An Axiomatic Foundation for the Expected Shortfall

References

• Acerbi C (2002) Spectral measures of risk: A coherent representation of subjective risk aversion. J. Banking Finance 26(7):1505–1518.Crossref, Google Scholar
• Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J. Banking Finance 26(7):1487–1503.Crossref, Google Scholar
• Acharya V, Engle R, Richardson M (2012) Capital shortfall: A new approach to ranking and regulating systemic risks. Amer. Econom. Rev. 102(3):59–64.Crossref, Google Scholar
• Acharya V, Pedersen LH, Philippon T, Richardson M (2017) Measuring systemic risk. Rev. Financial Stud. 30(1):2–47.Crossref, Google Scholar
• Adrian T, Brunnermeier MK (2016) CoVaR. Amer. Econom. Rev. 106(7):1705–1741.Crossref, Google Scholar
• Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math. Finance 9(3):203–228.Crossref, Google Scholar
• Azenes and Effixis (2018) Finma’s Swiss solvency test 2019: Major changes in the market risk standard model. Accessed July 2, 2020, https://effixis.ch/downloads/2018-10-31-finma-new-sst.pdf.Google Scholar
• BCBS (2016) Minimum capital requirements for market risk: January 2016. Report, Basel Committee on Banking Supervision, Bank for International Settlements, Basel, Switzerland.Google Scholar
• BCBS (2019) Minimum capital requirements for market risk: February 2019. Report, Basel Committee on Banking Supervision, Bank for International Settlements, Basel, Switzerland.Google Scholar
• Bellini F, Klar B, Müller A, Rosazza Gianin E (2014) Generalized quantiles as risk measures. Insurance Math. Econom. 54(1):41–48.Crossref, Google Scholar
• Cai Z, Wang X (2008) Nonparametric estimation of conditional VaR and expected shortfall. J. Econometrics 147:120–130.Crossref, Google Scholar
• Cerreia-Vioglio S, Maccheroni F, Marinacci M, Montrucchio L (2011) Uncertainty averse preferences. J. Econom. Theory 146(4):1275–1330.Crossref, Google Scholar
• Chambers CP (2009) An axiomatization of quantiles on the domain of distribution functions. Math. Finance 19(2):335–342.Crossref, Google Scholar
• Chen C, Iyengar G, Moallemi CC (2013) An axiomatic approach to systemic risk. Management Sci. 59(6):1373–1388.Link, Google Scholar
• Chen SX (2008) Nonparametric estimation of expected shortfall. J. Financial Econom. 6(1):87–107.Crossref, Google Scholar
• Cherny AS, Madan D (2009) New measures for performance evaluation. Rev. Financial Stud. 22(7):2571–2606.Crossref, Google Scholar
• Cont R, Deguest R, Scandolo G (2010) Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10(6):593–606.Crossref, Google Scholar
• Delbaen F (2007) Risk measures for non-integrable random variables. Math. Finance 19(2):329–333.Crossref, Google Scholar
• Delbaen F (2012) Monetary Utility Functions (Osaka University Press, Osaka, Japan).Google Scholar
• Denneberg D (1994) Non-additive Measure and Integral (Kluwer, Dordrecht, Netherlands).Crossref, Google Scholar
• Dhaene J, Denuit M, Goovaerts MJ, Kaas R, Vynche D (2002) The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31(1):3–33.Crossref, Google Scholar
• Du Z, Escanciano JC (2017) Backtesting expected shortfall: Accounting for tail risk. Management Sci. 63(4):940–958.Link, Google Scholar
• Embrechts P, Wang R (2015) Seven proofs for the subadditivity of Expected Shortfall. Dependence Model. 3(1):126–140.Crossref, Google Scholar
• Embrechts P, Liu H, Wang R (2018) Quantile-based risk sharing. Oper. Res. 66(4):936–949.Link, Google Scholar
• Embrechts P, Puccetti G, Rüschendorf L (2013) Model uncertainty and VaR aggregation. J. Banking Finance 37(8):2750–2764.Crossref, Google Scholar
• Embrechts P, Wang B, Wang R (2015) Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stochastics 19(4):763–790.Crossref, Google Scholar
• Embrechts P, Puccetti G, Rüschendorf L, Wang R, Beleraj A (2014) An academic response to Basel 3.5. Risks 2(1):25–48.Crossref, Google Scholar
• Emmer S, Kratz M, Tasche D (2015) What is the best risk measure in practice? A comparison of standard measures. J. Risk 18(2):31–60.Crossref, Google Scholar
• Fermanian JD, Scaillet O (2005) Sensitivity analysis of VaR and expected shortfall for portfolios under netting agreements. J. Banking Finance 29:927–958.Crossref, Google Scholar
• Fissler T, Ziegel JF (2016) Higher order elicitability and Osband’s principle. Ann. Statist. 44(4):1680–1707.Crossref, Google Scholar
• Föllmer H, Schied A (2002) Convex measures of risk and trading constraints. Finance Stochastics 6(4):429–447.Crossref, Google Scholar
