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CHAPTER 11
RISK AND RETURN: THE CAPITAL
ASSET PRICING MODEL (CAPM)
Answers to Concepts Review and Critical Thinking Questions
1. Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of
assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are
2. a. systematic
b. unsystematic
4. False. The variance of the individual assets is a measure of the total risk. The variance on a well-
diversified portfolio is a function of systematic risk only.
6. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be
7. The covariance is a more appropriate measure of a security’s risk in a well-diversified portfolio
because the covariance reflects the effect of the security on the variance of the portfolio. Investors are
8. If we assume that the market has not stayed constant during the past three years, then the lack in
movement of Southern Co.’s stock price only indicates that the stock either has a standard deviation
or a beta that is very near to zero. The large amount of movement in Texas Instruments’ (TIs’) stock
9. The wide fluctuations in the price of oil stocks do not indicate that these stocks are a poor investment.
If an oil stock is purchased as part of a well-diversified portfolio, only its contribution to the risk of
10. The statement is false. If a security has a negative beta, investors would want to hold the asset to reduce
the variability of their portfolios. Those assets will have expected returns that are lower than the risk-
free rate. To see this, examine the Capital Asset Pricing Model:
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
Basic
1. The portfolio weight of an asset is total investment in that asset divided by the total portfolio value.
First, we will find the portfolio value, which is:
Total value = 145($47) + 130($86)
2. The expected return of a portfolio is the sum of the weight of each asset times the expected return of
each asset. The total value of the portfolio is:
3. The expected return of a portfolio is the sum of the weight of each asset times the expected return of
4. Here we are given the expected return of the portfolio and the expected return of each asset in the
portfolio and are asked to find the weight of each asset. We can use the equation for the expected
return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%),
the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means:
E(Rp) = .112 = .127XX + .091(1 – XX)
We can now solve this equation for the weight of Stock X as:
5. The expected return of an asset is the sum of the probability of each state occurring times the rate of
return if that state occurs. So, the expected return of each stock asset is:
E(RA) = .25(.06) + .55(.07) + .20(.11)
E(RA) = .0755, or 7.55%
6. The expected return of an asset is the sum of the probability of each return occurring times the
probability of that return occurring. So, the expected return of the stock is:
E(RA) = .15(–.148) + .30(.031) + .45(.162) + .10(.348)
7. The expected return of a portfolio is the sum of the weight of each asset times the expected return of
each asset. So, the expected return of the portfolio is:
8. a. To find the expected return of the portfolio, we need to find the return of the portfolio in each
state of the economy. This portfolio is a special case since all three assets have the same weight.
To find the expected return in an equally weighted portfolio, we can sum the returns of each asset
and divide by the number of assets, so the expected return of the portfolio in each state of the
economy is:
Boom: Rp = (.06 + .16 + .33)/3
Rp = .1833, or 18.33%
b. This portfolio does not have an equal weight in each asset. We still need to find the return of the
portfolio in each state of the economy. To do this, we will multiply the return of each asset by its
portfolio weight and then sum the products to get the portfolio return in each state of the economy.
Doing so, we get:
Boom: Rp = .20(.06) +.20(.16) + .60(.33)
Rp = .2420, or 24.20%
9. a. This portfolio does not have an equal weight in each asset. We first need to find the return of the
portfolio in each state of the economy. To do this, we will multiply the return of each asset by its
portfolio weight and then sum the products to get the portfolio return in each state of the economy.
