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Seven Proofs for the Subadditivity of Expected Shortfall

Paul Embrechts / Ruodu Wang
  • Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada
  • Other articles by this author:
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Published Online: 2015-10-16 | DOI: https://doi.org/10.1515/demo-2015-0009


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

MSC:: Primary: 28A25; secondary: 60E15, 91B06


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About the article

Received: 2015-08-15

Accepted: 2015-10-07

Published Online: 2015-10-16

Citation Information: Dependence Modeling, Volume 3, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2015-0009.

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© 2015 Paul Embrechts and Ruodu Wang. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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