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Answers and Solutions: 25 - 1
Chapter 25
Portfolio Theory and Asset Pricing Models
ANSWERS TO END-OF-CHAPTER QUESTIONS
25-1 a. A portfolio is made up of a group of individual assets held in combination. An asset
that would be relatively risky if held in isolation may have little, or even no risk if
held in a well-diversified portfolio.
b. An indifference curve is the risk/return trade-off function for a particular investor and
reflects that investor's attitude toward risk. The indifference curve specifies an
investor's required rate of return for a given level of risk. The greater the slope of the
indifference curve, the greater is the investor's risk aversion.
The optimal portfolio for an investor is the point at which the efficient set of
Answers and Solutions: 25 - 2
c. The Capital Asset Pricing Model (CAPM) is a general equilibrium market model
developed to analyze the relationship between risk and required rates of return on
assets when they are held in well-diversified portfolios. The SML is part of the
CAPM.
d. The characteristic line for a particular stock is obtained by regressing the historical
returns on that stock against the historical returns on the general stock market. The
slope of the characteristic line is the stock's beta, which measures the amount by
which the stock's expected return increases for a given increase in the expected return
on the market.
25-2 Security A is less risky if held in a diversified portfolio because of its lower beta and
negative correlation with other stocks. In a single-asset portfolio, Security A would be
more risky because σA > σB and CVA > CVB.
Answers and Solutions: 25 - 3
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
25-1 bi = iM (i / M) = 0.70(0.40/0.20) = 1.4.
25-3 r
^p = wAr
^A + (1 − wA) r
^B
= 0.30(12%) + 0.70(18%) = 16.20%
25-4 a.
.)rr(rb)rr(rr
M
iiM
RFMRFiRFMRFi
−+=−+=
b. CML:
.
rr
rr p
M
RF
M
RF
p
−
+=
SML:
.r
rr
rr iiM
M
RFM
RFi
−
+=
With some arranging, the similarities between the CML and SML are obvious. When
25-5 a. A plot of the approximate regression line is shown in the following figure:
Answers and Solutions: 25 - 4
Using Excel, the regression equation estimates are: Beta = 0.56; Intercept = 0.037; R2
= 0.96.
10
15
20
25
30
r
X
(%)
Answers and Solutions: 25 - 5
b. The arithmetic average return for Stock X is calculated as follows:
%.6.10
7
)2.18...0.230.14(
rAvg =
+++−
=
The standard deviation of returns for the market portfolio is similarly determined to
be 22.6 percent. The results are summarized below:
Stock X Market Portfolio
Average return,
Avg
r
10.6% 12.1%
Standard deviation, σ 13.1 22.6
c. Since Stock X is in equilibrium and plots on the Security Market Line (SML), and
given the further assumption that
XX rr =
and
MM rr =
--and this assumption often
does not hold--then this equation must hold:
Answers and Solutions: 25 - 7
d. The SML is plotted below. Data on the risk-free security (bRF = 0,
rRF = 8.6%) and Security X (bX = 0.56,
X
r
= 10.6%) provide the two points through
which the SML can be drawn. rM provides a third point.
e. In theory, you would be indifferent between the two stocks. Since they have the same
beta, their relevant risks are identical, and in equilibrium they should provide the
same returns. The two stocks would be represented by a single point on the SML.
Stock Y, with the higher standard deviation, has more diversifiable risk, but this risk
k(%)
20
r(%)
Answers and Solutions: 25 - 8
25-6
a. The regression graph is shown below. Using a spreadsheet, we find b = 0.62.
b. Because b = 0.62, Stock Y is about 62 percent as volatile as the market; thus, its
relative risk is about 62 percent of that of an average firm.
c. 1. Total risk
)( 2
Y
would be greater because the second term of the firm's risk
equation,
2
eY
2
M
2
Y
2
Yb+=
, would be greater.
Answers and Solutions: 25 - 9
d. 1. The stock's variance would not change, but the risk of the stock to an investor
holding a diversified portfolio would be greatly reduced.
Answers and Solutions: 25 - 10
SOLUTION TO SPREADSHEET PROBLEM
Mini Case: 25 - 11
MINI CASE
Answer the following questions.
a. Suppose asset A has an expected return of 10 percent and a standard deviation
of 20 percent. Asset B has an expected return of 16 percent and a standard
deviation of 40 percent. If the correlation between A and B is 0.35, what are the
expected return and standard deviation for a portfolio comprised of 30 percent
asset A and 70 percent asset B?
