CHAPTER 9: THE CAPITAL ASSET PRICING MODEL
1. What must be the beta of a portfolio with E(rP) = 18%, if rf = 6% and E(rM) = 14%?
E(rP) = rf + βP [E(rM ) – rf ]
E(rP) – rf = βP [E(rM ) – rf ]
βP = [E(rP) – rf ]/[E(rM ) – rf ] = [0.18 – 0.06]/[0.14 – 0.06] = 0.12/0.08 = 1.5
2. The market price of a security is $50. Its expected rate of return is 14%. The risk-free rate is
6% and the market risk premium is 8.5%. What will be the market price of the security if its
correlation coefficient with the market portfolio doubles (and all other variables remain
unchanged)? Assume that the stock is expected to pay a constant dividend in perpetuity.
βi = σiM/σ2m and σiM = iMσiσM.
If iM doubles, then σiM doubles and βi doubles.
Therefore (according to the CAPM) the risk premium will also double:
E(ri) = rf + βi [E(rM ) – rf ]
E(ri) – rf = βi [E(rM ) – rf ]
If βi doubles, E(ri) – rf doubles.
Current Risk Premium = E(ri) – rf = 14% – 6% = 8%
New Risk Premium = E(ri) – rf = 16%
New discount rate (expected return) for the security = E(ri) = 16% + 6% = 22%
If the stock pays a constant perpetual dividend, and we know the price and the discount rate, we
can calculate that constant dividend. The dividend (D) must satisfy the equation for the present
value of a perpetuity:
Price = D/E(ri)
Solve for D given the original price and r:
D = (Price)E(ri) = ($50)(0.14) = $7.00
Now use the dividend (which does not change) and new E(ri) to solve for the new price:
Price = D/E(ri) = $7.00/0.22 = $31.82
The increase in stock’s risk has lowered its value by $31.82/$50 – 1 = 36.36%.
Note that now we can model the change in risk as a change in β and not a change in σ. This is
because we now know that total risk (σ) is not the “priced” relationship.
The relationship between the stock and all other stocks (the market) measured by β is important,
“priced” relationship.
In other words, according to the CAPM, you are only compensated for the portion of market risk
(measured by β) you chose to incur. You are not compensated for total risk (measured by σ).
3. Are the following true or false (according to the CAPM)? Explain.
(a) Stocks with a beta of zero offer an expected rate of return of zero.
False. The stocks with no market risk (β = 0) should offer the risk-free return:
E(r) = rf + 0[E(rM) – rf] = rf
1
(b) The CAPM implies that investors require a higher return to hold highly volatile
securities.
False. According to the CAPM, investors are compensated only for incurring un-diversifiable
or market risk, measured by β, not total risk or total volatility, measured by standard deviation
or variance.
(c) You can construct a portfolio with beta of .75 by investing .75 of the investment budget
in T-bills and the remainder in the market portfolio
False. Market’s beta is 1 and T-Bills beta is 0.
βP = W1 β1 + W2 β2 = WM βM + WT-Bills βT-Bills = 0.25(1) + 0.75(0) = 0.25
75% in the market and 25% in T-Bill results in βP = 0.75(1) + 0.25(0) = 0.75
4. Here are data on two companies. The T-bill rate is 4% and the market risk premium is 6%.
$1 Discount Store
Everything $5
Forecasted Return
12%
11%
Stdev of Returns
8%
10%
Beta
1.5
1.0
What would be the fair return for each company, according to the capital asset pricing
model (CAPM)?
According to the CAPM, holders of the stock are compensated only for market risk (measured by
β) and not total risk (measured by σ):
$1 Discount Store” E(r$1) = rf + β$1 [E(rM ) – rf ] = 0.04 + 1.5(0.10 – 0.04) = 13%
“Everything $5”
E(r$5) = rf + β$5 [E(rM ) – rf ] = 0.04 + 1.0(0.10 – 0.04) = 10%
5. Characterize each company in the previous problem as underpriced, overpriced, or properly
priced.
Since the CAPM return for the “$1 Discount Store” exceeds the market “forecast” return, the stock
of the “$1 Discount Store” is overpriced.
Since the CAPM return for “Everything $5” is less than the market “forecast” return, the stock of
“Everything $5” is underpriced.
To see this we can choose a pricing model and values for that pricing model to show the
relationship between the market price and the CAPM price. Note that the relationship between the
market price and CAPM price is not dependent on the pricing model we choose.
Use Constant Dividend Growth pricing model to calculate two prices for each stock using both the
forecast and CAPM returns for each stock.
Assume values for the next dividend and dividend growth: The next dividend for both companies
(D1) will be $2 and dividend growth for both companies (g) will be 5%.
