9: CHAPTER THE CAPITAL ASSET PRICING MODEL
CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
PROBLEM SETS
1.
E (rP ) = rf + P [ E (rM ) rf ]
.18 = .06 + P [.14 .06] P =
2.
.12
= 1.5
.08
If the securitys correlation coefficient with the market portfolio doubles (with all
other variables such as variances unchanged), then beta, and therefore the risk
premium, will also double. The current risk premium is: 14% 6% = 8%
The new risk premium would be 16%, and the new discount rate for the security
would be: 16% + 6% = 22%
If the stock pays a constant perpetual dividend, then we know from the original
data that the dividend (D) must satisfy the equation for the present value of a
perpetuity:
Price = Dividend/Discount rate
50 = D/0.14 D = 50 0.14 = $7.00
At the new discount rate of 22%, the stock would be worth: $7/0.22 = $31.82
The increase in stock risk has lowered its value by 36.36%.
3.
a.
b.
c.
4.
False. = 0 implies E(r) = rf , not zero.
False. Investors require a risk premium only for bearing systematic
(undiversifiable or market) risk. Total volatility includes diversifiable risk.
False. Your portfolio should be invested 75% in the market portfolio and
25% in T-bills. Then: P = (0.75 1) + (0.25 0) = 0.75
The expected return is the return predicted by the CAPM for a given level of
systematic risk.
E ( ri ) = rf + i [ E (rM ) rf ]
E ( r$1 Discount ) = .04 + 1.5 (.10 .04) = .13 = 13%
E ( rEverything $5 ) = .04 + 1.0 (.10 .04) = .10 = 10%
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
5.
According to the CAPM, $1 Discount Stores requires a return of 13% based on its
systematic risk level of = 1.5. However, the forecasted return is only 12%.
Therefore, the security is currently overvalued.
Everything $5 requires a return of 10% based on its systematic risk level of =
1.0. However, the forecasted return is 11%. Therefore, the security is currently
undervalued.
6.
The expected return of a stock with a = 1.0 must, on average, be the same as the
expected return of the market which also has a = 1.0.
7.
Beta is a measure of systematic risk. Since only systematic risk is rewarded, it is
safe to conclude that the expected return will be higher for Kaskins stock than for
Quinns stock.
8.
The appropriate discount rate for the project is:
rf + [E(rM ) rf ] = .08 + [1.8 (.16 .08)] = .224 = 22.4%
Using this discount rate:
10
$15
= $40 + [$15 Annuity factor (22.4%, 10 years)] = $18.09
t
t =1 1.224
NPV = $40 +
The internal rate of return (IRR) for the project is 35.73%. Recall from your
introductory finance class that NPV is positive if IRR > discount rate (or,
equivalently, hurdle rate). The highest value that beta can take before the hurdle
rate exceeds the IRR is determined by:
.3573 = .08 + (.16 .08) = .2773/.08 = 3.47
9.
a.
Call the aggressive stock A and the defensive stock D. Beta is the sensitivity
of the stocks return to the market return, i.e., the change in the stock return
per unit change in the market return. Therefore, we compute each stocks
beta by calculating the difference in its return across the two scenarios
divided by the difference in the market return:
A =
.02 .38
= 2.00
.05 .25
D =
9-3
.06 .12
= 0.30
.05 .25
CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
b.
With the two scenarios equally likely, the expected return is an average of
the two possible outcomes:
E(rA ) = 0.5 (.02 + .38) = .18 = 18%
E(rD ) = 0.5 (.06 + .12) = .09 = 9%
The SML is determined by the market expected return of [0.5 (.25 + .05)] =
15%, with M = 1, and rf = 6% (which has f = 0). See the following graph:
c.
Expected Return - Beta Relationship
40
SML
35
Expected Return
30
25
A
20
15
D
10
A
M
5
0
0
0.5
1
1.5
2
2.5
3
Beta
The equation for the security market line is:
E(r) = .06 + (.15 .06)
d.
