Robustness in the Optimization of Risk Measures

TL;DR

This paper investigates the robustness of risk measure optimization, introducing a new framework to assess whether solutions are stable against small perturbations, with a focus on VaR and Expected Shortfall in financial contexts.

Contribution

It develops a general methodology for evaluating robustness in risk measure optimization and compares the robustness of VaR and Expected Shortfall, providing new insights for financial regulation.

Findings

VaR leads to non-robust optimizers in general.

Convex risk measures tend to produce robust solutions.

The new robustness notion differs from traditional robust optimization.

Abstract

We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk measurement related optimization problem is robust, which we call "robustness against optimization". The new notion is studied for various classes of risk measures and expected utility and loss functions. Motivated by practical issues from financial regulation, special attention is given to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We establish that for a class of general optimization problems, VaR leads to non-robust optimizers whereas convex risk measures generally lead to robust ones. Our results offer extra insight on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are derived.