978-1259709685 Chapter 11 Solution Manual Part 1

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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

2. a. systematic

b. unsystematic

3. No to both questions. The portfolio expected return is a weighted average of the asset’s returns, so it

4. False. The variance of the individual assets is a measure of the total risk. The variance on a well-

5. Yes, the standard deviation can be less than that of every asset in the portfolio. However, p cannot

6. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be

equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the

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

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’ stock price

does not imply that the firm’s beta is high. Total volatility (the price fluctuation) is a function of both

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 the entire portfolio matters. This contribution is measured by systematic risk or beta.

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:

The portfolio weight for each stock is:

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:

So, the expected return of this portfolio is:

3. 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:

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:

So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or:

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:

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, and then add all of these up. The result is the variance. So, the variance

and standard deviation of each stock are:

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:

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, and then add all of these up. The result is the variance. So, the variance

and standard deviation are:

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:

If we own this portfolio, we would expect to earn a return of 11.35 percent.

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:

To find the expected return of the portfolio, we multiply the return in each state of the economy

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:

And the expected return of the portfolio is:

To find the variance, we find the squared deviations from the expected return. We then multiply

each possible squared deviation by its probability, and then add all of these up. The result is the

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:

Good: Rp = .30(.09) + .40(.10) + .30(.15)

Rp = .1120, or 11.20%

And the expected return of the portfolio is:

b. 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, and then add all of these up. The result is the variance. So,

the variance and standard deviation of the portfolio are:

10. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the

beta of the portfolio is:

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 one, we know the beta of our portfolio is one. We also need to remember that the beta of

the risk-free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the

beta of our portfolio, we get:

12. CAPM states the relationship between the risk of an asset and its expected return. CAPM is:

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 given the market risk premium. The market risk premium is the expected return of the

14. Here we need to find the expected return of the market using the CAPM. Substituting the values

given, and solving for the expected return of the market, we find:

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:

b. We need to find the portfolio weights that result in a portfolio with a  of .50. We know the  of

the risk-free asset is zero. 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:

And, the weight of the risk-free asset is:

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:

So, the  of the portfolio will be:

d. Solving for the  of the portfolio as we did in part b, we find:

The portfolio is invested 200% in the stock and –100% in the risk-free asset. This represents

borrowing at the risk-free rate to buy more of the stock.

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 .

Even though we are solving for the  and expected return of a portfolio of one stock and the risk-free

asset for different portfolio weights, we are really solving for the SML. Any combination of this

So, now we know the CAPM equation for any stock is:

The slope of the SML is equal to the market risk premium, which is .0658. Using these equations to

fill in the table, we get the following results:

XWE(Rp) ßp

0% .0400 0

25 .0598 .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:

It is given in the problem that the expected return of Stock Y is 12.7 percent, but according to the

CAPM, the return of the stock based on its level of risk should be 13.02 percent. This means the

stock return is too low, given its level of risk. Stock Y plots below the SML and is overvalued. In

The return given for Stock Z is 11.1 percent, but according to the CAPM the expected return of the

We can also answer this question using the reward-to-risk ratio. All assets must have the same

reward-to-risk ratio, that is, every asset must have the same ratio of the asset risk premium to its

beta. This follows from the linearity of the SML in Figure 11.11. The reward-to-risk ratio is the risk

The reward-to-risk ratio for Stock Y is too low, which means the stock plots below the SML, and the

stock is overvalued. Its price must decrease until its reward-to-risk ratio is equal to the market

reward-to-risk ratio. For Stock Z, we find:

19. We need to set the reward-to-risk ratios of the two assets equal to each other (see the previous

problem), which is:

(.127 – Rf) / 1.20 = (.111 – Rf) / .90

We can cross multiply to get: