Das, Bikramjit; Fasen-Hartmann, Vicky Conditional excess risk measures and multivariate regular variation. (English) Zbl 1434.60085 Stat. Risk. Model. 36, No. 1-4, 1-23 (2019). Summary: Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable. Cited in 2 Documents MSC: 60F10 Large deviations 60G50 Sums of independent random variables; random walks 60G70 Extreme value theory; extremal stochastic processes Keywords:copula models; expected shortfall; heavy-tails; hidden regular variation; mean excess; multivariate regular variation; systemic risk Software:QRM PDFBibTeX XMLCite \textit{B. Das} and \textit{V. Fasen-Hartmann}, Stat. Risk. Model. 36, No. 1--4, 1--23 (2019; Zbl 1434.60085) Full Text: DOI References: [1] V. V. Acharya, L. H. Pedersen, T. Philippon and M. P. Richardson, Measuring systemic risk, Rev. Financial Stud. 30 (2017), no. 1, 2-47. [2] P. L. Anderson and M. M. Meerschaert, Modeling river flows with heavy tails, Water Resour. Res. 34 (1998), no. 9, 2271-2280. [3] R. Ballerini, Archimedean copulas, exchangeability, and max-stability, J. Appl. Probab. 31 (1994), no. 2, 383-390. · Zbl 0812.60022 [4] M. Bargès, H. Cossette and E. Marceau, TVaR-based capital allocation with copulas, Insurance Math. Econom. 45 (2009), no. 3, 348-361. · Zbl 1231.91141 [5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl. 27, Cambridge University, Cambridge, 1989. · Zbl 0667.26003 [6] C. Brownlees and R. Engle, SRISK: A conditional capital shortfall index for systemic risk management, Rev. Financial Stud. 30 (2015), no. 1, 48-79. [7] J. Cai and H. Li, Conditional tail expectations for multivariate phase-type distributions, J. Appl. Probab. 42 (2005), no. 3, 810-825. · Zbl 1079.62022 [8] J.-J. Cai, J. H. J. Einmahl, L. de Haan and C. Zhou, Estimation of the marginal expected shortfall: The mean when a related variable is extreme, J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015), no. 2, 417-442. · Zbl 1414.91433 [9] J.-J. Cai and E. Musta, Estimation of the marginal expected shortfall under asymptotic independence, preprint (2017), https://arxiv.org/abs/1709.04285. [10] P. Capéraà, A.-L. Fougères and C. Genest, Bivariate distributions with given extreme value attractor, J. Multivariate Anal. 72 (2000), no. 1, 30-49. · Zbl 0978.62043 [11] A. Charpentier and J. Segers, Tails of multivariate Archimedean copulas, J. Multivariate Anal. 100 (2009), no. 7, 1521-1537. · Zbl 1165.62038 [12] A. Chiragiev and Z. Landsman, Multivariate Pareto portfolios: TCE-based capital allocation and divided differences, Scand. Actuar. J. (2007), no. 4, 261-280. · Zbl 1164.91028 [13] A. Cousin and E. Di Bernardino, On multivariate extensions of conditional-tail-expectation, Insurance Math. Econom. 55 (2014), 272-282. · Zbl 1296.91149 [14] M. Crovella, A. Bestavros and M. S. Taqqu, Heavy-tailed probability distributions in the world wide web, A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, Birkhäuser, Boston (1999), 3-25. · Zbl 0945.62130 [15] B. Das and V. Fasen-Hartmann, Risk contagion under regular variation and asymptotic tail independence, J. Multivariate Anal. 165 (2018), 194-215. · Zbl 1397.62168 [16] B. Das, A. Mitra and S. Resnick, Living on the multidimensional edge: Seeking hidden risks using regular variation, Adv. in Appl. Probab. 45 (2013), no. 1, 139-163. · Zbl 1276.60041 [17] B. Das and S. I. Resnick, Models with hidden regular variation: Generation and detection, Stoch. Syst. 5 (2015), no. 2, 195-238. · Zbl 1346.60073 [18] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extreme Events for Insurance and Finance, Springer, Berlin, 1997. · Zbl 0873.62116 [19] E. Hashorva, Approximation of some multivariate risk measures for Gaussian risks, J. Multivariate Anal. 169 (2019), 330-340. · Zbl 1404.60050 [20] H. Hoffmann, Multivariate conditional risk measures, PhD Thesis, Ludwig-Maximilians-Universität München, 2017. [21] H. Hoffmann, T. Meyer-Brandis and G. Svindland, Risk-consistent conditional systemic risk measures, Stochastic Process. Appl. 126 (2016), no. 7, 2014-2037. · Zbl 1414.91423 [22] L. Hua and H. Joe, Second order regular variation and conditional tail expectation of multiple risks, Insurance Math. Econom. 