Seven Proofs for the Subadditivity of Expected Shortfall

Embrechts Paul and Wang Ruodu
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Embrechts Paul: RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Wang Ruodu: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada

Dependence Modeling, 2015, vol. 3, issue 1, 15

Abstract: Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.

Keywords: Expected Shortfall, TVaR, subadditivity, comonotonicity, Value-at-Risk, risk management, education, Primary: 28A25; secondary: 60E15, 91B06, Expected Shortfall, TVaR, subadditivity, comonotonicity, Value-at-Risk, risk management, education, Primary: 28A25; secondary: 60E15, 91B06 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:3:y:2015:i:1:p:15:n:9

DOI: 10.1515/demo-2015-0009

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