Corporate Finance, Lecture Notes 2
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Portfolio Theory:
Mathematical Preliminaries
Professor Siyi Shen
Corporate Finance, Lecture Notes 2
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Introduction
➢ Typically, stocks are not held in isolation but as part of a
portfolio. It is overall return and risk of the entire
portfolio, which we care about. Therefore, the riskiness of
a stock is determined by its contribution to the overall risk
of the portfolio.
➢ Therefore, the risk premium of a stock would be a
function of the undiversifiable risk of the stock.
Corporate Finance, Lecture Notes 2
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Probability Distribution
➢ In probability and statistics, a random variable is a
variable whose values depend on outcomes of a random
phenomenon (e.g., AAPL’s stock return next year).
➢ The set of all possible values of a random variable, with
their associated probabilities, is called the probability
distribution of the random variable. All possible outcomes
are assigned values and probabilities, and thus the
probabilities are non-negative and must sum to 1.
Corporate Finance, Lecture Notes 2
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Expected Returns
➢ Suppose there are S possible states, and let us denote each possible
state as i, i = 1, 2, 3, …, S.
➢ The probability of the state s being realized is denoted as pi. These
states should be exclusive outcomes, i.e. only one of the possible
states i can occur. Further, these S states should exhaust all possible
outcomes. Thus these probabilities sum to 1.
➢ Consider a risky asset whose rate of return in each of these possible
S states are r1, r2, r3, …, rS. The rate of return on this risky asset
can be viewed as a random variable.
➢ The expected return E(r) of an asset is the probability weighted
average return in all scenarios
S
E[r ] = p1r1 + .... + pS rS =  pi ri
i =1
Corporate Finance, Lecture Notes 2
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Measures of Risk:
Variance and Standard Deviation
➢ What is a good measure of risk?
➢ We define the variance, a measure of risk, as the expected squared
deviation of the rate of return from its expectation:
S
 [r ] = E[r − E[r ]] =  pi [ri − E[r ]]
2
2
2
i =1
➢ To transform the variance into units of percentage return, we take
the square root of the variance, which gives us the standard
deviation.
Spring 2009
Corporate Finance, Lecture Notes 2
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Example 1
➢ Consider the following probability distributions of the returns on
XYZ Corp. and ABC Corp. Calculate the expected returns and
variances.
XYZ
ABC
Outcome
Probability
Value
Value
Boom
0.2
30%
60%
Normal
0.7
10%
10%
Recession
0.1
-20%
-40%
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Measures of Association:
Covariance and Correlation
➢ The covariance between returns is a measure of co-movement. The
covariance between stock A and stock B is defined as:
cov[rA , rB ] = E[rA − E[rA ]][rB − E[rB ]] =
S
=  pi [ rA,i − E[ rA ]][ rB ,i − E[ rB ]]
i =1
➢ Calculate the covariance between XYZ Corp. and ABC Corp.
Corporate Finance, Lecture Notes 2
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Measures of Association:
Covariance and Correlation – Cont’d
➢ How close do stocks move together? The correlation coefficient
provides the degree of intensity of the co-movement. The
correlation coefficient between the returns of stock A and stock B is
defined as:
 A, B =
cov[rA , rB ]
 A B
➢ Calculate the correlation between XYZ Corp. and ABC Corp.
Corporate Finance, Lecture Notes 2
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Basic Properties of Random Variables
➢ Consider random variables X and Y, and some constants a and b.
Expectation
E (a + X ) = a + E ( X )
E (aX ) = aE ( X )
E ( X + Y ) = E ( X ) + E (Y )
E (aX + bY ) = aE ( X ) + bE (Y )
Variance Var(a + X ) = Var( X )
Var(aX ) = a 2Var( X )
Cov(aX , bY ) = abCov( X , Y )
Var(aX + bY ) = a 2Var( X ) + b 2Var(Y ) + 2abCov( X , Y )
Corporate Finance, Lecture Notes 2
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Application to Portfolios
➢ Assume that the random variables X and Y correspond to the returns
of two risky assets, say rA and rB. The constants a and b correspond
to portfolio weights, say wA and wB. If you combine these two
assets into a portfolio P with the weights wA and wB, then the
expected return and variance of the portfolio are:
E (rp ) = E (wA rA + wB rB ) = wA E (rA ) + wB E (rB )
Var(rp ) = Var(wA rA + wB rB ) = wA2Var(rA ) + wB2Var(rB ) + 2wA wB Cov(rA , rB )
Corporate Finance, Lecture Notes 2
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Diversification with 2 Assets - Example
• Suppose we have two assets, US and JP, with:
mean
volatility
US
E[R1]=13.6%
σ1=15.4%
JP
E[R2]=15.0%
σ2=23.0%
and with correlation ρ12=27%.
• If an investor holds w1=60% in the US and w2=40% in JP what
is the mean and volatility of the portfolio?
Corporate Finance, Lecture Notes 2
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Risk, Return, and the
Mean-Variance Criterion
Corporate Finance, Lecture Notes 2
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Risk-Return Tradeoff - Introduction
➢ Suppose you could pick one of the stocks in each figure.
Which one would you pick?
