Adjusted Expected Shortfall

TL;DR

This paper proposes a new class of convex risk measures called adjusted Expected Shortfalls that refine traditional ES by controlling tail risks across different probability levels, allowing tailored risk assessment.

Contribution

It introduces adjusted Expected Shortfalls that incorporate risk profiles to refine tail risk measurement and links them to second-order stochastic dominance.

Findings

Adjusted ES measures can be tailored via risk profiles.

The measures are linked to second-order stochastic dominance.

They provide a more nuanced risk assessment than traditional ES.

Abstract

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position $X$ to ensure that Expected Shortfall $ES_p(X)$ does not exceed a pre-specified threshold $g(p)$ for every probability level $p\in[0,1]$. Through the choice of the benchmark risk profile $g$ one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.