Coherence and elicitability

TL;DR

This paper investigates the elicitability of various law-invariant spectral risk measures, revealing that most are not elicitable except for certain expectiles, impacting how risk forecasts can be verified and compared.

Contribution

It demonstrates that spectral risk measures like expected shortfall are not elicitable unless they simplify to the negative mean, and identifies which law-invariant coherent risk measures are elicitable.

Findings

Expected shortfall is not elicitable unless it reduces to minus the expected value.

Elicitability is limited to certain expectiles among law-invariant coherent risk measures.

Implications for risk forecast verification and comparison methods.

Abstract

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk measures such as value at risk are elicitable. In this paper we show that law-invariant spectral risk measures such as expected shortfall are not elicitable unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.