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Optimal stop-loss reinsurance: a dependence analysis

Year 2016, Volume: 45 Issue: 2, 497 - 519, 01.04.2016

Abstract

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it
presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of
the paper is to contribute to the analysis of the stop-loss contract in
one period from the point of view of the insurer and the reinsurer.
Firstly, the influence of the parameters of the reinsurance contract on
the correlation coefficient between the cost of the insurer and the cost
of the reinsurer is studied. Secondly, the optimal stop-loss contract is
obtained if the criterion used is the maximization of the joint survival
probability of the insurer and the reinsurer in one period.

References

  • Assa, H. On optimal reinsurance policy with distortion risk measures and premiums, Insur. Math. Econ. 61, 70-75, 2015.
  • Azcue, P. and Muler, N. Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Financ. 15 (2), 261-308, 2005.
  • Balbás, A., Balbás, B., Balbás, R. and Heras, A. Optimal reinsurance under risk and incertainty, Insur. Math. Econ. 60, 61-74, 2015.
  • Balbás, A., Balbás, R. and Heras, A. Optimal reinsurance with general risk measures, Insur. Math. Econ. 44 (3), 374-384, 2009.
  • Bohman, H. and Esscher, F. Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I, Scand. Actuar. J. 1963 (3-4), 173- 225, 1963.
  • Borch, K. An attempt to determine the optimum amount of stop loss reinsurance, Transactions of the 16th International Congress of Actuaries I, 597-610, 1960.
  • Borch, K. The optimal reinsurance treaty, Astin Bull. 5 (2), 293-297, 1969.
  • Borch, K. The mathematical theory of insurance (Lexington Books, Lexington, 1974).
  • Cai, J. and Tan, K. S. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, Astin Bull. 37 (1), 93-112, 2007.
  • Cai, J., Fang, Y., Li, Z. and Willmot, G. E. Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, J. Risk Insur. 80 (1), 145-168, 2013.
  • Cai, J., Tan, K. S., Weng, C. and Zhang, Y. Optimal reinsurance under VaR and CTE risk measures, Insur. Math. Econ. 43 (1), 185-196, 2008.
  • Castañer, A., Claramunt, M. M. and Lefèvre, C. Survival probabilities in bivariate risk models, with application to reinsurance, Insur. Math. Econ. 53 (3), 632-642, 2013.
  • Castañer, A., Claramunt, M. M. and Mármol, M. Ruin probability and time of ruin with a proportional reinsurance threshold strategy, Top 20 (3), 614-638, 2012.
  • Centeno, M. L. and Simões, O. Optimal reinsurance, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 103 (2), 387-404, 2009.
  • Cheung, K. C. Optimal reinsurance revisited - a geometric approach, Astin Bull. 40 (1), 221-239, 2010.
  • Cui, W., Yang, J. and Wu, L. Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insur. Math. Econ. 53 (1), 74-85, 2013
  • Daykin, C. D., Pentikainen, T. and Pesonen, M. Practical risk theory for actuaries (Chapman and Hall, London, 1994).
  • Denuit, M. and Charpentier, A. Mathématiques de l’assurance non-vie: Tome 1, Principes fondamentaux de théorie du risque (Economica, Paris, 2004).
  • Denuit, M. and Varmandele, C. Optimal reinsurance and stop-loss order, Insur. Math. Econ. 22 (1), 229-233, 1998.
  • Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. The concept of comonotonicity in actuarial science and finance: applications, Insur. Math. Econ. 31 (2), 133-161, 2002.
  • Dickson, D. C. M. Insurance risk and ruin (Cambridge University Press, United Kingdom, 2005).
  • Dimitrova, D. S. and Kaishev, V. K. Optimal joint survival reinsurance: an efficient frontier approach, Insur. Math. Econ. 47 (1), 27-35, 2010.
  • Fang, Y. and Qu, Z. Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability, IMA. J. Manag. Math. 25 (1), 89-103, 2014.
  • Gajek, L. and Zagrodny, D. Insurer’s optimal reinsurance strategies, Insur. Math. Econ. 27 (1), 105-112, 2000.