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Using the Forecast Returns: r$1 = 12% and r$5 = 11%
P0,$1 = D1/(r$1 –g) = $2/(0.12 – 0.05) = $28.57
P0,$5 = D1/(r$5 –g) = $2/(0.11 – 0.05) = $33.33
Using the CAPM Returns: r$1 = 13% and r$5 = 10%
P0,$1 = D1/(r$1 –g) = $2/(0.13 – 0.05) = $25.00
P0,$5 = D1/(r$5 –g) = $2/(0.10 – 0.05) = $40.00
According to the CAPM (and our pricing model), the price of “$1 Discount Store” stock should be
$25 but the market price is $28.57 – so it is overpriced.
According to the CAPM (and our pricing model), the price of “Everything $5” stock should be
$40 but the market price is $33.33 – so it is underpriced.
Note that the $25 and $28.57 prices for the “$1 Discount Store” stock will change if we choose a
different pricing model, but the relationship between the prices (the $28.57 price using forecast
return greater than the $25 price using the CAPM) will not change if the pricing model changes.
6. What is the expected rate of return for a stock that has a beta of 1.0 if the expected return on
the market is 15%?
According to the CAPM, the expected return of a stock with a β = 1.0 must be the same as the
expected return of the market which is given as 15%.
8. You are a consultant to a large manufacturing corporation that is considering a project with
the following net after-tax cash flows (in millions of dollars).
Years from Now
0
1-10
After-Tax Cash Flows
-40
15
The project's beta is 1.8. Assuming that rf = 8% and E(rM) = 16%, what is the net present
value of the project? What is the highest possible beta estimate for the project before its
NPV becomes negative? Assume the corporation has no debt so E(r) = WACC.
The appropriate discount rate for the company’s equity is:
E(r) = rf + β[E(rM ) – rf ] = 0.08 + [1.8  (0.16 – 0.08)] = 22.4%
NPV = -$40 + $15/(1.224) + $15/(1.224)2 + $15/(1.224)3 + … $15/(1.224)10 = $18.09
Using your calculator’s TVM function:
N = 10, I/Y = 22.4, PMT = 15, PV = -58.09  NPV = -40 + 58.09 = 18.09
Using your calculators CF Register:
CF0 = -40, CF1 = 15, N1 = 10, I/Y = 22.4, NPV = 18.09
NPV > 0 if IRR > discount rate.
To calculate the highest possible β estimate (and therefore the highest possible discount rate) for a
positive NPV, calculate the projects IRR. (Note that we can do this since the CFs do not change
signs.)
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Use the calculator’s CF register:
CF0 = -40, CF1 = 15, N1 = 10, IRR = 35.73
So the highest β before the hurdle rate exceeds the IRR is:
E(r) = rf + β[E(rM ) – rf ]  β = [E(r) – rf ]/[E(rM ) – rf ]
β = [0.3537 - 0.08]/[0.16 – 0.08] = 3.47
For Problems 10 to 16: If the simple CAPM is valid, which of the following situations are
possible? Explain. Consider each situation independently
10.
Portfolio
A
B
Expected Return
20%
30%
Beta
1.4
1.2
Not possible. Portfolio A has a higher beta but lower expected return than Portfolio B. The central
prediction of the CAPM is that investors are compensated only for incurring market risk as
measured by beta. Higher beta means higher return.
11.
Portfolio
A
B
Expected Return
30%
40%
Stdev
35%
25%
Possible. According to the CAPM, the expected rate compensates investors only for systematic
(aka market) risk, measured by beta, not for total risk (both market and idiosyncratic) measured by
standard deviation. Thus, Portfolio A’s return can be lower than B’s as long as its beta is lower
than B’s.
14.
Portfolio
Risk-Free
Market
A
Expected Return
10%
18%
16%
Beta
0
1.0
1.5
Not possible.
According to the CAPM, since βA > 1, the E(rA) must be greater than E(rM), but it is not.
E(rA) = rf + βA[E(rM ) – rf ] = 0.10 + 1.5[0.18 – 0.10] = 22% > 16%
Since the expected return for A is 16%, we can say that A = 16% – 22% = -6% or it plots
below the SML. Market participants will sell the stock until the price decreases enough so that
its expected return increases to 22% to compensate for the market risk measured by β A = 1.5.
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For Problems 17 to 19 assume that the risk-free rate of interest is 6% and the expected rate of
return on the market is 16%.
17. A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the
year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year (if
the CAPM holds)?
E(r) = rf + βA[E(rM ) – rf ] = 0.06 + 1.2[0.16 – 0.06] = 18%
E(r) = [E(P1) + E(D1)]/P1 – 1  P1 = [1 + E(r)]P0 – E(D1)
P1 = 1.18($50) – $6 = $59 – $6 = $53
18. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk.
If I think the beta of the firm is .5, when in fact the beta is really 1, how much more will I
offer for the firm than it is truly worth?
