Feasibility of Portfolio Optimization under Coherent Risk Measures

TL;DR

This paper demonstrates that portfolio optimization under coherent risk measures can become unstable when assets dominate others, leading to diverging risk assessments, especially illustrated through Expected Shortfall.

Contribution

It reveals the inherent instability in portfolio optimization using coherent risk measures, generalizing the issue beyond Expected Shortfall.

Findings

Portfolio risk diverges to minus infinity with dominating assets.

Instability occurs even in large samples with finite probability.

Expected Shortfall exemplifies this divergence phenomenon.

Abstract

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered on the special example of Expected Shortfall which is used here both as an illustration and as a prompt for generalization.