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TVaR-based capital allocation with copulas

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  • Bargès, Mathieu
  • Cossette, Hélène
  • Marceau, Étienne
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    Abstract

    Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 45 (2009)
    Issue (Month): 3 (December)
    Pages: 348-361

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    Handle: RePEc:eee:insuma:v:45:y:2009:i:3:p:348-361

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    Web page: http://www.elsevier.com/locate/inca/505554

    Related research

    Keywords: Capital allocation Tail value at risk Dependence models Copulas Discretization methods;

    References

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    1. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    2. Bauerle, Nicole & Muller, Alfred, 2006. "Stochastic orders and risk measures: Consistency and bounds," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 132-148, February.
    3. Carlo Acerbi & Claudio Nordio & Carlo Sirtori, 2001. "Expected Shortfall as a Tool for Financial Risk Management," Papers cond-mat/0102304, arXiv.org.
    4. Yeo, Keng Leong & Valdez, Emiliano A., 2006. "Claim dependence with common effects in credibility models," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 609-629, June.
    5. Søren Asmussen, 2000. "Matrix-analytic Models and their Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 27(2), pages 193-226.
    6. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    7. 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.
    8. Furman, Edward & Landsman, Zinoviy, 2005. "Risk capital decomposition for a multivariate dependent gamma portfolio," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 635-649, December.
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    Citations

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    Cited by:
    1. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    2. Cossette, Hélène & Landriault, David & Marceau, Etienne & Moutanabbir, Khouzeima, 2012. "Analysis of the discounted sum of ascending ladder heights," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 393-401.
    3. Willmot, Gordon E. & Woo, Jae-Kyung, 2012. "On the analysis of a general class of dependent risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 134-141.
    4. Giuseppe arbia, 2014. "Least quartic Regression Criterion with Application to Finance," Papers 1403.4171, arXiv.org.
    5. Cossette, Hélène & Côté, Marie-Pier & Marceau, Etienne & Moutanabbir, Khouzeima, 2013. "Multivariate distribution defined with Farlie–Gumbel–Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 560-572.
    6. Gijbels, Irène & Sznajder, Dominik, 2013. "Testing tail monotonicity by constrained copula estimation," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 338-351.
    7. Loisel, Stéphane & Trufin, Julien, 2014. "Properties of a risk measure derived from the expected area in red," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 191-199.
    8. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
    9. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
    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. Cossette, Hélène & Mailhot, Mélina & Marceau, Étienne, 2012. "TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 247-256.
    12. Areski Cousin & Elena Di Bernardinoy, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.

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