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Seven Proofs for the Subadditivity of Expected Shortfall

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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)

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.

Suggested Citation

  • Embrechts Paul & Wang Ruodu, 2015. "Seven Proofs for the Subadditivity of Expected Shortfall," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-15, October.
    • Handle: RePEc:vrs:demode:v:3:y:2015:i:1:p:15:n:9
    DOI: 10.1515/demo-2015-0009
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    References listed on IDEAS

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    Cited by:

    1. Engsner Hampus & Lindskog Filip, 2020. "Continuous-time limits of multi-period cost-of-capital margins," Statistics & Risk Modeling, De Gruyter, vol. 37(3-4), pages 79-106, July.
    2. Sebastian Fuchs & Ruben Schlotter & Klaus D. Schmidt, 2017. "A Review and Some Complements on Quantile Risk Measures and Their Domain," Risks, MDPI, vol. 5(4), pages 1-16, November.

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