×

Asymptotics for risk capital allocations based on conditional tail expectation. (English) Zbl 1228.91029

Summary: An investigation of the limiting behavior of a risk capital allocation rule based on the conditional tail expectation (CTE) risk measure is carried out. More specifically, with the help of general notions of extreme value theory (EVT), the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding value-at-risk (VaR) risk measure. The existing methodology acquired for VaR can therefore be applied to a somewhat less well-studied CTE. In the context of interest, the EVT approach is seemingly well-motivated by modern regulations, which openly strive for the excessive prudence in determining risk capitals.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G70 Extreme value theory; extremal stochastic processes
60E05 Probability distributions: general theory
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Albrecher, H.; Asmussen, S.; Kortschak, D., Tail asymptotics for the sum of two heavy-tailed dependent risks, Extremes, 9, 2, 107-130 (2006) · Zbl 1142.60009
[2] Alink, S.; Löwe, M.; Wüthrich, M. V., Analysis of the expected shortfall of aggregate dependent risks, ASTIN Bulletin, 35, 1, 25-43 (2005) · Zbl 1101.62092
[3] Asimit, A. V.; Badescu, A. L., Extremes on the discounted aggregate claims in a time dependent risk model, Scandinavian Actuarial Journal, 2, 93-104 (2010) · Zbl 1224.91041
[4] Asimit, A. V.; Jones, B. L., Asymptotic tail probabilities for large claims reinsurance of a portfolio of dependent risks, ASTIN Bulletin, 38, 1, 147-159 (2008) · Zbl 1169.91361
[5] Asmussen, S.; Rojas-Nandayapa, L., Asymptotics of sums of lognormal random variables with Gaussian copula, Statistics and Probability Letters, 78, 16, 2709-2714 (2008) · Zbl 1151.60009
[6] Balkema, A. A.; de Haan, L., Residual life time at great age, The Annals of Probability, 2, 5, 792-804 (1974) · Zbl 0295.60014
[7] Bühlmann, H., An economic premium principle, ASTIN Bulletin, 11, 1, 52-60 (1980)
[8] Cai, J.; Li, H., Conditional tail expectations for multivariate phase-type distributions, Journal of Applied Probability, 42, 3, 810-825 (2005) · Zbl 1079.62022
[9] Cai, J.; Tang, Q., On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications, Journal of Applied Probability, 41, 1, 117-130 (2004) · Zbl 1054.60012
[10] Charpentier, A.; Segers, J., Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis, 100, 7, 1521-1537 (2009) · Zbl 1165.62038
[11] Chiragiev, A.; Landsman, Z., Multivariate Pareto portfolios: TCE-based capital allocation and divided differences, Scandinavian Actuarial Journal, 2007, 4, 261-280 (2007) · Zbl 1164.91028
[12] Davis, R. A.; Resnick, S. I., Limit theory for bilinear processes with heavy-tailed noise, The Annals of Applied Probability, 6, 4, 1191-1210 (1996) · Zbl 0879.60053
[13] Dhaene, J.; Denuit, M.; Vanduffel, S., Correlation order, merging and diversification, Insurance: Mathematics and Economics, 45, 3, 325-332 (2009) · Zbl 1231.91175
[14] Dhaene, J.; Henrard, L.; Landsman, Z.; Vandendorpe, A.; Vanduffel, S., Some results on the CTE-based capital allocation rule, Insurance: Mathematics and Economics, 42, 2, 855-863 (2008) · Zbl 1152.91577
[15] Dhaene, J.; Tsanakas, A.; Valdez, E.; Vanduffel, E., Optimal capital allocation principles, Journal of Risk and Insurance, 78 (2011)
[16] Dhaene, J.; Vanduffel, S.; Goovaerts, M. J.; Kaas, R.; Tang, Q.; Vyncke, D., Risk measures and comonotonicity: a review, Stochastic Models, 22, 573-606 (2006) · Zbl 1159.91403
[17] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modeling Extremal Events for Insurance and Finance (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0873.62116
[18] Fisher, R. A.; Tippett, L. H.C., Limiting forms of the frequency distribution of the largest or smallest member of a sample, Mathematical Proceedings of the Cambridge Philosophical Society, 24, 2, 180-190 (1928) · JFM 54.0560.05
[19] Furman, E.; Landsman, Z., Risk capital decomposition for a multivariate dependent gamma portfolio, Insurance: Mathematics and Economics, 37, 3, 635-649 (2005) · Zbl 1129.91025
[20] Furman, E.; Landsman, Z., Tail variance premium with applications for elliptical portfolio of risks, ASTIN Bulletin, 36, 2, 433-462 (2006) · Zbl 1162.91373
[21] Furman, E.; Landsman, Z., Economic capital allocations for non-negative portfolios of dependent risks, ASTIN Bulletin, 38, 2, 601-619 (2008) · Zbl 1274.91379
[22] Furman, E.; Zitikis, R., Weighted premium calculation principles, Insurance: Mathematics and Economics, 42, 1, 459-465 (2008) · Zbl 1141.91509
