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Apr 3, 2024
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Uploaded by PresidentWaterDragonfly41
Baylor University
Hankamer School of Business
Department of Finance, Insurance & Real Estate
Risk Management
Name:
SOLUTIONS
Dr. Garven
Problem Set 4
Show your work and write as legibly as possible. Good luck!
Miles and John are considering two mutually exclusive risky investments, 1 and 2, with payoffs
given by:
W
1
,s
=
$20 with probability 20%
$60 with probability 50%
$100 with probability 30%
and
W
2
,s
=
$40 with probability 55%
$80 with probability 45%
Suppose that Miles’s initial wealth
W
0
= $0, and his utility
U
(
W
) =
W
.
5
.
A. Calculate Miles’s expected utility (
E
(
U
(
W
)) for both investments.
SOLUTION:
E
(
U
(
W
1
,s
)) =
.
2
√
20 +
.
5
√
60 +
.
3
√
100 = 7
.
76
E
(
U
(
W
2
,s
)) =
.
55
√
40 +
.
45
√
80 = 7
.
49
B. Now suppose that John also has initial wealth
W
0
= $0, but his utility function is
U
(
W
) =
ln
W
. Calculate John’s expected utility (
E
(
U
(
W
)) for both investments.
SOLUTION:
E
(
U
(
W
1
,s
)) =
.
2 ln(20) +
.
5 ln(60) +
.
3 ln(100) = 4
.
03
E
(
U
(
W
2
,s
)) =
.
55 ln(40) +
.
45 ln(80) = 4
.
00
C. Does either investment first order stochastically dominate the other? Explain why or why not.
SOLUTION: We check for first order stochastic dominance by comparing the cumulative
probabilities:
W
s
f
(
W
1
,s
)
F
(
W
1
,s
)
f
(
W
2
,s
)
F
(
W
2
,s
)
$
20
20%
20%
0%
0%
$
40
0%
20%
55%
55%
$
60
50%
70%
0%
55%
$
80
0%
70%
45%
100%
$
100
30%
100%
0%
100%
Since
F
(
W
1
,s
) initially exceeds
F
(
W
2
,s
) and is subsequently less than
F
(
W
2
,s
), there is no first
order stochastic dominance.
D. Compare these investments once again. Is there second order stochastic dominance? Explain
why or why not.
SOLUTION: We check for second order stochastic dominance by adding an additional column
to the table shown in part A where we calculate
n
∑
s
=1
(
F
(
W
2
,s
)
−
F
(
W
1
,s
)):
W
s
f
(
W
1
,s
)
F
(
W
1
,s
)
f
(
W
2
,s
)
F
(
W
2
,s
)
F
(
W
2
,s
)
−
F
(
W
1
,s
)
$
20
20%
20%
0%
0%
-20%
$
40
0%
20%
55%
55%
35%
$
60
50%
70%
0%
55%
-15%
$
80
0%
70%
45%
100%
30%
$
100
30%
100%
0%
100%
0%
n
∑
s
=1
(
F
(
W
2
,s
)
−
F
(
W
1
,s
)) = 30%
Thus, we find that investment 1 second order stochastically dominates investment 2.
E. Which investment should Miles choose? Explain why.
SOLUTION: Since investment 1 stochastically dominates investment 2, this ensures that Miles
will have higher expected utility from investment 1 than from investment 2 (as shown in Part A
of this problem). Therefore, Miles prefers (and will choose) investment 1 instead of investment
2.
F. Which investment should John choose? Explain why.
SOLUTION: Even though John is
more
risk averse than Miles (since as we showed in class,
logarithmic utility is more risk averse than square root utility), John prefers (and will choose)
investment 1 instead of investment 2 for the very same reason as Miles. Specifically, stochastic
dominance ensures that John will have higher expected utility from investment 1 than from
investment 2 (as shown in Part B of this problem).
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