Probability equivalent level of Value at Risk and higher-order Expected Shortfalls

TL;DR

This paper introduces and analyzes the probability equivalent level of the $n^{th}$-order Expected Shortfall (PEVEn), exploring its properties, calculations for specific distributions, asymptotic behavior, and its relation to Gini Shortfall.

Contribution

It extends the concept of PELVE to higher-order Expected Shortfalls, providing theoretical properties, explicit calculations, and asymptotic analysis.

Findings

PEVEn has well-defined finiteness and uniqueness properties.

Explicit PELVE_n values are computed for notable distributions.

Asymptotic behavior of PELVE_2 is characterized for regularly varying distributions.

Abstract

We investigate the probability equivalent level of Value at Risk and $n^{\mathrm{th}}$-order Expected Shortfall (called PELVE_n), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2022). We study the finiteness, uniqueness and several properties of PELVE_n, we calculate PELVE_n of some notable distributions, PELVE_2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE_2 of regularly varying distributions as the level tends to $0$. Some properties of $n^{\mathrm{th}}$-order Expected Shortfall are also investigated. Among others, it turns out that the Gini Shortfall at some level $p\in[0,1)$ corresponding to a (loading) parameter $\lambda\geq 0$ is the linear combination of the Expected Shortfall at level $p$ and the $2^{\mathrm{nd}}$-order Expected Shortfall at level $p$ with coefficients $1-2\lambda$ and $2\lambda$, respectively.