Distortion Risk Measures and Elicitability

TL;DR

This paper explores the mathematical properties of distortion risk measures, providing new characterizations and explaining why only certain measures like VaR are elicitable, which is important for statistical forecasting.

Contribution

It offers a novel, concise proof of the characterization of elicitable distortion risk measures and clarifies the conflict between elicitability and comonotonic additivity.

Findings

Only Value-at-Risk and the mean are elicitable among distortion risk measures.

A new axiomatic characterization of distortion risk measures is provided.

The paper explains the mathematical conflict between elicitability and comonotonic additivity.

Abstract

We discuss equivalent axiomatic characterizations of distortion risk measures, and give a novel and concise proof of the characterization of elicitable distortion risk measures. Elicitability has recently been discussed as a desirable criterion for risk measures, motivated by statistical considerations of forecasting. We reveal the mathematical conflict between the requirements of elicitability and comonotonic additivity which intuitively explains why only Value-at-Risk and the mean are elicitable distortion risk measures in a general sense.