Investments & Securities Chapter 7 Homework An example of this scenario would be an investment

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

CHAPTER 07

CAPITAL ASSET PRICING AND ARBITRAGE PRICING

THEORY

1. The required rate of return on a stock is related to the required rate of return on the

stock market via beta. Assuming the beta of Google remains constant, the increase in

2. An example of this scenario would be an investment in the SMB and HML. As of yet,

there are no vehicles (index funds or ETFs) to directly invest in SMB and HML. While

3. a. False. According to CAPM, when beta is zero, the “excess” return should be zero.

b. False. CAPM implies that the investor will only require risk premium for systematic

4. E(r) = rf + β [E(rM) – rf ] , rf = 4%, rM = 6%

7. Statement a is most accurate.

The flaw in statement b is that beta represents only the systematic risk. If the firm-

8. The APT may exist without the CAPM, but not the other way. Thus, statement a is

possible, but not b. The reason is that the APT accepts the principle of risk and return,

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

9. E(rp) = rf + β [E(rM) – rf ] Given rf = 5% and E(rM)= 15%, we can calculate 

10. If the beta of the security doubles, then so will its risk premium. The current risk

premium for the stock is: (13% – 7%) = 6%, so the new risk premium would be 12%,

and the new discount rate for the security would be: 12% + 7% = 19%

If the stock pays a constant dividend in perpetuity, then we know from the original data

that the dividend (D) must satisfy the equation for a perpetuity:

11. The cash flows for the project comprise a 10-year annuity of $10 million per year plus an

additional payment in the tenth year of $10 million (so that the total payment in the tenth

year is $20 million). The appropriate discount rate for the project is:

Using this discount rate:

10

The internal rate of return on the project is 49.55%. The highest value that beta can take

before the hurdle rate exceeds the IRR is determined by:

12.

a. The beta is the sensitivity of the stock’s return to the market return, or, the

change in the stock return per unit change in the market return. We denote the

aggressive stock A and the defensive stock D, and then compute each stock’s

beta by calculating the difference in its return across the two scenarios divided

by the difference in market return.

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

b. With the two scenarios equally likely, the expected rate of return is an average

of the two possible outcomes:

c. The SML is determined by the following: Expected return is the T-bill rate = 8%

when beta equals zero; beta for the market is 1.0; and the expected rate of return

for the market is:

Thus, we graph the SML as following:

The equation for the security market line is: E(r) = 8% + β(12.5% – 8%)

d. The aggressive stock has a fair expected rate of return of:

The security analyst’s estimate of the expected rate of return is also 17%.

Thus the alpha for the aggressive stock is zero. Similarly, the required return

for the defensive stock is:

The security analyst’s estimate of the expected return for D is only 8.75%, and

hence:

E(r)



1.0

2.0

A

SML

.7

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

e. The hurdle rate is determined by the project beta (i.e., 0.7), not by the firm’s

13. Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for

Portfolio A is lower.

14. Possible. If the CAPM is valid, the expected rate of return compensates only for

systematic (market) risk as measured by beta, rather than the standard deviation,

15. Not possible. The reward-to-variability ratio for Portfolio A is better than that of the

market, which is not possible according to the CAPM, since the CAPM predicts that the

market portfolio is the most efficient portfolio. Using the numbers supplied:

16. Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard

deviation with a higher expected return.

17. Not possible. Given these data, the SML is: E(r) = 10% + β(18% – 10%)

A portfolio with beta of 1.5 should have an expected return of:

18. Not possible. The SML is the same as in Problem 18. Here, the required expected

return for Portfolio A is: 10% + (0.9  8%) = 17.2%

This is still higher than 16%. Portfolio A is overpriced, with alpha equal to: –1.2%

19. Possible. Portfolio A’s ratio of risk premium to standard deviation is less attractive

a.

Ford GM Toyota S&P

Beta 5 years 1.81 0.86 0.71 1.00

Beta first two years 2.01 1.05 0.47 3.78 SD

Beta last two years 1.97 0.69 0.49

b. As a first pass, we note that large standard deviation of the beta estimates. None of

the subperiod estimates deviate from the overall period estimate by more than two

standard deviations. That is, the t-statistic of the deviation from the overall period is

not significant for any of the subperiod beta estimates. Looking beyond the

aforementioned observation, the differences can be attributed to different alpha

values during the subperiods. The case of Toyota is most revealing: The alpha

estimate for the first two years is positive and for the last two years negative (both

21. Since the stock’s beta is equal to 1.0, its expected rate of return should be equal to that

of the market, that is, 18%.

E(r) =

0

01

P

PPD −+

22. If beta is zero, the cash flow should be discounted at the risk-free rate, 8%:

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

PV = $1,000/0.08 = $12,500

If, however, beta is actually equal to 1, the investment should yield 18%, and the price

paid for the firm should be:

23. Using the SML: 6% = 8% + β(18% – 8%)  β= –2/10 = –0.2

24. We denote the first investment advisor 1, who has r1 = 19% and 1 = 1.5, and the

second investment advisor 2, as r2 = 16% and 2 = 1.0. In order to determine which

investor was a better selector of individual stocks, we look at the abnormal return,

which is the ex-post alpha; that is, the abnormal return is the difference between the

actual return and that predicted by the SML.

a. Without information about the parameters of this equation (i.e., the risk-free rate

b. If rf = 6% and rM = 14%, then (using alpha for the abnormal return):

α1 = 19% – [6% + 1.5  (14% – 6%)] = 19% – 18% = 1%

c. If rf = 3% and rM = 15%, then:

α1 =19% – [3% + 1.5  (15% – 3%)] = 19% – 21% = –2%

25. a. Since the market portfolio, by definition, has a beta of 1.0, its expected rate of

return is 12%.

