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Epstein-Zin Utility Maximization on a Random Horizon

Joshua Aurand, Yu-Jui Huang

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.

Epstein-Zin Utility Maximization on a Random Horizon

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.

Paper Structure

This paper contains 20 sections, 11 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Epstein-Zin Preferences on a Random Horizon
  3. The Consumption-Investment Problem
  4. The Markovian Setup
  5. The Ansatz
  6. Verification
  7. Examples
  8. The Default Problem: Bounded Time Horizon
  9. The Exit Problem: Unbounded Time Horizon
  10. Proofs for Section \ref{['EZ']}
  11. Proof of Proposition \ref{['P1']}
  12. Proof of Theorem \ref{['thm:EZ exists']}
  13. Proofs for Section \ref{['CI']}
  14. Derivation of Proposition \ref{['P5']}
  15. Derivation of Lemma \ref{['C4']}
  16. ...and 5 more sections

Key Result

Proposition 2.1

Suppose $\gamma,\psi>1$ and $c\in\mathcal{C}$. Then, EQ5 admits a unique solution $(Y,Z)$ in $\mathcal{B}_{2}$ with $Y\ge 0$ a.s.

Figures (3)

  • Figure 1: Optimal consumption and investment ratios at $t=0$ versus firm value. The upper panel fixes $\gamma=2$ and changes $\psi$; the lower one fixes $\psi=1.5$ and changes $\gamma$. The solid curves are computed via \ref{['optimal_u']}; the dotted lines are benchmark levels in the no-default case, computed via \ref{['optimal_no default']}. The upper-right plot has only one dotted line because the no-default level $\frac{\alpha-r}{\gamma\sigma^2}$ is independent of $\psi$.
  • Figure 2: Optimal strategies on the state space of $(Y_t,\mathcal{W}_t)$ (for $\varepsilon=0$ and $L=0.02$).
  • Figure 3: Optimal strategies on the state space of $(Y_t,\mathcal{W}_t)$ (for $\varepsilon =0.05$ and $L=0.02$, $0.08$).

Theorems & Definitions (35)

  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1
  • Remark 2.4
  • Remark 3.1
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 25 more