Download soln_ch_06_risk_avers

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2, 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) – .005A2 = E(r) – .0152
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) – .005A2 = E(r) – .02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).

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  A2 = E(r) – .0152 (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 A2 =
E(r) – .0252 (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
.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 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
.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 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