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On the Diversification Effect in Solvency II for Extremely Dependent Risks

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  • Yongzhao Chen

    (Department of Mathematics, Statistics and Insurance, The Hang Seng University of Hong Kong, Siu Lek Yuen, Shatin, Hong Kong, China
    Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong, China)

  • Ka Chun Cheung

    (Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong, China)

  • Sheung Chi Phillip Yam

    (Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong, China)

  • Fei Lung Yuen

    (School of Mathematics, Statistics and Actuarial Science, The University of Kent, Canterbury CT2 7NZ, UK)

  • Jia Zeng

    (Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong, China)

Abstract

In this article, we investigate the validity of diversification effect under extreme-value copulas, when the marginal risks of the portfolio are identically distributed, which can be any one having a finite endpoint or belonging to one of the three maximum domains of attraction. We show that Value-at-Risk (V@R) under extreme-value copulas is asymptotically subadditive for marginal risks with finite mean, while it is asymptotically superadditive for risks with infinite mean. Our major findings enrich and supplement the context of the second fundamental theorem of quantitative risk management in existing literature, which states that V@R of a portfolio is typically non-subadditive for non-elliptically distributed risk vectors. In particular, we now pin down when the V@R is super or subadditive depending on the heaviness of the marginal tail risk. According to our results, one can take advantages from the diversification effect for marginal risks with finite mean. This justifies the standard formula for calculating the capital requirement under Solvency II in which imperfect correlations are used for various risk exposures.

Suggested Citation

  • Yongzhao Chen & Ka Chun Cheung & Sheung Chi Phillip Yam & Fei Lung Yuen & Jia Zeng, 2023. "On the Diversification Effect in Solvency II for Extremely Dependent Risks," Risks, MDPI, vol. 11(8), pages 1-22, August.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:8:p:143-:d:1210898
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    References listed on IDEAS

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    1. Genest, Christian & Rivest, Louis-Paul, 1989. "A characterization of gumbel's family of extreme value distributions," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 207-211, August.
    2. Ser-Huang Poon, 2004. "Extreme Value Dependence in Financial Markets: Diagnostics, Models, and Financial Implications," The Review of Financial Studies, Society for Financial Studies, vol. 17(2), pages 581-610.
    3. Hua, Lei, 2017. "On a bivariate copula with both upper and lower full-range tail dependence," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 94-104.
    4. Pfeifer Dietmar & Ragulina Olena, 2021. "Generating unfavourable VaR scenarios under Solvency II with patchwork copulas," Dependence Modeling, De Gruyter, vol. 9(1), pages 327-346, January.
    5. Hua, Lei & Joe, Harry, 2012. "Tail comonotonicity: Properties, constructions, and asymptotic additivity of risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 492-503.
    6. Edward Frees & Emiliano Valdez, 1998. "Understanding Relationships Using Copulas," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 1-25.
    7. Ka Chun Cheung & Hok Kan Ling & Qihe Tang & Sheung Chi Phillip Yam & Fei Lung Yuen, 2019. "On additivity of tail comonotonic risks," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(10), pages 837-866, November.
    8. Gudendorf, Gordon & Segers, Johan, 2012. "Nonparametric estimation of multivariate extreme-value copulas," LIDAM Reprints ISBA 2012011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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