MINI CASE
You have been hired at the investment firm of Bowers & Noon. One of its clients doesn’t
understand the value of diversification or why stocks with the biggest standard deviations
don’t always have the highest expected returns. Your assignment is to address the client’s
concerns by showing the client how to 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:
%.2.14142.0
)16.0(7.0)1.0(3.0
r
ˆ
)w1(r
ˆ
wr
ˆ
BAAAP
306.0
)4.0)(2.0)(35.0)(7.0)(3.0(2)4.0(7.0)2.0(3.0
)W1(W2)W1(W
2222
BAAB
AA
2
B
2
A
2
A
2
Ap
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:
15% 20% 25% 30% 35% 4 0%
0%
5%
10%
15%
20%
pAB = +0.35: At t ainable Set of Risk/Ret urn Combinat ions
Risk, sigmap
Expected return
10% 20% 30% 4 0%
0%
5%
10%
15%
20%
rAB = +1.0: At t ainable Set of Risk/Ret urn Combinat ions
Risk, sp
Expect ed ret urn
0% 10% 20% 30% 4 0%
0%
5%
10%
15%
20%
rAB = -1.0: At t ainable Set of Risk/Ret urn Combinat ions
Risk, sp
Expect ed ret urn
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:
0% 5% 10 % 15 % 20%
0%
5%
10%
15%
At t ainable Set of Risk/Ret urn Combinat ions with Risk-Free Asset
Risk, sp
Expec t ed re t urn
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:
The figure above shows the feasible set of portfolios. The points B, C, D, and E
The boundary AB defines the efficient set of portfolios, which is also called the
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?
risk, P
risk, P
Expected Portfolio
Risk,
p
A
B
C
D
E
Return, k
p
Efficient Set
Feasible, or
Attainable, Set
(A,B)
^
Expected Portfolio
Return
^
rP
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
The optimal portfolio is found at the tangency point between the efficient set of
The investors choose different optimal portfolios because their risk aversion is
Expected Portfolio
Risk,
p
A
B
C
D
E
I
A3
I
A2
I
A1
I
B2
I
B1
Optimal
Portfolio
Investor B
Optimal
Portfolio
Investor A
Return, k
p
^
Expected Portfolio Return,
^
rp
f. 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
risky assets. This enables them to achieve any combination of risk and return that lies
Expected Portfolio
Risk,
p
A
B
Z
M
k
RF
Return, k
p
^
Expected Portfolio Return,
^
rp
rRFF
σp
g. 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:
CML:
.
rr
rr
p
M
RF
M
^
RF
p
The CML tells us that the expected rate of return on any efficient portfolio (that is,
risk premium is equal to
MRF
M
/)rr(
multiplied by the portfolio’s standard
σm.
The figure above shows a set of indifference curves (i1, i2, and i3), with i1 touching
The risky portfolio, m, must contain every asset in exact proportion to that asset’s
h. 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,
Investors have homogeneous expectations (that is, investors have identical
Expected Rate
k
RF
CML
I3I2I1
Optimal
Portfolio
Risk,
p
of Return, k
p
^
Expected Rate of Return,
^
rp
rRF
σp
All assets are perfectly divisible and perfectly marketable at the going price, and
there are no transactions costs.
The Security Market Line (SML) expresses a stock’s return as a function of the
risk-free rate and the stock’s beta:
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
The relationship between stock J’s total risk, market risk, and diversifiable risk
can be expressed as follows:
2
eJ
2
M
2
J
2
J
b
RISK BLEDIVERSIFIARISK MARKETVARIANCERISK TOTAL
Here
2
J
is the variance or total risk of stock j,
2
M
is the variance of the market, bj is
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
The second type of test is based on the slope of the SML. As we have seen, the
CAPM states that a linear relationship exists between a security’s required rate of
return and its beta. Further, when the SML is graphed, the vertical axis intercept
Roll questioned whether it is even conceptually possible to test the CAPM. Roll
showed that the linear relationship which prior researchers had observed in graphs
In general, evidence seems to support the CAPM model when it is applied to
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
on many fundamental factors such as the GNP growth, expected inflation, and