Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of copula. We also investigate the tail subadditivity for VaR and other distortion risk measures. In particular, we demonstrate that VaR is tail subadditive for the case where the support of risk is bounded. Various examples are also presented to illustrate the results.
Acerbi, C. (2002). Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion. Journal of Banking Finance, 26, 1505-1518. https://doi.org/10.1016/S0378-4266(02)00281-9
Alink, S., Löwe, M., & Wührich, M. V. (2004). Diversification of Aggregate Dependent Risks. Insurance Mathematics & Economics, 35, 77-95. https://doi.org/10.1016/j.insmatheco.2004.05.001
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9, 203-228. https://doi.org/10.1111/1467-9965.00068
Bannör, K. F., & Scherer, M. (2014). On the Calibration of Distortion Risk Measures to Bid-Ask Prices. Quantitative Finance, 14, 1217-1228. https://doi.org/10.1080/14697688.2014.887220
Belles-Sampera, J., Guillén, M., & Santolino, M. (2014a). Beyond Value-at-Risk: GlueVaR Distortion Risk Measures. Risk Analysis, 34, 121-134.
Belles-Sampera, J., Guillén, M., & Santolino, M. (2014b). GlueVaR Risk Measures in Capital Allocation Applications. Insurance: Mathematics and Economics, 58, 132-137.
Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular Variation. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511721434
Chen, D., Mao, T., Pan, X., & Hu, T. Z. (2012). Extreme Value Behavior of Aggregate Dependent Risks. Insurance: Mathematics and Economics, 50, 99-108.
Cherny, A. S. (2006). Weighted VaR and Its Properties. Finance and Stochastics, 10, 367-393. https://doi.org/10.1007/s00780-006-0009-1
Daníelsson, J., Jorgensen, B. N., Samorodnitsky, G., Sarmad, M., & Vries, C. G. (2013). Fat Tails, VaR and Subadditivity. Journal of Econometrics, 172, 283-291. https://doi.org/10.1016/j.jeconom.2012.08.011
Davis, R. A., & Resnick, S. I. (1996). Limit Theory for Bilinear Processes with Heavy-Tailed Noise. The Annals of Applied Probability, 6, 1191-1210. https://doi.org/10.1214/aoap/1035463328
Denneberg, D. (1994). Non-Additive Measure and Integral, Theory and Decision Library (Vol. 27). Dordrecht: Kluwer Academic Publilshers. https://doi.org/10.1007/978-94-017-2434-0
Denuit, M., Dhaene, J., Goovaerts, M., & Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. Hoboken, NJ: John Wiley & Sons, Ltd. https://doi.org/10.1002/0470016450
Dhaene, J., Kukush, A., Linders, D., & Tang, Q. (2012). Remarks on Quantiles and Distortion Risk Measures. European Actuarial Journal, 2, 319-328. https://doi.org/10.1007/s13385-012-0058-0
Dhaene, J., Vanduffel, S., Tang, Q., Goovaerts, M., Kaas, R., & Vyncke, D. (2006). Risk Measures and Comonotonicity: A Review. Stochastic Models, 22, 573-606. https://doi.org/10.1080/15326340600878016
Dolati, A., & Nezhad, A. D. (2014). Some Results on Convexity and Concavity of Multivariate Copulas. Iranian Journal of Mathematical Sciences and Informatics, 9, 87-100.
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.
Embrechts, P., Nesehová, J., & Wührich, M. V. (2009). Additivity Properties for Value-at-Risk under Archimedean Dependence and Heavy-Tailedness. Insurance: Mathematics and Economics, 44, 164-169.
Feller, W. (1971). An Introduction to Probability Theory and Its Applications (Vol. 2, 2nd ed.). New York, NY: Wiley.
Hardy, M. R., (2006). An Introduction to Risk Measures for Actuarial Applications. Schaumburg, IL: Society of Actuaries.
