STOCHASTIC

proportions to maximize expected utility. A modification of the Frank-Wolfe
method is developed in Exercise ME-25 to solve the problem. The objective
has the required concavity and differentiability properties to guarantee
convergence, and the algorithm should be quite efficient because the directionfinding
problem may be solved in a trivial way in each iteration.
It is important to know when investors will diversify their holdings. As is
shown in Exercise CR-10, it is sometimes optimal in mean-variance models to
invest entirely in one asset even if all assets have equal mean returns. Samuelson
considers quite general diversification problems in his second paper. He
utilizes an expected utility framework and assumes that the utility function is
strictly increasing, strictly concave and differentiable. The random investments
are assumed to possess equal means and positive but finite variances.
A natural question to pose is when is it desirable to invest equally in all
securities? Samuelson shows that such an allocation is optimal if the investments
have identical independent distributions or, more generally, if they have
nontrivial symmetric distribution functions. He also shows that equal diversification
will generally not be optimal unless some type of symmetry is present.
The finiteness assumption on the variances is crucial for equal allocation to
be the unique optimal solution. Indeed, as is shown in Exercise CR-8, all
possible allocations between independent identically distributed Cauchy
investments are equally good when expected utility is infinite. In fact for
symmetric distributions, an equal allocation policy is optimal but generally
not unique regardless of the finiteness of the means and variances.
Perhaps it is more basic to know when a particular investment will form a
part of the optimal portfolio. Utilizing the same framework Samuelson shows
that an investment whose return is distributed independently of the returns of
the remaining investments will always be purchased if its mean is not exceeded
by any of the remaining investments. Indeed if all investments have a common
mean and have independent distributions with finite positive variances, they
will enter positively in the optimal portfolio. When a mean-variance or meandispersion
analysis is optimal, the investment proportions are inversely related
to the variances or dispersions, respectively (Exercise CR-7). However, unless
the strict independence assumption holds, it is not true that every investment
in a group with equal means must enter the optimal portfolio.
Although it is natural to suppose that negative interdependence improves
the case for diversification, the appropriate measure of interdependence is not
clear. For example, if all investments have a common mean and have negative
correlations, then all enter into the optimal portfolio if the utility function is
quadratic or if the investments have a normal distribution. However, negative
correlation is neither necessary nor sufficient for complete or even any diversification
for general strictly concave functions (Brumelle, 1974) (Exercise
ME-27). In the two-asset case a concept of negative dependence that leads to
210 PART III STATIC PORTFOLIO SELECTION MODELS