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Using Aumann-Shapley Values to Allocate Insurance Risk

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  • Michael Powers

Abstract

The problem of allocating responsibility for risk among members of a portfolio arises in a variety of financial and risk-management contexts. Examples are particularly prominent in the insurance sector, where actuaries have long sought methods for distributing capital (net worth) across a number of distinct exposure units or accounts according to their relative contributions to the total “risk” of an insurer’s portfolio. Although substantial work has been done on this problem, no satisfactory solution has yet been presented for the case of inhomogeneous loss distributions— that is, losses X ∼ FX|λ such that FX|tλ (X) ≠ FtX|λ (X) for some t > 0. The purpose of this article is to show that the value-assignment method of nonatomic cooperative games proposed in 1974 by Aumann and Shapley may be used to solve risk-allocation problems involving losses of this type. This technique is illustrated by providing analytical solutions for a useful class of multivariatenormal loss distributions.

Suggested Citation

  • Michael Powers, 2007. "Using Aumann-Shapley Values to Allocate Insurance Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 113-127.
    • Handle: RePEc:taf:uaajxx:v:11:y:2007:i:3:p:113-127
    DOI: 10.1080/10920277.2007.10597470
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    Citations

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    Cited by:

    1. Seog S. Hun & Shin Sungwhee, 2009. "Comparison between Financial Theory and Cooperative Game Theory in Risk Capital Allocation," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 4(1), pages 1-18, November.
    2. Daniel Bauer & George Zanjani, 2016. "The Marginal Cost of Risk, Risk Measures, and Capital Allocation," Management Science, INFORMS, vol. 62(5), pages 1431-1457, May.
    3. George Zanjani, 2010. "An Economic Approach to Capital Allocation," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(3), pages 523-549, September.
    4. van Gulick, Gerwald & De Waegenaere, Anja & Norde, Henk, 2012. "Excess based allocation of risk capital," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 26-42.
    5. van Gulick, G. & De Waegenaere, A.M.B. & Norde, H.W., 2010. "Excess Based Allocation of Risk Capital," Other publications TiSEM f9231521-fea7-4524-8fea-8, Tilburg University, School of Economics and Management.
    6. Boonen, Tim J. & De Waegenaere, Anja & Norde, Henk, 2020. "A generalization of the Aumann–Shapley value for risk capital allocation problems," European Journal of Operational Research, Elsevier, vol. 282(1), pages 277-287.
    7. Gero Junike & Hauke Stier & Marcus C. Christiansen, 2022. "Profit and loss decomposition in continuous time and approximations," Papers 2212.06733, arXiv.org, revised Dec 2024.
    8. Christoph Frei, 2020. "A New Approach to Risk Attribution and Its Application in Credit Risk Analysis," Risks, MDPI, vol. 8(2), pages 1-13, June.
    9. 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).
    10. Kim, Joseph H.T. & Hardy, Mary R., 2009. "A capital allocation based on a solvency exchange option," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 357-366, June.
    11. Dorothea Diers & Martin Eling & Christian Kraus & Andreas Reuß, 2012. "Market-consistent embedded value in non-life insurance: how to measure it and why," Journal of Risk Finance, Emerald Group Publishing, vol. 13(4), pages 320-346, August.
    12. Stephen J. Mildenhall, 2017. "Actuarial Geometry," Risks, MDPI, vol. 5(2), pages 1-44, June.

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