Answer:
r
ˆ
)w1(r
ˆ
wr
ˆBAAAP
−+=
b. Plot the attainable portfolios for a correlation of 0.35. Now plot the attainable
portfolios for correlations of +1.0 and -1.0.
Answer:
20%
pAB = +0.35: Attainable Set of
Risk/Return Combinations
Mini Case: 25 - 12
20%
AB = +1.0: Attainable Set of Risk/Return
Combinations
20%
AB = -1.0: Attainable Set of Risk/Return
Combinations
Mini Case: 25 - 13
c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its
standard deviation is zero, and its correlation with any other asset is also zero. Using
only asset A and the risk-free asset, plot the attainable portfolios.
Answer:
15%
Attainable Set of Risk/Return
Combinations with Risk-Free Asset
Mini Case: 25 - 14
d. Construct a reasonable, but hypothetical, graph which shows risk, as measured
by portfolio standard deviation, on the x axis and expected rate of return on the
y axis. Now add an illustrative feasible (or attainable) set of portfolios, and show
what portion of the feasible set is efficient. What makes a particular portfolio
efficient? Don't worry about specific values when constructing the graph—
merely illustrate how things look with "reasonable" data.
Answer:
Expected Portfolio
B
C
Return, kp
Efficient Set
(A,B)
^
Expected Portfolio
Return
^
rP
Mini Case: 25 - 15
e. Now add a set of indifference curves to the graph created for part B. What do
these curves represent? What is the optimal portfolio for this investor? Finally,
add a second set of indifference curves which leads to the selection of a different
optimal portfolio. Why do the two investors choose different portfolios?
Answer:
The figure above shows the indifference curves for two hypothetical investors, A and
B. To determine the optimal portfolio for a particular investor, we must know the
Expected Portfolio
Risk,
p
A
B
C
D
E
IA3
IA2
IA1
IB2
IB1
Optimal
Portfolio
Investor B
Optimal
Portfolio
Investor A
Return, kp
^
Expected Portfolio
Return,
^
rp
risk, P
Mini Case: 25 - 16
f. What is the capital asset pricing model (CAPM)? What are the assumptions
that underlie the model?
Answer: The Capital Asset Pricing Model (CAPM) is an equilibrium model which specifies
the relationship between risk and required rates of return on assets when they are held
in well-diversified portfolios. The CAPM requires an extensive set of assumptions:
• All investors are single-period expected utility of terminal wealth maximizers,
who choose among alternative portfolios on the basis of each portfolio's expected
return and standard deviation.
Mini Case: 25 - 17
g. Now add the risk-free asset. What impact does this have on the efficient
frontier?
Answer: The risk-free asset by definition has zero risk, and hence σ = 0%, so it is plotted on
the vertical axis. Now, given the possibility of investing in the risk-free asset,
investors can create new portfolios that combine the risk-free asset with a portfolio of
Expected Portfolio
Risk,
p
A
B
Z
M
kRF
Return, kp
^
Expected Portfolio
Return,
^
rp
rR
F
σp
Mini Case: 25 - 18
h. Write out the equation for the capital market line (CML) and draw it on the
graph. Interpret the CML. Now add a set of indifference curves, and illustrate
how an investor's optimal portfolio is some combination of the risky portfolio
and the risk-free asset. What is the composition of the risky portfolio?
Answer: The line rRFmz in the figure above is called the capital market line (CML). It has an
intercept of rRF and a slope of
MRF
M/)rr( −
. Therefore the equation for the capital
market line may be expressed as follows:
Mini Case: 25 - 19
i. What is a characteristic line? How is this line used to estimate a stock’s beta
coefficient? Write out and explain the formula that relates total risk, market
risk, and diversifiable risk.
Answer: Betas are calculated as the slope of the characteristic line, which is the regression line
formed by plotting returns on a given stock on the y axis against returns on the
general stock market on the x axis. In practice, 5 years of monthly data, with 60
observations, would be used, and a computer would be used to obtain a least squares
Mini Case: 25 - 20
j. What are two potential tests that can be conducted to verify the CAPM? What
are the results of such tests? What is roll’s critique of CAPM tests?
Answer: Since the CAPM was developed on the basis of a set of unrealistic assumptions,
empirical tests should be used to verify the CAPM. The first test looks for stability in
historical betas. If betas have been stable in the past for a particular stock, then its
historical beta would probably be a good proxy for its ex-ante, or expected beta.
Mini Case: 25 - 21
k. Briefly explain the difference between the CAPM and the arbitrage pricing
theory (APT).
Answer: The CAPM is a single-factor model, while the Arbitrage Pricing Theory (APT) can
include any number of risk factors. It is likely that the required return is dependent
