STOCHASTIC

Choosing Investment Portfolios When the Returns
Have Stable Distributions* st
W. T. Ziemba
THE UNIVERSITY OF BRITISH COLUMBIA
This paper presents an efficient method for computing approximately
optimal portfolios when the returns have symmetric stable distributions and
there are many alternative investments. The procedure is valid, in particular,
for independent investments and for multivariate investments of the classes
introduced by Press and Samuelson. The algorithm is based on a two-stage
decomposition of the problem and is analogous to the procedure developed
by the author that is available for normally distributed investments utilizing
Lintner's reformulation of Tobin's separation theorem.
A tradeoff analysis between mean (/i) and dispersion (d) is valid since an
investment choice maximizes expected utility if and only if it lies on a \i-d
efficient curve. When a risk-free asset is available one may find the efficient
curve, which is a ray in fi-d space, by solving a fractional program. The
fractional program always has a pseudo-concave objective function and hence
may be solved by standard nonlinear programming algorithms. Its solution,
which is generally unique, provides optimal proportions for the risky assets
that are independent of the unspecified concave utility function. One must
then choose optimal proportions between the risk-free asset and a risky
composite asset utilizing a given utility function. The composite asset is stable
and consists of a sum of the random investments weighted by the optimal
proportions. This problem is a stochastic program having one random variable
and one decision variable. Symmetric stable distributions have known closedform
densities only when a, the characteristic exponent, is \ (the arc sine),
1 (the Cauchy), or 2 (the normal). Hence, there is no apparent algorithm that
will solve the stochastic program for general 1 < a < 2. However, one may
obtain a reasonably accurate approximate solution to this program utilizing
tables recently compiled by Fama and Roll. Standard nonlinear programming
* Presented by invitation at the NATO Advanced Study Institute on Mathematical
Programming in Theory and Practice, Figueira da Foz, Portugal, June 12-23, 1972. This
research was partially supported by the National Research Council of Canada Grant
NRC-A7-7147, the Samuel Bronfman Foundation, The Graduate School of Business,
Stanford University, and Atomic Energy Commission, Grant AT 04-3-326-PA #18.
t Reprinted from Mathematical Programming in Theory and Practice, P. L. Hammer and
G. Zoutendijk, eds. North-Holland Publishing Company, 1974.
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 243