Robust Markowitz mean-variance portfolio selection under ambiguous covariance matrix *

TL;DR

This paper develops a robust continuous-time Markowitz portfolio model accounting for covariance matrix ambiguity, providing explicit solutions and analyzing the impact on the efficient frontier and Sharpe ratios.

Contribution

It introduces a McKean-Vlasov dynamic programming approach to solve a min-max mean-variance problem with ambiguous covariance matrices, offering explicit strategies and bounds.

Findings

Explicit robust portfolio strategies derived

Closed-form robust efficient frontier obtained

Lower bound for Sharpe ratio established

Abstract

This paper studies a robust continuous-time Markowitz portfolio selection pro\-blem where the model uncertainty carries on the covariance matrix of multiple risky assets. This problem is formulated into a min-max mean-variance problem over a set of non-dominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed-form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. MSC Classification: 91G10, 91G80, 60H30