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Vector-Valued Tail Value-at-Risk and Capital Allocation

Abstract

Enterprise risk management, actuarial science or finance are practice areas in which risk measures are important to evaluate for heterogeneous classes of homogeneous risks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. They are based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al. (Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived. Notably, they are shown to be positive homogeneous, invariant under translation and subadditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sum of random pairs are presented, allowing to use a more accurate model for dependent classes of homogeneous risks.

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References

  1. Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9(3):203–228

    MathSciNet  Article  MATH  Google Scholar 

  2. Balbás A, Balbás R, Jiménez-Guerra P (2012) Vector risk functions. Mediterr J Math 9(3):563–574

  3. Barnett V (1976) The ordering of multivariate data. J R Stat Soc Ser A 139(3):318–355

    MathSciNet  Article  Google Scholar 

  4. Basel (2003) Trends in risk integration and aggregation, the joint forum, Bank for International Settlements. http://www.bis.org

  5. Basel (2004) International convergence of capital measurement and capital standards: a revised framework

  6. Bedoui R, Dbabis M (2009) Copulas and bivariate risk measures: an application to hedge funds. 26ème journée internationales d’économie monétaire et financière

  7. Belzunce F, Castaño A, Olvera-Cervantes A, Suárez-Llorens A (2007) Quantile curves and dependence structure for bivariate distributions. Comput Stat Data Anal 51(10):5112–5129

    MathSciNet  Article  MATH  Google Scholar 

  8. Cossette H, Mailhot M, Marceau M (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Math Econ 50(2):247–256

    MathSciNet  MATH  Google Scholar 

  9. Cousin A, Di Bernardino E (2013) On multivariate extensions of value-at-risk. J Multivar Anal 119:32–46

    MathSciNet  Article  MATH  Google Scholar 

  10. Cousin A, Di Bernardino E (2014) On multivariate extensions of conditional-tail-expectation. Insur Math Econ 55:272–282

    MathSciNet  Article  MATH  Google Scholar 

  11. Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley Online Library

  12. Di Bernardino E, Fernández-Ponce JM, Palacios Rodríguez F, Rodrígez-Griñolo MR (2015) On multivariate extensions of the conditional Value-at-Risk measure. Insur Math Econ 61:1–16

    MathSciNet  Article  MATH  Google Scholar 

  13. Embrechts P, Puccetti G (2006) Bounds for functions of multivariate risks. J Multivar Anal 97(2):526–547

    MathSciNet  Article  MATH  Google Scholar 

  14. Fernández-Ponce JM, Suárez-Llorens A (2002) Central regions for bivariate distributions. Austrian J Stat 31:2–3

    Google Scholar 

  15. Joe H (1997) Multivariate models and dependence concepts. Chapman and Hall, London

    Book  MATH  Google Scholar 

  16. Jouini E, Meddeb M, Touzi N (2004) Vector-valued coherent risk measures. Finance Stochast 8(4):531–552

    MathSciNet  Article  MATH  Google Scholar 

  17. Mainik G, Schaanning E (2012) On dependence consistency of CoVaR and some other systemic risk measures, arXiv preprint

  18. McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton

    MATH  Google Scholar 

  19. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, New York

    MATH  Google Scholar 

  20. Ruschendorf L (1982) Random variables with maximum sums. Adv Appl Probab 14(3):623–632

    MathSciNet  Article  MATH  Google Scholar 

  21. Shaked M, Shanthikumar JG (2007) Stochastic orders and their applications, 2nd edn. Springer series in Statistics, Springer, New York

    MATH  Google Scholar 

  22. Serfling R (2002) Quantile functions for multivariate analysis: approaches and applications. Stat Neerlandica 56(2):214–232

    MathSciNet  Article  MATH  Google Scholar 

  23. Sordo MA, Suárez-Llorens A, Bello AJ (2015) Comparison of conditional distributions in portfolios of dependent risks. Insur Math Econ 61:62–69

    MathSciNet  Article  MATH  Google Scholar 

  24. Tahar IB, Lépinette E (2012) Vector-valued risk measure processes, Working Paper

  25. Tahar IB, Lépinette E (2014) Vector-valued coherent risk measure processes. Int J Theor Appl Finance 17(02)

  26. Tasche D (1999) Risk contributions and performance measurement. Report of the Lehrstuhl für mathematische Statistik, TU München

  27. Tibiletti L (1993) On a new notion of multidimensional quantile. Metron 51(3–4):77–83

    MathSciNet  MATH  Google Scholar 

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Correspondence to Mélina Mailhot.

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Cossette, H., Mailhot, M., Marceau, É. et al. Vector-Valued Tail Value-at-Risk and Capital Allocation. Methodol Comput Appl Probab 18, 653–674 (2016). https://doi.org/10.1007/s11009-015-9444-9

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Keywords

  • Bivariate Tail Value-at-Risk
  • Multivariate risk measures
  • Capital allocation
  • Copulas
  • Bounds

Mathematics Subject Classification (2010)

  • 62P05
  • 91B30