The series of $1,000 payments is a perpetuity, so the price today is: P0 = CF/r.
Price the cash flows at a discount rate using β = 0.5 β = 1:
E(r) = rf + β [E(rM ) – rf ] = 0.06 + 0.5[0.16 – 0.06] = 11%
P0 = CF/E(r) = $1,000/0.11 = $9,091
E(r) = rf + β [E(rM ) – rf ] = 0.06 + 1[0.16 – 0.06] = 16%
P0 = CF/E(r) = $1,000/0.16 = $6,250
$9,091 – $6,250 = $2,841 is the amount you will overpay if you underestimate the risk of the cash
flows by assuming that beta is 0.5 rather than 1.
19. A stock has an expected rate of return of 4%. What is its beta?
E(r) = rf + β [E(rM ) – rf ]  β = [E(r) – rf]/[E(rM ) – rf ]
β = [0.04 – 0.06]/[0.16 – 0.06] = -0.02/0.10 = -0.2
20. Two investment advisers are comparing performance. One averaged a 19% rate of return
and the other a 16% rate of return. However, the beta of the first investor was 1.5, whereas
that of the second was 1.
(a) Can you tell which investor was a better selector of individual stocks (aside from the
issue of general movements in the market)?
Compare the advisor’s realized return to what the return should have been according the
CAPM. The difference is the advisor’s α.
A positive α indicates a positive risk-adjusted abnormal return: after correcting for market
risk, the advisor “beat the market.”
A negative α indicates a negative risk-adjusted abnormal return: after correcting for market
risk, the advisor did not “beat the market.”
Note that we will show shortly (in Chapter 11 – Multifactor Models) that the single factor
CAPM is not sufficient for correctly assessing risk-adjusted abnormal returns (α).
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(b) If the T-bill rate were 6% and the market return during the period were 14%, which
investor would be the superior stock selector?
E(r1) = rf + β1[E(rM ) – rf ] = 0.06 + 1.5[0.14 – 0.06] = 18%
Realized r1 = 19%
α1 = 19% – 18% = 1%
E(r2) = rf + β2[E(rM ) – rf ] = 0.06 + 1[0.14 – 0.06] = 14%
Realized r2 = 16%
α2 = 16% – 14% = 2%
Although both advisors earned positive CAPM risk-adjusted excess returns, the second
advisor is the superior stock picker. The first advisor achieved a greater return than the second
advisor only by incurring more risk (higher β), not through “skill.”
In other words: According to the CAPM, the first advisor should have earned 18% but earned
19%, for a risk-adjusted excess return of 1%. The second advisor should have earned 14% but
earned 16%, for a risk-adjusted excess return of 2%.
(c) What if the T-bill rate were 3% and the market return were 15%?
E(r1) = rf + β1[E(rM ) – rf ] = 0.03 + 1.5[0.15 – 0.03] = 21%
Realized r1 = 19%
α1 = 19% – 21% = -2%
E(r2) = rf + β2[E(rM ) – rf ] = 0.03 + 1[0.15 – 0.03] = 15%
Realized r2 = 16%
α2 = 16% –15% = 1%
Now only the second advisor earned a positive CAPM risk-adjusted excess return. So the
second advisor is the superior stock picker. Again, the first advisor achieved a greater realized
return than the second advisor only by incurring more risk (higher β) and not through “skill.”
21. Suppose the rate of return on short-term government securities (perceived to be risk-free) is
about 5%. Suppose also that the expected rate of return required by the market for a
portfolio with a beta of 1 is 12%. According to the capital asset pricing model:
(a) What is the expected rate of return on the market portfolio?
Portfolios with β = 1 earn the market rate. Therefore E(rM) = 12%
Show this:
E(r1) = rf + β[E(rM ) – rf ]  E(rM ) = [E(r) – rf ]/β – 1 = [0.12 – 0.05]/1 + 0.05 = 12%
(b) What would be the expected rate of return on a stock with β = 0?
Stocks with β = 0 have no market risk. The CAPM implies that investors are only
compensated for incurring market risk. Therefore E(r) = 5%.
Show this: E(r1) = rf + β1[E(rM ) – rf ] = 0.05 + 0[0.12 – 0.05] = 5%
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(c) Suppose you consider buying a share of stock at $40. The stock is expected to pay $3
dividends next year and you expect it to sell then for $41. The stock risk has been
evaluated at β = -0.5. Is the stock overpriced or underpriced?
According to the CAPM:
E(r) = rf + β[E(rM ) – rf ] = 0.05 + -0.5[0.12 – 0.05] = 1.5%
According to the CAPM, the price should be:
P0 = [E(P1) + E(D1)]/[1 + E(R)] = [$41 + $3]/(1.015) = $43.35
Since the market price is $40, the stock is underpriced. Therefore market participants will buy
the stock and drive up the price until it reaches $43.35 so that holders of the stock are
compensated only for the market risk they incur.
