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Impact of multimodality of distributions on VaR and ES calculations

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  • Dominique Guegan

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, Labex ReFi - UP1 - Université Paris 1 Panthéon-Sorbonne)

  • Bertrand Hassani

    (Grupo Santander, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, Labex ReFi - UP1 - Université Paris 1 Panthéon-Sorbonne)

  • Kehan Li

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, Labex ReFi - UP1 - Université Paris 1 Panthéon-Sorbonne)

Abstract

Unimodal probability distribution has been widely used for Value-at-Risk (VaR) computation by investors, risk managers and regulators. However, financial data may be characterized by distributions having more than one modes. Using a unimodal distribution may lead to bias for risk measure computation. In this paper, we discuss the influence of using multimodal distributions on VaR and Expected Shortfall (ES) calculation. Two multimodal distribution families are considered: Cobb's family and distortion family. We provide two ways to compute the VaR and the ES for them: an adapted rejection sampling technique for Cobb's family and an inversion approach for distortion family. For empirical study, two data sets are considered: a daily data set concerning operational risk and a three month scenario of market portfolio return built five minutes intraday data. With a complete spectrum of confidence levels from 0001 to 0.999, we analyze the VaR and the ES to see the interest of using multimodal distribution instead of unimodal distribution.

Suggested Citation

  • Dominique Guegan & Bertrand Hassani & Kehan Li, 2017. "Impact of multimodality of distributions on VaR and ES calculations," Post-Print halshs-01491990, HAL.
    • Handle: RePEc:hal:journl:halshs-01491990
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01491990
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    References listed on IDEAS

    as
    1. Dominique Guégan & Bertrand Hassani, 2015. "Distortion Risk Measure or the Transformation of Unimodal Distributions into Multimodal Functions," International Series in Operations Research & Management Science, in: Alain Bensoussan & Dominique Guegan & Charles S. Tapiero (ed.), Future Perspectives in Risk Models and Finance, edition 127, pages 71-88, Springer.
    2. Gao, Fuqing & Wang, Shaochen, 2011. "Asymptotic behavior of the empirical conditional value-at-risk," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 345-352.
    3. Dominique Guegan & Bertrand Hassani, 2015. "Distorsion Risk Measure or the Transformation of Unimodal Distributions into Multimodal Functions," Post-Print hal-01310467, HAL.
    4. Pérignon, Christophe & Smith, Daniel R., 2010. "The level and quality of Value-at-Risk disclosure by commercial banks," Journal of Banking & Finance, Elsevier, vol. 34(2), pages 362-377, February.
    5. Hang Chan, Ngai & Deng, Shi-Jie & Peng, Liang & Xia, Zhendong, 2007. "Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 137(2), pages 556-576, April.
    6. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    7. Dominique Guegan & Bertrand Hassani, 2015. "Distorsion Risk Measure or the Transformation of Unimodal Distributions into Multimodal Functions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01310467, HAL.
    8. Bertrand K. Hassani & Wei Yang, 2016. "The Lila distribution and its applications in risk modelling," Documents de travail du Centre d'Economie de la Sorbonne 16068, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Bertrand K. Hassani & Wei Yang, 2016. "The Lila distribution and its applications in risk modelling," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01400186, HAL.
    10. Bertrand K. Hassani & Wei Yang, 2016. "The Lila distribution and its applications in risk modelling," Post-Print halshs-01400186, HAL.
    11. W. R. Gilks & P. Wild, 1992. "Adaptive Rejection Sampling for Gibbs Sampling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 41(2), pages 337-348, June.
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    Keywords

    Risks; Multimodal distributions; Value-at-Risk; Expected Shortfall; Moments method; Adapted rejection sampling; Regulation;
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