Seven Proofs for the Subadditivity of Expected Shortfall

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.

[1]  Jan Dhaene,et al.  Risk Measures and Comonotonicity: A Review , 2006, Stochastic Models.

[2]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[3]  Paul Embrechts,et al.  A note on generalized inverses , 2013, Math. Methods Oper. Res..

[4]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[5]  Shaun S. Wang,et al.  Axiomatic characterization of insurance prices , 1997 .

[6]  Jan Dhaene,et al.  Modern Actuarial Risk Theory , 2001 .

[7]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[8]  L. Rüschendorf.  Mathematical Risk Analysis , 2013 .

[9]  C. Goodhart,et al.  An academic response to Basel II , 2001 .

[10]  Alexander J. McNeil,et al.  Quantitative Risk Management: Concepts, Techniques and Tools Revised edition , 2015 .

[11]  W. R. van Zwet,et al.  A Strong Law for Linear Functions of Order Statistics , 1980 .

[12]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[13]  J. Norris.  Appendix: probability and measure , 1997 .

[14]  P. Embrechts,et al.  An Academic Response to Basel 3.5 , 2014 .

[15]  Qihe Tang,et al.  Some New Classes of Consistent Risk Measures , 2004 .

[16]  Jan Dhaene,et al.  Modern Actuarial Risk Theory: Using R , 2008 .

[17]  Isaac Meilijson,et al.  Convex majorization with an application to the length of critical paths , 1979, Journal of Applied Probability.

[18]  Haim Levy,et al.  Ordering Uncertain Options with Borrowing and Lending , 1978 .

[19]  J. Dhaene,et al.  Can a Coherent Risk Measure Be Too Subadditive? , 2008 .

[20]  Ludger Rüschendorf,et al.  Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios , 2013 .

[21]  Abaxbank,et al.  Spectral Measures of Risk : a Coherent Representation of Subjective Risk Aversion , 2002 .

[22]  Rama Cont,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2010 .

[23]  Jon A. Wellner,et al.  A Glivenko-Cantelli Theorem and Strong Laws of Large Numbers for Functions of Order Statistics , 1977 .

[24]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[25]  M. Fréchet.  Sur les tableaux de correlation dont les marges sont donnees , 1951 .

[26]  Freddy Delbaen,et al.  Monetary utility functions , 2012 .

[27]  Romain Deguest,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2008 .

[28]  Alexander J. McNeil,et al.  Quantitative Risk Management: Concepts, Techniques and Tools : Concepts, Techniques and Tools , 2015 .

[29]  Jan Dhaene,et al.  Comonotonicity, correlation order and premium principles , 1998 .

[30]  Sarah Eichmann,et al.  Non Additive Measure And Integral , 2016 .

[31]  Lei Si Ni Ke Resnick.S.I..  Extreme values. regular variation. and point processes , 2011 .

[32]  Stochastic Orders , 2008 .

[33]  P. Embrechts,et al.  Quantitative Risk Management: Concepts, Techniques, and Tools , 2005 .

[34]  D. Tasche,et al.  On the coherence of expected shortfall , 2001, cond-mat/0104295.

[35]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[36]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[37]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[38]  Jan Dhaene,et al.  The Concept of Comonotonicity in Actuarial Science and Finance: Theory , 2002, Insurance: Mathematics and Economics.

[39]  G. Choquet.  Theory of capacities , 1954 .