Doing so, we get:
Boom: Rp = .30(.24) + .40(.45) + .30(.33)
Rp = .3510, or 35.10%
10. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta
11. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio
is as risky as the market it must have the same beta as the market. Since the beta of the market is 1.0,
12. CAPM states the relationship between the risk of an asset and its expected return. CAPM is:
E(Ri) = Rf + [E(RM) – Rf] × i
13. We are given the values for the CAPM except for the of the stock. We need to substitute these values
into the CAPM, and solve for the of the stock. One important thing we need to realize is that we are
14. Here we need to find the expected return of the market using the CAPM. Substituting the values given,
15. Here we need to find the risk-free rate using the CAPM. Substituting the values given, and solving for
the risk-free rate, we find:
16. a. We have a special case where the portfolio is equally weighted, so we can sum the returns of
each asset and divide by the number of assets. The expected return of the portfolio is:
E(Rp) = (.116 + .036)/2
E(Rp) = .0760, or 7.60%
b. We need to find the portfolio weights that result in a portfolio with a of .50. We know the of
c. We need to find the portfolio weights that result in a portfolio with an expected return of 10
percent. We also know the weight of the risk-free asset is one minus the weight of the stock since
the portfolio weights must sum to one, or 100 percent. So:
d. Solving for the of the portfolio as we did in part b, we find:
p = 2.16 = XS(1.08) + (1 – XS)(0)
XS = 2.16/1.08
XS = 2
17. First, we need to find the of the portfolio. The of the risk-free asset is zero, and the weight of the
risk-free asset is one minus the weight of the stock, so the of the portfolio is:
ßp = XW(1.2) + (1 – XW)(0) = 1.2XW
So, to find the of the portfolio for any weight of the stock, we multiply the weight of the stock times
its .
The slope of the SML is equal to the market risk premium, which is .0692. Using these equations to
fill in the table, we get the following results:
XW E(Rp) ßp
0% .0400 0
25 .0608 .300
18. There are two ways to correctly answer this question. We will work through both. First, we can use
the CAPM. Substituting in the value we are given for each stock, we find:
E(RY) = .045 + .071(1.15)
E(RY) = .1267, or 12.67%
It is given in the problem that the expected return of Stock Y is 11.8 percent, but according to the
19. We need to set the reward-to-risk ratios of the two assets equal to each other (see the previous
problem), which is:
(.118 – Rf)/1.15 = (.107 – Rf)/.85
20. For a portfolio that is equally invested in large-company stocks and long-term bonds:
Return = (12.1% + 6.0%)/2
21. We know that the reward-to-risk ratios for all assets must be equal (See Question 19). This can be
expressed as:
[E(RA) – Rf]/A = [E(RB) – Rf]/B
22. a. We need to find the return of the portfolio in each state of the economy. To do this, we will
multiply the return of each asset by its portfolio weight and then sum the products to get the
portfolio return in each state of the economy. Doing so, we get:
Boom: Rp = .4(.25) + .4(.35) + .2(.40)
Rp = .3200, or 32.00%
To calculate the standard deviation, we first need to calculate the variance. To find the variance,
we find the squared deviations from the expected return. We then multiply each possible squared
deviation by its probability, then add all of these up. The result is the variance. So, the variance
and standard deviation of the portfolio are:
b. The risk premium is the return of a risky asset minus the risk-free rate. T-bills are often used as
the risk-free rate, so:
RPi = E(Rp) – Rf
c. The approximate expected real return is the expected nominal return minus the inflation rate, so:
Approximate expected real return = .1215 – .035
Approximate expected real return = .0865, or 8.65%
To find the exact real return, we will use the Fisher equation. Doing so, we get:
And using the Fisher effect for the exact real risk-free rate, we find:
1 + E(Ri) = (1 + h)[1 + e(ri)]
1.038 = (1.0350)[1 + e(ri)]
23. We know the total portfolio value and the investment of two stocks in the portfolio, so we can find the
weight of these two stocks. The weights of Stock A and Stock B are:
XA = $190,000/$1,000,000
XA = .19
XB = $325,000/$1,000,000
24. We are given the expected return of the assets in the portfolio. We also know the sum of the weights
of each asset must be equal to one. Using this relationship, we can express the expected return of the
portfolio as:
E(Rp) = .127 = XX(.114) + XY(.0868)
.127 = XX(.114) + (1 – XX)(.0868)
And the weight of Stock Y is:
XY = 1 – 1.47794
25. The expected return of an asset is the sum of the probability of each state occurring times the rate of
return if that state occurs. So, the expected return of each stock is:
E(RA) = .33(.099) + .33(.113) + .33(.059)
E(RA) = .0903, or 9.03%
