STOCHASTIC

(a) Determine necessary and sufficient conditions for stochastic dominance in U„, that
is, for Eu(x) £ Eu(Y) for all ue U„. [Hint: Find the appropriate G„ "= U„ such that
£«(*) § Eu(Y) for all u e U„ iff £K(X) & £«(y) for all w e G„.]
(b) Find lim„ G„ = G^, and give necessary and sufficient conditioning for stochastic
dominance over the set U„ = {u\u m £0, k odd and i/ k> SO, k even, fc = 1,2,...}.
21. This problem is concerned with the qualitative behavior of portfolios consisting of a
blend of one safe and one risky asset. The portfolio may be characterized in the following
three ways: (1) by the percentage of its value held in the safe asset; (2) by its statistical
properties, e.g., variance, etc.; and (3) by certainty equivalents, i.e., the rate of return that
would make the investor indifferent between his risky portfolio and a safe asset.
Suppose that the investor maximizes expected utility of terminal wealth w, where his
utility function u is strictly increasing, strictly concave, and thrice differentiable. Initial
wealth is w0. This risky asset yields a return per dollar of pe if the state of the world is 0,
while the safe asset returns px for all 6. Assume that Epe > pi and that the risky asset does
not dominate the safe asset, i.e., mm.0pB < pt. Let a, and 1 — ai denote the fractions of initial
wealth invested in the safe and risky assets, respectively. The investor may maximize expected
utility by solving
U* = max £«[(«!/>! + (\-al)pg)w0]ne
where ng is the probability that state 8 will occur.
(a) Show that the necessary and sufficient condition for optimality is
£w'|>o(«iPi + (l-tfi)Pe)]0>i-A>) = 0.
Let R denote the index of relative risk aversion — u"w/u'.
(b) Show that
dat S dR ^
= 0 as — = 0.
dw0 = dw S
That is, the wealth elasticity of the demand for the riskless asset is greater than, equal to,
or less than unity as relative risk aversion is an increasing, constant, or decreasing function
of wealth. [Hint: Write the expression for dal/dw0 in terms of dR/dw.]
There are many statistics that may be used to characterize a portfolio, such as the mean,
variance, and range. Let the mean return per dollar invested be r = a^i + SeO — ai)pen6.
Then the variance of return per dollar invested is a 2 = Y.e{"iPi + (i—ai)Pa — r) 2 ne. The
range is [rL,rv] = [min9re, maxece] where r„ = a,p! + (1 —a^pe-
(c) Show that
dr g da 2 & drL g dru § dR S
= 0, = 0, — = 0, and = 0 as — = 0.
dw0 ^ dw0 S dwa ^ dw0 ^ dw S:
That is, the mean, variance, and range of the rate of return on the portfolio as a whole are
increasing, constant, or decreasing functions of wealth as relative risk aversion is a decreasing,
constant, or increasing function of wealth. [Hint: Utilize (b).]
We say that a random variable X with distribution function F is "more variable" than
another Y with distribution G if all risk averters prefer Y to X, i.e. J u(x) dF{x) < $u(y) dG(y).
196 PART II. QUALITATIVE ECONOMIC RESULTS