Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 3 | 1 |
Tytuł artykułu

Seven Proofs for the Subadditivity of Expected Shortfall

Treść / Zawartość
Warianty tytułu
Języki publikacji
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.
Opis fizyczny
  • RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
  • Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON
    N2L 3G1, Canada
  • [1] Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. J. Bank. Finance, 26(7),1505–1518.
  • [2] Acerbi, C. and Tasche, D. (2002). On the coherence of expected shortfall. J. Bank. Finance, 26(7), 1487–1503.
  • [3] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance, 9(3), 203–228.[Crossref]
  • [4] BCBS (2012). Consultative Document May 2012. Fundamental review of the trading book. Basel Committee on Banking Supervision.Basel: Bank for International Settlements.
  • [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.
  • [6] BCBS (2014). Consultative Document December 2014. Fundamental review of the trading book: Outstanding issues. BaselCommittee on Banking Supervision. Basel: Bank for International Settlements.
  • [7] Billingsley, P. (1995). Probability and Measure. Third Edition. Wiley.
  • [8] Choquet, G. (1953). Theory of capacities. Ann. Inst. Fourier, 5, 121–293.
  • [9] Cont, R., Deguest, R. and Scandolo, G. (2010). Robustness and sensitivity analysis of risk measurement procedures. Quant.Finance, 10(6), 593–606.[WoS][Crossref]
  • [10] Delbaen, F. (2012). Monetary Utility Functions. Osaka University Press.
  • [11] Denneberg, D. (1994). Non-additive Measure and Integral. Springer.
  • [12] Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, R. (2005). Actuarial Theory for Dependent Risks. Wiley.
  • [13] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vynche, D. (2002). The concept of comonotonicity in actuarial scienceand finance: Theory. Insur. Math. Econ. 31(1), 3-33.[Crossref]
  • [14] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Tang, Q. and Vynche, D. (2006). Risk measures and comonotonicity: areview. Stoch. Models, 22, 573–606.
  • [15] Dhaene, J., Laeven, R. J., Vanduffel, S., Darkiewicz, G. and Goovaerts, M. J. (2008). Can a coherent risk measure be toosubadditive? J. Risk Insur., 75(2), 365–386.[WoS]
  • [16] Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Oper. Res., 77(3), 423–432.
  • [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.
  • [18] Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Third Edition.
  • [19] Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon. Sect. A., 14, 53–77.
  • [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.[Crossref]
  • [21] Hoeffding, W. (1940). Massstabinvariante Korrelationstheorie. Schriften Math. Inst. Univ. Berlin, 5(5), 181–233.
  • [22] Huber, P. J. (1980). Robust Statistics. First ed., Wiley Series in Probability and Statistics. Wiley, New Jersey.
  • [23] Huber, P. J. and Ronchetti E. M. (2009). Robust Statistics. Second ed., Wiley Series in Probability and Statistics. Wiley, NewJersey. First ed.: Huber, P. (1980).
  • [24] IAIS (2014). Consultation Document December 2014. Risk-based global insurance capital standard. International Associationof Insurance Supervisors.
  • [25] Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008). Modern Actuarial Risk Theory: Using R. Springer.
  • [26] Kusuoka, S. (2001). On law invariant coherent risk measures. Adv. Math. Econ., 3, 83–95.[Crossref]
  • [27] Levy, H. and Kroll, Y. (1978). Ordering uncertain options with borrowing and lending. J. Finance, 33(2), 553-574.[Crossref]
  • [28] McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton UniversityPress.
  • [29] McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools. RevisedEdition. Princeton University Press.
  • [30] Meilijson, I. and A. Nádas (1979). Convexmajorization with an application to the length of critical paths. J. Appl. Prob. 16(3),671–677.[Crossref]
  • [31] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer.
  • [32] Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. J. Bank. Finance, 26(7),1443–1471.
  • [33] Rüschendorf, L. (2013).Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer.
  • [34] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer.
  • [35] Van Zwet, W.R. (1980). A strong law for linear functions of order statistics. Ann. Probab., 8, 986–990.[Crossref]
  • [36] Wang, S. and Dhaene, J. (1998). Comonotonicity, correlation order and premium principles. Insur. Math. Econ., 22(3), 235–242.[Crossref]
  • [37] Wang, S., Young, V. R. and Panjer, H. H. (1997). Axiomatic characterization of insurance prices. Insur. Math. Econ., 21(2),173–183.[Crossref]
  • [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.[Crossref]
  • [39] Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55(1), 95–115.[Crossref]
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.