STOCHASTIC

diversification is that both conditional means are strictly decreasing (Exercise
ME-28). Diversification is also optimal if the conditional cumulative distribution
functions are strictly increasing, as shown in Exercise CR-28. This is
the opposite of the condition in Samuelson's Theorem IV (so that the sign
in the expression above Theorem IV should be reversed). The reader is asked
to investigate the diversification effects of a three-asset problem in Exercise
CR-27. Hadar and Russell (1971, 1974) show how to derive the Samuelson
results and some extensions using stochastic dominance concepts.
An important diversification problem arises in the study of optimal financial
intermediation. Financial intermediaries issue claims on themselves and then
use the proceeds to purchase other assets. Hence it is of interest to determine
when the firm will sell deposits (short) and purchase loans (long). Exercise
ME-6 considers this problem when there is also a risk-free asset. If the random
investments are independent, there is intermediation if the mean loan return
exceeds the risk-free rate, which in turn exceeds the mean deposit return rate.
Such an ordering is sufficient but not necessary if the investments have increasing
conditional means. If a mean-variance approach is optimal, one can
determine the optimal allocations explicitly in terms of means and variances.
Exercise ME-7 develops these results and additional sharper conditions for
intermediation.
As an alternative to the maximization of the expected utility criterion, an
investor may wish to minimize the expected disutility of his regret. Given a
particular realization of the random returns, the regret of a particular decision
is the maximum return that could have been obtained given that realization
minus the return obtained with the given decision. Exercise ME-11 is concerned
with diversification effects involved in such a formulation when the disutility
is a strictly monotic and strictly increasing function of regret. If the distribution
of returns is symmetric, then equal investment in each security is the unique
minimax strategy and it also is the unique minimizer of expected disutility of
regret. In the two-asset case it is always optimal to diversify if one utilizes
the disutility approach and if the mean returns are equal regardless of the
joint distribution of returns. In Pye's paper in Part V it is shown how this
criterion can be used to explain the phenomenon of dollar cost averaging.
Specification requirements escalate extremely rapidly as one attempts to
determine the joint distribution function when there are more and more
securities in an investor's "universe." Hence an important concern of many
investors relates to the choice of a proper size of the "universe" in order to
provide a satisfactory level of diversification and expected utility. Exercise
CR-15 develops a mean-variance-inspired method to experiment with
different-sized security universes in a way that is computationally feasible.
Investment returns are often made under limited liability conditions in the
sense that the investor can lose no more than his initial outlay. Hence gross
INTRODUCTION 211