STOCHASTIC

INTRODUCTION
In the second part of the book we are concerned with the qualitative analysis
of portfolio choice. The material covered in this section is single-period
portfolio analysis, in which case the investor's goal is to maximize the expected
utility of terminal (end-of-period) wealth. The investor will normally have a
number of options available from which he can freely choose. Typically, he
can invest his wealth in a risk-free asset with known nonnegative rate of
return, and he can invest in a number of risky assets with random rates of
return. It is assumed that the investor knows the joint probability distribution
of returns on the risky assets. The investor can control the fraction of his
wealth to be invested in each of the available assets. In a realistic case there
may be borrowing or short-sale constraints that limit the investor's choice of
control variables. For any given feasible allocation, the return on the investment
will be a random variable, and the investor's problem is to choose an
allocation which maximizes the expected utility of gross portfolio return.
The relevant mathematical tools included in this part are concerned with
stochastic ordering (comparison of random variables) and risk-aversion
measures. Both of these tools will continue to be useful throughout the remainder
of the book. The qualitative analysis of portfolio choice is limited in this
part primarily to the question of the aggregation of risky assets which enable
decentralized decision-making via the separation theorems. This topic is
discussed using both the expected utility and the mean-variance (or safetyfirst)
criteria. A deeper treatment of the quantitative and computational
aspects of portfolio choice is presented in Part III.
I. Stochastic Dominance
The Hanoch and Levy article introduces the idea of stochastic ordering of
random variables. The paper is concerned with the following question: Given
two random variables, when can it be said that the expected utility of one is at
least as great as the expected utility of the other for all utility functions in
some general class? The paper discusses this question for two utility function
classes of relevance to finance: the class of nondecreasing utility functions, and
the class of nondecreasing concave utility functions. It is shown that a necessary
and sufficient condition for ordering with respect to the class of nondecreasing
utility functions is that the cumulative distribution functions of the random
variables be ordered. That is, a random variable X has consistently larger
expected utility than a random variable Y if and only if the distribution
function of X\s everywhere less than or equal to that of Y. This type of ordering
is commonly referred to as first-degree stochastic dominance. The result stated
above is known in the statistics literature, and was proved by Lehmann (1955).
INTRODUCTION 81