Tail Conditional Expectation for the Multivariate Pareto Distribution of the Second Kind: Another Approach

  • Raluca Vernic


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.


Multivariate Pareto distribution of the second kind Tail conditional expectation 

Mathematics Subject Classifications (2000)

60E05 62P05 91B30 


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  1. Arnold BC (1983) Pareto distributions. International Cooperative Publ. House, FairlandzbMATHGoogle Scholar
  2. Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228zbMATHCrossRefMathSciNetGoogle Scholar
  3. Buch A, Dorfleitner G (2008) Coherent risk measures, coherent capital allocations and the gradient allocation principle. Insur Math Econ 42:235–242zbMATHCrossRefMathSciNetGoogle Scholar
  4. Chiragiev A, Landsman Z (2007) Multivariate Pareto portfolios: TCE-based capital allocation and divided differences. Scand Actuar J 4:261–280CrossRefMathSciNetGoogle Scholar
  5. Dhaene J, Goovaerts MJ, Kaas R (2003) Economic capital allocation derived from risk measures. N Am Actuar J 7:44–59zbMATHMathSciNetGoogle Scholar
  6. Embrechts P, Resnick S, Samorodnitsky G (1999) Extreme value theory as a risk management tool. N Am Actuar J 3:30–41zbMATHMathSciNetGoogle Scholar
  7. Furman E, Zitikis R (2008) Weighted premium calculation principles. Insur Math Econ 42:459–465zbMATHCrossRefMathSciNetGoogle Scholar
  8. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions. Wiley, New YorkzbMATHGoogle Scholar
  9. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions, vol 1. Models and applications. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  10. Landsman Z, Valdez EA (2005) Tail conditional expectation for exponential dispersion models. Astin Bull 35:189–209zbMATHCrossRefMathSciNetGoogle Scholar
  11. Panjer HH (2002) Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. In: 27th international congress of actuaries, CancunGoogle Scholar
  12. Pareto V (1897) Cours d’Economie politique. Rouge et Cie, ParisGoogle Scholar
  13. Yeh HC (2000) Two multivariate Pareto distributions and their related inferences. Bull Inst Math Acad Sin 28(2):71–86zbMATHGoogle Scholar
  14. Yeh HC (2004) Some properties and characterizations for generalized multivariate Pareto distributions. J Multivar Anal 88:47–60zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceOvidius University of ConstantaConstantaRomania

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