Computing Discrete Expected Utility Maximizing Portfolios

Abstract

Maximizing the expected logarithmic utility, or equivalently, the geometric mean, of a portfolio is a well-known yet controversially discussed objective. Nonetheless, it is an often-used objective function for computing real-world portfolios, and it has generated a great amount of interest in the alternative investment industry, in particular. In the purely continuous case, the resulting portfolio optimization problem can be solved using methods from convex optimization rather efficiently. However, in reality, we often face discrete decisions, e.g., setting up a new venture, or engaging in acquisitions and mergers, where approximation by a continuous variable is inappropriate. We focus on how to solve the utility maximization problem in the presence of discrete decisions, allowing for the inclusion of transaction costs and fixed charges. We demonstrate the efficiency of our approach by computational experiments for large-scale portfolios showing the applicability, and we present a brief backtesting experiment conducted for the German stock market as well as the U.S. stock market. Our approach generalizes to other utility functions satisfying some mild requirements.