IDEAS home Printed from https://ideas.repec.org/p/mse/cesdoc/17008r.html
   My bibliography  Save this paper

A novel multivariate risk measure: the Kendall VaR

Author

Listed:

Abstract

The definition of multivariate Value at Risk is a challenging problem, whose most common solutions are given by the lower- and upper-orthant VaRs, which are based on copulas: the lower-orthant VaR is indeed the quantile of the multivariate distribution function, whereas the upper-orthant VaR is the quantile of the multivariate survival function. In this paper we introduce a new multivariate Value at Risk, referred to as the Kendall Value at Risk, which linkd the copula approach to an alternative definition of multivariate quantiles, known as the quantile surface, which is not used in finance, to our knowledge. We more precisely transform the notion of orthant VaR tanks to the Kendall function so as to get a multivariate VaR, that is to say a set of loss vectors, with some advantageous properties compared to the standard orthant VaR: i/ it is based on a total order, ii/ the probability level of the VaR is consistent with the probability measure of the set of the more severe loss vectors, iii/ the d-dimensional Vars based on the distribution function or on the survival function have vectors in common, which conciliate both upper- and lower-orthant approaches. We quantify the differences between this new Kendall VaR and orthant VaRs. In particular, we show that Kendall VaRs are less (respectively more) conservative than lower-orthant (resp. upper-orthant) VaRs. the definition and the properties of the Kendall VaR are illustrated using Gumbel and Clayton copulas with lognormal marginal distributions and several levels of risk

Suggested Citation

  • 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 17008r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Apr 2018.
  • Handle: RePEc:mse:cesdoc:17008r
    as