1.2
0.6
1.2
1
1
0.5
0.8
0.8
0.4
0.6
0.6
0.3
0.4
0.2
0.2
0.4
0.2
0
0.1
0
-0.2
0
-0.2
-0.4
-0.1
0
5
10
15
20
25
30
35
40
45
50
-0.6
-0.4
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
Mean 0.15 0.35
Mean 0.20 0.20
Mean 0.20 0.40
Stdev 0.10 0.10
Stdev 0.10 0.40
Stdev 0.10 0.40
50
Corporate Finance, Lecture Notes 2
Risk-Return Tradeoff
• Recall two of the Finance Axioms:
– Investors prefer more to less
– Investors are risk-averse
• This means that investors prefer an investment :
– with a higher expected return E(Ri)
– with a lower variance and standard deviation, i
• Investors must optimally tradeoff risk and return in order to
maximize their expected utility.
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Corporate Finance, Lecture Notes 2
Indifference Curves: Review
• A person likes 2 goods: bread (B) and chocolate (C).
• An indifference curve gives all the combinations of B and C
that give the same utility level U0 = U(B,C).
• People like to be on the highest possible indifference curve
(people prefer more to less).
B
U0
C
15
Corporate Finance, Lecture Notes 2
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Indifference Curves in Finance
Indifference curve: A set of (E(Rp),p) combinations that
give an investor the same expected utility
0.45
High utility
0.4
Medium utility
0.35
Low utility
Exp. Return
0.3
0.25
Series1
Series2
Series3
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
Standard Deviation
0.25
0.3
0.35
Corporate Finance, Lecture Notes 2
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Risk Aversion and Utility
➢ A simple, widely used form of utility function (quadratic) is:
U = E(r) – 0.5A2
➢ What is A? Risk aversion coefficient
➢ What are we assuming about the distribution of returns?
Corporate Finance, Lecture Notes 2
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Mean-Variance Criterion
➢ The mean-variance criterion is defined as follows: For a risk
averse, expected utility maximizing investor, Asset A dominates
Asset B:
if E(rA)  E(rB) and A2 < B2
or
if E(rA) > E(rB) and A2  B2
Corporate Finance, Lecture Notes 2
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Asset Allocation across Risky
and Risk-Free Portfolios
Corporate Finance, Lecture Notes 2
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Introduction
➢ Asset allocation decision refers to a choice among broad investment
classes, such as stocks, long-term T-bonds, and short-term T-bills.
➢ In this section, we shall study the most basic asset allocation
choice: the choice of how to allocate one’s investment fund
between risk-free securities and risk asset classes.
Corporate Finance, Lecture Notes 2
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Example 1
➢ Suppose a portfolio of risky stocks P is your “optimal” risky
portfolio (we shall discuss what it means to be optimal later).
E(rp)=15%, and p=22%. The risk-free rate of return rf (T-bill rate)
is 7%. You have $1,000 to invest and want 12% expected return for
your investment. How should you combine the portfolio P and the
risk-free asset?
➢ What is the standard deviation of the return on your investment?
Corporate Finance, Lecture Notes 2
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Example 1 – Cont’d
➢ Suppose you think the standard deviation of 13.75% is too risky,
and want 10% standard deviation of return for your investment.
What should be the allocation in this case? What is the expected
return of this portfolio?
Corporate Finance, Lecture Notes 2
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Capital Allocation Line (CAL)
➢ In addition to the two particular allocation choices above, we can examine the
risk-return combinations of all possible allocation choices.
Weight
Portfolio P
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Weight
T-Bill
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
Expected Standard
Return Deviation
0.07
0
0.078
0.022
0.086
0.044
0.094
0.066
0.102
0.088
0.11
0.11
0.118
0.132
0.126
0.154
0.134
0.176
0.142
0.198
0.15
0.22
0.158
0.242
0.166
0.264
0.174
0.286
0.182
0.308
➢ The cases where the weight on T-bill is negative are where you borrow at the
risk-free rate and invest the borrowed amount in the risky portfolio P.
Corporate Finance, Lecture Notes 2
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Capital Allocation Line – Cont’d
➢ The Capital Allocation Line (CAL) is simply a plot of these combinations in a
mean-standard deviation space: CAL depicts the risk-return combinations
available by varying asset allocation between a risk-free asset and a risky
portfolio.
Capital
Allocation
Line
0.2
0.18
0.16
Expected return
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
Standard deviation
0.25
0.3
0.35
Corporate Finance, Lecture Notes 2
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Sharpe Ratio
➢ The slope of the CAL, called the Sharpe ratio (or reward-to-risk
ratio), equals the increase in expected return that can be obtained
per unit of additional standard deviation. It is a measure of the riskreturn trade-off (extra return per extra risk), and is give by
S=
E (rp ) − r f
p
➢ Thus, the Sharpe ratio is a ratio of risk premium to standard
deviation.
Corporate Finance, Lecture Notes 2
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Risk Aversion, Utility
➢ Given all the choices on the CAL, what is the optimal allocation for the investor?
What is the allocation that provides the highest expected utility?
➢ Recall our representative investor’s preferences provided by the utility function
U = E(r) – 0.5A2
➢ We can calculate and plot the utility levels as a function of the weight on the
risky portfolio P as follows.
0.1
0.09
0.08
0.07
Utility
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Weight on the risk portfolio P
A=4
A=6
0.7
0.8
0.9
1