payments remaining to maturity, the price becomes:
( ) ( ) ( ) ( )1 2 11 1 1 1
No. of days between settlement and next coupon paymentNo. of days in the coupon period
w w w T wC C C C FP
y y y ywhere :
w
+ + − +
+= + + + +
+ + + +
=
The price calculated in this way is called the dirty price because it reflects the total cash flow the buyer will receive.
In this illustration, the dirty price has been moved without any economic reasons, such as changes in interest rate. When we look at the dynamics of the dirty price, the price gradually rises and then suddenly drops every period. The rises reflect the accrued interest to the seller and the drop is the result of ex-coupon.
Therefore, the dirty price can be disaggregated into clean price and accrued interest. In short, Dirty Price = Clean Price + Accrued Interest.
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To compute the accrued interest, No. of days from last coupon payment to settlement date
No. of days in coupon period
accrued interestcoupon
AI C
where :AIC
=
==
The accrued interest calculation for a bond is dependent on the day-count basis. Below are different versions of day-count conventions:
Convention Definition Securities Actual / Actual The actual number of
days between two dates is used.
Leap years are 366 days, non-leap years are 365 days.
US Treasury bonds and notes
Actual / 365 The actual number of days between two dates is used in numerator.
All years are assumed to have 365 days.
Eurobonds, and Euro-floating rate notes (FRNs)
Foreign government bonds
Actual / 360 The actual number of days between two dates is used in numerator.
A year is assumed to have 12 months of 30 days each.
Eurodollar deposits, commercial paper, banker’s acceptance
Repo, FRNs and LIBOR-based transactions
30 / 360 All months are assumed to have 30 days, resulting in a 360 day year.
If the first date falls on the 31st, it is changed to the 30th.
If the second date falls on the 31st, it is changed to the 30th, but only if the first date falls on the 30th or the 31st.
Corporate bonds, US agency securities, municipal bonds, mortgages
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Example 9.3 Suppose that a corporate bond with a coupon rate of 10% maturing March 1, 2006 is purchased with a settlement date of July 17, 2000. What would the price of this bond be if it is priced to yield 6.5%?
For corporate bond, the day count convention is 30 / 360. That is, each month is assumed to have 30 days and each year 360 days. The number of days between July 17 and September 1 is:
July 13 days August 30 days September 1 days 44 days
There are 44 days between the settlement date and the next coupon date. The number of days in the coupon period is 180. Therefore:
44 0.2444180
w = =
The number of coupon payments remaining is 12. The semiannual interest rate is 3.25%. Suppose the par value is $1,000, the dirty price is:
( ) ( ) ( ) ( )1 2 1
0 2444 1 2444 2 2444 11 2444
1 1 1 150 50 50 1050
1 0325 1 0325 1 0325 1 03251200 28
w w w T w
. . . .
C C C C FPy y y y
. . . ..
+ + − +
+= + + + +
+ + + +
= + + + +
=
The number of days from the last coupon payment date (March 1, 2000) to the settlement date is 180 – 44 = 136. The accrued interest per $1,000 of par value is:
No. of days from last coupon payment to settlement dateNo. of days in coupon period
13650180
37 78
AI C
.
=
=
=
The clean price is the dirty price minus accrued interest, 1200.28 – 37.78 = 1162.5.
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9.4 Conventional Yield Measures
An investor who purchases a bond can expect to receive a dollar return from one or more of the following sources: • The coupon interest payments made by the issuer. • Any capital gain or loss when the bond matures, is called or is sold. • Income from reinvestment of the coupon interest payments (interest-on-interest).
Three yield measures are commonly used by market participants to measure the three potential sources of return.
9.4.1 Current Yield
The current yield relates the annual coupon interest to the market price.
Annual dollar coupon interestCurrent yieldPrice
=
Example 9.4
The current yield for an 18-year, 6% coupon bond selling for $700.89 per $1,000 par value is:
60Current yield700.890.0856
=
=
The current yield does not account for the return from interest-on-interest.
9.4.2 Yield-to-Maturity
Yield-to-maturity (YTM) is the average rate of return that will be earned on a bond if it is bought now and held until maturity.
