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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
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(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%.
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
