STOCHASTIC

COMPUTATIONAL AND REVIEW EXERCISES
1. Suppose that gross returns per $100 of investment for eight alternative investment
opportunities were
Investment
Year-\^^
1963
1964
1965
1966
1967
1968
1969
1970
Mean net return
A
90
110
110
110
110
90
110
110
5.00
B
70
90
70
90
130
70
130
70
-10.00
C
160
170
170
170
170
170
160
170
67.50
D
145
145
120
180
120
145
190
145
50.00
E
60
60
20
40
60
70
120
70
-37.50
F
110
140
140
160
160
160
160
160
48.75
G
150
170
170
160
170
170
180
170
67.50
Assume that the investor attaches equal probability to past outcomes as predictors of
future returns.
(a) Plot the cumulative probability distributions.
(b) Utilize the plots in (a) to show that investments C, D, and G dominate the others for
all monotone nondecreasing utility functions.
(c) Show that investment C is preferred if the investor has a logarithmic utility function.
(d) Show that investment G is preferred if the investor has the quadratic utility function
u(w) = w 2 .
(e) Construct a utility function for which investment D is preferred.
2. (a) Consider an investment problem in which it is desired to discount the wealth
level before taking the expected utility. Show that the stochastic dominance theorems in
the Hanoch-Levy paper apply to a reinterpreted discounted wealth.
(b) Show that the stochastic dominance theorems are independent of initial wealth,
if the investment returns are independent of initial wealth.
3. Suppose investments X and Y have the following returns
Outcome (z) PT(X = z) Pr (Y = z)
0 i i
i i 0
1 0 i
H i 0
2 i 0
(a) Show that investment X is preferred to investment Y for all monotone nondecreasing
utility functions even though the variance of X is larger than the variance of Y.
(b) Suppose that Pr(A"=z) has entries (i,i,0,0,i). Show that X is not preferred to Y
for all monotone nondecreasing utility functions.
(c) Referring to (b) show that X is preferred to Y for all monotone nondecreasing concave
utility functions, even though the variance of X is larger than the variance of Y.
COMPUTATIONAL AND REVIEW EXERCISES
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