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

c. Using the SML, the fair rate of return for a stock with β = –0.5 is:

E(r) = 4% + (–0.5)  (12% – 4%) = 0.0%

26. The data can be summarized as follows:

Expected Return Beta

Standard

Deviation

Portfolio A 11% 0.8 10%

a. Using the SML, the expected rate of return for any portfolio P is:

Substituting for portfolios A and B:

Hence, Portfolio A is desirable and Portfolio B is not.

b. The slope of the CAL supported by a portfolio P is given by:

Computing this slope for each of the three alternative portfolios, we have:

S (S&P 500) = (12% − 6%)/20% = 6/20

27. Since the beta for Portfolio F is zero, the expected return for Portfolio F equals the

risk-free rate.

For Portfolio A, the ratio of risk premium to beta is: (10 − 4)/1 = 6

The ratio for Portfolio E is higher: (9 − 4)/(2/3) = 7.5

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

This implies that an arbitrage opportunity exists. For instance, by taking a long position

in Portfolio E and a short position in Portfolio F (that is, borrowing at the risk-free rate

and investing the proceeds in Portfolio E), we can create another portfolio which has

the same beta (1.0) but higher expected return than Portfolio A. For the beta of the new

portfolio to equal 1.0, the proportion (w) of funds invested in E must be: 3/2 = 1.5.

Portfolio Weight

In Asset

Contribution to

β

Contribution to Excess

Return

1.5

Portfolio E

1.5 x βE = 1.0

1.5 x (9% – 4%) = 7.5%

Portfolio F

28. Substituting the portfolio returns and betas in the mean-beta relationship, we obtain two

equations in the unknowns, the risk-free rate (rf) and the factor return (F):

From the first equation we find that F = 14%. Substituting this value for F into the second

equation, we get:

a. Shorting equal amounts of the 10 negative-alpha stocks and investing the proceeds

equally in the 10 positive-alpha stocks eliminates the market exposure and creates a

zero-investment portfolio. Using equation 7.5 and denoting the market factor as RM,

the expected dollar return is [noting that the expectation of residual risk (e) in

equation 7.8 is zero]:

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

b. If n = 50 stocks (i.e., 25 long and 25 short), $40,000 is placed in each position,

and the variance of dollar returns is:

50  [(40,000  0.30)2] = 7,200,000,000

30. Any pattern of returns can be “explained” if we are free to choose an indefinitely large

number of explanatory factors. If a theory of asset pricing is to have value, it must

31. The APT factors must correlate with major sources of uncertainty, i.e., sources of

uncertainty that are of concern to many investors. Researchers should investigate

factors that correlate with uncertainty in consumption and investment opportunities.

32. The revised estimate of the expected rate of return of the stock would be the old

estimate plus the sum of the unexpected changes in the factors times the sensitivity

coefficients, as follows:

33. Equation 7.11 applies here:

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

We need to find the risk premium for these two factors:

To find these values, we solve the following two equations with two unknowns:

40% = 7% + 1.81 + 2.12

34. The first two factors (the return on a broad-based index and the level of interest rates)

are most promising with respect to the likely impact on Jennifer’s firm’s cost of capital.

35. Since the risk free rate is not given, we assume a risk free rate of 0%. The APT required

(i.e., equilibrium) rate of return on the stock based on rf and the factor betas is:

CFA 1

Answer:

CFA 2

Answer:

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

b.

i. For an investor who wants to add this stock to a well-diversified equity

portfolio, Kay should recommend Stock X because of its positive alpha,

ii. For an investor who wants to hold this stock as a single-stock portfolio, Kay

should recommend Stock Y, because it has higher forecasted return and

lower standard deviation than Stock X. Stock Y’s Sharpe ratio is:

CFA 3

Answer:

a. McKay should borrow funds and invest those funds proportionally in Murray’s

existing portfolio (i.e., buy more risky assets on margin). In addition to

b. McKay should substitute low beta stocks for high beta stocks in order to reduce

the overall beta of York’s portfolio. By reducing the overall portfolio beta,

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

CFA 4

Answer:

a. “Both the CAPM and APT require a mean-variance efficient market portfolio.”

b. “The CAPM assumes that one specific factor explains security returns but APT

CFA 5

Answer:

CFA 6

Answer:

d. The expect return on the market, rM:

CFA 7

Answer:

CFA 8

Answer:

CFA 9

Answer:

Under the CAPM, the only risk that investors are compensated for bearing is the risk

CFA 10

Answer:

Chapter 07 – Capital Asset Pricing and Arbitrage Pricing Theory

CFA 11

Answer:

CFA 12

Answer:

CFA 13

Answer:

CFA 14

Answer: