The Robust Merton Problem of an Ambiguity Averse Investor

TL;DR

This paper derives a closed-form portfolio optimization rule for ambiguity-averse investors with CRRA utility, using ellipsoidal uncertainty sets for mean returns and a worst-case approach based on a max-min HJB equation.

Contribution

It introduces a novel ellipsoidal uncertainty model for mean returns and derives a simple, explicit optimal portfolio policy extending the classical Merton problem.

Findings

Optimal portfolio depends on a rescaled market Sharpe ratio under worst-case volatility.

Provides a closed-form solution for ambiguity-averse investors with CRRA utility.

Extends the classical Merton problem to incorporate ambiguity aversion through a max-min PDE.

Abstract

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simple and concise analysis, as the optimal portfolio allocation policy is shaped by a rescaled market Sharpe ratio, computed under the worst case volatility. The result is based on a max-min Hamilton-Jacobi-Bellman-Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.