  • Gajek, L. and Zagrodny, D. Reinsurance arrangements maximizing insurer’s survival probability, J. Risk Insur. 71 (3), 421-435, 2004.
  • Gendron, M. and Crepeau, H. On the computation of the aggregate claim distribution when individual claims are inverse gaussian, Insur. Math. Econ. 8 (3), 251-258, 1989.
  • Gerber, H. U. The impact of reinsurance on the insurer’s risk, in: F. De Vylder, M. Goovaerts, J. Haezendonck. Premium calculation in insurance (D. Reidel Publishing Company, Dordrecht, Holland, 1984), 171-181.
  • Guerra, M. and Centeno, M. D. L. Optimal reinsurance policy: the adjustment coefficient and the expected utility criteria, Insur. Math. Econ. 42 (2), 529-539, 2008.
  • Hesselager, O. Some results on optimal reinsurance in terms of the adjustment coefficient, Scand. Actuar. J. 1990 (1), 80-95, 1990.
  • Hoøjgaard, B. and Taksar, M. Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J. 1998 (2), 166-180, 1998.
  • Hoøjgaard, B. and Taksar, M. Optimal proportional reinsurance policies for diffusion models with transaction costs, Insur. Math. Econ. 22 (1), 41-51, 1998.
  • Hürlimann, W. Non-optimality of a linear combination of proportional and non-proportional reinsurance, Insur. Math. Econ. 24 (3), 219-227, 1999.
  • Hürlimann, W. Optimal reinsurance revisited - point of view of cedent an reinsurer, Astin Bull. 41 (2), 547-574, 2011.
  • Ignatov, Z. G., Kaishev, V. K. and Krachunov, R. S. Optimal retention levels, given the joint survival of cedent and reinsurer, Scand. Actuar. J. 2004 (6), 401-430, 2004.
  • Kaas, R. How to (and how not to) compute stop-loss premiums in practice, Insur. Math. Econ. 13 (3), 241-254, 1993.
  • Kaas, R., Goovaerts, M. J., Dhaene, J. and Denuit, M. Modern actuarial risk theory: using R (Springer, Heidelberg, 2008).
  • Kaishev, V. K. and Dimitrova, D. S. Excess of loss reinsurance under joint survival optimality, Insur. Math. Econ. 39 (3), 376-389, 2006.
  • Kaluszka, M. Optimal reinsurance under mean-variance premium principles, Insur. Math. Econ. 28 (1), 61-67, 2001.
  • Liang, Z. and Guo, J. Optimal proportional reinsurance and ruin probability, Stoch. Model. 1984 (2), 65-90, 2007.
  • Pesonen, M. I. Optimal Reinsurance, Scand. Actuar. J. 1990 (1), 80-95, 1984.
  • Salcedo-Sanz, S., Carro-Calvo, L., Claramunt, M. M., Castañer, A. and Mármol, M. Effectively tackling reinsurance problems by using evolutionary and swarm intelligence algorithms, Risks 2 (2), 132-145, 2014.
  • Seal, H. L. Approximations to risk theory’s F(x,t) by means of the gamma distribution, Astin Bull. 9 (1-2), 213-218, 1977.
  • Schmidli, H. Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J. 2001 (1), 55-68, 2001. 519
  • Schmidli, H. On Cramer-Lundberg approximations for ruin probabilities under optimal excess of loss reinsurance, Working Paper 193. Laboratory of Actuarial Mathematics, University of Copenhagen, 2004.
  • Tan, K. S., Weng, C. and Zhang, Y. VaR and CTE criteria for optimal quota-share and stop-loss reinsurance, N. Am. Actuar. J. 13 (4), 459-482, 2009.
  • Van Heerwaarden, A. E., Kaas, R. and Goovaerts, M. J. Optimal reinsurance in relation to ordering of risks, Insur. Math. Econ. 8 (1), 11-17, 1989.
  • Van Wouwe, M., De Vylder, F. and Goovaerts, M. The influence of reinsurance limits on infinite time of ruin probabilities, in: F. De Vylder, M. Goovaerts, J. Haezendonck. Premium calculation in insurance (D. Reidel Publishing Company, Dordrecht, Holland, 1984), 493- 504.
  • Zheng, Y. and Cui, W. Optimal reinsurance with premium constraint under distortion risk measures, Insur. Math. Econ. 59, 109-120, 2014.
  • Zheng, Y., Cui, W. and Yang, J. Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer, J. Syst. Sci. Complex. 28 (1), 122-143, 2015.
  • Zhu, Y., Chi, Y. and Weng, C. Multivariate reinsurance designs for minimizing an insurer’s capital requirement, Insur. Math. Econ. 59, 144-155, 2014.
Year 2016, Volume: 45 Issue: 2, 497 - 519, 01.04.2016

