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Diversification in heavy-tailed portfolios: properties and pitfalls

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  • Mainik, Georg
  • Embrechts, Paul

Abstract

We discuss risk diversification in multivariate regularly varying models and provide explicit formulas for Value-at-Risk asymptotics in this case. These results allow us to study the influence of the portfolio weights, the overall loss severity, and the tail dependence structure on large portfolio losses. We outline sufficient conditions for the sub- and superadditivity of the asymptotic portfolio risk in multivariate regularly varying models and discuss the case when these conditions are not satisfied. We provide several examples to illustrate the resulting variety of diversification effects and the crucial impact of the tail dependence structure in infinite mean models. These examples show that infinite means in multivariate regularly varying models do not necessarily imply negative diversification effects. This implication is true if there is no loss-gain compensation in the tails, but not in general. Depending on the loss-gain compensation, asymptotic portfolio risk can be subadditive, superadditive, or neither.

Suggested Citation

  • 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.
    • Handle: RePEc:cup:anacsi:v:7:y:2013:i:01:p:26-45_00
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    Cited by:

    1. Di Lascio, F. Marta L. & Giammusso, Davide & Puccetti, Giovanni, 2018. "A clustering approach and a rule of thumb for risk aggregation," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 236-248.
    2. John H. J. Einmahl & Fan Yang & Chen Zhou, 2021. "Testing the Multivariate Regular Variation Model," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 907-919, October.
    3. Mainik, Georg & Mitov, Georgi & Rüschendorf, Ludger, 2015. "Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz," Journal of Empirical Finance, Elsevier, vol. 32(C), pages 115-134.
    4. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    5. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Discussion Paper 2023-001, Tilburg University, Center for Economic Research.
    6. A. Ford Ramsey & Barry K. Goodwin, 2019. "Value-at-Risk and Models of Dependence in the U.S. Federal Crop Insurance Program," JRFM, MDPI, vol. 12(2), pages 1-21, April.
    7. Peter Tankov, 2014. "Tails of weakly dependent random vectors," Papers 1402.4683, arXiv.org, revised Jan 2016.
    8. Fan Yang & Yi Zhang, 2024. "Asymptotics of Sum of Heavy-tailed Risks with Copulas," Papers 2411.09657, arXiv.org.
    9. Georg Mainik & Georgi Mitov & Ludger Ruschendorf, 2015. "Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz," Papers 1505.04045, arXiv.org.
    10. Andreas Mühlbacher & Thomas Guhr, 2018. "Extreme Portfolio Loss Correlations in Credit Risk," Risks, MDPI, vol. 6(3), pages 1-25, July.
    11. Thilo A. Schmitt & Rudi Schafer & Thomas Guhr, 2016. "Credit risk: Taking fluctuating asset correlations into account," Papers 1601.03015, arXiv.org.
    12. Einmahl, John & Krajina, Andrea, 2023. "Empirical Likelihood Based Testing for Multivariate Regular Variation," Other publications TiSEM 261583f5-c571-48c6-8cea-9, Tilburg University, School of Economics and Management.
    13. Andreas Mühlbacher & Thomas Guhr, 2018. "Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations," Risks, MDPI, vol. 6(2), pages 1-25, April.
    14. Packham, Natalie & Kalkbrener, Michael & Overbeck, Ludger, 2018. "Default probabilities and default correlations under stress," IRTG 1792 Discussion Papers 2018-037, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Alexandru V. Asimit & Raluca Vernic & Ricardas Zitikis, 2016. "Background Risk Models and Stepwise Portfolio Construction," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 805-827, September.
    16. 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.
    17. Andreas Muhlbacher & Thomas Guhr, 2018. "Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations," Papers 1803.00261, arXiv.org.
    18. Xia Han & Liyuan Lin & Ruodu Wang, 2023. "Diversification quotients based on VaR and ES," Papers 2301.03517, arXiv.org, revised May 2023.
    19. Xia Han & Liyuan Lin & Ruodu Wang, 2022. "Diversification quotients: Quantifying diversification via risk measures," Papers 2206.13679, arXiv.org, revised Jul 2024.
    20. 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.
    21. 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.

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