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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
2. An example of this scenario would be an investment in the SMB and HML. As of yet,
3. a. False. According to CAPM, when beta is zero, the “excess” return should be zero.
4. E(r) = rf + β [E(rM) – rf ] , rf = 4%, rM = 6%
5. $1 Discount Store is overpriced; Everything $5 is underpriced.
7. Statement a is most accurate.
8. A. A long position in a portfolio (P) composed of portfolios A and B will offer an
expected return-beta trade-off lying on a straight line between points A and B.
Therefore, we can choose weights such that βP = βC but with expected return higher
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
than that of portfolio C. Hence, combining P with a short position in C will create an
arbitrage portfolio with zero investment, zero beta, and positive rate of return.
b. No, arbitrage opportunities would be taken advantage of quickly selling short
portfolio C will cause a rise in the E(r) of C.
c. The argument in part (a) leads to the proposition that the coefficient of β2 must be
zero in order to preclude arbitrage opportunities.
9. E(rp) = rf + β [E(rM) – rf ] Given rf = 5% and E(rM)= 15%, we can calculate
20% = 5% + (15% – 5%) = 1.5
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%
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:
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.
b. With the two scenarios equally likely, the expected rate of return is an average
of the two possible outcomes:
E(rA) = 0.5 (2% + 32%) = 17%
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:
E(r)
8%
12.5%
1.0
2.0
A
SML
M
.7
D
D
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
d. The aggressive stock has a fair expected rate of return of:
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:
1016 =
−
deviation with a higher expected return.
17. Not possible. Given these data, the SML is: E(r) = 10% + β(18% – 10%)
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
E(r) = 10% + 1.5 (18% – 10%) = 22%
The expected return for Portfolio A is 16% so that Portfolio A plots below the SML
(i.e., has an alpha of –6%), and hence is an overpriced portfolio. This is inconsistent
with the CAPM.
18. Not possible. The SML is the same as in Problem 18. Here, the required expected
19. Possible. Portfolio A's ratio of risk premium to standard deviation is less attractive
20.
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
SE of residual 12.01 8.34 5.14
SE beta 5 years 0.42 0.29 0.18
Intercept 5 years -0.93 -1.44 0.45
Intercept first two years -2.37 -1.82 1.80
Intercept last two years 0.81 -3.41 -1.91
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
large). Following a good performance in the "normal" years prior to the crisis,
Toyota surprised investors with a negative performance, beyond what could be
expected from the index. This suggests that a beta of around 0.5 is more reliable.
The shift of the intercepts from positive to negative when the index moved to
largely negative returns, explains why the line is steeper when estimated for the
overall period. Draw a line in the positive quadrant for the index with a slope of 0.5
and positive intercept. Then draw a line with similar slope in the negative quadrant
of the index with a negative intercept. You can see that a line that reconciles the
observations for both quadrants will be steeper. The same logic explains part of the
behavior of subperiod betas for Ford and GM.
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%.
01
PPD −+
100
22. If beta is zero, the cash flow should be discounted at the risk-free rate, 8%:
PV = $1,000/0.08 = $12,500
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
and the market rate of return), we cannot determine which investment adviser is
the better selector of individual stocks.
b. If rf = 6% and rM = 14%, then (using alpha for the abnormal return):
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
α2 = 16% – [3%+ 1.0 (15% – 3%)] = 16% – 15% = 1%
Here, not only does the second investment adviser appear to be a better stock
selector, but the first adviser's selections appear valueless (or worse).
25. a. Since the market portfolio, by definition, has a beta of 1.0, its expected rate of
return is 12%.
b. β = 0 means the stock has no systematic risk. Hence, the portfolio's expected
26. The data can be summarized as follows:
Expected Return Beta
Standard
Deviation
Portfolio A 11% 0.8 10%
Portfolio B 14% 1.5 31%
S & P 500 12% 120%
T-bills 6% 00%
a. Using the SML, the expected rate of return for any portfolio P is:
E(rP) = rf + [E(rM) –rf ]
Substituting for portfolios A and B:
E(rA) = 6% + 0.8 (12% – 6%) = 10.8% < 11%
E(rB) = 6% + 1.5 (12% – 6%) = 15.0% > 14%
Hence, Portfolio A is desirable and Portfolio B is not.
b. The slope of the CAL supported by a portfolio P is given by:
S = E(rP) － rf
P
Computing this slope for each of the three alternative portfolios, we have:
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
S (S&P 500) = (12% − 6%)/20% = 6/20
S (A) = (11% − ) = 5/10 > S(S&P 500)
S (B) = (14% − ) = 8/31 < S(S&P 500)
Hence, portfolio A would be a good substitute for the S&P 500.
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
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
equations in the unknowns, the risk-free rate (rf) and the factor return (F):
14.0% = rf + 1 (F – rf )
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
29.
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. The expected dollar return is [noting that the expectation
of residual risk (e) is zero]:
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:
30. Any pattern of returns can be "explained" if we are free to choose an indefinitely large
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
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
determine risk premiums. In particular, industrial production (IP) is a good indicator of
changes in the business cycle. Thus, IP is a candidate for a factor that is highly
correlated with uncertainties related to investment and consumption opportunities in the
economy.
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 7.7 applies here:
E(rP) = rf + P1 [E(r1) − rf] + P2 [E(r2) – rf]
We need to find the risk premium for these two factors:
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
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
CFA 1
Answer:
CFA 2
Answer:
a. E(rX) = 5% + 0.8 (14% – 5%) = 12.2%
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,
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
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
increased expected return, the alternative portfolio on the capital market line
CFA 4
Answer:
a. “Both the CAPM and APT require a mean-variance efficient market portfolio.”
CFA 5
Answer:
CFA 6
Answer:
CFA 7
Chapter 07 - Capital Asset Pricing and Arbitrage Pricing Theory
CFA 8
CFA 9
Answer:
Under the CAPM, the only risk that investors are compensated for bearing is the risk
that cannot be diversified away (i.e., systematic risk). Because systematic risk
CFA 10
Answer:
CFA 11
CFA 12
CFA 13
Answer:
CFA 14
