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Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution

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  • Enrique Calder'in-Ojeda
  • Yuyu Chen
  • Soon Wei Tan

Abstract

Capital allocation is a procedure used to assess the risk contributions of individual risk components to the total risk of a portfolio. While the conditional tail expectation (CTE)-based capital allocation is arguably the most popular capital allocation method, its inability to reflect important tail behaviour of losses necessitates a more accurate approach. In this paper, we introduce a new capital allocation method based on the tail central moments (TCM), generalising the tail covariance allocation informed by the tail variance. We develop analytical expressions of the TCM as well as the TCM-based capital allocation for the class of normal mean-variance mixture distributions, which is widely used to model asymmetric and heavy-tailed data in finance and insurance. As demonstrated by a numerical analysis, the TCM-based capital allocation captures several significant patterns in the tail region of equity losses that remain undetected by the CTE, enhancing the understanding of the tail risk contributions of risk components.

Suggested Citation

  • Enrique Calder'in-Ojeda & Yuyu Chen & Soon Wei Tan, 2026. "Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution," Papers 2601.00568, arXiv.org.
    • Handle: RePEc:arx:papers:2601.00568
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    References listed on IDEAS

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    1. Yannick Hoga, 2019. "Extreme Conditional Tail Moment Estimation under Serial Dependence," Journal of Financial Econometrics, Oxford University Press, vol. 17(4), pages 587-615.
    2. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    3. Denuit, Michel, 2020. "Size-Biased Risk Measures of Compound Sums," LIDAM Reprints ISBA 2020034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Esmat Jamshidi Eini & Hamid Khaloozadeh, 2021. "Tail conditional moment for generalized skew-elliptical distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 48(13-15), pages 2285-2305, November.
    5. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    6. Michael Kalkbrener, 2005. "An Axiomatic Approach To Capital Allocation," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 425-437, July.
    7. Joseph Kim, 2010. "Conditional Tail Moments of the Exponential Family and Its Related Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(2), pages 198-216.
    8. Gómez, Fabio & Tang, Qihe & Tong, Zhiwei, 2022. "The gradient allocation principle based on the higher moment risk measure," Journal of Banking & Finance, Elsevier, vol. 143(C).
    9. PAVLO A. Krokhmal, 2007. "Higher moment coherent risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 373-387.
    10. Ignatieva, Katja & Landsman, Zinoviy, 2019. "Conditional tail risk measures for the skewed generalised hyperbolic family," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 98-114.
    11. Kim, Joseph H.T. & Kim, So-Yeun, 2019. "Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 145-157.
    12. Landsman, Zinoviy & Pat, Nika & Dhaene, Jan, 2013. "Tail Variance premiums for log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 441-447.
    13. Dirk Tasche, 2007. "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle," Papers 0708.2542, arXiv.org, revised Jun 2008.
    14. Virginie Konlack Socgnia & Diane Wilcox, 2014. "A Comparison of Generalized Hyperbolic Distribution Models for Equity Returns," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    15. Jan Dhaene & Andreas Tsanakas & Emiliano A. Valdez & Steven Vanduffel, 2012. "Optimal Capital Allocation Principles," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 79(1), pages 1-28, March.
    16. Vernic, Raluca, 2006. "Multivariate skew-normal distributions with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 413-426, April.
    17. 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, December.
    18. Ignatieva, Katja & Landsman, Zinoviy, 2025. "Tail variance for generalised hyper-elliptical models," ASTIN Bulletin, Cambridge University Press, vol. 55(1), pages 144-167, January.
    19. Paul A. Samuelson, 1970. "The Fundamental Approximation Theorem of Portfolio Analysis in terms of Means, Variances and Higher Moments," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 37(4), pages 537-542.
    20. Jiandong Ren, 2022. "Tail Moments of Compound Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 26(3), pages 336-350, August.
    21. Campbell Harvey & John Liechty & Merrill Liechty & Peter Muller, 2010. "Portfolio selection with higher moments," Quantitative Finance, Taylor & Francis Journals, vol. 10(5), pages 469-485.
    22. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    23. Dirk Tasche, 2001. "Conditional Expectation as Quantile Derivative," Papers math/0104190, arXiv.org.
    24. Michel Denuit, 2020. "Size-Biased Risk Measures of Compound Sums," North American Actuarial Journal, Taylor & Francis Journals, vol. 24(4), pages 512-532, October.
    25. Virginie Konlack Socgnia & Diane Wilcox, 2014. "A Comparison of Generalized Hyperbolic Distribution Models for Equity Returns," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-15, June.
    26. 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.
    27. Ignatieva, Katja & Landsman, Zinoviy, 2015. "Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 172-186.
    28. Ignatieva, Katja & Landsman, Zinoviy, 2021. "A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 437-465.
    29. Necula, Ciprian, 2009. "Modeling Heavy-Tailed Stock Index Returns Using the Generalized Hyperbolic Distribution," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 6(2), pages 118-131, June.
    30. Laeven, Roger J. A. & Goovaerts, Marc J., 2004. "An optimization approach to the dynamic allocation of economic capital," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 299-319, October.
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