Expected Shortfall and Beyond

TL;DR

This paper examines Expected Shortfall as a coherent risk measure, highlighting its advantages over Value-at-Risk, and proposes methods for risk attribution to portfolio components, including extensions to higher moments.

Contribution

It characterizes Expected Shortfall as a coherent risk measure, explores its properties, and introduces a general approach for attributing risk contributions within portfolios.

Findings

Expected Shortfall is the smallest coherent risk measure dominating VaR.

Expected Shortfall satisfies properties like coherence and law invariance.

A method for attributing risk contributions to portfolio components is proposed.

Abstract

Financial institutions have to allocate so-called "economic capital" in order to guarantee solvency to their clients and counter parties. Mathematically speaking, any methodology of allocating capital is a "risk measure", i.e. a function mapping random variables to the real numbers. Nowadays "value-at-risk", which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not "subadditive". In the search for a suitable alternative to value-at-risk, "Expected Shortfall" (or "conditional value-at-risk" or "tail value-at-risk") has been characterized as the smallest "coherent" and "law invariant" risk measure to dominate value-at-risk. We discuss these and some other properties of Expected Shortfall as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute Expected Shortfall "risk contributions" to portfolio components. Key words: Expected Shortfall; Value-at-Risk; Spectral Risk Measure; coherence; risk contribution.