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Excess based allocation of risk capital

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  • van Gulick, Gerwald
  • De Waegenaere, Anja
  • Norde, Henk

Abstract

In this paper we propose a new rule to allocate risk capital to portfolios or divisions within a firm. Specifically, we determine the capital allocation that minimizes the excesses of sets of portfolios in a lexicographical sense. The excess of a set of portfolios is defined as the expected loss of that set of portfolios in excess of the amount of risk capital allocated to them. The underlying idea is that large excesses are undesirable, and therefore the goal is to determine the allocation for which the largest excess is as small as possible. We show that this allocation rule yields a unique allocation, and that it satisfies some desirable properties. We also show that the allocation can be determined by solving a series of linear programming problems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:26-42
    DOI: 10.1016/j.insmatheco.2011.09.003
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    1. Dhaene, Jan & Goovaerts, Marc J., 1996. "Dependency of Risks and Stop-Loss Order1," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 201-212, November.
    2. Stevens, R.S.P., 2011. "Longevity risk in life insurance products," Other publications TiSEM 6533a354-2d66-4219-8086-f, Tilburg University, School of Economics and Management.
    3. Kaas, Rob & Laeven, Roger J.A. & Nelsen, Roger B., 2009. "Worst VaR scenarios with given marginals and measures of association," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 146-158, April.
    4. Maschler, M. & Potters, J.A.M. & Tijs, S.H., 1992. "The general nucleolus and the reduced game property," Other publications TiSEM ab187dab-1b5b-40c3-a673-8, Tilburg University, School of Economics and Management.
    5. Fischer, T., 2003. "Risk capital allocation by coherent risk measures based on one-sided moments," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 135-146, February.
    6. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    7. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Tsanakas, Andreas, 2004. "Dynamic capital allocation with distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 223-243, October.
    9. Tsanakas, Andreas & Barnett, Christopher, 2003. "Risk capital allocation and cooperative pricing of insurance liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 33(2), pages 239-254, October.
    10. Carter, Lawrence R. & Lee, Ronald D., 1992. "Modeling and forecasting US sex differentials in mortality," International Journal of Forecasting, Elsevier, vol. 8(3), pages 393-411, November.
    11. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    12. Pitacco, Ermanno & Denuit, Michel & Haberman, Steven & Olivieri, Annamaria, 2009. "Modelling Longevity Dynamics for Pensions and Annuity Business," OUP Catalogue, Oxford University Press, number 9780199547272.
    13. Anja De Waegenaere & Bertrand Melenberg & Ralph Stevens, 2010. "Longevity Risk," De Economist, Springer, vol. 158(2), pages 151-192, June.
    14. 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.
    15. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    16. Michael Powers, 2007. "Using Aumann-Shapley Values to Allocate Insurance Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(3), pages 113-127.
    17. Gary Venter, 2004. "Capital Allocation Survey with Commentary," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(2), pages 96-107.
    18. Michael Sherris, 2006. "Solvency, Capital Allocation, and Fair Rate of Return in Insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(1), pages 71-96, March.
    19. Stevens, R.S.P. & De Waegenaere, A.M.B. & Melenberg, B., 2011. "Longevity Risk and Natural Hedge Potential in Portfolios Of Life Insurance Products : The Effect of Investment Risk," Discussion Paper 2011-036, Tilburg University, Center for Economic Research.
    20. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    21. Jan Dhaene & Mark Goovaerts & Rob Kaas, 2003. "Economic Capital Allocation Derived from Risk Measures," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(2), pages 44-56.
    22. Michael Kalkbrener, 2005. "An Axiomatic Approach To Capital Allocation," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 425-437, July.
    23. Dhaene, J. & Henrard, L. & Landsman, Z. & Vandendorpe, A. & Vanduffel, S., 2008. "Some results on the CTE-based capital allocation rule," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 855-863, April.
    24. Jennifer L. Wang & H.C. Huang & Sharon S. Yang & Jeffrey T. Tsai, 2010. "An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(2), pages 473-497, June.
    25. Marc Goovaerts & Eddy Van den Borre & Roger Laeven, 2005. "Managing Economic and Virtual Economic Capital Within Financial Conglomerates," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(3), pages 77-89.
    26. Tsanakas, Andreas, 2009. "To split or not to split: Capital allocation with convex risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 268-277, April.
    27. Louis J. Billera & David C. Heath, 1982. "Allocation of Shared Costs: A Set of Axioms Yielding A Unique Procedure," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 32-39, February.
    28. 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.
    29. Peter C. Fishburn, 1974. "Exceptional Paper--Lexicographic Orders, Utilities and Decision Rules: A Survey," Management Science, INFORMS, vol. 20(11), pages 1442-1471, July.
    30. Maschler, M & Potters, J A M & Tijs, S H, 1992. "The General Nucleolus and the Reduced Game Property," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(1), pages 85-106.
    31. 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.
    32. 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.
    33. Koster, M.A.L., 1999. "Cost sharing in production situations and network exploitation," Other publications TiSEM 87f45f30-1cc6-48e3-b37a-3, Tilburg University, School of Economics and Management.
    34. Tsai, Jeffrey T. & Wang, Jennifer L. & Tzeng, Larry Y., 2010. "On the optimal product mix in life insurance companies using conditional value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 235-241, February.
    35. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
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    Cited by:

    1. 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.
    2. Jens Leth Hougaard & Aleksandrs Smilgins, 2014. "Risk Capital Allocation: The Lorenz Set," MSAP Working Paper Series 03_2014, University of Copenhagen, Department of Food and Resource Economics.
    3. Jilber Urbina & Miguel Santolino & Montserrat Guillen, 2021. "Covariance Principle for Capital Allocation: A Time-Varying Approach," Mathematics, MDPI, vol. 9(16), pages 1-13, August.
    4. Wei Wang & Huifu Xu, 2023. "Preference robust distortion risk measure and its application," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 389-434, April.
    5. Matthias Fischer & Thorsten Moser & Marius Pfeuffer, 2018. "A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations," Risks, MDPI, vol. 6(4), pages 1-28, December.
    6. Hougaard, Jens Leth & Smilgins, Aleksandrs, 2016. "Risk capital allocation with autonomous subunits: The Lorenz set," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 151-157.
    7. Belles-Sampera, Jaume & Guillén, Montserrat & Santolino, Miguel, 2014. "GlueVaR risk measures in capital allocation applications," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 132-137.
    8. Wang, Wei & Xu, Huifu & Ma, Tiejun, 2023. "Optimal scenario-dependent multivariate shortfall risk measure and its application in risk capital allocation," European Journal of Operational Research, Elsevier, vol. 306(1), pages 322-347.
    9. Boonen, Tim J. & De Waegenaere, Anja & Norde, Henk, 2017. "Redistribution of longevity risk: The effect of heterogeneous mortality beliefs," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 175-188.
    10. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    11. Balog, Dóra & Bátyi, Tamás László & Csóka, Péter & Pintér, Miklós, 2017. "Properties and comparison of risk capital allocation methods," European Journal of Operational Research, Elsevier, vol. 259(2), pages 614-625.
    12. Boonen, T.J. & De Waegenaere, A.M.B. & Norde, H.W., 2012. "A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems," Other publications TiSEM 2c502ef8-76f0-47f5-ab45-1, Tilburg University, School of Economics and Management.

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    More about this item

    Keywords

    Risk capital; Capital allocation; Excesses; Lexicographic minimum;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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