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(r P) = r f f + βP [E(r M ) – r r f ] E(r P) – r r f f = βP [E(r M ) – r r f ] βP = [E(r P) – r r f f ]/[E(r r f ] = [0.18 – [0.18 – 0.06]/[0.14 – 0.06]/[0.14 – 0.06] 0.06] = 0.12/0.08 = 1.5 M ) – r 2. The market price of a security is $50. Its expected rate of return is 14%. The risk-free risk-f ree rate is 6% and the market risk premium is 8.5%. What Wh at 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(r i) = r f f + βi [E(r M ) – r r f ] E(r i) – r r f f = βi [E(r M ) – r r f ] If βi doubles, E(r i) – r r f f doubles. Current Risk Premium = E(r i) – r r f f = 14% – 14% – 6% 6% = 8% New Risk Premium = E(r i) – r r f f = 16% New discount rate (expected (expected return) for the security = E(r i) = 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(r i) Solve for D given the original price and r: D = (Price)E(r i) = ($50)(0.14) = $7.00 Now use the dividend (which (which does not change) and new new E(ri) to solve for the new price: price: Price = D/E(r i) = $7.00/0.22 = $31.82 The increase in stock ’s risk ’s risk has lowered its value by $31.82/$50 – $31.82/$50 – 1 1 = 36.36%. Note that now we can model the the change in risk as a change in β and not a change in σ. This is because we now know that total total risk (σ) is not the “priced” relationship relationship.. The relationship between the stock and all other stocks (the market) measured by β is 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. incur. You are not compensated for total 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) = r f + 0[E(r M) – r r f f] = r f 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 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 T- Bill results in β P = 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 Discount Store” E(r $1 r f ] = 0.04 + 1.5(0.10 – 1.5(0.10 – 0.04) 0.04) = 13% $1) = r f f + β$1 [E(r M ) – r “Everything $5”
E(r $5 r f ] = 0.04 + 1.0(0.10 – 1.0(0.10 – 0.04) 0.04) = 10% $5) = r f f + β$5 [E(r M ) – r
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 stoc k 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: Returns: r $1 $1 = 12% and r $5 $5 = 11% P0,$1 = D1/(r $1 g) = $2/(0.12 – $2/(0.12 – 0.05) 0.05) = $28.57 $1 – g) P0,$5 = D1/(r $5 g) = $2/(0.11 – $2/(0.11 – 0.05) 0.05) = $33.33 $5 – g) Using the CAPM Returns: r $1 $1 = 13% and r $5 $5 = 10% P0,$1 = D1/(r $1 – g) g ) = $2/(0.13 – $2/(0.13 – 0.05) 0.05) = $25.00 $1 P0,$5 = D1/(r $5 g) = $2/(0.10 – $2/(0.10 – 0.05) 0.05) = $40.00 $5 – g) According to the CAPM (and our pricing model), model) , the price of “$1 Discount Store” stock should be $25 but the market price is $28.57 – $28.57 – so so it is overpriced. According to the CAPM (and our pricing model), the price pri ce of “Everything $5” stock should be $40 but the market price is $33.33 – $33.33 – so so it is underpriced. Note that the $25 and $28.57 $28.57 prices for the “$1 Discount Discount Store” stock will 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 r eturn for a stock that has a beta of 1.0 if the expected return on the market is 15%?
According According to the CAPM, the expected return of a stock with a β = 1.0 must be the same a s 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
After-Tax Cash Flows
0
-40
1-10
15
The project's beta is 1.8. Assuming that r f = 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? negative? Assume the corporation has has no debt so E(r) = WACC.
The appropriate discount rate for the company’s equity is: E(r) = r f f + β[E(r M ) – r r f ] = 0.08 + [1.8 (0.16 – (0.16 – 0.08)] 0.08)] = 22.4% NPV = -$40 + $15/(1.224) + $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 -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 the projects IRR. (Note that we can do this since since the CFs do not change signs.) 3
Use the calculator’s CF register: CF0 = -40, CF1 = 15, N1 = 10, 10, IRR = 35.73 So the highest β before the hurdle rate exceeds the IRR is: E(r) = r f f + β[E(r M ) – r r f ] β = [E(r) – r r f f ]/[E(r r f ] M ) – r β = [0.3537 - 0.08]/[0.16 – 0.08]/[0.16 – 0.08] 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
Expected Return
Beta
A
20%
1.4
B
30%
1.2
Not possible. Portfolio A has a higher beta but lower expected expected return than Portfolio B. The central prediction of the CAPM is that that investors are compensated compensated only for incurring market market risk as measured by beta. Higher beta means higher higher return. 11.
Portfolio
Expected Return
Stdev
A
30%
35%
B
40%
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. T hus, Portfolio A’s return can be lower than B’s as long as its beta is lower than B’s. 14.
