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Diversification quotients based on VaR and ES

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  • Han, Xia
  • Lin, Liyuan
  • Wang, Ruodu

Abstract

The diversification quotient (DQ) is recently introduced for quantifying the degree of diversification of a stochastic portfolio model. It has an axiomatic foundation and can be defined through a parametric class of risk measures. Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most prominent risk measures widely used in both banking and insurance, we investigate DQ constructed from VaR and ES in this paper. In particular, for the popular models of elliptical and multivariate regular varying (MRV) distributions, explicit formulas are available. The portfolio optimization problems for the elliptical and MRV models are also studied. Our results further reveal favorable features of DQ, both theoretically and practically, compared to traditional diversification indices based on a single risk measure.

Suggested Citation

  • Han, Xia & Lin, Liyuan & Wang, Ruodu, 2023. "Diversification quotients based on VaR and ES," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 185-197.
    • Handle: RePEc:eee:insuma:v:113:y:2023:i:c:p:185-197
    DOI: 10.1016/j.insmatheco.2023.08.006
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    References listed on IDEAS

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    1. Gilles Boevi Koumou & Georges Dionne, 2022. "Coherent Diversification Measures in Portfolio Theory: An Axiomatic Foundation," Risks, MDPI, vol. 10(11), pages 1-19, October.
    2. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    3. Jean-Gabriel Lauzier & Liyuan Lin & Ruodu Wang, 2023. "Pairwise counter-monotonicity," Papers 2302.11701, arXiv.org, revised May 2023.
    4. Johanna F. Ziegel, 2016. "Coherence And Elicitability," Mathematical Finance, Wiley Blackwell, vol. 26(4), pages 901-918, October.
    5. Dhaene, Jan & Denuit, Michel, 1999. "The safest dependence structure among risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 11-21, September.
    6. Paul Embrechts & Haiyan Liu & Tiantian Mao & Ruodu Wang, 2017. "Quantile-Based Risk Sharing with Heterogeneous Beliefs," Swiss Finance Institute Research Paper Series 17-65, Swiss Finance Institute, revised Jan 2018.
    7. Lee, Woojoo & Ahn, Jae Youn, 2014. "On the multidimensional extension of countermonotonicity and its applications," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 68-79.
    8. Bürgi, Roland & Dacorogna, Michel M & Iles, Roger, 2008. "Risk aggregation, dependence structure and diversification benefit," MPRA Paper 10054, University Library of Munich, Germany.
    9. Rockafellar, R.T. & Royset, J.O. & Miranda, S.I., 2014. "Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk," European Journal of Operational Research, Elsevier, vol. 234(1), pages 140-154.
    10. Aharon Ben‐Tal & Marc Teboulle, 2007. "An Old‐New Concept Of Convex Risk Measures: The Optimized Certainty Equivalent," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 449-476, July.
    11. Mainik, Georg & Embrechts, Paul, 2013. "Diversification in heavy-tailed portfolios: properties and pitfalls," Annals of Actuarial Science, Cambridge University Press, vol. 7(1), pages 26-45, March.
    12. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    13. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    14. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    15. Paul Embrechts & Giovanni Puccetti & Ludger Rüschendorf & Ruodu Wang & Antonela Beleraj, 2014. "An Academic Response to Basel 3.5," Risks, MDPI, vol. 2(1), pages 1-24, February.
    16. Ibragimov, Rustam & Jaffee, Dwight & Walden, Johan, 2011. "Diversification disasters," Journal of Financial Economics, Elsevier, vol. 99(2), pages 333-348, February.
    17. Cui, Hengxin & Tan, Ken Seng & Yang, Fan & Zhou, Chen, 2022. "Asymptotic analysis of portfolio diversification," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 302-325.
    18. Paul Embrechts & Bin Wang & Ruodu Wang, 2015. "Aggregation-robustness and model uncertainty of regulatory risk measures," Finance and Stochastics, Springer, vol. 19(4), pages 763-790, October.
    19. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    20. Ruodu Wang & Ričardas Zitikis, 2021. "An Axiomatic Foundation for the Expected Shortfall," Management Science, INFORMS, vol. 67(3), pages 1413-1429, March.
    21. Georg Mainik & Ludger Rüschendorf, 2010. "On optimal portfolio diversification with respect to extreme risks," Finance and Stochastics, Springer, vol. 14(4), pages 593-623, December.
    22. Susanne Emmer & Marie Kratz & Dirk Tasche, 2013. "What is the best risk measure in practice? A comparison of standard measures," Papers 1312.1645, arXiv.org, revised Apr 2015.
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    More about this item

    Keywords

    Value-at-Risk; Expected Shortfall; Diversification quotient; Elliptical models; Regular varying models;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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