Given its maturity, the principal and the coupon rate, there is a one to one mapping between the price of a bond and its YTM.
( ) ( )1 1 1
Tt
t Tt
C FPYTM YTM=
= ++ +
∑
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Example 9.5 Consider a 30-year bond with $1,000 face value has an 8% coupon. Suppose the bond sells for $1,276.76, the YTM:
( ) ( )60
601
40 10001276.761 12 2
tt YTM YTM=
= ++ +
∑
Solve for the discount rate, YTM = 6%.
The yield-to-maturity considers the return from interest-on-interest. It assumes that the coupon interest can be reinvested at YTM.
Suppose we deposit $1,000 for 4 years which earns 10% p.a. After 4 years, we receive:
( )41000 1 1 1464 1. .× =
What if we invest in a coupon bond that matures in 4 years has a par value of $1,000? The bond pays 10% coupon annually and the market yield is also 10%.
After 4 years, the total cash flow we receive is $1,400. The difference 1464.1 – 1400 = 64.1 is the interest earned from the reinvestment at the rate equal to the YTM.
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Example 9.6 Consider the T-bills maturing on April 5, 2007, the asked quote is 4.9%. That is, the T-bills is selling at a discount d = 4.9%. There are n = 90 days to maturity.
The price you actually pay:
( )( )
10,000 1 360
10,000 1 0.049 90 360
9,877.5
P d n= − × = − × =
Rate of return for 90 days:
10,000 9,877.59,877.5
0.01241.24%
r −=
==
The ask yield: 1.24% 365 90 5.03%× =
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9.5 Default Risk
Fixed income securities have promised payoffs of fixed amount at fixed times. Excluding government bonds, other fixed income securities carry the risk of failing to pay off as promised.
Default risk refers to the risk that a debt issuer fails to make the promised payments – interest or principal.
To gauge the default risk, rating agencies, like Moody’s and S&P provide indications of the likelihood of default by each issuer.
Moody’s S&P Investment Grade Gilt-edge Aaa AAA Very high grade Aa AA Upper medium grade A A Lower medium grade Baa BBB Below Investment Grade Low grade Ba BB
Investment grade bonds are generally more appropriate for conservative clients. These bonds typically provide the highest degree of principal and interest payment protection, and they are generally the least likely to default: • Moody’s – Aaa to Baa • S&P – AAA to BBB
Speculative (junk) bonds may be suitable for more aggressive clients willing to accept greater degrees of credit risk in exchange for significantly higher yields: • Moody’s – Ba or below • S&P – BB or below
9.5.1 Traditional Credit Analysis
Traditional credit analysis for corporate bond default or potential downgrade has focused on the calculation of a series of ratios historically associated with fixed income investments.
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The Altman Z-score is a metric that gives insights into the likelihood of a firm going bankrupt in the next 2 years:
EBIT Net sales Market value of equity3 3 1 0 6Total assets Total assets Total liabilitiesRetained earnings Working capital1 4 1 2
Total assets Total assets
Z . .
. .
= × + × + ×
+ × + ×
The interpretation of Altman Z-score: • Z > 3.00 – not likely to go bankrupt • 2.70 < Z < 2.99 – on alert • 1.80 < Z < 2.70 – likely to go bankrupt within 2 years • Z < 1.80 – financial catastrophe
9.6 Interest Rate Risk
The fundamental principle of bonds is that the price changes in the opposite direction of the change in the yield for the bond. An increase (decrease) in the yield decreases (increases) the present value of its future cash flow, and therefore, the bond’s price.
9.6.1 Price Volatility and Bond Characteristics
The characteristics of a bond that affect its price volatility are: • Maturity • Coupon rate
As an illustration, consider the four hypothetical bonds with par value of $1,000 but different maturity and coupon rate.