Abstract

References

  • Assa, H. On optimal reinsurance policy with distortion risk measures and premiums, Insur. Math. Econ. 61, 70-75, 2015.
  • Azcue, P. and Muler, N. Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Financ. 15 (2), 261-308, 2005.
  • Balbás, A., Balbás, B., Balbás, R. and Heras, A. Optimal reinsurance under risk and incertainty, Insur. Math. Econ. 60, 61-74, 2015.
  • Balbás, A., Balbás, R. and Heras, A. Optimal reinsurance with general risk measures, Insur. Math. Econ. 44 (3), 374-384, 2009.
  • Bohman, H. and Esscher, F. Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I, Scand. Actuar. J. 1963 (3-4), 173- 225, 1963.
  • Borch, K. An attempt to determine the optimum amount of stop loss reinsurance, Transactions of the 16th International Congress of Actuaries I, 597-610, 1960.
  • Borch, K. The optimal reinsurance treaty, Astin Bull. 5 (2), 293-297, 1969.
  • Borch, K. The mathematical theory of insurance (Lexington Books, Lexington, 1974).
  • Cai, J. and Tan, K. S. Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, Astin Bull. 37 (1), 93-112, 2007.
  • Cai, J., Fang, Y., Li, Z. and Willmot, G. E. Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, J. Risk Insur. 80 (1), 145-168, 2013.
  • Cai, J., Tan, K. S., Weng, C. and Zhang, Y. Optimal reinsurance under VaR and CTE risk measures, Insur. Math. Econ. 43 (1), 185-196, 2008.
  • Castañer, A., Claramunt, M. M. and Lefèvre, C. Survival probabilities in bivariate risk models, with application to reinsurance, Insur. Math. Econ. 53 (3), 632-642, 2013.
  • Castañer, A., Claramunt, M. M. and Mármol, M. Ruin probability and time of ruin with a proportional reinsurance threshold strategy, Top 20 (3), 614-638, 2012.
  • Centeno, M. L. and Simões, O. Optimal reinsurance, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 103 (2), 387-404, 2009.
  • Cheung, K. C. Optimal reinsurance revisited - a geometric approach, Astin Bull. 40 (1), 221-239, 2010.
  • Cui, W., Yang, J. and Wu, L. Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insur. Math. Econ. 53 (1), 74-85, 2013
  • Daykin, C. D., Pentikainen, T. and Pesonen, M. Practical risk theory for actuaries (Chapman and Hall, London, 1994).
  • Denuit, M. and Charpentier, A. Mathématiques de l’assurance non-vie: Tome 1, Principes fondamentaux de théorie du risque (Economica, Paris, 2004).
  • Denuit, M. and Varmandele, C. Optimal reinsurance and stop-loss order, Insur. Math. Econ. 22 (1), 229-233, 1998.
  • Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. The concept of comonotonicity in actuarial science and finance: applications, Insur. Math. Econ. 31 (2), 133-161, 2002.
  • Dickson, D. C. M. Insurance risk and ruin (Cambridge University Press, United Kingdom, 2005).
  • Dimitrova, D. S. and Kaishev, V. K. Optimal joint survival reinsurance: an efficient frontier approach, Insur. Math. Econ. 47 (1), 27-35, 2010.
  • Fang, Y. and Qu, Z. Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability, IMA. J. Manag. Math. 25 (1), 89-103, 2014.
  • Gajek, L. and Zagrodny, D. Insurer’s optimal reinsurance strategies, Insur. Math. Econ. 27 (1), 105-112, 2000.
  • Gajek, L. and Zagrodny, D. Reinsurance arrangements maximizing insurer’s survival probability, J. Risk Insur. 71 (3), 421-435, 2004.
  • Gendron, M. and Crepeau, H. On the computation of the aggregate claim distribution when individual claims are inverse gaussian, Insur. Math. Econ. 8 (3), 251-258, 1989.
  • Gerber, H. U. The impact of reinsurance on the insurer’s risk, in: F. De Vylder, M. Goovaerts, J. Haezendonck. Premium calculation in insurance (D. Reidel Publishing Company, Dordrecht, Holland, 1984), 171-181.
  • Guerra, M. and Centeno, M. D. L. Optimal reinsurance policy: the adjustment coefficient and the expected utility criteria, Insur. Math. Econ. 42 (2), 529-539, 2008.
  • Hesselager, O. Some results on optimal reinsurance in terms of the adjustment coefficient, Scand. Actuar. J. 1990 (1), 80-95, 1990.
  • Hoøjgaard, B. and Taksar, M. Optimal proportional reinsurance policies for diffusion models, Scand. Actuar. J. 1998 (2), 166-180, 1998.
  • Hoøjgaard, B. and Taksar, M. Optimal proportional reinsurance policies for diffusion models with transaction costs, Insur. Math. Econ. 22 (1), 41-51, 1998.
  • Hürlimann, W. Non-optimality of a linear combination of proportional and non-proportional reinsurance, Insur. Math. Econ. 24 (3), 219-227, 1999.
  • Hürlimann, W. Optimal reinsurance revisited - point of view of cedent an reinsurer, Astin Bull. 41 (2), 547-574, 2011.
  • Ignatov, Z. G., Kaishev, V. K. and Krachunov, R. S. Optimal retention levels, given the joint survival of cedent and reinsurer, Scand. Actuar. J. 2004 (6), 401-430, 2004.
  • Kaas, R. How to (and how not to) compute stop-loss premiums in practice, Insur. Math. Econ. 13 (3), 241-254, 1993.
  • Kaas, R., Goovaerts, M. J., Dhaene, J. and Denuit, M. Modern actuarial risk theory: using R (Springer, Heidelberg, 2008).
  • Kaishev, V. K. and Dimitrova, D. S. Excess of loss reinsurance under joint survival optimality, Insur. Math. Econ. 39 (3), 376-389, 2006.
  • Kaluszka, M. Optimal reinsurance under mean-variance premium principles, Insur. Math. Econ. 28 (1), 61-67, 2001.
  • Liang, Z. and Guo, J. Optimal proportional reinsurance and ruin probability, Stoch. Model. 1984 (2), 65-90, 2007.
  • Pesonen, M. I. Optimal Reinsurance, Scand. Actuar. J. 1990 (1), 80-95, 1984.
  • Salcedo-Sanz, S., Carro-Calvo, L., Claramunt, M. M., Castañer, A. and Mármol, M. Effectively tackling reinsurance problems by using evolutionary and swarm intelligence algorithms, Risks 2 (2), 132-145, 2014.
  • Seal, H. L. Approximations to risk theory’s F(x,t) by means of the gamma distribution, Astin Bull. 9 (1-2), 213-218, 1977.
  • Schmidli, H. Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J. 2001 (1), 55-68, 2001. 519
  • Schmidli, H. On Cramer-Lundberg approximations for ruin probabilities under optimal excess of loss reinsurance, Working Paper 193. Laboratory of Actuarial Mathematics, University of Copenhagen, 2004.
  • Tan, K. S., Weng, C. and Zhang, Y. VaR and CTE criteria for optimal quota-share and stop-loss reinsurance, N. Am. Actuar. J. 13 (4), 459-482, 2009.
  • Van Heerwaarden, A. E., Kaas, R. and Goovaerts, M. J. Optimal reinsurance in relation to ordering of risks, Insur. Math. Econ. 8 (1), 11-17, 1989.
  • Van Wouwe, M., De Vylder, F. and Goovaerts, M. The influence of reinsurance limits on infinite time of ruin probabilities, in: F. De Vylder, M. Goovaerts, J. Haezendonck. Premium calculation in insurance (D. Reidel Publishing Company, Dordrecht, Holland, 1984), 493- 504.
  • Zheng, Y. and Cui, W. Optimal reinsurance with premium constraint under distortion risk measures, Insur. Math. Econ. 59, 109-120, 2014.
  • Zheng, Y., Cui, W. and Yang, J. Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer, J. Syst. Sci. Complex. 28 (1), 122-143, 2015.
  • Zhu, Y., Chi, Y. and Weng, C. Multivariate reinsurance designs for minimizing an insurer’s capital requirement, Insur. Math. Econ. 59, 144-155, 2014.
There are 50 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Anna Castañer This is me

