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Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Author

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  • Dobrislav Dobrev∗

    () (Board of Governors of the Federal Reserve System, 20th St. and Constitution Ave. N.W., Washington, DC 20551, USA)

  • Travis D. Nesmith

    () (Board of Governors of the Federal Reserve System, 20th St. and Constitution Ave. N.W., Washington, DC 20551, USA)

  • Dong Hwan Oh

    () (Board of Governors of the Federal Reserve System, 20th St. and Constitution Ave. N.W., Washington, DC 20551, USA)

Abstract

We provide an accurate closed-form expression for the expected shortfall of linear portfolios with elliptically distributed risk factors. Our results aim to correct inaccuracies that originate in Kamdem (2005) and are present also in at least thirty other papers referencing it, including the recent survey by Nadarajah et al. (2014) on estimation methods for expected shortfall. In particular, we show that the correction we provide in the popular multivariate Student t setting eliminates understatement of expected shortfall by a factor varying from at least four to more than 100 across different tail quantiles and degrees of freedom. As such, the resulting economic impact in financial risk management applications could be significant. We further correct such errors encountered also in closely related results in Kamdem (2007 and 2009) for mixtures of elliptical distributions. More generally, our findings point to the extra scrutiny required when deploying new methods for expected shortfall estimation in practice.

Suggested Citation

  • Dobrislav Dobrev∗ & Travis D. Nesmith & Dong Hwan Oh, 2017. "Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 10(1), pages 1-14, February.
  • Handle: RePEc:gam:jjrfmx:v:10:y:2017:i:1:p:5-:d:89239
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    References listed on IDEAS

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    1. Paindaveine, Davy & Siman, Miroslav, 2011. "On directional multiple-output quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 193-212, February.
    2. Yamai, Yasuhiro & Yoshiba, Toshinao, 2002. "Comparative Analyses of Expected Shortfall and Value-at-Risk: Their Estimation Error, Decomposition, and Optimization," Monetary and Economic Studies, Institute for Monetary and Economic Studies, Bank of Japan, vol. 20(1), pages 87-121, January.
    3. Liu, Yong-Jun & Zhang, Wei-Guo, 2013. "Fuzzy portfolio optimization model under real constraints," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 704-711.
    4. Mbairadjim Moussa, A. & Sadefo Kamdem, J. & Terraza, M., 2014. "Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns," Economic Modelling, Elsevier, vol. 39(C), pages 247-256.
    5. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook0501.
    6. Sadefo Kamdem, J. & Genz, A., 2008. "Approximation of multiple integrals over hyperboloids with application to a quadratic portfolio with options," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3389-3407, March.
    7. Sun, Chuanwang & Zhang, Yifan & Peng, Shuijun & Zhang, Wencheng, 2015. "The inequalities of public utility products in China: From the perspective of the Atkinson index," Renewable and Sustainable Energy Reviews, Elsevier, vol. 51(C), pages 751-760.
    8. Kotz,Samuel & Nadarajah,Saralees, 2004. "Multivariate T-Distributions and Their Applications," Cambridge Books, Cambridge University Press, number 9780521826549.
    9. repec:eee:apmaco:v:282:y:2016:i:c:p:187-203 is not listed on IDEAS
    10. O. Scaillet, 2004. "Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 115-129.
    11. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
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    13. Bormetti, Giacomo & Cisana, Enrica & Montagna, Guido & Nicrosini, Oreste, 2007. "A non-Gaussian approach to risk measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 532-542.
    14. Saralees Nadarajah & Bo Zhang & Stephen Chan, 2014. "Estimation methods for expected shortfall," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 271-291, February.
    15. Makdissi, Paul & Sylla, Daouda & Yazbeck, Myra, 2013. "Decomposing health achievement and socioeconomic health inequalities in presence of multiple categorical information," Economic Modelling, Elsevier, vol. 35(C), pages 964-968.
    16. Robert Elliott & Hong Miao, 2009. "VaR and expected shortfall: a non-normal regime switching framework," Quantitative Finance, Taylor & Francis Journals, vol. 9(6), pages 747-755.
    17. G. Bormetti & V. Cazzola & G. Livan & G. Montagna & O. Nicrosini, 2009. "A Generalized Fourier Transform Approach to Risk Measures," Papers 0909.3978, arXiv.org, revised May 2012.
    18. Mbairadjim Moussa, A. & Sadefo Kamdem, J. & Shapiro, A.F. & Terraza, M., 2014. "CAPM with fuzzy returns and hypothesis testing," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 40-57.
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    1. repec:gam:jrisks:v:6:y:2018:i:3:p:92-:d:168425 is not listed on IDEAS
    2. repec:gam:jsusta:v:11:y:2019:i:3:p:720-:d:201942 is not listed on IDEAS

    More about this item

    Keywords

    expectedshortfall; ellipticaldistributions; multivariateStudent t distribution; mixturesof elliptical distributions; accurate closed-form expression;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • E - Macroeconomics and Monetary Economics
    • F2 - International Economics - - International Factor Movements and International Business
    • F3 - International Economics - - International Finance
    • G - Financial Economics

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