Yield 6% / 5 Year 6% / 20 Year 9% / 5 Year 9% / 20 Year4.00% 1,089.83 1,273.55 1,224.56 1,683.89 5.00% 1,043.76 1,125.51 1,175.04 1,502.06 5.50% 1,021.60 1,060.20 1,151.20 1,421.37 5.90% 1,004.28 1,011.65 1,132.56 1,361.19 6.00% 1,000.00 1,000.00 1,127.95 1,346.72 6.01% 999.57 998.85 1,127.49 1,345.29 6.10% 995.75 988.54 1,123.37 1,332.47 6.50% 978.94 944.48 1,105.28 1,277.61 7.00% 958.42 893.22 1,083.17 1,213.55 8.00% 918.89 802.07 1,040.55 1,098.96
Price
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An examination of this exhibit reveals that: • Although the price moves in the opposite direction from the change in required yield, the
percentage price change is not the same for all bonds. • For small changes in the required yield, the percentage price change for a given bond is
roughly the same, whether the required yield increases or decreases. • For large changes in required yield, the percentage price change is not the same for an
increase in required yield as it is for a decrease in required yield. • For a given large change in basis points in the required yield, the percentage price
increase is greater than the percentage price decrease.
Yield 6% / 5 Year 6% / 20 Year 9% / 5 Year 9% / 20 Year4.00% 8.98% 27.36% 8.57% 25.04%5.00% 4.38% 12.55% 4.17% 11.53%5.50% 2.16% 6.02% 2.06% 5.54%5.90% 0.43% 1.17% 0.41% 1.07%5.99% 0.43% 1.17% 0.41% 1.07%6.01% -0.04% -0.12% -0.04% -0.11%6.10% -0.43% -1.15% -0.41% -1.06%6.50% -2.11% -5.55% -2.01% -5.13%7.00% -4.16% -10.68% -3.97% -9.89%8.00% -8.11% -19.79% -7.75% -18.40%
Percentage Price Change (Initial Yield = 6.00%)
In short, The impact of maturity: • The longer the bond’s maturity, the greater the bond’s price sensitivity to changes in
interest rates, holding all other factors constant. The impact of coupon rate: • The lower the coupon rate, the greater the bond’s price sensitivity to changes in interest
rates. • An implication is that zero-coupon bonds have greater price sensitivity to interest rate
changes than same maturity bonds bearing a coupon rate and trading at the same yield. With the background about the price volatility characteristics of a bond, we can now turn to an alternate approach to full valuation: the duration/convexity approach.
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9.6.2 Duration Recall that: • When interest rate goes up, bond price drops. • Long-term bond is more sensitive to interest rate movement. For zero coupon bond, the maturity is well defined. For coupon bond, however, there are many payments and each has its own “maturity date”. Therefore, we need a measure for the average maturity of the bond’s cash flows, or effective maturity. Macaulay’s Duration Macaulay’s duration is the average maturity of the portfolio of mini-zeros. Let wt denotes the weight associated with cash flow made at time t (CFt), then:
( )1
:YTM
= Bond Price
tt
t
CF yw
Pwhere
yP
+=
=
The weights sums to one because the sum of cash flows discounted at yield to maturity is equal to the bond price. The Macaulay’s duration formula is given by:
1
T
tt
D t w=
= ×∑
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Example 9.7 Suppose we have 8% coupon and zero coupon bond, each with 2 years to maturity. Assume the YTM is 10% on each bond or 5% semiannually.
t tCF ( )1 ttCF y+ tw tt w×
8% Coupon Bond
0.5 40 38.095 0.0395 0.0197 1.0 40 36.281 0.0376 0.0376 1.5 40 34.554 0.0358 0.0537 2.0 1,040 855.611 0.8871 1.7741
∑ 964.540 1.0000 1.8852
Zero Coupon Bond
0.5 0 0.000 0.0000 0.0000 1.0 0 0.000 0.0000 0.0000 1.5 0 0.000 0.0000 0.0000 2.0 1,000 822.702 1.0000 2.0000
∑ 822.702 1.0000 2.0000 The duration of the zero coupon bond is 2 years. The duration of the 2-year coupon bond is 1.8852 years < 2 years. In general, duration is a measure of the approximate sensitivity of a bond’s value to interest rate changes. Remember that the price of a bond is the present value of all of its future cash flows.
