Choosing Investment Portfolios When the Returns Have Stable Distributions

Abstract

This chapter 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 Paul A. Samuelson. The algorithm is based on a two-stage decomposition of the problem and is analogous to the procedure that is available for normally distributed investments utilizing John Lintner's reformulation of Tobin's separation theorem. A tradeoff analysis between mean (μ) and dispersion (d) is valid, because an investment choice maximizes expected utility if and only if it lies on a μ-d efficient curve. When a risk-free asset is available, the efficient curve that is a ray in μ-d space can be found 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 that is generally unique provides optimal proportions for the risky assets that are independent of the unspecified concave utility function. Optimal proportions must then be chosen 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.