Chapter 9 Fin
Risk-free E(r)10 0STDV Market E(r)18 24STDV A E(r)16 12STDV
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) / .12 = 0.5 SM= (.18-.10) / .24 = .33 Portfolio A provides a better risk-reward tradeoff than the market portfolio
Risk-free E(r)10 0 Beta Market E(r)18 1.0 Beta A E(r)16 0.9 Beta
15. Not possible. The SML is the same as in Problem 14. Here, Portfolio A's required return is: . 10 + (0.9 × .08) = 17.2% This is greater than 16%. Portfolio A is overpriced with a negative alpha: alpha A = -1.2%
Risk-free E(r)10 0 STDV Market E(r)18 24 STDV A E(r)16 22 STDV
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.
Characterize each company in the previous problem as underpriced, overpriced, or properly priced. ============== Company 1 E(company1) = .04+1.5 x (.10-.04) = .13 or 13% Company 2 E(company2) = .04+1 x (.10- .04) = .10 or 10%
According to the CAPM, $1 Discount Stores requires a return of 13% based on its systematic risk level of β = 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 β = 1.0. However, the forecasted return is 11%. Therefore, the security is currently undervalued.
Outline how you would incorporate the following into the CCAPM: Nontraded assets. (Do you have to worry about labor income?)
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.
Market Return =========== 5% 25% Aggressive Stock ============ -2% 38% Defensive Stock ============ 6% 12% What are the betas of the two stocks?
Ba= (-.02 - .38) / (.05 -.25) = 2 Bd = (.06 -.12) /(.05-.25) = 0.30
Market Return =========== 5% 25% Aggressive Stock ============ -2% 38% Defensive Stock ============ 6% 12% Plot the two securities on the SML graph. What are the alphas of each
Based on its risk, the aggressive stock has a required expected return of: E(rA ) = .06 + 2.0 × (.15 - .06) = .24 = 24% αA = 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% αD = actually expected return - required return (given risk) = .09 - .087 = +0.003 = +0.3%
Kaskin, Inc., stock has a beta of 1.2 and Quinn, Inc., stock has a beta of .6. Which of the following statements is most accurate?
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 Kaskin's stock than for Quinn's stock.
Current risk Premium
E(r) -rf
The equation for the security market line is:
E(r) =rf + β × (E(rm) - rf)
Market Return =========== 5% 25% Aggressive Stock ============ -2% 38% Defensive Stock ============ 6% 12% What is the expected rate of return on each stock if the market return is equally likely to be 5% or 25%?
E(rA ) = 0.5 x (-.02 + .38) = .18 = 18% E(rD) = 0.5 x (.06 +.12) = 0.09 = 9%
actually expected return
E(rA ) = Market return x (A1 + A2) A1 and A2 are two returns for the stock
required expected return of:
E(rA ) = rf + b × (SML(a) - rf ) = answer
A mutual fund with beta of .8 has an expected rate of return of 14%. If r f = 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(rP) = rf + βP × [E(rM ) - rf ] = 5% + 0.8 (15% − 5%) = 13% alpha = 14% - 13% = 1% You should invest in this fund because alpha is positive.
The expected return is the return predicted by the CAPM for a given level of systematic risk.
E(ri) = rf +Bi x [E(rm) -rf]<---- formula E(rm) = rf+Er rf/E(rm) = are decimals
Here are data on two companies. The T-bill rate is 4% and the market risk premium is 6%. A) Company 1 Forecasted return 12% Standard deviation of returns 8% Beta 1.5 ============================= B)Company 2 Forecasted return 11% Standard deviation of returns 10% Beta 1.0 ============================== ============================== What would be the fair return for each company, according to the capital asset pricing model (CAPM)?
E(ri) = rf +Bi x [E(rm) -rf]<---- formula E(rm) = rf+Er 10% rf/E(rm) = are decimals ============= Company 1 E(company1) = .04+1.5 x (.10-.04) = .13 or 13% Company 2 E(company2) = .04+1 x (.10- .04) = .10 or 10%
Find beta
E(rp) = rf +Betap x [E(rm)-rf] E(rp) = given rf = given E(rm) given Betap = Not given .
The CAPM implies that investors require a higher return to hold highly volatile securities.(true or false)
False. Investors require a risk premium only for bearing systematic (undiversifiable or market) risk. Total volatility includes diversifiable risk.
