STOCHASTIC

Show that the x p s that minimize r{x) are the same as the minimizers of x 1 'Q.1 J x J
separately for each/'.
(d) Utilize the result in (c) and that of Exercise 9 to show that
and that
Q~ l p
x* = ———— . where Q =
x
where ps is the portion of p corresponding to class a,-.
(e) Suppose K — 2, n = 4, £i = (J J) and E2 = (f f), and pi = ps for all ij. Show
that** = ^(8,16,5,10).
11. Consider an investor with a disutility function — Kover regret R. Suppose disutility
increases with regret (— V > 0) and at an increasing rate reflecting risk aversion (— V" > 0).
Let x = (xi,...,x„)' denote the investment allocations where x £ 0 and e'x = l, where
e = (1, ...,1) and initial wealth is 1. The investment returns are p = (pt, ...,p„), and have
the joint distribution function F(p). For a given x,
«!>
R(x,p) = (max/?'*) — p'x = maxfpj — p'x
[pes j I
(assume that the pt are nonnegative and E is compact).
(a) Suppose that F is symmetric as defined in Samuelson's paper on diversification.
Show that the solution x = {\jn,...,\jn)' is a minimax strategy, i.e., S(x) —
min, m 0, x2* > 0 if x* solves
(1). [Hint: Eliminate x2 from (1) and investigate dW(xi)jdxi at x^ = 0.]
*(e) Attempt to generalize the result in (d) to the case n~ n, pi = • • • = p„.
*(f) Attempt to prove that xt* > 0 if pt a {min/5j 11 gygn, j ^ ;'} when F is general
and when the pt are independent.
(g) Compare the results in (c)-(f) with those in Samuelson's paper.
12. Consider a portfolio problem where the only actions available are bets on the occurrence
of states of nature. Let v, be the return per dollar invested in / if state / occurs; investment
in i produces no return if j occurs (i # j). Assume that the number of states of nature
is finite. Let xt be the amount bet on the occurrence of (' out of initial wealth one dollar.
Suppose that the investor wishes to maximize expected utility of final wealth, and that the
utility function is concave and differentiable.
MIND-EXPANDING EXERCISES 347