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FINAL EXAM [50 pts] QUESTION 1 [5 pts] Suppose you are presented with the following regression design to test the empirical success of the CAPM: Ri,t− r f ,t = αi +βi[Rm,t− r f ,t ]+ εi,t (1) Ri,t− r f ,t = γ0 + γ1β̂i + γ2σ2ε,i (2) where in equation (1), Ri,t is the return to portfolio i in period (month) t, r f ,t is the risk-free rate, Rm,t is the return on the market portfolio, and εi,t is the residual of the regression; and in equation (2) Ri,t is the average return to portfolio i over the sample, β̂i is the estimated CAPM beta from equation (1) (for portfolio i), and σ2ε,i is the variance of portfolio i′s residuals in equation (1). (a) What does the CAPM predict should be the values of γ0, γ1, and γ2. Explain your answers. (b) Suppose that in the sample, the risk premium on the market portfolio is Rm,t− r f ,t = 6.5%, but the estimate from equation (2) is γ̂1 = 4%. Is the empirical Security Market Line (the SML implied by the data) too steep or too flat? Explain. QUESTION 2 [5 pts] Describe the Fama-French 3-factor model. Which are the factors and how are they (generally) constructed? What are some possible reasons that these factors earn positive expected returns? QUESTION 3 [5 pts] Suppose you own a house worth $500. However, there is a risk the house could burn down. If the house burns down, it will only be worth $25. There is a 5% chance the house burns down. However, you can buy insurance that will pay you if in the event the house burns down. Call the amount of insurance purchased K. The premium you have to pay for K dollars of insurance is 0.05×K. So, if hypothetically you wanted $100 of insurance, the premium would be $5. Assume you have log-utility u(x) = ln(x). What is the optimal amount of insurance, K∗? (Note: the premium must be paid whether the house burns down or not.) QUESTION 4 [5 pts] Suppose you have log-utility u(x) = ln(x). You are faced with the following gamble: a 50% chance of getting $100 and a 50% chance of getting $200. You have no wealth at the start of the gamble, so your wealth after the gamble will either be $100 or $200. 1 What is the certainty equivalent of the gamble? QUESTION 5 [5 pts] Imagine you are choosing between two portfolios, and you believe the CAPM is the correct pricing model. Portfolio A has a market beta of βA = 0.6. Portfolio B has a market beta of βB = 0.9. The average excess return for A is 6%, while for B the average excess return is 8.5%. The average excess return on the market is 7%. The standard deviation of A’s return is 12%, and for B is 18%. (a) Which portfolio, A or B, has a higher alpha? (b) Which portfolio has a higher Sharpe ratio? (c) If you could borrow as much as you want at the risk-free rate (say r f = 1%), and you wanted your investment portfolio to have a market beta of βp = 1, how would you construct such a portfolio? Explain. QUESTION 6 [5 pts] Consider the following figure: 11-1 Figure 11.1 Cumulative Abnormal Returns Before Takeover Attempts: Target Companies 2 The figure shows the cumulative abnormal return plotted against the days to a merger announce- ment. State whether each of the three forms of the Efficient Market Hypothesis are supported by or can be rejected by the figure, and explain your answer: (a) Weak form (b) Semi-strong form (c) Strong form (Notes: (1) abnormal return means the unexpected return, like the difference between the actual return and the return predicted by the Fama French 3-factor model; (2) it is a common observation that firms that get acquired (taken over, merged, etc.) are often acquired at a very high premium above their current market price, so we would expect at the time of a merger/takeover announcement that the return for the target company would increase significantly) QUESTION 7 [5 pts] XYZ stock price and dividend history are as follows: Year Beginning-of-year-price Dividend Paid at Year End 2015 $ 100 $4 2016 $ 120 $4 2017 $ 90 $4 2018 $ 100 $4 An investment manager buys three shares of XYZ at the beginning of 2015, buys two more shares at the beginning of 2016, sells one share at the beginning of 2017, and sells all four remaining shares at the beginning of 2018. (a) What are the arithmetic and geometric average returns the manager earned with this strategy? (b) What is the dollar-weighted return? QUESTION 8 [5 pts] Consider stocks A and B below, with performance estimated according to the CAPM. The risk-free rate over the period was 6%, and the market’s average return was 14%. Stock A Stock B CAPM regression estimates 1%+1.2(rm− r f ) 2%+0.8(rm− r f ) Residual standard deviation, σ(e) 10.3% 19.1% Standard deviation of excess returns 21.6% 24.9% (a) Calculate the following statistics for each stock: 1. Information ratio 3 2. Sharpe ratio 3. M2 measure 4. Treynor ratio (b) Which stock is best if it is the only risky asset held by the investor? (c) Which stock is best if it will be mixed with the rest of the investor’s portfolio, which is currently composed of holding only the market portfolio? QUESTION 9 [5 pts] Mark all that are true, and explain your answers: The tangency portfolio has: (a) the maximum Sharpe ratio (b) a market beta equal to 0 (if CAPM is true) (c) the minimum possible variance of all risky portfolios QUESTION 10 [5 pts] Suppose we are in a standard APT factor model type world, with two factors F1 and F2. Suppose a well-diversified portfolio A has expected return of 11%, with betas of 0.4 on the first factor and 0.6 on the second factor. Also assume that two factor portfolios exist (corresponding to the two factors we have), F1 has an expected return of 7%, and F2 has an expected return of 8%. (a) Is there arbitrage here? If so, how would you construct the arbitrage portfolio? If not, why not? 4