Problem 1 (BKM, Q3 of Chapter 7) (10 points1) What must be the beta of a portfolio with E( rP ) = 20.0%, if the risk free rate is 5.0% and the expected return of the market is E( rM ) = 15.0%? Answer: We use E( rP ) = β P *(E( rM ) – r f ) + r f . We then have: 0.20 = β P *(0.15-0.05) + 0.05. Solving for the beta we get: β P =1.5.
Problem 2 (BKM, Q4 of Chapter 7) (20 points) The market price of a security is $40. Its expected rate of return is 13%. The risk-free rate is 7%, and the market risk premium is 8%. What will the market price of the security be if its beta doubles (and all other variables remain unchanged)? Assume that the stock is expected to pay a constant dividend in perpetuity. Hint: Use zero-growth Dividend Discount Model to calculate the intrinsic value, which is the market price. Answer: First, we need to calculate the original beta before it doubles from the CAPM. Note that: β = (the security’s risk premium)/(the market’s risk premium) = 6/8 = 0.75. Second, when its beta doubles to 2*0.75 = 1.5, then its expected return becomes: 7% + 1.5*8% = 19%. (Alternatively, we can find the expected return after the beta doubles in the following way.
If the beta of the security doubles, then so will its risk premium. The current risk premium for the stock is: (13% – 7%) = 6%, so the new risk premium would be 12%, and the new discount rate for the security would be: 12% + 7% = 19%.) Third, we find out the implied constant dividend payment from its current market price of $40. If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount rate 40 = D/0.13 ⇒ D = 40 * 0.13 = $5.20 Last, at the new discount rate of 19%, the stock would be worth: $5.20/0.19 = $27.37. The increase in stock risk has lowered the value of the stock by 31.58%. Problem 3 (BKM, Q16 of Chapter 7) (10 points)
A share of stock is now selling for $100. It will pay a dividend of $9 per share at the end of the year. Its beta is 1.0. What do investors expect the stock to sell for at the end of the year if the market expected return is18% and the risk free rate for the year is 8%? Answer: Since the stock’s beta is equal to 1, its expected rate of return should be equal to that of D + P1 − P0 , therefore, we can solve for P1 as the market, that is, 18%. Note that: E(r) = P0 9 + P1 − 100 the following: 0.18 = ⇒ P1 = $109. 100 Problem 4 (15 points) Assume two stocks, A and B. One has that E( rA ) = 12% and E( rB ) = 15.%. The beta for stock A is 0.8 and the beta for B is 1.2. If the expected returns of both stocks lie in the SML line, what is the expected return of the market and what is the risk-free rate? What is the beta of a portfolio made of these two assets with equal weights?
Answer: Since both stocks lie in the SML line, we can immediately find its slope or the risk premium of the market. Slope = (E(rM) – rF) = ( E(r2) – E(r1))/( β2- β1) = (0.15-0.12)/(1.2-0.8) = 0.03/0.4= 0.075. Putting these values in E(r2) = β2*(E(rM) – rF) + rF one gets: 0.15 = 1.2*0.075 + rF or rF =0.06=6.0%. The Expected return of the market is then given by (E(rM) – 0.06) = 0.075 giving: E(rM) = 13.5%. If you create a portfolio with these two assets putting equals amounts of money in them (equally weighted), the beta will be βP = w1*β1+w2*β2= 0.5*1.2+0.5*0.8 = 1.0. Problem 5 (15 points) You have an asset A with annual expected return, beta, and volatility given by: E( rA ) = 20%, β A =1.2, σ A =25%, respectively. If the annual risk-free rate is r f =2.5% and the expected annual return and volatility of the market are E( rM )=10%, σ A =15%, what is the alpha of asset A? Answer: In order to find the alpha, α A , of asset A we need to find out the difference between the expected return of the asset E( rA ) and the expected return implied by the CAPM which is r f + β A (E(rM) – r f ).
That is, express its expected return as: α A = E( rA ) – r f + β A (E( rM ) – r f )). Since we know the expected return of the market, the beta of the asset with respect to the market, and the risk-free rate, alpha is given by: α A = E( rA ) – β A (E( rM ) – r f ) – r f = 0.20 – 1.2(0.1 – 0.025) – 0.025
= 0.085 = 8.5%.
Problem 6 (BKM, Q23 of Chapter 7) (20 points) Consider the following data for a one-factor economy. All portfolios are well diversified. _______________________________________ Portfolio E(r) Beta ———————————————————-A 10% 1.0 F 4% 0 ———————————————————-Suppose another portfolio E is well diversified with a beta of 2/3 and expected return of 9%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be? Answer: You can create a Portfolio G with beta equal to 1.0 (the same as the beta for Portfolio A) by taking a long position in Portfolio E and a short position in Portfolio F (that is, borrowing at the risk-free rate and investing the proceeds in Portfolio E). For the beta of G to equal 1.0, the proportion (w) of funds invested in E must be: 3/2 = 1.5
The expected return of G is then: E(rG) = [(−0.50) × 4%] + (1.5 × 9%) = 11.5% βG = 1.5 × (2/3) = 1.0 Comparing Portfolio G to Portfolio A, G has the same beta and a higher expected return. This implies that an arbitrage opportunity exists. Now, consider Portfolio H, which is a short position in Portfolio A with the proceeds invested in Portfolio G: βH = 1βG + (−1)βA = (1 × 1) + [(−1) × 1] = 0 E(rH) = (1 × rG) + [(−1) × rA] = (1 × 11.5%) + [(− 1) × 10%] = 1.5% The result is a zero investment portfolio (all proceeds from the short sale of Portfolio A are invested in Portfolio G) with zero risk (because β = 0 and the portfolios are well diversified), and a positive return of 1.5%. Portfolio H is an arbitrage portfolio.
Problem 7 (10 points) Compare the CAPM theory with the APT theory, explain the difference between these two theories? Answer: APT applies to well-diversified portfolios and not necessarily to individual stocks. It is possible for some individual stocks not to be on the SML. CAPM assumes rational behavior for all investors; APT only requires some rational investors: APT is more general in that its factor does not have to be the market portfolio. Both models give the expected return-beta relationship. 3
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