COHERENCE AND ELICITABILITY

TLDR

Risk of a financial position is usually summarized by a risk measure, and in statistical decision theory, risk measures that allow verification and comparison are called elicitable; quantile‑based measures such as value‑at‑risk are elicitable, while the class of elicitable law‑invariant coherent risk measures does not reduce to minus the expected value. The study aims to extend the known non‑elicitability of expected shortfall to all law‑invariant spectral risk measures unless they reduce to minus the expected value, highlighting the importance of verifying and comparing estimation procedures from historical data. The authors extend the non‑elicitability result to all law‑invariant spectral risk measures except those equivalent to minus the expected value. The extension shows that forecast verification or comparison is unclear, but the class of elicitable law‑invariant coherent risk measures consists of certain expectiles.

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, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures 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.