Non-Concave Utility Maximization with Transaction Costs

TL;DR

This paper develops a theoretical framework for portfolio optimization with non-concave utility and transaction costs, revealing how such costs influence optimal risky asset holdings at maturity.

Contribution

It introduces a novel approach using discontinuous viscosity solutions to analyze non-concave utility maximization with transaction costs, overcoming limitations of the concavification principle.

Findings

Transaction costs can lead to larger long positions in risky assets.

Transaction costs can cause investors to hold large short positions despite positive risk premiums.

The framework provides insights into optimal trading behavior near maturity.

Abstract

This paper studies a finite-horizon portfolio selection problem with non-concave terminal utility and proportional transaction costs, in which the commonly used concavification principle for terminal value is no longer applicable. We establish a proper theoretical characterization of this problem via a two-step procedure. First, we examine the asymptotic terminal behavior of the value function, which implies that any transaction close to maturity only provides a marginal contribution to the utility. Second, we establish the theoretical foundation in terms of the discontinuous viscosity solution, incorporating the proper characterization of the terminal condition. Via extensive numerical analyses involving several types of utility functions, we find that the introduction of transaction costs into non-concave utility maximization problems can make it optimal for investors to hold on to a larger long position in the risky asset compared to the frictionless case, or hold on to a large short position in the risky asset despite a positive risk premium.