[23] Furman, E.; Zitikis, R., Weighted risk capital allocations, Insurance: Mathematics and Economics, 43, 2, 263-269 (2008) · Zbl 1189.62163
[24] Geluk, J.; Tang, Q., Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, 22, 4, 871-882 (2009) · Zbl 1177.62017
[25] Genest, C.; Ghoudi, K.; Rivest, L. P., Discussions of understanding relationships using copulas, North American Actuarial Journal, 2, 3, 143-149 (1998)
[26] Gnedenko, B. V., Sur la distribution limité du terme maximum d’une série aléatoaire, Annals of Mathematics, 44, 423-453 (1943) · Zbl 0063.01643
[27] Goldie, C. M.; Resnick, S., Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution, Advances in Applied Probability, 20, 4, 706-718 (1988) · Zbl 0659.60028
[28] Hashorva, E.; Pakes, A. G.; Tang, Q., Asymptotics of random contractions, Insurance: Mathematics and Economics, 47, 3, 405-414 (2010) · Zbl 1231.91196
[29] Joe, H., Li, L., 2011. Tail risk of multivariate regular variation, Methodology and Computing in Applied Probability, in press (doi:10.1007/s11009-010-9183-x; Joe, H., Li, L., 2011. Tail risk of multivariate regular variation, Methodology and Computing in Applied Probability, in press (doi:10.1007/s11009-010-9183-x · Zbl 1239.62060
[30] Juri, A.; Wüthrich, M. V., Tail dependence from a distributional point of view, Extremes, 6, 3, 213-246 (2003) · Zbl 1049.62055
[31] Kallenberg, O., Random Measures (1983), Akademie-Verlag: Akademie-Verlag Berlin · Zbl 0288.60053
[32] Khoudraji, A., 1995. Contributions à l’étude des copules et à la modélasion des valeurs extrêmes bivariées, Ph.D. thesis, Université Laval, Québec, Canada.; Khoudraji, A., 1995. Contributions à l’étude des copules et à la modélasion des valeurs extrêmes bivariées, Ph.D. thesis, Université Laval, Québec, Canada.
[33] Klüppelberg, C.; Resnick, S. I., The Pareto copula, aggregation of risks, and the emperor’s socks, Journal of Applied Probability, 45, 1, 67-84 (2008) · Zbl 1144.62037
[34] Ko, B.; Tang, Q., Sums of dependent nonnegative random variables with subexponential tails, Journal of Applied Probability, 45, 1, 85-94 (2008) · Zbl 1137.62310
[35] Kortschak, D.; Albrecher, H., Asymptotic results for the sum of dependent non-identically distributed random variables, Methodology and Computing in Applied Probability, 11, 3, 279-306 (2009) · Zbl 1171.60348
[36] Landsman, Z.; Valdez, E., Tail conditional expectation for elliptical distributions, North American Actuarial Journal, 7, 4, 55-71 (2003) · Zbl 1084.62512
[37] Li, J.; Tang, Q.; Wu, R., Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Advances in Applied Probability, 42, 4, 1126-1146 (2010) · Zbl 1205.62061
[38] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management (2005), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1089.91037
[39] Mitra, A.; Resnick, S. I., Aggregation of rapidly varying risks and asymptotic independence, Advances in Applied Probability, 41, 3, 797-828 (2009) · Zbl 1181.60019
[40] Nelsen, R. B., An Introduction to Copulas (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0909.62052
[41] Panjer, H.H., Jia, J., 2001. Solvency and capital allocation. Institute of Insurance and Pension Research Report 01-14. Waterloo, Ontario: University of Waterloo.; Panjer, H.H., Jia, J., 2001. Solvency and capital allocation. Institute of Insurance and Pension Research Report 01-14. Waterloo, Ontario: University of Waterloo.
[42] Resnick, S. I., Extreme Values, Regular Variation and Point Processes (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0633.60001
[43] Resnick, S. I., Heavy-Tail Phenomena: Probabilistic and Statistical Modeling (2007), Springer-Verlag: Springer-Verlag New York · Zbl 1152.62029
[44] Sklar, A., 1959. Fonctions de répartion à n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Université de Paris, 8, p. 229-231.; Sklar, A., 1959. Fonctions de répartion à n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Université de Paris, 8, p. 229-231. · Zbl 0100.14202
[45] Valdez, E.; Chernih, A., Wang’s capital allocation formula for elliptically contoured distributions, Insurance: Mathematics and Economics, 33, 3, 517-532 (2003) · Zbl 1103.91375
[46] Vernic, R., Multivariate skew-normal distributions with applications in insurance, Insurance: Mathematics and Economics, 38, 2, 413-426 (2006) · Zbl 1132.91501
[47] Vernic, R., Tail conditional expectation for the multivariate Pareto distribution of the second kind: another approach, Methodology and Computing in Applied Probability, 13, 1, 121-137 (2011) · Zbl 1208.60014
[48] Wang, S. S., Premium calculation by transforming the layer premium density, ASTIN Bulletin, 26, 71-92 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.