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On Multivariate Extensions of Value-at-Risk


  • Areski Cousin

    () (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Elena Di Bernadino

    () (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)


In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.

Suggested Citation

  • Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
  • Handle: RePEc:hal:wpaper:hal-00638382
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    References listed on IDEAS

    1. Christian Genest & Jean-François Quessy & Bruno Rémillard, 2006. "Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 337-366.
    2. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    3. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    4. Imen Bentahar, 2006. "Tail Conditional Expectation for vector-valued Risks," SFB 649 Discussion Papers SFB649DP2006-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. repec:dau:papers:123456789/353 is not listed on IDEAS
    6. Wei, Gang & Hu, Taizhong, 2002. "Supermodular dependence ordering on a class of multivariate copulas," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 375-385, May.
    7. Bühlmann, H. & Gagliardi, B. & Gerber, H. U. & Straub, E., 1977. "Some Inequalities for Stop-Loss Premiums," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 9(1-2), pages 75-83, January.
    8. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    9. Dirk Tasche, 2007. "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle," Papers 0708.2542,, revised Jun 2008.
    10. Mathieu Bargès & Hélène Cossette & Etienne Marceau, 2009. "TVaR-based capital allocation with copulas," Working Papers hal-00431265, HAL.
    11. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232.
    12. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    13. Barbe, Philippe & Genest, Christian & Ghoudi, Kilani & Rémillard, Bruno, 1996. "On Kendall's Process," Journal of Multivariate Analysis, Elsevier, vol. 58(2), pages 197-229, August.
    14. Bargès, Mathieu & Cossette, Hélène & Marceau, Étienne, 2009. "TVaR-based capital allocation with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 348-361, December.
    15. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
    16. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 391-399, July.
    17. Celine Gauthier & Alfred Lehar & Moez Souissi, 2010. "Macroprudential Regulation and Systemic Capital Requirements," Staff Working Papers 10-4, Bank of Canada.
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    Cited by:

    1. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2017. "A novel multivariate risk measure: the Kendall VaR," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01467857, HAL.
    2. Elena Di Bernardino & Didier Rullière, 2017. "A note on upper-patched generators for Archimedean copulas," Post-Print hal-01347869, HAL.
    3. Ra'ul Torres & Rosa E. Lillo & Henry Laniado, 2015. "A Directional Multivariate Value at Risk," Papers 1502.00908,
    4. Klaus Herrmann & Marius Hofert & Melina Mailhot, 2017. "Multivariate Geometric Expectiles," Papers 1704.01503,, revised Jan 2018.
    5. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007,, revised May 2016.
    6. Di Bernardino, E. & Fernández-Ponce, J.M. & Palacios-Rodríguez, F. & Rodríguez-Griñolo, M.R., 2015. "On multivariate extensions of the conditional Value-at-Risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 1-16.
    7. Sordo, Miguel A., 2016. "A multivariate extension of the increasing convex order to compare risks," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 224-230.
    8. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
    9. Véronique Maume-Deschamps & Didier Rullière & Khalil Said, 2017. "Multivariate Extensions Of Expectiles Risk Measures," Working Papers hal-01367277, HAL.
    10. Torres, Raúl & Lillo, Rosa E. & Laniado, Henry, 2015. "A directional multivariate value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 111-123.
    11. Elena Di Bernardino & Didier Rullière, 2012. "Distortions of multivariate risk measures: a level-sets based approach," Working Papers hal-00756387, HAL.
    12. V'eronique Maume-Deschamps & Didier Rulli`ere & Khalil Said, 2015. "Impact of dependence on some multivariate risk indicators," Papers 1507.01175,
    13. Maume-Deschamps Véronique & Said Khalil & Rullière Didier, 2017. "Multivariate extensions of expectiles risk measures," Dependence Modeling, Sciendo, vol. 5(1), pages 20-44, January.
    14. Elena Bernardino & Thomas Laloë & Rémi Servien, 2015. "Estimating covariate functions associated to multivariate risks: a level set approach," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(5), pages 497-526, July.
    15. repec:hal:wpaper:hal-01171395 is not listed on IDEAS
    16. Merve Merakli & Simge Kucukyavuz, 2017. "Vector-Valued Multivariate Conditional Value-at-Risk," Papers 1708.01324,
    17. Bazovkin, Pavel, 2014. "Geometrical framework for robust portfolio optimization," Discussion Papers in Econometrics and Statistics 01/14, University of Cologne, Institute of Econometrics and Statistics.
    18. Areski Cousin & Elena Di Bernardinoy, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.
    19. Zhu, Yunzhou & Chi, Yichun & Weng, Chengguo, 2014. "Multivariate reinsurance designs for minimizing an insurer’s capital requirement," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 144-155.
    20. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2017. "A novel multivariate risk measure: the Kendall VaR," Documents de travail du Centre d'Economie de la Sorbonne 17008, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    21. repec:hal:wpaper:hal-00750873 is not listed on IDEAS

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    Multivariate Risk Measures; Level sets of distribution functions; Kendall distributions; Copulas;

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