Portfolio
Expected Return
Beta
Risk-Free
10%
0
Market
18%
1.0
A
16%
1.5
Not possible. According to the CAPM, since βA > 1, the E(r A) must be greater than E(r M), but it is not. E(r A) = r f f + βA[E(r M ) – r r f ] = 0.10 + 1.5[0.18 – 1.5[0.18 – 0.10] 0.10] = 22% > 16% Since the expected return for A is 16%, we can say that A = 16% – 16% – 22% 22% = -6% or it plots below the SML. Market participants will will sell the stock until 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 int erest 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) = r f f + βA[E(r M ) – r r f ] = 0.06 + 1.2[0.16 – 1.2[0.16 – 0.06] 0.06] = 18% E(r) = [E(P1) + E(D1)]/P1 – 1 1 P1 = [1 + E(r)]P0 – E(D E(D1) P1 = 1.18($50) – 1.18($50) – $6 $6 = $59 – $59 – $6 $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 b eta 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: P 0 = CF/r. Price the cash flows at a discount rate using β = 0.5 β = 1: E(r) = r f f + β [E(r M ) – r r f ] = 0.06 + 0.5[0.16 – 0.5[0.16 – 0.06] 0.06] = 11% P0 = CF/E(r) = $1,000/0.11 = $9,091 E(r) = r f f + β [E(r M ) – r r f ] = 0.06 + 1[0.16 – 1[0.16 – 0.06] 0.06] = 16% P0 = CF/E(r) = $1,000/0.16 = $6,250 $9,091 – $9,091 – $6,250 $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) = r f f + β [E(r M ) – r r f ] β = [E(r) – [E(r) – r r f f]/[E(r r f ] M ) – r β = [0.04 – [0.04 – 0.06]/[0.16 – 0.06]/[0.16 – 0.06] 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 is the advisor’s advisor’s α. A positive α indicates a positive risk -adjusted risk -adjusted abnormal return: after correcting for market risk, the advisor “beat the market.” A negative α indicates a negative risk -adjusted 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 – 11 – Multifactor Multifactor Models) that the single factor CAPM is not sufficient for correctly c orrectly assessing risk-adjusted risk- adjusted abnormal returns (α).
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(b) If the T-bill rate were 6% and the t he market return during the period were 14%, which investor would be the superior stock selector?
E(r 1) = r f f + β1[E(r M ) – r r f ] = 0.06 + 1.5[0.14 – 1.5[0.14 – 0.06] 0.06] = 18% Realized r 1 = 19% α1 = 19% – 19% – 18% 18% = 1% E(r 2) = r f f + β2[E(r M ) – r r f ] = 0.06 + 1[0.14 – 1[0.14 – 0.06] 0.06] = 14% Realized r 2 = 16% α2 = 16% – 16% – 14% 14% = 2% Although both advisors earned positive CAPM risk-adjusted excess returns, the second advisor is the superior superior stock picker. The first advisor achieved 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(r 1) = r f f + β1[E(r M ) – r r f ] = 0.03 + 1.5[0.15 – 1.5[0.15 – 0.03] 0.03] = 21% Realized r 1 = 19% α1 = 19% – 19% – 21% 21% = -2% E(r 2) = r f f + β2[E(r M ) – r r f ] = 0.03 + 1[0.15 – 1[0.15 – 0.03] 0.03] = 15% Realized r 2 = 16% α2 = 16% – 16% – 15% 15% = 1% Now only the second advisor advisor earned a positive CAPM risk-adjusted risk-adjusted excess return. So the second advisor is the the superior stock picker. Again, the first advisor 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 market rate. Therefore E(r M) = 12% Show this: E(r 1) = r f f + β[E(r M ) – r r f ] E(r M ) = [E(r) – [E(r) – r r f ]/β – 1 1 = [0.12 – [0.12 – 0.05]/1 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 market risk. The CAPM implies that investors investors are only compensated for incurring incurring market risk. risk. Therefore E(r) = 5%. Show this: E(r 1) = r f f + β1[E(r M ) – r r f ] = 0.05 + 0[0.12 – 0[0.12 – 0.05] 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) = r f f + β[E(r M ) – r r f ] = 0.05 + -0.5[0.12 – -0.5[0.12 – 0.05] 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 stock is underpriced. Therefore market participants will 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(r MF r f ] = 0.05 + 0.8(0.15 – 0.8(0.15 – 0.05) 0.05) = 13% MF) = r f f + βMF [E(r M ) – r 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 f und equals the alpha from part (a).
Another way to ask this question: The market portfolio earns 15% a nd the risk-free rate is 5%. A mutual fund is available available that has a β of 0.8 and earns 14%. Construct a portfolio portfolio of the market and the risk-free risk-free (called a “passive portfolio”) that has the same risk as the mutual fund. What is the expected return of of the passive portfolio? βP = 0.8 = WM βM + WRF βRF = WM (1) + WRF (0) WM = 0.80 and W RF = 0.2 E(r P) = WM E(r) + WRF r f f = 0.8(0.15) + 0.2(0.05) = 13% E(r MF E(r P) = 0.14 – 0.14 – 0.13 0.13 = 1% = MF) – E(r
MF
This is an important result. It shows (in theory) with well-informed investors 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, opportunity, the investor will buy the mutual fund for $100 and earn 14%. The investor will also also sell short $80 of the market portfolio 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 7
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 β MF = 0.8 and the short portfolio position as having β P = -0.8 -0.8 for a combined β = 0. No risk but a positive positive return, so there this is 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 the risk-free rate. Therefore we cannot take advantage advantage of potentially profitable profitable investment opportunities. opportunities. But major investment 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 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 CM L creates a portfolio possibility line that is higher than all points on the efficient frontier e xcept 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 individual stocks in a portfolio portfolio that is diversified 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 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 c onnecting 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 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) = r f f + β × [E(r M ) − r f f ] Fuhrman Labs: E(r) = 5 + 1.5 × [11.5 [ 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 – return – Required Required return = 13.25% − 14.75% = −1.50% 8
Garten Testing: Forecast return – return – Required 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 risk that investors are compensated for bearing is the risk that cannot be diversified away (systematic (systematic risk). Because systematic risk (measured by beta) is equal to 1.0 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 risky assets on margin). In addition to increased 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 substitute low beta stocks for high high beta stocks in order to reduce the overall beta of York’s portfolio. By 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) (S ML) 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 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. 9
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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