STOCHASTIC

Let Pi (A) = Pi + A, for any constant A.
(a) Suppose Pi and p2 have symmetric distributions but are not positively correlated,
and Ael=[a-b,b~a\. Show that A > 0 implies k* > i, A = 0 implies k* = |, and
A < 0 implies A:* < i, where A* maximizes expected utility. [Hint: Investigate the firstorder
conditions at k = £.]
(b) Show that dk*/dA\A=0 > 0 in (a) but that dk*\dA is not generally positive. However,
show that dk*\dA is strictly positive over / if u displays constant absolute risk aversion.
(c) The result in (a) indicates that for the case of an additive shift the investor will hold
more than half his portfolio in the stochastically dominant asset. It is of interest to
determine whether or not strict first-degree stochastic dominance will imply k* > i.
Show that this is not the case by considering the following example. Let w0 = 1, u(w) =
log(w—0.2), and pi and p2 have the joint probability distribution
\ \ Pi
P2~*^
-0.9
1
-0.9
0
0.5
1 + 5
where B > 0. Show that p, strictly dominates pz for all strictly increasing utility functions.
Show that
1/3^61 + 1 ^ 1
- 2\3.61 + 1.95/ 2'
(d) Show that essentially all the results above are valid if the assumption that the random
returns are bounded is replaced by the assumption that k is bounded and expected utility
is bounded.
(e) Suppose u displays constant relative risk aversion, where the constant e (0,1].
Suppose Pi and p2 have symmetric distributions, a = — \. Let Pt(C) — (l + C)Pi + C,
where C e (-1, oo). Show that dk*/dC > 0, C > 0, implies k* > i, C = 0 implies A:* = },
and C < 0 implies k* < i- (Note that strict inequalities require that pi and p2 satisfy the
regularity assumption that Pr [pt = — 1] < 1.)
(f) The assumption that the relative risk-aversion constant e (0,1] is crucial for dk*/dC
to be positive. Let w0 — i and «(w) = — w" (c- ", c> 1, where c is the risk-aversion
constant. Suppose pi and/>2 have the joint probability distribution
\ Pi
P 2 \ ^
0
2
0
0
0.5
Show that dk*ldA\A = 0 < 0 whenever c> 4. Investigate the behavior of k* for this example.
13. (Concavity of indifference curves) Let u(w) be an investor's utility function over
wealth w and suppose f(w; p,d) is the distribution over wealth. Assume that / depends
only on mean (p.) and standard deviation (a) or independent functions of these parameters.
176 PART II QUALITATIVE ECONOMIC RESULTS
0.5
0
2
0.5
0