He, X. D., Jin, H., & Zhou, X. Y. (2015). Dynamic Portfolio Choice When Risk Is Measured by Weighted VaR. Mathematics of Operations Research, 40, 773-796. https://doi.org/10.1287/moor.2014.0695
Hua, L., & Joe, H. (2012). Tail Comonotonicity: Properties, Constructions, and Asymptotic Additivity of Risk Measures. Insurance: Mathematics and Economics, 51, 492-503.
Jang, J., & Jho, J. H. (2011). Asymptotic Super (Sub) Additivity of Value-at-Risk of Regularly Varying Dependent Variables. Journal of Risk Management, 22, 181-202. https://doi.org/10.21480/tjrm.22.1.201106.008
Joe, H. (1997). Multivariate Models and Dependence Concepts. London: Chapman & Hall.
Kriele, M., & Wolf, J. (2014). Value-Oriented Risk Management of Insurance Companies. London: Springer-Verlag.
Lv, W., Pan, X., & Hu, T. (2013). Asymptotics of the Risk Concentration Based on the Tail Distortion Risk Measure. Statistics & Probability Letters, 83, 2703-2710. https://doi.org/10.1016/j.spl.2013.09.006
Mao, T., & Hu, T. (2013). Second-Order Properties of Risk Concentrations without the Condition of Asymptotic Smoothness. Extremes, 16, 383-405. https://doi.org/10.1007/s10687-012-0164-z
Mao, T., Lv, W., & Hu, T. (2012). Second-Order Expansions of the Risk Concentration Based on CTE. Insurance: Mathematics & Economics, 51, 449-456. https://doi.org/10.1016/j.insmatheco.2012.07.002
Nelsen, R. B. (1999). An Introduction to Copulas. New York, NY: Springer-Verlag.
Tsukahara, H. (2009). One-Parameter Families of Distortion Risk Measures. Mathematical Finance, 19, 691-705. https://doi.org/10.1111/j.1467-9965.2009.00385.x
Wang, S. S. (1996). Premium Calculation by Transforming the Layer Premium Density. ASTIN Bulletin, 26, 71-92. https://doi.org/10.2143/AST.26.1.563234
Wang, S. S. (2000). A Class of Distortion Operators for Pricing Financial and Insurance Risks. Journal of Risk and Insurance, 67, 15-36. https://doi.org/10.2307/253675
Wang, S., & Dhaene, J. (1998). Comonotonicity, Correlation Order and Premium Principles. Insurance: Mathematics and Economics, 22, 235-242.
Wang, S., & Young, V. R. (1998). Ordering Risks. Expected Utility Theory versus Yaari Dual Theory of Risk. Insurance: Mathematics Economics, 22, 145-161.
Wei, P. (2017). Risk Management with Weighted VaR. Mathematical Finance, 1-41.
Wirch, J. L., & Hardy, M. R. (1999). A Synthesis of Risk Measures for Capital Adequacy. Insurance: Mathematics and Economics, 25, 337-347.
Yaari, M. E. (1987). The Dual Theory of Choice under Risk. Econometrica, 55, 95-115. https://doi.org/10.2307/1911158
Yang, F. (2015). First- and Second-Order Asymptotics for the Tail Distortion Risk Measure of Extreme Risks. Communications in Statistics-Theory and Methods, 44, 520-532. https://doi.org/10.1080/03610926.2012.751116
Yang, J. P., Cheng, S., & Zhang, L. H. (2006). Bivariate Copula Decomposition in Terms of Comonotonicity, Countermonotonicity and Independence. Insurance: Mathematics and Economics, 39, 267-284. https://doi.org/10.1016/j.insmatheco.2006.02.015
Zhang, Y., Shen, X., & Weng, C. (2009). Approximation of the Tail Probability of Randomly Weighted Sums and Applications. Stochastic Processes and their Applications, 119, 655-675. https://doi.org/10.1016/j.spa.2008.03.004
Zhu, L., & Li, H. (2012). Tail Distortion Risk and Its Asymptotic Analysis. Insurance: Mathematics & Economics, 51, 115-121. https://doi.org/10.1016/j.insmatheco.2012.03.010