## Tail conditional expectation for the multivariate Pareto distribution of the second kind: Another approach.(English)Zbl 1208.60014

Summary: In risk analysis, the Tail Conditional Expectation (TCE) describes the expected amount of risk that can be experienced given that the risk exceeds a threshold value. Thus, TCE provides an important measure of the right-tail risk. In this paper, we present TCE formulas for the multivariate Pareto distribution of the second kind. Because of the complex form of this distribution, the formulas for the $$n$$-variate case are expressed recursively, in terms of the $$(n - 1)$$-variate case.

### MSC:

 60E05 Probability distributions: general theory 91B30 Risk theory, insurance (MSC2010)
Full Text:

### References:

 [1] Arnold BC (1983) Pareto distributions. International Cooperative Publ. House, Fairland · Zbl 1169.62307 [2] Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228 · Zbl 0980.91042 [3] Buch A, Dorfleitner G (2008) Coherent risk measures, coherent capital allocations and the gradient allocation principle. Insur Math Econ 42:235–242 · Zbl 1141.91490 [4] Chiragiev A, Landsman Z (2007) Multivariate Pareto portfolios: TCE-based capital allocation and divided differences. Scand Actuar J 4:261–280 · Zbl 1164.91028 [5] Dhaene J, Goovaerts MJ, Kaas R (2003) Economic capital allocation derived from risk measures. N Am Actuar J 7:44–59 · Zbl 1084.91515 [6] Embrechts P, Resnick S, Samorodnitsky G (1999) Extreme value theory as a risk management tool. N Am Actuar J 3:30–41 · Zbl 1082.91530 [7] Furman E, Zitikis R (2008) Weighted premium calculation principles. Insur Math Econ 42:459–465 · Zbl 1141.91509 [8] Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions. Wiley, New York · Zbl 0811.62001 [9] Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions, vol 1. Models and applications. Wiley, New York · Zbl 0946.62001 [10] Landsman Z, Valdez EA (2005) Tail conditional expectation for exponential dispersion models. Astin Bull 35:189–209 · Zbl 1099.62122 [11] Panjer HH (2002) Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. In: 27th international congress of actuaries, Cancun [12] Pareto V (1897) Cours d’Economie politique. Rouge et Cie, Paris [13] Yeh HC (2000) Two multivariate Pareto distributions and their related inferences. Bull Inst Math Acad Sin 28(2):71–86 · Zbl 0998.62051 [14] Yeh HC (2004) Some properties and characterizations for generalized multivariate Pareto distributions. J Multivar Anal 88:47–60 · Zbl 1032.62045
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.