Based on its risk, the aggressive stock has a required expected return of:
E(rA ) = .06 + 2.0 (.15 .06) = .24 = 24%
The analysts forecast of expected return is only 18%. Thus the stocks alpha
is:
A = actually expected return required return (given risk)
= 18% 24% = 6%
Similarly, the required return for the defensive stock is:
E(rD) = .06 + 0.3 (.15 .06) = 8.7%
The analysts forecast of expected return for D is 9%, and hence, the stock
has a positive alpha:
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D = actually expected return required return (given risk)
= .09 .087 = +0.003 = +0.3%
The points for each stock plot on the graph as indicated above.
e. The hurdle rate is determined by the project beta (0.3), not the firms beta. The
correct discount rate is 8.7%, the fair rate of return for stock D.
10.
Not possible. Portfolio A has a higher beta than Portfolio B, but the expected
return for Portfolio A is lower than the expected return for Portfolio B. Thus,
these two portfolios cannot exist in equilibrium.
11.
Possible. If the CAPM is valid, the expected rate of return compensates only for
systematic (market) risk, represented by beta, rather than for the standard
deviation, which includes nonsystematic risk. Thus, Portfolio As lower rate of
return can be paired with a higher standard deviation, as long as As beta is less
than Bs.
12.
Not possible. The reward-to-variability ratio for Portfolio A is better than that
of the market. This scenario is impossible according to the CAPM because the
CAPM predicts that the market is the most efficient portfolio. Using the
numbers supplied:
SA =
.16 .10
= 0.5
.12
SM =
.18 .10
= 0.33
.24
Portfolio A provides a better risk-reward tradeoff than the market portfolio.
13.
Not possible. Portfolio A clearly dominates the market portfolio. Portfolio A has
both a lower standard deviation and a higher expected return.
14.
Not possible. The SML for this scenario is: E(r) = 10 + (18 10)
Portfolios with beta equal to 1.5 have an expected return equal to:
E(r) = 10 + [1.5 (18 10)] = 22%
The expected return for Portfolio A is 16%; that is, Portfolio A plots below the
SML ( A = 6%), and hence, is an overpriced portfolio. This is inconsistent
with the CAPM.
15.
Not possible. The SML is the same as in Problem 14. Here, Portfolio As required
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
return is: .10 + (0.9 .08) = 17.2%
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This is greater than 16%. Portfolio A is overpriced with a negative alpha:
A = 1.2%
16.
Possible. The CML is the same as in Problem 12. Portfolio A plots below the
CML, as any asset is expected to. This scenario is not inconsistent with the
CAPM.
17.
Since the stocks beta is equal to 1.2, its expected rate of return is:
.06 + [1.2 (.16 .06)] = 18%
E (r ) =
18.
D1 + P P0
P $50 + $6
1
0.18 = 1
P = $53
1
P0
$50
The series of $1,000 payments is a perpetuity. If beta is 0.5, the cash flow should
be discounted at the rate:
.06 + [0.5 (.16 .06)] = .11 = 11%
PV = $1,000/0.11 = $9,090.91
If, however, beta is equal to 1, then the investment should yield 16%, and the price
paid for the firm should be:
PV = $1,000/0.16 = $6,250
The difference, $2,840.91, is the amount you will overpay if you erroneously
assume that beta is 0.5 rather than 1.
19.
Using the SML: .04 = .06 + (.16 .06) = .02/.10 = 0.2
20.
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 1
a. To determine which investor was a better selector of individual stocks we look
at 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.
Without information about the parameters of this equation (risk-free rate and
market rate of return) we cannot determine which investor was more
accurate.
b.
If rf = 6% and rM = 14%, then (using the notation alpha for the abnormal
return):
1 = .19 [.06 + 1.5 (.14 .06)] = .19 .18 = 1%
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2 = .16 [.06 + 1 (.14 .06)] = .16 .14 = 2%
Here, the second investor has the larger abnormal return and thus appears to
be the superior stock selector. By making better predictions, the second
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investor appears to have tilted his portfolio toward underpriced stocks.
c.
If rf = 3% and rM = 15%, then:
1 = .19 [.03 + 1.5 (.15 .03)] = .19 .21 = 2%
2 = .16 [.03+ 1 (.15 .03)] = .16 .15 = 1%
21.
a.