    Download full text from publisher

    File URL: ftp://mse.univ-paris1.fr/pub/mse/CES2017/17008R.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nilay Noyan & Gábor Rudolf, 2013. "Optimization with Multivariate Conditional Value-at-Risk Constraints," Operations Research, INFORMS, vol. 61(4), pages 990-1013, August.
    2. Abdous, B. & Theodorescu, R., 1992. "Note on the spatial quantile of a random vector," Statistics & Probability Letters, Elsevier, vol. 13(4), pages 333-336, March.
    3. Dominique Guégan & Wayne Tarrant, 2012. "On the necessity of five risk measures," Annals of Finance, Springer, vol. 8(4), pages 533-552, November.
    4. Dominique Guegan & Bertrand Hassani, 2015. "Risk or Regulatory Capital? Bringing distributions back in the foreground," Post-Print halshs-01169268, HAL.
    5. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    6. Matthieu Garcin & Dominique Guegan, 2012. "Extreme values of random or chaotic discretization steps," Documents de travail du Centre d'Economie de la Sorbonne 12033, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    7. Dominique Guegan & Matthieu Garcin, 2012. "Extreme values of random or chaotic discretization steps and connected networks," Post-Print halshs-00750231, HAL.
    8. Dominique Guegan & Bertrand Hassani, 2015. "Risk or Regulatory Capital? Bringing distributions back in the foreground," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01169268, HAL.
    9. Cousin, Areski & Di Bernardino, Elena, 2013. "On multivariate extensions of Value-at-Risk," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 32-46.
    10. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    11. 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.
    12. Dominique Guegan & Pierre-André Maugis, 2010. "An Econometric Study of Vine Copulas," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00492124, HAL.
    13. Dominique Guegan & Pierre-André Maugis, 2010. "An Econometric Study of Vine Copulas," Post-Print halshs-00492124, HAL.
    14. 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, May.
    15. Dominique Gu�gan & Bertrand Hassani, 2015. "Risk or Regulatory Capital? Bringing distributions back in the foreground," Working Papers 2015:18, Department of Economics, University of Venice "Ca' Foscari".
    16. Dominique Guegan & Pierre-André Maugis, 2008. "New prospects on vines," Documents de travail du Centre d'Economie de la Sorbonne b08095, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Mar 2010.
    17. Dominique Guegan & Wayne Tarrant, 2012. "On the Necessity of Five Risk Measures," Post-Print halshs-00721339, HAL.
    18. Dominique Guegan & Bertrand K. Hassani, 2016. "More Accurate Measurement for Enhanced Controls: VaR vs ES?," Documents de travail du Centre d'Economie de la Sorbonne 16015, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    19. Dominique Guegan & Bertrand Hassani, 2013. "Multivariate VaRs for operational risk capital computation: a vine structure approach," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00645778, HAL.
    20. repec:hal:wpaper:halshs-00721339 is not listed on IDEAS
    21. Dominique Guegan & Bertrand K Hassani, 2015. "Risk or Regulatory Capital? Bringing distributions back in the foreground," Documents de travail du Centre d'Economie de la Sorbonne 15046, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    22. Dominique Guegan & Bertrand Hassani, 2013. "Multivariate VaRs for operational risk capital computation: a vine structure approach," Post-Print halshs-00645778, HAL.
    23. Dominique Guegan & Matthieu Garcin, 2012. "Extreme values of random or chaotic discretization steps and connected networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00750231, HAL.
    24. Dominique Guegan & Bertrand Hassani, 2013. "Multivariate VaRs for operational risk capital computation: a vine structure approach," PSE-Ecole d'économie de Paris (Postprint) halshs-00645778, HAL.
    25. Elena Di Bernardino & Didier Rullière, 2016. "On tail dependence coefficients of transformed multivariate Archimedean copulas," Post-Print hal-00992707, HAL.
    26. Yebin Cheng & Jan G. de Gooijer, 2004. "On the u-th Geometric Conditional Quantile," Tinbergen Institute Discussion Papers 04-072/4, Tinbergen Institute.
    27. Dominique Guegan & Bertrand Hassani, 2016. "More Accurate Measurement for Enhanced Controls: VaR vs ES?," Post-Print halshs-01281940, HAL.
    28. Dominique Guegan & Bertrand Hassani, 2016. "More Accurate Measurement for Enhanced Controls: VaR vs ES?," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01281940, HAL.
    29. Matthieu Garcin & Dominique Guegan, 2012. "Extreme values of random or chaotic discretization steps," Post-Print hal-00706825, HAL.
    30. Dominique Guegan & Pierre-André Maugis, 2010. "New Prospects on Vines," Post-Print halshs-00348884, HAL.
    31. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
    32. repec:dau:papers:123456789/353 is not listed on IDEAS
    33. Dominique Guegan & Wayne Tarrant, 2012. "On the Necessity of Five Risk Measures," PSE-Ecole d'économie de Paris (Postprint) halshs-00721339, HAL.
    34. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 391-399, July.
    35. Dominique Guegan & Matthieu Garcin, 2012. "Extreme values of random or chaotic discretization steps and connected networks," PSE-Ecole d'économie de Paris (Postprint) halshs-00750231, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2018. "A novel multivariate risk measure: the Kendall VaR," Post-Print halshs-01467857, HAL.
    2. 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.
    3. Matthieu Garcin & Dominique Guegan & Bertrand Hassani, 2018. "A novel multivariate risk measure: the Kendall VaR," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01467857, HAL.
    4. Dominique Guegan & Bertrand K. Hassani, 2019. "Risk Measurement," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02119256, HAL.
    5. Klaus Herrmann & Marius Hofert & Melina Mailhot, 2017. "Multivariate Geometric Expectiles," Papers 1704.01503, arXiv.org, revised Jan 2018.
    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. Elena Di Bernardino & Clémentine Prieur, 2014. "Estimation of multivariate conditional-tail-expectation using Kendall's process," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(2), pages 241-267, June.
    8. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00800997, HAL.
    9. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007, arXiv.org, revised May 2016.
    10. 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.
    11. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
    12. Cousin, Areski & Di Bernardino, Elena, 2013. "On multivariate extensions of Value-at-Risk," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 32-46.
    13. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    14. Matthieu Garcin & Maxime L. D. Nicolas, 2021. "Nonparametric estimator of the tail dependence coefficient: balancing bias and variance," Papers 2111.11128, arXiv.org, revised Jul 2023.
    15. Hélène Cossette & Mélina Mailhot & Étienne Marceau & Mhamed Mesfioui, 2016. "Vector-Valued Tail Value-at-Risk and Capital Allocation," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 653-674, September.
    16. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Post-Print hal-00800997, HAL.
    17. Dominique Guegan & Bertrand K. Hassani, 2016. "Risk Measures At Risk- Are we missing the point? Discussions around sub-additivity and distortion," Post-Print halshs-01318093, HAL.
    18. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    19. Dominique Guegan & Bertrand K. Hassani, 2016. "Risk Measures At Risk- Are we missing the point? Discussions around sub-additivity and distortion," Documents de travail du Centre d'Economie de la Sorbonne 16039, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    20. Dominique Guegan & Bertrand K. Hassani, 2016. "Risk Measures At Risk- Are we missing the point? Discussions around sub-additivity and distortion," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01318093, HAL.

    More about this item

    Keywords

    Value at Risk; multivariate quantile; risk measure; Kendall function; copula; total order;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:mse:cesdoc:17008r. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Lucie Label (email available below). General contact details of provider: https://edirc.repec.org/data/cenp1fr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.