On Quantile Risk Measures and Their Domain

TL;DR

This paper explores quantile risk measures, their domain, and their relation to spectral and distortion risk measures, providing theoretical insights and results, especially regarding the expected shortfall.

Contribution

It introduces a general framework for quantile risk measures, extending spectral risk measures, and analyzes their domain and properties, including the expected shortfall.

Findings

Quantile risk measures cannot attain +infinity within their domain.

The domain of quantile risk measures is characterized and constrained.

Expected shortfall is examined within this framework.

Abstract

In the present paper we study quantile risk measures and their domain. Our starting point is that, for a probability measure $ Q $ on the open unit interval and a wide class $ \mathcal{L}_Q $ of random variables, we define the quantile risk measure $ \varrho_Q $ as the map which integrates the quantile function of a random variable in $ \mathcal{L}_Q $ with respect to $ Q $. The definition of $ \mathcal{L}_Q $ ensures that $ \varrho_Q $ cannot attain the value $ +\infty $ and cannot be extended beyond $ \mathcal{L}_Q $ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view at the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall.