A Remark on the Structure of Expectiles

TL;DR

This paper explores the mathematical structure of expectiles, a class of coherent risk measures, by analyzing their scenario sets and identifying a most severe commonotonic risk measure that is dominated by them.

Contribution

It provides a detailed description of the scenario set of expectiles and introduces the concept of a most severe commonotonic risk measure related to expectiles.

Findings

Identified the scenario set of expectiles.

Established the existence of a most severe commonotonic risk measure.

Showed that this risk measure is smaller than the expectile.

Abstract

Expectiles were defined using a minimisation principle. They form a special class of coherent risk measures. We will describe the scenario set and we will show that there is a most severe commonotonic risk measure that is smaller than the given expectile.