Here, not only does the second investor appear to be the superior stock
selector, but the first investors predictions appear valueless (or worse).
Since the market portfolio, by definition, has a beta of 1, its expected rate of
return is 12%.
b.
= 0 means no systematic risk. Hence, the stocks expected rate of
return in market equilibrium is the risk-free rate, 5%.
c.
Using the SML, fair the expected rate of return for a stock with = 0.5 is:
E ( r ) = 0.05 + [(0.5) (0.12 0.05)] = 1.5%
The actually expected rate of return, using the expected price and dividend
for next year is:
E (r ) =
$41 + $3
1 = 0.10 = 10%
$40
Because the actually expected return exceeds the fair return, the stock is
underpriced.
22.
In the zero-beta CAPM the zero-beta portfolio replaces the risk-free rate, and thus:
E(r) = 8 + 0.6(17 8) = 13.4%
23.
a.
E(rP) = rf + P [E(rM ) rf ] = 5% + 0.8 (15% 5%) = 13%
= 14% 13% = 1%
You should invest in this fund because alpha is positive.
b.The passive portfolio with the same beta as the fund should be invested 80% in
the market-index portfolio and 20% in the money market account. For this
portfolio:
E(rP) = (0.8 15%) + (0.2 5%) = 13%
14% 13% = 1% =
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
24.
a.
We would incorporate liquidity into the CCAPM in a manner analogous to
the way in which liquidity is incorporated into the conventional CAPM. In
the latter case, in addition to the market risk premium, expected return is also
dependent on the expected cost of illiquidity and three liquidity-related betas
which measure the sensitivity of: (1) the securitys illiquidity to market
illiquidity; (2) the securitys return to market illiquidity; and, (3) the
securitys illiquidity to the market return. A similar approach can be used for
the CCAPM, except that the liquidity betas would be measured relative to
consumption growth rather than the usual market index.
b.
As in part (a), non-traded assets would be incorporated into the CCAPM in a
fashion similar to part (a). Replace the market portfolio with consumption
growth. The issue of liquidity is more acute with non traded-assets such as
privately-held businesses and labor income.
While ownership of a privately-held business is analogous to ownership of
an illiquid stock, expect a greater degree of illiquidity for the typical private
business. If the owner of a privately-held business is satisfied with the
dividends paid out from the business, then the lack of liquidity is not an
issue. If the owner seeks to realize income greater than the business can pay
out, then selling ownership, in full or part, typically entails a substantial
liquidity discount. The illiquidity correction should be treated as suggested
in part (a).
The same general considerations apply to labor income, although it is
probable that the lack of liquidity for labor income has an even greater
impact on security market equilibrium values. Labor income has a major
impact on portfolio decisions. While it is possible to borrow against labor
income to some degree, and some of the risk associated with labor income
can be ameliorated with insurance, it is plausible that the liquidity betas of
consumption streams are quite significant, as the need to borrow against
labor income is likely cyclical.
CFA PROBLEMS
1.
a.
Agree; Regans conclusion is correct. By definition, the market portfolio lies on
the capital market line (CML). Under the assumptions of capital market theory,
all portfolios on the CML dominate, in a risk-return sense, portfolios that lie on
the Markowitz efficient frontier because, given that leverage is allowed, the
CML creates a portfolio possibility line that is higher than all points on the
efficient frontier except for the market portfolio, which is Rainbows portfolio.
Because Eagles portfolio lies on the Markowitz efficient frontier at a point other
than the market portfolio, Rainbows portfolio dominates Eagles portfolio.
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
b.
Nonsystematic risk is the unique risk of individual stocks in a portfolio that is
diversified away by holding a well-diversified portfolio. Total risk is composed
of systematic (market) risk and nonsystematic (firm-specific) risk.
Disagree; Wilsons remark is incorrect. Because both portfolios lie on the
Markowitz efficient frontier, neither Eagle nor Rainbow has any nonsystematic
risk. Therefore, nonsystematic risk does not explain the different expected
returns. The determining factor is that Rainbow lies on the (straight) line (the
CML) connecting the risk-free asset and the market portfolio (Rainbow), at the
point of tangency to the Markowitz efficient frontier having the highest return
per unit of risk. Wilsons remark is also countered by the fact that, since
nonsystematic risk can be eliminated by diversification, the expected return for
bearing nonsystematic is zero. This is a result of the fact that well-diversified
investors bid up the price of every asset to the point where only systematic risk
earns a positive return (nonsystematic risk earns no return).
2.
E(r) = rf + [E(r M ) rf ]
Furhman Labs: E(r) = .05 + 1.5 [.115 .05] = 14.75%
Garten Testing: E(r) = .05 + 0.8 [.115 .05] = 10.20%
If the forecast rate of return is less than (greater than) the required rate of return,
then the security is overvalued (undervalued).
Furhman Labs: Forecast return Required return = 13.25% 14.75% = 1.50%
Garten Testing: Forecast return Required return = 11.25% 10.20% = 1.05%
Therefore, Furhman Labs is overvalued and Garten Testing is undervalued.
3.
a.
4.
d.
From CAPM, the fair expected return = 8 + 1.25 (15 8) = 16.75%
Actually expected return = 17%
= 17 16.75 = 0.25%
5.
d.
6.
c.
7.
d.
8.
d.
[You need to know the risk-free rate]
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CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
9.
d.
10.
Under the CAPM, the only risk that investors are compensated for bearing is the
risk that cannot be diversified away (systematic risk). Because systematic risk
(measured by beta) is equal to 1.0 for both portfolios, an investor would expect the
same rate of return from both portfolios A and B. Moreover, since both portfolios
are well diversified, it doesnt matter if the specific risk of the individual securities
is high or low. The firm-specific risk has been diversified away for both
portfolios.
11.
a.
McKay should borrow funds and invest those funds proportionately in
Murrays existing portfolio (i.e., buy more risky assets on margin). In
addition to increased expected return, the alternative portfolio on the capital
market line will also have increased risk, which is caused by the higher
proportion of risky assets in the total portfolio.
b.
McKay should substitute low beta stocks for high beta stocks in order to
reduce the overall beta of Yorks portfolio. By reducing the overall portfolio
beta, McKay will reduce the systematic risk of the portfolio, and therefore
reduce its volatility relative to the market. The security market line (SML)
suggests such action (i.e., moving down the SML), even though reducing beta
may result in a slight loss of portfolio efficiency unless full diversification is
maintained. Yorks primary objective, however, is not to maintain
efficiency, but to reduce risk exposure; reducing portfolio beta meets that
objective. Because York does not want to engage in borrowing or lending,
McKay cannot reduce risk by selling equities and using the proceeds to buy
risk-free assets (i.e., lending part of the portfolio).
12.
[You need to know the risk-free rate]
a.
Stock X
Stock Y
b.
Expected Return
5% + 0.8 (14% 5%) = 12.2%
5% + 1.5 (14% 5%) = 18.5%
Alpha
14.0% 12.2% = 1.8%
17.0% 18.5% = 1.5%
i. Kay should recommend Stock X because of its positive alpha, compared to
Stock Y, which has a negative alpha. In graphical terms, the expected
return/risk profile for Stock X plots above the security market line (SML),
while the profile for Stock Y plots below the SML. Also, depending on the
individual risk preferences of Kays clients, the lower beta for Stock X may
have a beneficial effect on overall portfolio risk.
ii. Kay should recommend Stock Y because it has higher forecasted return
and lower standard deviation than Stock X. The respective Sharpe ratios for
Stocks X and Y and the market index are:
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Stock X:
(14% 5%)/36% = 0.25
Stock Y:
(17% 5%)/25% = 0.48
Market index:
(14% 5%)/15% = 0.60
The market index has an even more attractive Sharpe ratio than either of the
individual stocks, but, given the choice between Stock X and Stock Y, Stock
Y is the superior alternative.
When a stock is held as a single stock portfolio, standard deviation is the
relevant risk measure. For such a portfolio, beta as a risk measure is
irrelevant.
Although holding a single asset is not a typically recommended investment
strategy, some investors may hold what is essentially a single-asset portfolio
when they hold the stock of their employer company. For such investors, the
relevance of standard deviation versus beta is an important issue.
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