Epstein-Zin Utility Maximization on a Random Horizon

TL;DR

This paper addresses Epstein-Zin utility maximization on unbounded random horizons in incomplete markets, providing a novel characterization of optimal strategies via backward stochastic differential equations, with significant differences from fixed-horizon cases.

Contribution

It introduces a framework for solving utility maximization on unbounded random horizons without fixed upper bounds, extending prior fixed-horizon models.

Findings

Optimal strategies are characterized by BSDEs with superlinear growth.

Changing from fixed to random horizon significantly impacts optimal strategies.

The model accommodates unbounded, observable, but not necessarily tradable, random horizons.

Abstract

This paper solves the consumption-investment problem under Epstein-Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed-horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies.