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Unformatted text preview: Chapter 06  Risk Aversion and Capital Allocation to Risky Assets 61 CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e) 2. (b) A higher borrowing is a consequence of the risk of the borrowers’ default. In perfect markets with no additional cost of default, this increment would equal the value of the borrower’s option to default, and the Sharpe measure, with appropriate treatment of the default option, would be the same. However, in reality there are costs to default so that this part of the increment lowers the Sharpe ratio. Also, notice that answer (c) is not correct because doubling the expected return with a fixed riskfree rate will more than double the risk premium and the Sharpe ratio. 3. Assuming no change in risk tolerance, that is, an unchanged risk aversion coefficient (A), then higher perceived volatility increases the denominator of the equation for the optimal investment in the risky portfolio (Equation 6.12). The proportion invested in the risky portfolio will therefore decrease. 4. a. The expected cash flow is: (0.5 × $70,000) + (0.5 × 200,000) = $135,000 With a risk premium of 8% over the riskfree 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 for $118,421, and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is derived as follows: $118,421 × [1 + E(r)] = $135,000 Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate or return with the required rate of return. c. If the risk premium over Tbills is now 12%, then the required return is: 6% + 12% = 18% The present value of the portfolio is now: $135,000/1.18 = $114,407 Chapter 06  Risk Aversion and Capital Allocation to Risky Assets 62 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. 5. When we specify utility by U = E(r) – 0.5A σ 2 , the utility level for Tbills is: 0.07 The utility level for the risky portfolio is: U = 0.12 – 0.5A(0.18) 2 = 0.12 – 0.0162A In order for the risky portfolio to be preferred to bills, the following inequality must hold: 0.12 – 0.0162A > 0.07 ⇒ A < 0.05/0.0162 = 3.09 A must be less than 3.09 for the risky portfolio to be preferred to bills. 6. Points on the curve are derived by solving for E(r) in the following equation: U = 0.05 = E(r) – 0.5A σ 2 = E(r) – 1.5 σ 2 The values of E(r), given the values of σ 2 , are therefore: σ σ 2 E(r) 0.00 0.0000 0.05000 0.05 0.0025 0.05375 0.10 0.0100 0.06500 0.15 0.0225 0.08375 0.20 0.0400 0.11000 0.25 0.0625 0.14375 The bold line in the following graph (labeled Q6, for Question 6) depicts the indifference curve. Chapter 06  Risk Aversion and Capital Allocation to Risky Assets 63 7. Repeating the analysis in Problem 6, utility is now: U = E(r) – 0.5A σ 2 = E(r) – 2.0= E(r) – 2....
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 Spring '09
 WILSONCHOI
 Utility, representative, Risk premium

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