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.(true or false)
False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills. Then: βp=(0.75 x 1) + (0.25 x 0 ) = 0.75
Stocks with a beta of zero offer an expected rate of return of zero.(true or false)
False. β = 0 implies E(r) = rf , not zero
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. What if the T-bill rate were 3% and the market return were 15%?
If rf = 3% and rM = 15%, then: α1 = .19 - [.03 + 1.5 × (.15 - .03)] = .19 - .21 = -2% α2 = .16 - [.03+ 1 × (.15 - .03)] = .16 - .15 = 1% Here, not only does the second investor appear to be the superior stock selector, but the first investor's predictions appear valueless (or worse).
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. If the T-bill rate were 6% and the market return during the period were 14%, which investor would be the superior stock selector?
If rf = 6% and rM = 14%, then (using the notation alpha for the abnormal return): α1 = .19 - [.06 + 1.5 × (.14 - .06)] = .19 - .18 = 1% α 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
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.
If the security's 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 x 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%.
Suppose that borrowing is restricted so that the zero-beta version of the CAPM holds. The expected return on the market portfolio is 17%, and on the zero-beta portfolio it is 8%. What is the expected return on a portfolio with a beta of .6?
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%
Risk-free E(r)10 0STDV Market E(r)18 24STDV A E(r)20 22STDV
Not possible. Portfolio A clearly dominates the market portfolio. Portfolio A has both a lower standard deviation and a higher expected return.
Portfolio Expected Return Beta A 20% Beta 1.4 B 25% Beta 1.2
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
Risk-free E(r)10 0 Beta Market E(r)18 1.0 Beta A E(r)16 1.5 Beta
Not possible. The SML for this scenario is: E(r) = 10 + β × (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 (alpah A = -6%), and hence, is an overpriced portfolio. This is inconsistent with the CAPM
A E(r)30% 35 STDV B E(r)40% 25 STDV
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 A's lower rate of return can be paired with a higher standard deviation, as long as A's beta is less than B's.
present value of a perpetuity
Price = Dividend/Discount rate dividend =find Discount rate = E(r) is decimal form
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: What is the expected rate of return on the market portfolio?
Since the market portfolio, by definition, has a beta of 1, its expected rate of return is 12%.
rf =6% E(rm) =16% A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year
Since the stock's beta is equal to 1.2, its expected rate of return is .06 + [1.2 x (.16 - .06)] = 18% E(r)= (D1 +P1-P0 / P0) = 0.18 (P1 -50$+6$ / $50 ) = P1 = $53
Market Return =========== 5% 25% Aggressive Stock ============ -2% 38% Defensive Stock ============ 6% 12% If the T-bill rate is 6% and the market return is equally likely to be 5% or 25%, draw the SML for this economy
The SML is determined by the market expected return of [0.5 × (.25 + .05)] = 15%, with βM = 1, and rf = 6% (which has βf = 0). See the following graph: E(r) = .06 + β × (.15 - .06)
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%?
The expected return of a stock with a β = 1.0 must, on average, be the same as the expected return of the market which also has a β = 1.0.
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 previous question
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% = alpha
rf =6% E(rm) =16% 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. 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.
rf =6% E(rm) =16% A stock has an expected rate of return of 4%. What is its beta?
Using the SML: .04 = .06 + β × (.16 - .06) -> β = -.02/.10 = -0.2
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: 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 .5. Is the stock overpriced or underpriced?
Using the SML, the fair expected rate of return for a stock with β = -0.5 is: E(r)= 0.05 +[(-0.5) x (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)/ 40 - 1 = 0.10 or 10% Because the actually expected return exceeds the fair return, the stock is underpriced.
Outline how you would incorporate the following into the CCAPM: Liquidity.
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 security's illiquidity to market illiquidity; (2) the security's return to market illiquidity; and, (3) the security's 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.
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: What would be the expected rate of return on a stock with 0?
b. β = 0 means no systematic risk. Hence, the stock's expected rate of return in market equilibrium is the risk-free rate, 5%.
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. Can you tell which investor was a better selector of individual stocks (aside from the issue of general movements in the market)?
r1 = 19%; r2 = 16%; β1 = 1.5; β2 = 1 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.
discount rate
rf + β × [E(rM ) - rf ]
The project's beta is 1.8. Assuming that rf= 8% and E ( r M ) 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?
rf + β × [E(rM ) - rf ] = .08 + [1.8 (.16 - .08)] = .224 = 22.4%
to find alpha for A or any i
αA = actually expected return - required return (given risk)
