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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) – .005A2, the utility from bills is 7%, while that from the risky portfolio is U = 12 – .005A 182 = 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) – .005A2 = E(r) – .0152 The necessary value of E(r), given the value of 2, is therefore: 0% 5 10 15 20 25 2 0 25 100 225 400 625 E(r) 5.0% 5.375 6.5 8.375 11.0 14.375 6-1 The indifference curve is depicted by the bold line in the following graph (labeled Q3, for Question 3). E(r) U(Q4,A=4) U(Q3,A=3) 5 U(Q5,A=0) 4 U(Q6,A<0) 4. Repeating the analysis in Problem 3, utility is: U = E(r) – .005A2 = E(r) – .022 = 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). 0% 5 10 15 20 25 2 0 25 100 225 400 625 E(r) 4.00% 4.50 6.00 8.50 12.00 16.50 The indifference curve in Problem 4 differs from that in Problem 3 in both slope and intercept. When A increases from 3 to 4, the higher risk aversion results in a greater slope for the indifference curve since more expected return is needed to compensate for additional . The lower level of utility assumed for Problem 4 (4% rather than 5%), shifts the vertical intercept down by 1%. 6-2 5. The coefficient of risk aversion of a risk neutral investor is zero. The corresponding utility is simply equal to the portfolio's expected return. The corresponding indifference curve in the expected return-standard deviation plane is a horizontal line, drawn in the graph of Problem 3, and labeled Q5. 6. A risk lover, rather than penalizing portfolio utility to account for risk, derives greater utility as variance increases. This amounts to a negative coefficient of risk aversion. The corresponding indifference curve is downward sloping, as drawn in the graph of Problem 3, and labeled Q6. 7. c [Utility for each portfolio = E(r) – .005 4 2. We choose the portfolio with the highest utility value.) 8. d [When investors are risk neutral, A = 0, and the portfolio with the highest utility is the one with the highest expected return.] 9. b 10. The portfolio expected return can be computed as follows: Portfolio Portfolio Return Exp. return expected standard deviation Wbills on bills + Wmarket on market return (= Wmarket 20%) __________________________________________________________________ 0.0 .2 .4 .6 .8 1.0 5% 5 5 5 5 5 1.0 .8 .6 .4 .2 0.0 13.5% 13.5 13.5 13.5 13.5 13.5 6-3 13.5% 11.8 10.1 8.4 6.7 5.0 20% 16 12 8 4 0 11. Computing the utility from U = E(r) – .005 A2 = E(r) – .0152 (because A = 3), we arrive at the following table. Wbills Wmarket E(r) 2 U(A=3) U(A=5) ___________________________________________________________________ 0.0 .2 .4 .6 .8 1.0 1.0 .8 .6 .4 .2 0.0 13.5% 11.8 10.1 8.4 6.7 5.0 20 16 12 8 4 0 400 256 144 64 16 0 7.5 7.96 7.94 7.44 6.46 5.0 3.5 5.4 6.5 6.8 6.3 5.0 The utility column implies that investors with A = 3 will prefer a position of 80% in the market and 20% in bills over any of the other positions in the table. 12. The column labeled U(A = 5) in the table above is computed from U = E(r) – .005 A2 = E(r) – .0252 (since A = 5). It shows that the more risk averse investors will prefer the position with 40% in the market index portfolio, rather than the 80% market weight preferred by investors with A = 3. 13. SugarKane is now less of a hedge, and the entire probability distribution is: Normal Sugar Crop Sugar Crisis Bullish Stock Market Bearish Stock Market Probability .5 .3 .2 Stock Best Candy 25% 10% –25% SugarKane 10 –5 20 Humanex's Portfolio 17.5 2.5 – 2.5 Using the portfolio rate of return distribution, its expected return and standard deviation can be calculated as follows: E(rp) = .5 17.5 + .3 2.5 + .2 (–2.5) = 9% p = [.5(17.5 – 9)2 + .3(2.5 – 9)2 + .2(–2.5 – 9)2]1/2 = 8.67% While the expected return has even improved slightly, the standard deviation is significantly greater and only marginally better than investing half in T-bills. 6-4 14. The expected return of Best is 10.5% and its standard deviation 18.9%. The mean and standard deviation of SugarKane are now: E(rSK) = .5 10 + .3 (–5) + .2 20 = 7.5% SK = [ .5(10 – 7.5)2 – .3(–5 – 7.5)2 + .2(20 – 7.5)2 ]1/2 = 9.01% and its covariance with Best is Cov = .5 (10 – 7.5)(25 – 10.5) + .3(–5 – 7.5)(10 – 10.5) + .2(20 –7.5)(–25 – 10.5) = –68.75 15. From the calculations in (14), the portfolio expected rate of return is E(rp) = .5 10.5 + .5 7.5 = 9% Using the portfolio weights wB = wSK = .5 and the covariance between the stocks, we can compute the portfolio standard deviation from rule 5. 2 2 2 2 p = [ wB B + wSK SK+ 2wBwSKCov(rB,rSK) ]1/2 = [ .52 18.92 + .52 9.012 + 2 .5 .5 (–68.75) ]1/2 = 8.67% 6-5 CHAPTER 6: APPENDIX A 1. The current price of Klink stock is $12. Thus, the rates of return in each scenario and their deviations from the mean are given by: Probability Rate of Return (%) Deviation from Mean (%) .10 .20 .40 .25 .05 –100.00 –81.25 20.00 71.67 157.08 –107.52 –88.77 12.48 64.15 149.56 Mean = 7.52% Std Dev = 70.30% a. Mean = 7.52% Median = 20.00% Mode = 20.00% b. Std. Dev. = 70.30% MAD = sPr(s) Abs[r(s) – E(r)] = 57.01% c. The first moment is the mean (7.52%), the second moment around the mean is the variance (70.302) and the third moment is: M3 = sPr(s) [r(s) – E(r)]3 = –30,157.82 Therefore the probability distribution is negatively (left) skewed. 6-6 CHAPTER 6: APPENDIX B 1. Your $50,000 investment will grow to $50,000(1.06) = $53,000 by year end. Without insurance your wealth will then be: No fire: Fire: Probability .999 .001 Wealth $253,000 $ 53,000 which gives expected utility .001loge(53,000) + .999loge(253,000) = 12.439582 and a certainty equivalent wealth of exp(12.439582) = $252,604.85 With fire insurance at a cost of $P, your investment in the risk-free asset will be only $(50,000 – P). Your year-end wealth will be certain (since you are fully insured) and equal to (50,000 – P) 1.06 + 200,000. Setting this expression equal to $252,604.85 (the certainty equivalent of the uninsured house) results in P = $372.78. This is the most you will be willing to pay for insurance. Note that the expected loss is "only" $200, meaning that you are willing to pay quite a risk premium over the expected value of losses. The main reason is that the value of the house is a large proportion of your wealth. 2. a. With 1/2 coverage, your premium is $100, your investment in the safe asset is $49,900 which grows by year end to $52,894. If there is a fire, your insurance proceeds are only $100,000. Your outcome will be: Fire No fire Probability .001 .999 Wealth $152,894 $252,894 Expected utility is .001loge(152,894) + .999 loge(252,894) = 12.440222 and WCE = exp(l2.440222) = $252,767 6-7 b. With full coverage, costing $200, end-of-year wealth is certain, and equal to (50,000 – 200) 1.06 + 200,000 = $252,788 Since wealth is certain, this is also certainty equivalent wealth of the fully insured position. c. With over-insurance, the insurance costs $300, and pays off $300,000 in the event of a fire. The outcomes are Event fire no fire Probability .001 .999 Wealth $352,682 = (50,000 – 300) 1.06 + 300,000 $252,682 = (50,000 – 300) 1.06 + 200,000 Expected utility is .001 loge(352,682) + .999 loge(252,682) = 12.4402205 and WCE = exp(l2.4402205) = 252,766 Therefore, full insurance dominates both over- and under-insurance. Over-insuring creates a gamble (you actually gain when the house burns down). Risk is minimized when you insure exactly the value of the house. 6-8