B
AYLOR
U
NIVERSITY
H
ANKAMER
S
CHOOL OF
B
USINESS
D
EPARTMENT OF
F
INANCE
,
I
NSURANCE
&
R
EAL
E
STATE
Dr. Garven
Name ___SOLUTIONS_________
Problem Set #3
1.
A worker whose utility function
U
(
W
) =
W
has received a job offer which pays $80,000
with a bonus. The bonus is equally likely to be $0, $10,000, $20,000, $30,000, $40,000,
$50,000, or $60,000. Assume that initial wealth is $0.
A.
What is the expected value of this pay package?
Solution:
To compute the expected value of this pay package, we must solve the following
equation:
1
(
)
n
s
s
s
E W
p W
= (1/7)(80,000)
+ (1/7)(90,000)
+ (1/7)(100,000)
+
(1/7)(110,000)
+ (1/7)(120,000) + (1/7)(130,000)
+ (1/7)(140,000)
= $110,000.
B.
What is the certainty equivalent of this pay package?
Solution:
There are two ways to compute the certainty equivalent.
One method is to first
compute the expected utility of this job offer, set the utility of the certainty equivalent equal
to expected utility, and then solve for the certainty equivalent.
The other method requires
computing the risk premium using the ArrowPratt absolute risk aversion coefficient.
First Method:
1
(
(
))
(
)
n
s
s
s
E U W
p U W
= (1/7)(80,000)
.5
+ (1/7)(90,000)
.5
+ (1/7)(100,000)
.5
+
(1/7)(110,000)
.5
+ (1/7)(120,000)
.5
+ (1/7)(130,000)
.5
+ (1/7)(140,000)
= 330.27.
Therefore, the certainty equivalent is
2
(
(
))
CEQ
W
E U W
= 330.27
2
= $109,075.82.
Second Method:
First, compute variance:
2
2
1
(
)
n
W
s
s
s
p
W
E W
= (2/7)(30,000)
2
+ (2/7)(20,000)
2
+
(2/7)(10,000)
2
= 400,000,000.
Next, compute the ArrowPratt absolute risk aversion coefficient:
R
A
(
W
) = 
U
WW
/
U
W
= .25
W

1.5
/.5
W

.5
= .5/
W
.
Next, compute the risk premium:
2
(
)
.5
(
)
A
W
R
W
= 400,000,000(.5)(.5)/110,000 =
$909.09.
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P
ROBLEM
S
ET
#3
SOLUTIONS
2
Finally, compute the certainty equivalent:
(
)
(
)
CEQ
W
E W
W
= $110,000  $909.09
= $109,090.91.
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