A note on robust convex risk measures

TL;DR

This paper refines and generalizes formulas for worst-case convex risk measures under various uncertainty sets, providing explicit solutions and analyzing their impact on financial decision-making through numerical simulations.

Contribution

It introduces generalized closed-form solutions for robust convex risk measures with new uncertainty set characterizations and analyzes their practical implications.

Findings

Explicit closed forms for risk measures under Wasserstein and moment constraints

Characterization of argmax in worst-case risk problems

Numerical insights into robustness effects on capital and portfolio optimization

Abstract

In this paper, we refine and generalize closed forms for worst-case law invariant convex risk measures with uncertainty sets based on: i) closed balls under $p$-norms and Wasserstein distance; and ii) moment constraints involving mean and variance. We also characterize the argmax of the worst-case problem in both settings. From such general results, we illustrate our framework by developing explicit closed forms for concrete examples of convex risk measures. Furthermore, we use extensive numerical simulations in order to assess the impact of robustness on capital determination and portfolio optimization.