M. Mercè Claramunt This is me

Publication Date April 1, 2016
Published in Issue Year 2016 Volume: 45 Issue: 2

Cite

APA Castañer, A., & Claramunt, M. M. (2016). Optimal stop-loss reinsurance: a dependence analysis. Hacettepe Journal of Mathematics and Statistics, 45(2), 497-519.
AMA Castañer A, Claramunt MM. Optimal stop-loss reinsurance: a dependence analysis. Hacettepe Journal of Mathematics and Statistics. April 2016;45(2):497-519.
Chicago Castañer, Anna, and M. Mercè Claramunt. “Optimal Stop-Loss Reinsurance: A Dependence Analysis”. Hacettepe Journal of Mathematics and Statistics 45, no. 2 (April 2016): 497-519.
EndNote Castañer A, Claramunt MM (April 1, 2016) Optimal stop-loss reinsurance: a dependence analysis. Hacettepe Journal of Mathematics and Statistics 45 2 497–519.
IEEE A. Castañer and M. M. Claramunt, “Optimal stop-loss reinsurance: a dependence analysis”, Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, pp. 497–519, 2016.
ISNAD Castañer, Anna - Claramunt, M. Mercè. “Optimal Stop-Loss Reinsurance: A Dependence Analysis”. Hacettepe Journal of Mathematics and Statistics 45/2 (April 2016), 497-519.
JAMA Castañer A, Claramunt MM. Optimal stop-loss reinsurance: a dependence analysis. Hacettepe Journal of Mathematics and Statistics. 2016;45:497–519.
MLA Castañer, Anna and M. Mercè Claramunt. “Optimal Stop-Loss Reinsurance: A Dependence Analysis”. Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, 2016, pp. 497-19.
Vancouver Castañer A, Claramunt MM. Optimal stop-loss reinsurance: a dependence analysis. Hacettepe Journal of Mathematics and Statistics. 2016;45(2):497-519.