Coherence and Elicitability

18 Pages Posted: 20 Sep 2016

Date Written: October 2016


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.

Keywords: coherent risk measures, decision theory, elicitability, expected shortfall, expectiles, law‐invariant risk measures, spectral risk measures

Suggested Citation

Ziegel, Johanna, Coherence and Elicitability (October 2016). Mathematical Finance, Vol. 26, Issue 4, pp. 901-918, 2016, Available at SSRN: or

Johanna Ziegel (Contact Author)

University of Bern

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