Chap006 - Chapter 06 Risk Aversion and Capital Allocation...

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Chapter 06 - Risk Aversion and Capital Allocation to Risky Assets 6-1 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 risk-free 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 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 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 T-bills 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
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Chapter 06 - Risk Aversion and Capital Allocation to Risky Assets 6-2 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 T-bills 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.
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Chapter 06 - Risk Aversion and Capital Allocation to Risky Assets 6-3 7. Repeating the analysis in Problem 6, utility is now: U = E(r) – 0.5A σ 2 = E(r) – 2.0 σ 2 = 0.04 The equal-utility combinations of expected return and standard deviation are presented in the table below. The indifference curve is the upward sloping line in the graph above, labeled Q7 (for Question 7).
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