( ) ( )
( )
1 22
1
1 1 1
1
TT
Tt
tt
CF CF CFPy y y
CFy=
= + + ++ + +
=+
∑
Mathematically, to represent the sensitivity of a bond’s value to changes in the yield:
( )
( )
1
1
1
11 1
Tt
tt
Tt
tt
CFPy y y
CFty y
=
=
∂ ∂=
∂ ∂ +
= − × + +
∑
∑
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Dividing by P gives:
( )1
11 1
11
Tt
tt
CFP ty yy
P P
Dy
=
∂ − ×
+ + ∂ =
= − ×+
∑
We can switch to discrete changes:
11
11
1
Py D
P yPy D
P yP y D
P y
∂∂ = − ×
+∆∆ ≈ − ×
+∆ ∆
⇒ = − ×+
Modified Duration The modified duration is defined as:
*
1DD
y=
+
With slightly modification,
*P D yP∆
= − ×∆
The percentage change in bond price is just the product of modified duration and the change in the bond’s yield to maturity. Modified duration is a natural measure of the bond’s exposure to changes in interest rates.
137
Example 9.8 The 2-year 8% coupon bond sells at $964.54 at half-year discount rate of 5%. If the discount rate rises to 5.01% (1 basis point), then the bond price will drop by:
*
2 1.8852 0.01%1 0.05
0.0359%
P D yP∆
= − ×∆
×= − ×
+= −
The bond price becomes:
( )$964.54 1 0.0359% $964.19× − =
Notice here the discount rate 5% is a semiannual rate. Therefore the corresponding duration is 2(1.8852) = 3.7704 half year periods. Effective Duration The Macaulay’s / modified duration assumed that yield changes do not change the expected cash flows. For bonds that have embedded options, such as puttable and callable bonds, the Macaulay’s / modified duration will not correctly approximate the price move for a change in yield. The effective duration is a measure in which recognition is given to the fact that yield changes may change the expected cash flows. The effective duration of a bond is estimated as follows:
( )( )0
0
2:
change in yield in decimalinitial priceprice if yields decline by price if yields increase by
effective V VDV y
wherey
VV yV y
− +
−
+
−=
∆
∆ === ∆= ∆
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Example 9.9 Consider a 9% coupon 20-year option-free bond selling at 1,346.72 to yield 6%. Suppose the yield changed by 20 basis points, what is the effective duration? With 20 basis points up and down,
0
0.0021,346.721,375.891,318.44
yVVV−
+
∆ ====
( )( )
( )( )
021,375.89 1,318.442 1,346.72 0.00210.66
effective V VDV y− +−
=∆
−=
=
The duration of 10.66 means that the approximate change in price for this bond is 10.66% for a 100 basis point change in rates.
9.6.3 The Determinants of Duration 1. The duration of a zero coupon bond equals its time to maturity. 2. Holding maturity constant, a bond’s duration is lower when the coupon rate is higher.
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3. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. • For bonds selling at par or at a premium, duration always increases with maturity. • For deep discount bonds, duration can decrease with maturity.
4. Holding other factors constant, the duration of a coupon bond is higher when the bond’s
yield to maturity is lower.
140
5. The duration of a level perpetuity is: 1 y
y+
• This rule makes it clear that maturity and duration can differ substantially. • For y = 10%, a level perpetuity paying $100 forever will have a duration of 1.1/0.1 =
11 years. But the time to maturity is infinite. 6. The duration of a level annuity is:
( )1
1 1:
the no of paymentsthe annuity's yield per payment period
Ty T
y ywhereTy
+−
+ −
==
• The duration of a 10-year annual annuity with a yield of 8% is:
10
1.08 10 4.870.08 1.08 1
years− =−
7. The duration of a coupon bond equals:
( ) ( )( )
111 1
:coupon rate per payment periodthe no of paymentsthe annuity's yield per payment period
T
y T c yyy c y y
wherecTy
+ + −+−
+ − +
===
For coupon bond selling at par (i.e., c = y), the duration simplifies to:
( )1 11
1 Ty
y y
+−
+
Example 9.10 A 10% coupon bond with 20 years to maturity pays semiannual coupons. The annualized yield to maturity is 8%. What is its duration? Note here c = 5%, T = 40 and y = 4%, the outcome will be the half-year period!
