Non-concave expected utility optimization with uncertain time horizon

TL;DR

This paper investigates optimal investment strategies under non-concave utility functions with uncertain time horizons, establishing conditions for optimality, highlighting limitations of existing methods, and proposing a recursive dynamic programming approach.

Contribution

It introduces a necessary and sufficient condition for optimality in non-concave utility maximization with uncertain horizons and develops a recursive procedure for cases where traditional methods fail.

Findings

Non-concave portfolio distributions are right-skewed with long right tails.

Random time horizon portfolios are multimodal, offering flexibility for investors.

The distribution of the time horizon significantly affects the portfolio's modal structure.

Abstract

We consider an expected utility maximization problem where the utility function is not necessarily concave and the time horizon is uncertain. We establish a necessary and sufficient condition for the optimality for general non-concave utility function in a complete financial market. We show that the general concavification approach of the utility function to deal with non-concavity, while being still applicable when the time horizon is a stopping time with respect to the financial market filtration, leads to sub-optimality when the time horizon is independent of the financial risk, and hence can not be directly applied. For the latter case, we suggest a recursive procedure which is based on the dynamic programming principle. We illustrate our findings by carrying out a multi-period numerical analysis for optimal investment problem under a convex option compensation scheme with random time horizon. We observe that the distribution of the non-concave portfolio in both certain and uncertain random time horizon is right-skewed with a long right tail, indicating that the investor expects frequent small losses and a few large gains from the investment. While the (certain) average time horizon portfolio at a premature stopping date is unimodal, the random time horizon portfolio is multimodal distributed which provides the investor a certain flexibility of switching between the local maximizers, depending on the market performance. The multimodal structure with multiple peaks of different heights can be explained by the concavification procedure, whereas the distribution of the time horizon has significant impact on the amplitude between the modes.