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Dependence Modeling

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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:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-10-16 | DOI: https://doi.org/10.1515/demo-2015-0009

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

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

References

  • [1] Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. J. Bank. Finance, 26(7), 1505–1518. Google Scholar

  • [2] Acerbi, C. and Tasche, D. (2002). On the coherence of expected shortfall. J. Bank. Finance, 26(7), 1487–1503. Google Scholar

  • [3] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance, 9(3), 203–228. CrossrefGoogle Scholar

  • [4] BCBS (2012). Consultative Document May 2012. Fundamental review of the trading book. Basel Committee on Banking Supervision. Basel: Bank for International Settlements. Google Scholar

  • [5] BCBS (2013). Consultative Document October 2013. Fundamental reviewof the trading book: A revisedmarket risk framework. Basel Committee on Banking Supervision. Basel: Bank for International Settlements. Google Scholar

  • [6] BCBS (2014). Consultative Document December 2014. Fundamental review of the trading book: Outstanding issues. Basel Committee on Banking Supervision. Basel: Bank for International Settlements. Google Scholar

  • [7] Billingsley, P. (1995). Probability and Measure. Third Edition. Wiley. Google Scholar

  • [8] Choquet, G. (1953). Theory of capacities. Ann. Inst. Fourier, 5, 121–293. Google Scholar

  • [9] Cont, R., Deguest, R. and Scandolo, G. (2010). Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance, 10(6), 593–606. Web of ScienceCrossrefGoogle Scholar

  • [10] Delbaen, F. (2012). Monetary Utility Functions. Osaka University Press. Google Scholar

  • [11] Denneberg, D. (1994). Non-additive Measure and Integral. Springer. Google Scholar

  • [12] Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, R. (2005). Actuarial Theory for Dependent Risks. Wiley. Google Scholar

  • [13] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vynche, D. (2002). The concept of comonotonicity in actuarial science and finance: Theory. Insur. Math. Econ. 31(1), 3-33. CrossrefGoogle Scholar

  • [14] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Tang, Q. and Vynche, D. (2006). Risk measures and comonotonicity: a review. Stoch. Models, 22, 573–606. Google Scholar

  • [15] Dhaene, J., Laeven, R. J., Vanduffel, S., Darkiewicz, G. and Goovaerts, M. J. (2008). Can a coherent risk measure be too subadditive? J. Risk Insur., 75(2), 365–386. Web of ScienceGoogle Scholar

  • [16] Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Oper. Res., 77(3), 423–432. Google Scholar

  • [17] Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014). An academic response to Basel 3.5. Risks, 2(1), 25-48. Google Scholar

  • [18] Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Third Edition. Google Scholar

  • [19] Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon. Sect. A., 14, 53–77. Google Scholar

  • [20] Goovaerts, M. J., Kaas, R., Dhaene, J. and Tang, Q. (2004). Some new classes of consistent risk measures. Insur.Math. Econ., 34(3), 505–516. CrossrefGoogle Scholar

  • [21] Hoeffding, W. (1940). Massstabinvariante Korrelationstheorie. Schriften Math. Inst. Univ. Berlin, 5(5), 181–233. Google Scholar

  • [22] Huber, P. J. (1980). Robust Statistics. First ed., Wiley Series in Probability and Statistics. Wiley, New Jersey. Google Scholar

  • [23] Huber, P. J. and Ronchetti E. M. (2009). Robust Statistics. Second ed., Wiley Series in Probability and Statistics. Wiley, New Jersey. First ed.: Huber, P. (1980). Google Scholar

  • [24] IAIS (2014). Consultation Document December 2014. Risk-based global insurance capital standard. International Association of Insurance Supervisors. Google Scholar

  • [25] Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008). Modern Actuarial Risk Theory: Using R. Springer. Google Scholar

  • [26] Kusuoka, S. (2001). On law invariant coherent risk measures. Adv. Math. Econ., 3, 83–95. CrossrefGoogle Scholar

  • [27] Levy, H. and Kroll, Y. (1978). Ordering uncertain options with borrowing and lending. J. Finance, 33(2), 553-574. CrossrefGoogle Scholar

  • [28] McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press. Google Scholar

  • [29] McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools. Revised Edition. Princeton University Press. Google Scholar

  • [30] Meilijson, I. and A. Nádas (1979). Convexmajorization with an application to the length of critical paths. J. Appl. Prob. 16(3), 671–677. CrossrefGoogle Scholar

  • [31] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer. Google Scholar

  • [32] Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. J. Bank. Finance, 26(7), 1443–1471. Google Scholar

  • [33] Rüschendorf, L. (2013).Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer. Google Scholar

  • [34] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer. Google Scholar

  • [35] Van Zwet, W.R. (1980). A strong law for linear functions of order statistics. Ann. Probab., 8, 986–990. CrossrefGoogle Scholar

  • [36] Wang, S. and Dhaene, J. (1998). Comonotonicity, correlation order and premium principles. Insur. Math. Econ., 22(3), 235– 242. CrossrefGoogle Scholar

  • [37] Wang, S., Young, V. R. and Panjer, H. H. (1997). Axiomatic characterization of insurance prices. Insur. Math. Econ., 21(2), 173–183. CrossrefGoogle Scholar

  • [38] Wellner, J. A. (1977). A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics. Ann. Stat., 5(3), 473–480. CrossrefGoogle Scholar

  • [39] Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55(1), 95–115. CrossrefGoogle Scholar

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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