Bayes Risk, Elicitability, and the Expected Shortfall

Mathematical Finance, forthcoming

32 Pages Posted: 24 Nov 2020 Last revised: 23 Feb 2022

See all articles by Paul Embrechts

Paul Embrechts

Swiss Federal Institute of Technology Zurich; Swiss Finance Institute

Tiantian Mao

University of Science and Technology of China (USTC) - Department of Statistics and Finance

Qiuqi Wang

University of Waterloo - Department of Statistics and Actuarial Science

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: October 9, 2020

Abstract

Motivated by recent advances on elicitability of risk measures and practical considerations of risk optimization, we introduce the notions of Bayes pairs and Bayes risk measures. Bayes risk measures are the counterpart of elicitable risk measures, extensively studied in the recent literature. The Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. One of our central results is that under a continuity condition, ES is the only class of coherent Bayes risk measures. We further show that entropic risk measures are the only risk measures which are both elicitable and Bayes. Several other theoretical properties and open questions on Bayes risk measures are discussed.

Keywords: Bayes risk, quantiles, Expected Shortfall, elicitability, entropic risk measures

JEL Classification: C1, G00

Suggested Citation

Embrechts, Paul and Mao, Tiantian and Wang, Qiuqi and Wang, Ruodu, Bayes Risk, Elicitability, and the Expected Shortfall (October 9, 2020). Mathematical Finance, forthcoming, Available at SSRN: https://ssrn.com/abstract=3708379 or http://dx.doi.org/10.2139/ssrn.3708379

Paul Embrechts

Swiss Federal Institute of Technology Zurich ( email )

ETH-Zentrum
CH-8092 Zurich
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Tiantian Mao (Contact Author)

University of Science and Technology of China (USTC) - Department of Statistics and Finance ( email )

96, Jinzhai Road
Hefei, Anhui 230026
China

Qiuqi Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

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