STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
univariate stable it is shown here that the fi-d analysis is valid as long as the
mean vector exists.
The calculation of the y.-d curve is generally quite difficult because one must
solve a parametric concave program. However, when a risk-free asset exists
the efficient surface is a ray in \i-d space and a generalization of Tobin's
separation theorem [31] obtains. One may then calculate the optimal proportions
of the risky assets by solving a fractional program. The character of the
fractional program depends, of course, on the assumptions made about the
joint distribution of the stable random variables. However, in fairly general
circumstances the fractional program has a pseudo-concave objective function
and hence may be solved via a standard nonlinear programming algorithm.
Typically the optimal solution is unique. This calculation comprises stage 1
of a two-stage procedure that will efficiently solve the portfolio problem when
there are many random investments. In the second stage one introduces the
investor's utility function and an optimal ratio between a stable composite
asset and the risk-free asset must be chosen. The composite asset is a sum of
the random investments weighted by the optimal proportions found in stage 1.
The problem to solve is a stochastic program having a single random variable
and a single decision variable. Such problems are generally easy to solve if
the density of the random variable is known (as it is in the normal distribution
case). Unfortunately, the density of the stable composite asset is known only
in a few special cases. However, Fama and Roll [7], using series approximations
due to Bergstrom, have tabulated, at discrete points, the density and
cumulative distributions of a standardized symmetric stable distribution.
These tables may be used to obtain a very good nonlinear programming
approximation to the stochastic program. The solution of the nonlinear
program generally provides a good approximation to the optimal solution of
the stochastic program and hence of the portfolio problem.
Section II discusses the case when the random returns are independent.
Some sufficient conditions for the expected utility and expected marginal utility
to be bounded are given. The fractional program in this case is shown to have
a strictly pseudo-concave objective, and it has a unique solution. The nonlinear
programming approximation of the stochastic program is also described.
Section III considers a class of multivariate stable distributions introduced by
Press [22, 23]. This class generalizes the independent case and decomposes the
dependence of the random variables into several independent subsets. Thus
partial and full dependence may be handled in a convenient fashion. The
optimization procedure described in Section II may be utilized for this class
and a class of multivariate stable random variables introduced by Samuelson
[29] as well. Section IV shows how one may find a portfolio that approximately
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 245