Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T13:35:40.777Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTORTION RISKMETRICS ON GENERAL SPACES

Published online by Cambridge University Press:  11 June 2020

Qiuqi Wang Open the ORCID record for Qiuqi Wang [Opens in a new window] ,
Ruodu Wang  and
Yunran Wei
Qiuqi Wang*
Affiliation:
Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L3G1, Canada
Ruodu Wang
Affiliation:
Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L3G1, Canada E-mail: wang@uwaterloo.ca
Yunran Wei
Affiliation:
Department of Statistics and Actuarial Science Northern Illinois University DeKalb, IL 60115, United States E-mail: ywei1@niu.edu
*
E-mail: q428wang@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The class of distortion riskmetrics is defined through signed Choquet integrals, and it includes many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.

Keywords

ComonotonicityChoquet integralsconvexityconvex ordercontinuityC6D8G00
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 827 - 851
Copyright
© 2020 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking and Finance, 26(7), 1505–1518.CrossRefGoogle Scholar
Acerbi, C. and Szekely, B. (2017) General properties of backtestable statistics. SSRN: 2905109.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
Belles-Sampera, J., Guilln, M. and Santolino, M. (2014). Beyond value-at-risk: GlueVaR distortion risk measures. Risk Analysis, 34(1), 121–134.CrossRefGoogle ScholarPubMed
Cheridito, P. and Li, T. (2009) Risk measures on Orlicz hearts. Mathematical Finance, 19(2), 189–214.CrossRefGoogle Scholar
Cherny, A.S. and Madan, D. (2009) New measures for performance evaluation. Review of Financial Studies, 22(7), 2571–2606.CrossRefGoogle Scholar
Choquet, G. (1954) Theory of capacities. Annales de l'institut Fourier, 5, 131–295.Google Scholar
Delbaen, F. (2002) Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics: Essays in Honor of Dieter Sondermann (ed. K. Sandmann and P.J. Schönbucher), pp. 1–37. Berlin: Springer.Google Scholar
Denneberg, D. (D). Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, 20, 181–190.CrossRefGoogle Scholar
Denneberg, D. (1994). Non-additive Measure and Integral. Springer Science & Business Media, Dordrecht: Springer.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. Chichester, UK: Wiley.CrossRefGoogle Scholar
Dhaene, J., Kukush, A., Linders, D. and Tang, Q. (2012) Remarks on quantiles and distortion risk measures. European Actuarial Journal, 2(2), 319–328.CrossRefGoogle Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M.J., Kaas, R., Tang, Q. and Vynche, D. (2006) Risk measures and comonotonicity: A review. Stochastic Models, 22, 573–606.CrossRefGoogle Scholar
Embrechts, P. and Puccetti, G. (2006). Bounds for functions of multivariate risks. Journal of Multivariate Analysis, 97(2), 526–547.CrossRefGoogle Scholar
Filipović, D. and Svindland, G. (2012) The canonical model space for law-invariant convex risk measures is L1. Mathematical Finance, 22(3), 585–589.CrossRefGoogle Scholar
Fissler, T. and Ziegel, J.F. (2016) Higher order elicitability and Osband's principle. Annals of Statistics, 44(4), 1680–1707.CrossRefGoogle Scholar
Frittelli, M. and Rosazza Gianin, E. (2002) Putting order in risk measures. Journal of Banking and Finance, 26, 1473–1486.CrossRefGoogle Scholar
FÖllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6(4), 429–447.CrossRefGoogle Scholar
FÖllmer, H. and Schied, A. (2016) Stochastic Finance: An Introduction in Discrete Time. Forth Edition. Berlin: Walter de Gruyter.Google Scholar
Frongillo, R. and Kash, I.A. (2018) Elicitation complexity of statistical properties. arXiv:1506.07212v2.Google Scholar
Furman, E., Wang, R. and Zitikis, R. (2017) Gini-type measures of risk and variability: Gini shortfall, capital allocation and heavy-tailed risks. Journal of Banking and Finance, 83, 70–84.CrossRefGoogle Scholar
Gerber, H.U. (1974) On additive premium calculation principles. ASTIN Bulletin, 7(3), 215–222.CrossRefGoogle Scholar
Grechuk, B., Molyboha, A. and Zabarankin, M. (2009) Maximum entropy principle with general deviation measures. Mathematics of Operations Research, 34(2), 445–467.CrossRefGoogle Scholar
Gneiting, T. (2011) Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494), 746–762.CrossRefGoogle Scholar
Huber, P.J. and Ronchetti, E.M. (2009) Robust Statistics. Second Edition, Wiley Series in Probability and Statistics. New Jersey: Wiley.CrossRefGoogle Scholar
Kou, S. and Peng, X. (2016) On the measurement of economic tail risk. Operations Research, 64(5), 1056–1072.CrossRefGoogle Scholar
Krätschmer, V., Schied, A. and Zähle, H. (2014) Comparative and quantitiative robust-ness for law-invariant risk measures. Finance and Stochastics, 18(2), 271–295.CrossRefGoogle Scholar
Kusuoka, S. (S). On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.CrossRefGoogle Scholar
Lambert, N., Pennock, D.M. and Shoham, Y. (2008) Eliciting properties of probability distributions. Proceedings of the 9th ACM Conference on Electronic Commerce, pp. 129–138.Google Scholar
Liebrich, F.-B. and Svindland, G. (2017) Model spaces for risk measures. Insurance: Mathematics and Economics, 77, 150–165.Google Scholar
Liu, F., Cai, J., Lemieux, C. and Wang, R. (2020) Convex risk functionals: Representation and applications. Insurance: Mathematics and Economics, 90, 66–79.Google Scholar
Liu, P., Schied, A. and Wang, R. (2020) Distributional transforms, probability distortions, and their applications. Mathematics of Operations Research (forthcoming). SSRN: 3419388.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques and Tools. Revised Edition. Princeton, NJ: Princeton University Press.Google Scholar
Pichler, A. (2013) The natural Banach space for version independent risk measures. Insurance: Mathematics and Economics, 53(2), 405–415.Google Scholar
Rockafellar, R.T., Uryasev, S. and Zabarankin, M. (2006) Generalized deviation in risk analysis. Finance and Stochastics, 10, 51–74.CrossRefGoogle Scholar
Rudin, W. (1987) Real and Complex Analysis. New York, NY: Tata McGraw-Hill Education.Google Scholar
Rüschendorf, L. (2013) Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Heidelberg: Springer.CrossRefGoogle Scholar
Ruszczyński, A. and Shapiro, A. (2006) Optimization of convex risk functions. Mathematics of Operations Research, 31(3), 433–452.CrossRefGoogle Scholar
Schmeidler, D. (1986) Integral representation without additivity. Proceedings of the American Mathematical Society, 97(2), 255–261.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. Springer Series in Statistics, New York, NY: Springer.CrossRefGoogle Scholar
Wang, R. and Wei, Y. (2020) Risk functionals with convex level sets. Mathematical Finance. DOI:10.1111/mafi.12270.CrossRefGoogle Scholar
Wang, R., Wei, Y. and Willmot, G. (2020) Characterization, robustness and aggrega-tion of signed Choquet integrals. Mathematics of Operations Research. DOI:10.1287/moor.2019.1020.CrossRefGoogle Scholar
Wang, S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21(2), 173–183.Google Scholar
Williamson, R.E. (1956) Multiply monotone functions and their Laplace transforms. Duke Mathematical Journal, 23(2), 189–257.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55(1), 95–115.CrossRefGoogle Scholar