Soln Ch 06 Risk Avers

7/29/2019 Soln Ch 06 Risk Avers

http://slidepdf.com/reader/full/soln-ch-06-risk-avers 1/8

CHAPTER 6: RISK AND RISK AVERSION

1. a. The expected cash flow is: .5 × 70,000 + .5 × 200,000 = $135,000. With a risk premiumof 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, thepresent 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 therequired rate of return.

c. If the risk premium over bills is now 12%, the required return is 6 + 12 = 18%. Thepresent 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 atlower prices. The extra discount from expected value is a penalty for risk.

2. When we specify utility by U = E(r) – .005Aσ2, the utility from bills is 7%, while that

from the risky portfolio is U = 12 – .005A × 182 = 12 – 1.62A. For the portfolio to bepreferred 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) – .005Aσ2 = E(r) – .015σ2

The necessary value of E(r), given the value of σ2, is therefore:

σ σ2 E(r)

0% 0 5.0%5 25 5.375

10 100 6.515 225 8.37520 400 11.025 625 14.375

6-1



The indifference curve is depicted by the bold line in the following graph (labeled Q3, forQuestion 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) – .005Aσ2 = E(r) – .02σ2 = 4

leading to the equal-utility combinations of expected return and standard deviationpresented in the table below. The indifference curve is the upward sloping line appearingin the graph of Problem 3, labeled Q4 (for Question 4).

σ σ2 E(r)

0% 0 4.00%5 25 4.50

10 100 6.0015 225 8.5020 400 12.0025 625 16.50

The indifference curve in Problem 4 differs from that in Problem 3 in both slope andintercept. When A increases from 3 to 4, the higher risk aversion results in a greater slopefor the indifference curve since more expected return is needed to compensate for additional

σ. 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 correspondingutility is simply equal to the portfolio's expected return. The corresponding indifferencecurve in the expected return-standard deviation plane is a horizontal line, drawn in thegraph of Problem 3, and labeled Q5.

6. A risk lover, rather than penalizing portfolio utility to account for risk, derives greater utilityas variance increases. This amounts to a negative coefficient of risk aversion. Thecorresponding 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 × σ2. 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 isthe one with the highest expected return.]

9. b

10. The portfolio expected return can be computed as follows:

Portfolio Portfolio

Wbills × + Wmarket × =

__________________________________________________________________

0.0 5% 1.0 13.5% 13.5% 20%.2 5 .8 13.5 11.8 16.4 5 .6 13.5 10.1 12.6 5 .4 13.5 8.4 8.8 5 .2 13.5 6.7 4

1.0 5 0.0 13.5 5.0 0

6-3



11. Computing the utility from U = E(r) – .005 × Aσ2 = E(r) – .015σ2 (because A = 3), we

arrive at the following table.

Wbills Wmarket E(r) σ σ2 U(A=3) U(A=5)

___________________________________________________________________

0.0 1.0 13.5% 20 400 7.5 3.5.2 .8 11.8 16 256 7.96 5.4.4 .6 10.1 12 144 7.94 6.5.6 .4 8.4 8 64 7.44 6.8.8 .2 6.7 4 16 6.46 6.3

1.0 0.0 5.0 0 0 5.0 5.0

The utility column implies that investors with A = 3 will prefer a position of 80% in themarket 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 Aσ2 =

E(r) – .025σ2 (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 weightpreferred by investors with A = 3.

13. SugarKane is now less of a hedge, and the entire probability distribution is:

Normal Sugar Crop Sugar CrisisBullish Stock Market Bearish Stock Market

Probability .5 .3 .2

Stock

Best Candy 25% 10% –25%SugarKane 10 –5 20Humanex's Portfolio 17.5 2.5 – 2.5

Using the portfolio rate of return distribution, its expected return and standard deviationcan be calculated as follows:

E(rp) = .5 × 17.5 + .3 × 2.5 + .2 × (–2.5) = 9%

σp = [.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 issignificantly 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 andstandard deviation of SugarKane are now:

E(rSK) = .5 × 10 + .3 × (–5) + .2 × 20 = 7.5%

σSK = [ .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.

σp = [ wσ+ wσ+ 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 andtheir deviations from the mean are given by:

Probability Rate of Return (%) Deviation fromMean (%)

.10 –100.00 –107.52

.20 –81.25 –88.77

.40 20.00 12.48

.25 71.67 64.15

.05 157.08 149.56

Mean = 7.52%Std Dev = 70.30%

a. Mean = 7.52%Median = 20.00%Mode = 20.00%

b. Std. Dev. = 70.30%

MAD = ΣsPr(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 = ΣsPr(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:

Probability Wealth

No fire: .999 $253,000Fire: .001 $ 53,000

which gives expected utility

.001 × loge(53,000) + .999 × loge(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 thatthe expected loss is "only" $200, meaning that you are willing to pay quite a risk premiumover 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 $49,900which grows by year end to $52,894. If there is a fire, your insurance proceeds are only$100,000. Your outcome will be:

Probability Wealth

Fire .001 $152,894No fire .999 $252,894

Expected utility is

.001 × loge(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 afire. The outcomes are

Event Probability Wealth

fire .001 $352,682 = (50,000 – 300) × 1.06 + 300,000no fire .999 $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 createsa gamble (you actually gain when the house burns down). Risk is minimized when youinsure exactly the value of the house.

6-8

Soln Ch 06 Risk Avers

Documents

Transcript of Soln Ch 06 Risk Avers