Brahim, Brahimi; Fatah, Benatia; Djabrane, Yahia Copula conditional tail expectation for multivariate financial risks. (English) Zbl 1413.91035 Arab J. Math. Sci. 24, No. 1, 82-100 (2018). Summary: Our goal in this paper is to propose an alternative risk measure which takes into account the fluctuations of losses and possible correlations between random variables. This new notion of risk measures, that we call copula conditional tail expectation describes the expected amount of risk that can be experienced given that a potential bivariate risk exceeds a bivariate threshold value, and provides an important measure for right-tail risk. An application to real financial data is given. Cited in 4 Documents MSC: 91B30 Risk theory, insurance (MSC2010) 62P05 Applications of statistics to actuarial sciences and financial mathematics 62H20 Measures of association (correlation, canonical correlation, etc.) 91B26 Auctions, bargaining, bidding and selling, and other market models 62H05 Characterization and structure theory for multivariate probability distributions; copulas Keywords:conditional tail expectation; positive quadrant dependence; copulas; dependence measure; risk management; market models PDFBibTeX XMLCite \textit{B. Brahim} et al., Arab J. Math. Sci. 24, No. 1, 82--100 (2018; Zbl 1413.91035) Full Text: DOI References: [1] Artzner, P. H.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Math. Finance, 9, 3, 203-228 (1999) · Zbl 0980.91042 [2] Benes̆, V.; S̆tĕpán, J., Distributions with given marginals and moment problems, (Proceedings of the Conference Held in Prague, September 1996 (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) [3] Brahimi, B.; Meraghni, D.; Necir, A., Distortion risk measures for sums of dependent losses, J. Afr. Stat., 5, 260-267 (2010) · Zbl 1241.91061 [4] Clayton, D. G., A model for association in bivariate life tables and its application in epidemi-ological studies of familial tendency in chronic disease incidence, Biometrika, 65, 141-151 (1978) · Zbl 0394.92021 [5] Cook, R. D.; Johnson, M. E., A family of distributions for modeling non-elliptically symmetric multivariate data, J. R. Stat. Soc. Ser. B Stat. Methodol., 43, 210-218 (1981) · Zbl 0471.62046 [6] Cook, R. D.; Johnson, M. E., Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, 28, 123-131 (1986) [7] Cossette, H.; Mailhot, M.; Marceau, E.; Mesfioui, M., Vector-valued tail valueat-risk and capital allocation, Methodol. Comput. Appl. Probab., 18, 653-674 (2016) · Zbl 1349.91319 [8] Cox, D. R.; Oakes, D., Analysis of Survival Data (1984), Chapman & Hall: Chapman & Hall London [9] (Cuadras, C. M.; Fortiana, J.; Rodriguez-Lallena, J. A., Distributions with Given Marginals and Statistical Modelling (2002), Kluwer: Kluwer Dordrecht) · Zbl 1054.62002 [10] Dall’Aglio, G.; Kotz, S.; Salinetti, G., (Advances in Probability Distributions with Given Marginals. Advances in Probability Distributions with Given Marginals, Mathematics and its Applications, vol. 67 (1991), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht) · Zbl 0722.00031 [11] Dalla Valle, L., Bayesian copulae distributions, with application to operational risk management, Methodol. Comput. Appl. Probab., 11, 1, 95-115 (2009) · Zbl 1293.62016 [12] Denuit, M.; Dhaene, J.; Goovaerts, M.; Kaas, R., Actuarial Theory for Dependent Risks: Measures, Orders and Models (2006), John Wiley & Sons [13] Denuit, M.; Dhaene, J.; Ribas, C., Does positive dependence between individual risks increase stop-loss premiums?, Insurance Math. Econom., 28, 3, 305-308 (2001) · Zbl 1055.91046 [14] Dhaene, J.; Goovaerts, M. J., On the dependency of risks in the individual life model, Insurance Math. Econom., 19, 3, 243-253 (1997) · Zbl 0931.62089 [15] Dhaene, J.; Kolev, N.; Morettin, P., Proceedings of the First Brazilian Conference on Statistical Modelling in Insurance and Finance (2003), Institute of Mathematics and Statistics, University of São Paulo [16] Di Clemente, A.; Romano, C., Measuring and optimizing portfolio credit risk: A copula-based approach, Econ. Notes, 33, 3, 325-357 (2004) [17] Embrechts, P.; Höing, A.; Juri, A., Using copulae to bound the value-at-risk for functions of dependent risks, Finance Stoch., 7, 2, 145-167 (2003) · Zbl 1039.91023 [18] Embrechts, P.; Lindskog, F.; McNeil, A., Modelling dependence with copulas and applications to risk management, (Rachev, S., Handbook of Heavy Tailed Distributions in Finance (2003), Elsevier: Elsevier New York), 329-384 [19] Farlie, D. J.G., The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47, 307-323 (1960) · Zbl 0102.14903 [20] Frees, E. W.; Valdez, E. A., Understanding relationships using copulas, N. Am. Actuar. J., 2, 1, 1-25 (1998) · Zbl 1081.62564 [21] Genest, C.; Huang, W.; Dufour, J.-M., A regularized goodness-of-fit test for copulas, J. Soc. Franç. Statist., 154, 64-77 (2013) · Zbl 1316.62075 [22] Genest, C.; MacKay, R. J., Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Canad. J. Statist., 14, 145-159 (1986) · Zbl 0605.62049 [23] Gumbel, E. J., Distributions à plusieurs variables dont les marges sont données, C. R. Acad. Sci. Paris, 246, 2717-2719 (1958) · Zbl 0084.35803 [24] Gumbel, E. J., Bivariate exponential distributions, J. Amer. Statist. Assoc., 55, 698-707 (1960) · Zbl 0099.14501 [25] Hougaard, P., A class of multivariate failure time distributions, Biometrika, 73, 671-678 (1986) · Zbl 0613.62121 [26] T.P. Hutchinson, C.D. Lai, Continuous Bivariate Distributions, Emphasizing Applications. Rumsby, Sydney, Australia, 1990.; T.P. Hutchinson, C.D. Lai, Continuous Bivariate Distributions, Emphasizing Applications. Rumsby, Sydney, Australia, 1990. [27] Joe, H., (Multivariate Models and Dependence Concepts. Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability, vol. 73 (1997), Chapman and Hall: Chapman and Hall London) · Zbl 0990.62517 [28] Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601 [29] Mailhot, M.; Mesfioui, M., Multivariate TVaR-based risk decomposition for vector-valued portfolios, Risks, 4, 4, 33 (2016) [30] Morgenstern, D., Einfache Beispiele zweidimensionaler Verteilungen, Mitteilungsbl. Math. Statist., 8, 234-235 (1956), (in German) · Zbl 0070.36202 [31] Nelsen, R. B., An Introduction to Copulas (2006), Springer: Springer New York · Zbl 1152.62030 [32] Oakes, D., A model for association in bivariate survival data, J. R. Stat. Soc. Ser. B Stat. Methodol., 44, 414-422 (1982) · Zbl 0503.62035 [33] Oakes, D., Semiparametric inference in a model for association in bivariate survival data, Biometrika, 73, 353-361 (1986) · Zbl 0604.62035 [34] Overbeck, L.; Sokolova, M., Risk Measurement with Spectral Capital Allocation, Applied Quantitative Finance, II, 139-159 (2008), Springer · Zbl 1258.91109 [35] Resnick, S. I., (Extreme Values, Regular Variation and Point Processes. Extreme Values, Regular Variation and Point Processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4 (1987), Springer-Verlag: Springer-Verlag New York) · Zbl 0633.60001 [36] Seneta, E., (Regularly Varying Functions. Regularly Varying Functions, Lecture Notes in Mathematics, vol. 508 (1976), Springer-Verlag: Springer-Verlag Berlin-New York) · Zbl 0324.26002 [37] Sklar, A., Fonctions de répartition à \(n\) dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202 [38] Wirch, J.; Hardy, M., A synthesis of risk measures for capital adequacy, Insurance Math. Econom., 25, 337-347 (1999) · Zbl 0951.91032 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.