Exponential utility maximization in small/large financial markets

TL;DR

This paper derives closed-form solutions for exponential utility maximization in markets with normal mean-variance mixture returns, demonstrating convergence of small market utilities to large market limits and providing a simplified numerical approach for general utilities.

Contribution

It introduces explicit formulas for optimal portfolios in markets with non-normal return distributions and shows utility convergence in large markets, also simplifying utility maximization for general utility functions.

Findings

Closed-form solutions for exponential utility in normal mean-variance mixture markets.

Demonstration that small market utilities converge to large market utility.

A reduced-dimensional numerical method for general utility maximization.

Abstract

Obtaining utility maximizing optimal portfolios in closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this note, we give closed form expressions, in markets based on finitely many assets, for optimal portfolios that maximize the expected exponential utility when the return vector follows normal mean-variance mixture models. We then consider large financial markets based on normal mean-variance mixture models also and show that, under exponential utility, the optimal utilities based on small markets converge to the optimal utility in the large financial market. This result shows, in particular, that to reach optimal utility level investors need to diversify their portfolios to include infinitely many assets into their portfolio and with portfolios based on any set of only finitely many assets, they never be able to reach optimum level of utility. In this paper, we also consider portfolio optimization problems with more general class of utility functions and provide an easy-to-implement numerical procedure for locating optimal portfolios. Especially, our approach in this part of the paper reduces a high dimensional problem in locating optimal portfolio into a three dimensional problem for a general class of utility functions.