( )40
1.04 40 0.05 0.041.04 19.740.04 0.05 1.04 1 0.04
+ −− =
− +
19.74 half-year = 9.87 years.
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9.6.4 Convexity Consider a 4-year T-note with face value $100 and 7% coupon, selling at $103.5, yielding 6%. For T-notes, coupons are paid semiannually. Using 6-month intervals, the coupon rate is 3.5% and the yield is 3%.
t CF PV(CF) t*PV(CF) 1 3.5 3.40 3.40 2 3.5 3.30 6.60 3 3.5 3.20 9.60 4 3.5 3.11 12.44 5 3.5 3.02 15.10 6 3.5 2.93 17.59 7 3.5 2.85 19.92 8 103.5 81.70 653.63 103.50 738.28
Duration in half year periods is:
738.28 103.50 7.13D = = Modified duration is:
* 7.13 6.921 1.03
DDy
= = =+
As the yield changes, the bond price also changes:
Yield Price Using D* Difference 0.040 96.63 96.30 0.33 0.035 100.00 99.90 0.10 0.031 102.79 102.78 0.01 0.030 103.50 - - 0.029 104.23 104.22 0.01 0.025 107.17 107.08 0.08 0.020 110.98 110.70 0.28
For small yield changes, pricing by modified duration is accurate. However, for large yield changes, pricing by modified duration is in accurate.
142
The reason is that bond price is not a linear function of the yield. For large yield changes, the effect of curvature (nonlinearity) becomes important.
The mathematical definition of convexity can be derived by applying Taylor series expansion of a function. The Taylor series expansion of a function ( )f y h+ in the region of y as h approaches zero is:
( ) ( ) ( ) ( ) ( )2 ( )
1! 2! !
n nf y h f y h f y hf y h f y
n′ ′′
+ = + + + +
Define ( )P y as the price of a bond at a yield y. Then, writing the price of the bond at a new
yield ( )y y+ ∆ using the Taylor series expansion results:
( ) ( ) ( ) ( ) 2
1! 2!P y y P y y
P y y P y′ ′′∆ ∆
+ ∆ = + +
The price of the bond is:
( )( )1 1
Tt
tt
CFP yy=
=+
∑
The first derivative with respect to y:
( )( )1
11 1
Tt
tt
CFP y ty y=
′ = − ×
+ + ∑
The second derivative is:
( )( )
( )( )2
1
1 11 1
Tt
tt
CFP y t ty y=
′′ = + ×
+ + ∑
Then the return due to the change in yield is:
143
( ) ( )( )
( )( )
( )( )
( )( )
( )( )
( )( )
2
21 1 2
* 2
12
1 1 11 1 1 11
212
T Tt t
t tt t
P y y P y P y y P y yP y P y P y
CF CFt t ty y y yP y y
P P y P y
D y Convexity y
= =
′ ′′+ ∆ − ∆ ∆= + ×
− × + ×
+ + + + ∆ = ×∆ + × ×∆
= − ×∆ + × ×∆
∑ ∑
Where convexity is the curvature of the bond price as a function of the yield:
( ) ( )( )2
21
11 1
Tt
tt
CFConvexity t tP y y=
= +
× + + ∑
9.6.5 Immunization Bank’s assets and liabilities are subject to interest rate risk. Their assets are loans and their liabilities are the deposits. Both will fluctuate when interest rate moves.
By properly adjusting the maturity of their portfolios, banks can shed their interest rate risk.
Immunization refers to strategies to shield their overall financial status from exposure to interest rate fluctuations.
For assets with equal yields, the duration of a portfolio is the weighted average of the durations of the assets comprising the portfolio. Suppose you have an obligation of $19,487 due in 7 years. At 10% rate, the PV of the amount is $10,000.
You want to immunize the obligation by holding a portfolio of the 3-year zero coupon bond and a perpetuity with y = 10% (i.e., paying $100 forever, with a face value of F = $1,000).