49 (2011), no. 3, 537-546. · Zbl 1228.91039 [23] L. Hua and H. Joe, Tail order and intermediate tail dependence of multivariate copulas, J. Multivariate Anal. 102 (2011), no. 10, 1454-1471. · Zbl 1221.62079 [24] L. Hua and H. Joe, Tail comonotonicity and conservative risk measures, Astin Bull. 42 (2012), no. 2, 601-629. · Zbl 1277.62251 [25] L. Hua and H. Joe, Intermediate tail dependence: A review and some new results, Stochastic Orders in Reliability and Risk, Lect. Notes Stat. 208, Springer, New York (2013), 291-311. · Zbl 1312.62062 [26] L. Hua and H. Joe, Strength of tail dependence based on conditional tail expectation, J. Multivariate Anal. 123 (2014), 143-159. · Zbl 1278.62074 [27] L. Hua, H. Joe and H. Li, Relations between hidden regular variation and the tail order of copulas, J. Appl. Probab. 51 (2014), no. 1, 37-57. · Zbl 1294.62131 [28] H. Hult and F. Lindskog, Regular variation for measures on metric spaces, Publ. Inst. Math. (Beograd) (N. S.) 80(94) (2006), 121-140. · Zbl 1164.28005 [29] P. Jaworski, On uniform tail expansions of multivariate copulas and wide convergence of measures, Appl. Math. (Warsaw) 33 (2006), no. 2, 159-184. · Zbl 1102.62053 [30] A. H. Jessen and T. Mikosch, Regularly varying functions, Publ. Inst. Math. (Beograd) (N. S.) 80(94) (2006), 171-192. · Zbl 1164.60301 [31] H. Joe and H. Li, Tail risk of multivariate regular variation, Methodol. Comput. Appl. Probab. 13 (2011), no. 4, 671-693. · Zbl 1239.62060 [32] O. Kley, C. Klüppelberg and G. Reinert, Risk in a large claims insurance market with bipartite graph structure, Oper. Res. 64 (2016), no. 5, 1159-1176. · Zbl 1378.91100 [33] O. Kley, C. Klüppelberg and G. Reinert, Conditional risk measures in a bipartite market structure, Scand. Actuar. J. (2018), no. 4, 328-355. · Zbl 1416.91194 [34] R. Kulik and P. Soulier, Heavy tailed time series with extremal independence, Extremes 18 (2015), no. 2, 273-299. · Zbl 1333.60102 [35] Z. M. Landsman and E. A. Valdez, Tail conditional expectations for elliptical distributions, N. Am. Actuar. J. 7 (2003), no. 4, 55-71. · Zbl 1084.62512 [36] A. W. Ledford and J. A. Tawn, Modelling dependence within joint tail regions, J. Roy. Statist. Soc. Ser. B 59 (1997), no. 2, 475-499. J. B. Levy and M. S. Taqqu, Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards, Bernoulli 6 (2000), no. 1, 23-44. · Zbl 0954.60071 [37] H. Li, Operator tail dependence of copulas, Methodol. Comput. Appl. Probab. 20 (2018), no. 3, 1013-1027. · Zbl 1405.62054 [38] H. Li and L. Hua, Higher order tail densities of copulas and hidden regular variation, J. Multivariate Anal. 138 (2015), 143-155. · Zbl 1321.62015 [39] F. Lindskog, S. I. Resnick and J. Roy, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, Probab. Surv. 11 (2014), 270-314. · Zbl 1317.60007 [40] K. Maulik and S. Resnick, Characterizations and examples of hidden regular variation, Extremes 7 (2004), no. 1, 31-67. · Zbl 1088.62066 [41] A. J. McNeil, R. Frey and P. Embrechts, Quantitative Risk Management. Concepts, Techniques and Tools, Princeton Ser. Finance, Princeton University, Princeton, 2005. · Zbl 1089.91037 [42] R. B. Nelsen, An Introduction to Copulas, 2nd ed., Springer Ser. Statist., Springer, New York, 2006. · Zbl 1152.62030 [43] R.-D. Reiss, Approximate Distributions of Order Statistics. With Applications to Nonparametric Statistics, Springer Ser. Statist., Springer, New York, 1989. · Zbl 0682.62009 [44] S. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extremes 5 (2002), no. 4, 303-336. · Zbl 1035.60053 [45] S. I. Resnick, Heavy-Tail Phenomena. Probabilistic and Statistical Modeling, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2007. · Zbl 1152.62029 [46] S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2008. · Zbl 1136.60004 [47] F. Salmon, Recipe for disaster: The formula that killed Wall Street, Wired Mag. (2009), https://www.wired.com/2009/02/wp-quant/. [48] M. Sibuya, Bivariate extreme statistics. I, Ann. Inst. Statist. Math. Tokyo 11 (1960), 195-210. · Zbl 0095.33703 [49] R. L. Smith, Statistics of extremes, with applications in environment, insurance and finance, Extreme Values in Finance, Telecommunications, and the Environment, Chapman & Hall, London (2003), 1-78. [50] J. L. Wadsworth and J. A. Tawn, A new representation for multivariate tail probabilities, Bernoulli 19 (2013), no. 5B, 2689-2714. · Zbl 1284.60107 [51] G. B. Weller and D. Cooley, A sum characterization of hidden regular variation with likelihood inference via expectation-maximization, Biometrika 101 (2014), no. 1, 17-36. · Zbl 1285.62057 [52] C. Zhou, Are banks too big to fail? Measuring systemic importance of financial institutions, Int. J. Central Banking 6(34) (2010), 205-250. [53] L. Zhu and H. Li, Asymptotic analysis of multivariate tail conditional expectations, N. Am. Actuar. J. 16 (2012), no. 3, 350-363. · Zbl 1291.60108 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.