Actuarial Science 363 DE
Solution Set
Assignment #1
June 12
th
2009
1.
An individual has a fractional power utility function whereby:
u(w) = w
0.8
where
"w"
is wealth.
She is offered a wager (bet) where there is a 30% probability of
winning a sound system worth $10,000 (plus the return of the wager) and a
70% probability of losing the wager.
Her initial wealth is $30,000.
a)
Show that this person is risk averse.
u (w) = w^0.8 >0
if w>0
u’(w) = 0.8*w^(0.2) >0 if w>0
u”(w)= 0.16w^(1.2) <0
So we have decreasing marginal utility which indicates that the individual is
risk averse.
b)
What is the maximum bet she will make?
Let G be the amount of the wager.
1)
No wager, wealth is 30,000 and expected utility position is
u (30,000) = 30,000^0.8=3816.77891.
(2) If winning a sound system worth $10,000, then wealth is 40,000.
(3) If losing the wager, then wealth is 30,000G.
Therefore, the expected utility position is:
0.3 *( 40,000^0.8)+0.7*[(30,000G)^0.8]
Then, we can get the equation:
0.3 *( 40,000^0.8)+0.7*[(30,000G)^0.8] = 3816.77891
1441.349321 + 0.7*[(30,000G)^0.8] = 3816.77891
(30,000G)^0.8 = 3393.470841
30,000G = 3393.470841 ^ 1.25
G = 4099.678665
Therefore, the maximum bet she will make is $4099.678665.
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 Spring '09
 robertbrown
 Economics, Utility, St. Petersburg paradox, Economics of uncertainty, utility position

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