Expectile Quadrangle and Applications

TL;DR

This paper investigates the properties of Expectile Quadrangles within the Fundamental Risk Quadrangle framework, emphasizing their role as both a statistic and a risk measure, and compares them to VaR and CVaR-based quadrangles.

Contribution

It provides a rigorous analysis of Expectile Quadrangles, highlighting their theoretical properties and potential applications in risk management.

Findings

Expectile Quadrangles extend the FRQ framework.

They share properties with VaR and CVaR quadrangles.

The paper clarifies the dual role of expectile as statistic and risk measure.

Abstract

The paper explores the concept of the \emph{expectile risk measure} within the framework of the Fundamental Risk Quadrangle (FRQ) theory. According to the FRQ theory, a quadrangle comprises four stochastic functions associated with a random variable: ``error'', ``regret'', ``risk'', and ``deviation''. These functions are interconnected through a stochastic function known as the ``statistic''. Expectile is a risk measure that, similar to VaR (quantile) and CVaR (superquantile), can be employed in risk management. While quadrangles based on VaR and CVaR statistics are well-established and widely used, the paper focuses on the recently proposed quadrangles based on expectile. The aim of this paper is to rigorously examine the properties of these Expectile Quadrangles, with particular emphasis on a quadrangle that encompasses expectile as both a statistic and a measure of risk.