• Föllmer H, Schied A (2016) Stochastic Finance: An Introduction in Discrete Time, 4th ed. (Walter de Gruyter, Berlin).Crossref, Google Scholar
• Frittelli M, Rosazza Gianin E (2002) Putting order in risk measures. J. Banking Finance 26(7):1473–1486.Crossref, Google Scholar
• Furman E, Wang R, Zitikis R (2017) Gini-type measures of risk and variability: Gini shortfall, capital allocation and heavy-tailed risks. J. Banking Finance 83:70–84.Crossref, Google Scholar
• Gao N, Leung DH, Munari C, Xanthos F (2018) Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces. Finance Stochastics 22(2):395–415.Crossref, Google Scholar
• Gilboa I (2009) Theory of Decision Under Uncertainty (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• Gilboa I, Maccheroni F, Marinacci M, Schmeidler D (2010) Objective and subjective rationality in a multiple prior model. Econometrica 78(2):755–770.Crossref, Google Scholar
• Gilboa I, Postlewaite A, Samuelson L (2019) What are axiomatizations good for? Theory Decision 86(3–4):339–359.Crossref, Google Scholar
• Gneiting T (2011) Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106(494):746–762.Crossref, Google Scholar
• He X, Peng X (2018) Surplus-invariant, law-invariant, and conic acceptance sets must be the sets induced by value-at-risk. Oper. Res. 66(5):1268–1276.Link, Google Scholar
• Kou S, Peng X (2016) On the measurement of economic tail risk. Oper. Res. 64(5):1056–1072.Link, Google Scholar
• Krätschmer V, Schied A, Zähle H (2014) Comparative and quantitiative robustness for law-invariant risk measures. Finance Stochastics 18(2):271–295.Crossref, Google Scholar
• Kusuoka S (2001) On law invariant coherent risk measures. Adv. Math. Econom. 3:83–95.Crossref, Google Scholar
• Liu F, Wang R (2020) A theory for measures of tail risk. Math. Oper. Res. Forthcoming.Google Scholar
• Maccheroni F, Marinacci M, Rustichini A (2006) Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74(6):1447–1498.Crossref, Google Scholar
• Marinacci M, Montrucchio L (2004) Introduction to the mathematics of ambiguity. Gilboa I, ed. Uncertainty in Economic Theory (Routledge, New York), 46–107.Crossref, Google Scholar
• McNeil AJ, Frey R, Embrechts P (2015) Quantitative Risk Management: Concepts, Techniques and Tools, revised ed. (Princeton University Press, Princeton, NJ).Google Scholar
• Nelsen R (2006) An Introduction to Copulas, 2nd ed. (Springer, New York).Google Scholar
• Newey W, Powell J (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847.Crossref, Google Scholar
• Patton AJ, Ziegel JF, Chen R (2019) Dynamic semiparametric models for expected shortfall (and Value-at-Risk). J. Econometrics 211(2):388–413.Crossref, Google Scholar
• Quiggin J (1982) A theory of anticipated utility. J. Econom. Behav. Organ. 3(4):323–343.Crossref, Google Scholar
• Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J. Banking Finance 26(7):1443–1471.Crossref, Google Scholar
• Rothschild M, Stiglitz JE (1970) Increasing risk, I: A definition. J. Econom. Theory 2(3):225–243.Crossref, Google Scholar
• Rudin W (1991) Functional Analysis, 2nd ed. (McGraw-Hill, New York).Google Scholar
• Rüschendorf L (2013) Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios (Springer, Heidelberg, Germany).Crossref, Google Scholar
• Scaillet O (2004) Nonparametric estimation and sensitivity analysis of expected shortfall. Math. Finance 14(1):115–129.Crossref, Google Scholar
• Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57(3):571–587.Crossref, Google Scholar
• Shaked M, Shanthikumar JG (2007) Stochastic Orders (Springer, New York).Crossref, Google Scholar
• Wakker PP (2010) Prospect Theory: For Risk and Ambiguity (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• Wang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull. 26(1):71–92.Crossref, Google Scholar
• Wang R, Peng L, Yang J (2013) Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stochastics 17(2):395–417.Crossref, Google Scholar
• Wang R, Wei Y (2020) Risk functionals with convex level sets. Math. Finance, ePub ahead of print May 27, https://doi.org/10.1111/mafi.12270.Google Scholar
• Wang R, Zitikis R (2020) Weak comonotonicity. European J. Oper. Res. 282:386–397.Crossref, Google Scholar
• Yaari ME (1987) The dual theory of choice under risk. Econometrica 55(1):95–115.Crossref, Google Scholar
• Ziegel JF (2016) Coherence and elicitability. Math. Finance 26(4):901–918.Crossref, Google Scholar