solutions_ch_06 - CHAPTER 6 RISK AND RISK AVERSION 1 a The...

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CHAPTER 6: RISK AND RISK AVERSION 1. a. The expected cash flow is: .5 × 70,000 + .5 × 200,000 = $135,000. With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, the present value of the portfolio is 135,000/1.14 = $118,421 b. If the portfolio is purchased at $118,421, and provides an expected payoff of $135,000, then the expected rate of return, E(r), is derived as follows: $118,421 × [1 + E(r)] = $135,000 so that E(r) = 14%. The portfolio price is set to equate the expected return with the required rate of return. c. If the risk premium over bills is now 12%, the required return is 6 + 12 = 18%. The present value of the portfolio is now $135,000/1.18 = $114,407. d. For a given expected cash flow, portfolios that command greater risk premia must sell at lower prices. The extra discount from expected value is a penalty for risk. 2. When we specify utility by U = E(r) – .005A σ 2 , the utility from bills is 7%, while that from the risky portfolio is U = 12 – .005A × 18 2 = 12 – 1.62A. For the portfolio to be preferred to bills, the following inequality must hold: 12 – 1.62A > 7, or, A < 5/1.62 = 3.09. A must be less than 3.09 for the risky portfolio to be preferred to bills. 3. Points on the curve are derived as follows: U = 5 = E(r) – .005A σ 2 = E(r) – .015 σ 2 The necessary value of E(r), given the value of σ 2 , is therefore: σ σ 2 E(r) 0% 0 5.0% 5 25 5.375 10 100 6.5 15 225 8.375 20 400 11.0 25 625 14.375 6-1
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The indifference curve is depicted by the bold line in the following graph (labeled Q3, for Question 3). E(r) σ   5    4 U(Q3,A=3) U(Q4,A=4) U(Q5,A=0) U(Q6,A<0) 4. Repeating the analysis in Problem 3, utility is: U = E(r) – .005A σ 2 = E(r) – .02 σ 2 = 4 leading to the equal-utility combinations of expected return and standard deviation presented in the table below. The indifference curve is the upward sloping line appearing in the graph of Problem 3, labeled Q4 (for Question 4).
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