23. Consider the following:
(a) A mutual fund with beta of .8 has an expected rate of return of 14%. If rf = 5%, and you
expect the rate of return on the market portfolio to be 15%, should you invest in this
fund? What is the fund's alpha?
E(rMF) = rf + βMF [E(rM ) – rf ] = 0.05 + 0.8(0.15 – 0.05) = 13%
MF = 14%  13% = 1%
Invest in this Mutual Fund because its alpha is positive and therefore it is earning a positive
risk-adjusted excess return.
(b) What passive portfolio comprised of a market-index portfolio and a money market
account would have the same beta as the fund? Show that the difference between the
expected rate of return on this passive portfolio and that of the fund equals the alpha
from part (a).
Another way to ask this question: The market portfolio earns 15% and the risk-free rate is 5%.
A mutual fund is available that has a β of 0.8 and earns 14%. Construct a portfolio of the
market and the risk-free (called a “passive portfolio”) that has the same risk as the mutual
fund. What is the expected return of the passive portfolio?
βP = 0.8 = WM βM + WRF βRF = WM (1) + WRF (0)  WM = 0.80 and WRF = 0.2
E(rP) = WM E(r) + WRF rf = 0.8(0.15) + 0.2(0.05) = 13%
E(rMF) – E(rP) = 0.14 – 0.13 = 1% = MF
This is an important result. It shows (in theory) with well-informed investors and the ability
to borrow and lend at the risk-free rate, positive (or negative) ’s cannot persist.
Assume a size $100. Given this opportunity, the investor will buy the mutual fund for $100
and earn 14%. The investor will also sell short $80 of the market portfolio and borrow
another $20. The net cash flow is $100 - $100 = $0.
The net return will be:
A inflow from the mutual fund = $100(0.14) = $14
An out flow from the 80-20 passive portfolio:
Short Market = (0.8)($100)(15%) = $12
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Borrowing = (0.2)($100)(0.05) = $1
So these investments provide $14 inflow and $13 outflow for a net inflow of $1with
(according to the CAPM), no risk.
Think about the long mutual fund position as having βMF = 0.8 and the short portfolio position
as having βP = -0.8 for a combined β = 0. No risk but a positive return, so there this is not in
equilibrium.
This is not in equilibrium because market participants will continue to buy the mutual fund
(driving the price up and lowing its return) and selling the market portfolio (driving its price
down and increasing its return) and borrowing money at the risk-free rate thereby increasing
that rate) until the positive  available from the mutual fund investment disappears.
A quick note on borrowing and lending at the risk-free and investment opportunities that rely
on this:
You and I cannot borrow at the risk-free rate. Therefore we cannot take advantage of
potentially profitable investment opportunities. But major investment banks can. These are
the market participants that are ensuring that these opportunities do not persist.
CFA PROBLEMS
1.
2.
a.
Agree; Regan’s 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 Rainbow’s
portfolio. Because Eagle’s portfolio lies on the Markowitz efficient frontier at a point other
than the market portfolio, Rainbow’s portfolio dominates Eagle’s portfolio.
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; Wilson’s 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. Wilson’s 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).
E(r) = rf + β × [E(r M ) − rf ]
Fuhrman Labs: E(r) = 5 + 1.5 × [11.5 − 5.0] = 14.75%
Garten Testing: E(r) = 5 + 0.8 × [11.5 − 5.0] = 10.20%
If the forecast rate of return is less than (greater than) the required rate of return, then the
security is overvalued (undervalued).
Fuhrman Labs: Forecast return – Required return = 13.25% − 14.75% = −1.50%
8
Garten Testing: Forecast return – Required return = 11.25% − 10.20% = 1.05%
Therefore, Fuhrman Labs is overvalued and Garten Testing is undervalued.
3.
a.
4.
d.
5.
d.
6.
c.
7.
d.
8.
d.
[You need to know the risk-free rate]
9.
d.
[You need to know the risk-free rate]
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 doesn’t 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 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 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 York’s 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. York’s 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.
From CAPM, the fair expected return = 8 + 1.25(15  8) = 16.75%
Actually expected return = 17%
 = 17  16.75 = 0.25%
a.
b.
Expected Return
Alpha
Stock X
5% + 0.8(14%  5%) = 12.2%
14.0%  12.2% = 1.8%
Stock Y
5% + 1.5(14%  5%) = 18.5%
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 Kay’s clients, the lower beta
for Stock X may have a beneficial effect on overall portfolio risk.
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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:
Stock X:
Stock Y:
Market index:
(14%  5%)/36% = 0.25
(17%  5%)/25% = 0.48
(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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