Relative Bound and Asymptotic Comparison of Expectile with Respect to Expected Shortfall

TL;DR

This paper explores the relationship between expectile and expected shortfall, deriving bounds, characterizations, and asymptotic behaviors, and compares their estimation requirements, especially for heavy-tailed distributions at high confidence levels.

Contribution

It establishes bounds and characterizations of expectile in relation to expected shortfall, and analyzes their asymptotic behavior and estimation properties for heavy tails.

Findings

Expectile bounds are derived in terms of expected shortfall.

Asymptotic behavior of expectile is characterized as confidence level approaches 1.

Heavy tail distributions require larger sample sizes for accurate risk estimation.

Abstract

Expectile bears some interesting properties in comparison to the industry wide expected shortfall in terms of assessment of tail risk. We study the relationship between expectile and expected shortfall using duality results and the link to optimized certainty equivalent. Lower and upper bounds of expectile are derived in terms of expected shortfall as well as a characterization of expectile in terms of expected shortfall. Further, we study the asymptotic behavior of expectile with respect to expected shortfall as the confidence level goes to $1$ in terms of extreme value distributions. We use concentration inequalities to illustrate that the estimation of value at risk requires larger sample size than expected shortfall and expectile for heavy tail distributions when $\alpha$ is close to $1$. Illustrating the formulation of expectile in terms of expected shortfall, we also provide explicit or semi-explicit expressions of expectile and some simulation results for some classical distributions.