Let w be the weight for the zero’s weight and (1 – w) be the perpetuity’s weight. The duration of the zero is 3 years and the duration of the perpetuity is 1.1/0.1 = 11 years.
The desired duration is 7 years, what is the weight?
( )3 1 11 712
w w
w
× + − × =
⇒ =
144
You therefore invest $5,000 in the zero coupon bond and $5,000 in the perpetuity.
Next year, even if the interest rate does not change, rebalance is necessary!
Because now the zero coupon bond has a duration of 2 years, while the perpetuity’s duration is still 11 years. The obligation’s duration is 6 years.
Therefore, the new weight is given by:
( )2 1 11 659
w w
w
× + − × =
⇒ =
Since now you have $11,000 after 1 year of investment at 10%, you will invest 5/9*$11,000 = $6,111 in the 2-year zero bond and $11,000 – $6,111 = $4,889 in the perpetuity. You need to put the entire $500 perpetuity payment in the zero and sell an additional $111 of the perpetuity.
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10 Topic 10 – Term Structure of Interest Rates Recall that from bond pricing, we have assumed that the interest rate is constant over all future periods. In reality, interest rates vary through time. The term structure of interest rates refers to the relation between the interest rate and the maturity or horizon of the investment. The term structure can be described using the yield curve.
10.1 The Yield Curve The yield curve is a plot of yield to maturity as a function of time to maturity.
The slope of the yield curve depends on the difference between yields on longer and shorter maturity bonds.
Upward sloping yield curve: • Short term interest rates are below long term interest rates. • Reflect the higher inflation risk premium that investors demand for longer term bonds. Downward sloping yield curve: • Long term interest rates are below short term interest rates. • The market expects interest rate to fall. • An inverted curve may indicate a worsening economic situation in future.
146
10.1.1 Using the Yield Curve to Price a Bond Consider a five-year Treasury bonds, the coupon rate is 12%. The cash flow for the bond per $100 par value for the 10 six-month periods to maturity would be:
Period Cash Flow 1 – 9 $6 10 $106
1 2 3 10
$6 $6 $6 $106
Because of different cash flow patterns, it is not appropriate to use the same interest rate to discount all cash flows. Instead, each cash flow should be discounted at a unique interest rate that is appropriate for the time period in which the cash flow will be received.
( ) ( ) ( )2 9 101 2 9 10
6 6 6 1061 1 1 1
Pz z z z
= + + + ++ + + +
But what should be the interest rate for each period? We can view the bond as 10 zero-coupon instruments: One with a maturity value of $6 maturing six months from now, a second with a maturity value of $6 maturing one year from now, and so on. The final zero-coupon instrument matures 10 six-month periods from now has a maturity value of $106. To determine the value of each zero-coupon instrument, it is necessary to know the yield on a zero-coupon Treasury with that same maturity. This yield is called the spot rate.
10.1.2 Constructing the Theoretical Spot-Rate Curve Consider the six-month Treasury in the following:
Maturity Coupon Rate Annualized Yield Price 0.5 0.0000 0.0800 96.15 1.0 0.0000 0.0830 92.19 1.5 0.0850 0.0890 99.45 2.0 0.0900 0.0920 99.64
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What is the spot rate for a (theoretical) 1.5-year zero coupon Treasury? The price of a theoretical 1.5-year Treasury should equal the present value of three cash flows from an actual 1.5-year coupon Treasury. Using $100 as par, the cash flow is:
t CF 0.5 0.085*$100*0.5 = $4.25 1.0 0.085*$100*0.5 = $4.25 1.5 0.085*$100*0.5 + $100 = $104.25
The present value:
( ) ( ) ( )1 2 31 2 3
4.25 4.25 104.251 1 1
Pz z z
= + ++ + +
Because the 6-month & 1-year spot rate are 8% and 8.3%, and the price of the 1.5-year coupon Treasury is $99.45, the following relationship must hold:
( ) ( ) ( )1 2 33
4.25 4.25 104.2599.451.04 1.0415 1 z
= + ++
We can solve for the theoretical 1.5-year spot rate:
( ) ( ) ( )
( )
1 2 33
33
3
4.25 4.25 104.2599.451.04 1.0415 1
104.254.08654 3.918051
0.04465
z
zz
= + ++
= + ++
⇒ =
Doubling this yield, the theoretical 1.5-year spot rate is 8.93%.
148
10.2 Spot and Forward Interest Rates The spot rate yt is the annualized interest rate for maturity date t. Example 10.1 On 1/8/2001, the spot interest rates for different maturities are:
The set of spot interest rates for different dates gives the term structure of spot interest rates, which refers to the relation between spot rates and their maturities.
The forward rate ft,T is the rate of return for investing that is set today. It is the rate for a transaction between two future dates, for instance, t and T. Consider a 2-year investment horizon and the following instruments: a 2-year spot rate y2
(from time 0 to 2), a 1-year spot rate y1 (from time 0 to 1) and a forward rate from year 1 to year 2, which we denote f1,2.
149
An investor with a 2-year investment horizon has two choices: 1. Invest at the 2-year spot rate y2. 2. Invest at the 1-year spot rate y1 and roll over the deposit with the forward rate f1,2. Since all rates y1, y2, and f1,2 are known today, the two investments can be compared:
( ) ( )( )22 1 1,21 1 1y y f+ = + +
In general, the forward interest rate between time t – 1 and t is:
( ) ( ) ( )
( ) ( )( )
11 1,
1, 11
1 1 1
11
1
t tt t t t
tt
t t tt
y y f
or
yf
y
−− −
− −−
+ = + +
++ =
+
Example 10.2 Suppose the discount bond prices are as follows:
t 1 2 3 4 Price 0.9524 0.8900 0.8278 0.7629 YTM 0.050 0.060 0.065 0.070
A customer would like to have a forward contract to borrow $20 million three years from now for one year. What is the quote rate for this forward loan?
( )( )
44
3,4 33
4
3
11
1
1.070 11.0658.51%
yf
y+
= −+
= −
=
10.3 Theories of the Term Structure What determines the shape of the term structure? • The expectation hypothesis • Liquidity preference
150
10.3.1 The Expectation Hypothesis The expectation hypothesis states that the forward rate is a prediction of the future spot rate. Recall that:
( ) ( )( )22 1 1,21 1 1y y f+ = + +
We can restate this as:
( ) ( ) ( )( )22 1 1,21 1 1y y E r+ = + +
Suppose that y1 is 8%, y2 is 9% and E(r1,2) is 6%, investors with two-year horizons can either: 1. Buy a two-year bond or invest one-year spot and one-year forward. The expected two-
year return is:
( ) ( )( )21.09 1.08 1.1001 1.188= =
2. Invest for one year and take whatever r1,2 happens to be at time 1. The expected two-year
return is: ( )( )1.08 1.06 1.1448=
Given the investors’ belief, (1) is more attractive, thus, they will buy the two-year bond. Therefore, the price of two-year bond will go up, and y2 will drop, therefore, f1,2 decreases. The adjustment stops when f1,2 and E(r1,2) are approximately equal.
The pure expectation theory tells us: • An upward sloping yield curve (the forward rates are higher than the current spot rates
and therefore) implies that the market is expecting higher spot rates in the future. • A downward sloping (inverted) yield curve implies that the market is expecting lower
spot rates in the future. When f1,2 = E(r1,2), the theory tells us that there would be no buying or selling pressure, and hence prices and yields would be in equilibrium. When expectations are revised, the yield curve changes its shape accordingly.
151
10.3.2 Liquidity Preference In the market, there are short-term and long-term investors.
Short-term investors will be unwilling to hold long-term bonds unless the forward rate exceeds the expected short interest rate, i.e., f1,2 > E(r1,2).
Long-term investors will be unwilling to hold short bonds unless f1,2 < E(r1,2).
The liquidity preference of the term structure believes that short-term investors dominate the market so that the forward rate will generally exceed the expected short rate. That is: f1,2 = E(r1,2) + L where: L is the liquidity premium at 2 years horizon
