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OPTIMAL PROPORTIONAL REINSURANCE UNDER TWO CRITERIA: MAXIMIZING THE EXPECTED UTILITY AND MINIMIZING THE VALUE AT RISK

Part of: Mathematical economics

Published online by Cambridge University Press:  06 January 2011

ZHIBIN LIANG  and
JUNYI GUO
ZHIBIN LIANG*
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210046, PR China (email: liangzhibin111@hotmail.com)
JUNYI GUO
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China
*
✉For correspondence; e-mail: liangzhibin111@hotmail.com
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Abstract

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We consider the optimal proportional reinsurance from an insurer’s point of view to maximize the expected utility and minimize the value at risk. Under the general premium principle, we prove the existence and uniqueness of the optimal strategies and Pareto optimal solution, and give the relationship between the optimal strategies. Furthermore, we study the optimization problem with the variance premium principle. When the total claim sizes are normally distributed, explicit expressions for the optimal strategies and Pareto optimal solution are obtained. Finally, some numerical examples are presented to show the impact of the major model parameters on the optimal results.

Keywords

proportional reinsuranceexpected utilityvalue at riskpremium principlePareto optimal solution

MSC classification

Secondary: 91B30: Risk theory, insurance
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 4 , April 2010 , pp. 449 - 463
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Basak, S. and Shapiro, A., “Value-at-risk-based risk management: optimal policies and asset prices”, Rev. Financ. Stud. 14 (2001) 371–405.CrossRefGoogle Scholar
[2]Bernard, C. and Tian, W., “Optimal reinsurance arrangements under tail risk measures”, J. Risk Insurance 76 (2009) 709–725.CrossRefGoogle Scholar
[3]Cai, J. and Tan, K., “Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures”, Astin Bull. 37 (2007) 93–112.CrossRefGoogle Scholar
[4]Cai, J., Tan, K., Weng, C. and Zhang, Y., “Optimal reinsurance under VaR and CTE risk measures”, Insurance Math. Econom. 43 (2008) 85–196.CrossRefGoogle Scholar
[5]Centeno, M. L., “Measuring the effects of reinsurance by the adjustment coefficient”, Insurance Math. Econom. 5 (1986) 169–182.CrossRefGoogle Scholar
[6]Dickson, D. C. M., Insurance risk and ruin (Cambridge University Press, Cambridge, 2004).Google Scholar
[7]Ehrgott, M., Multicriteria optimization (Springer, Berlin, 2000).CrossRefGoogle Scholar
[8]Embrechts, P., Klüppelberg, C. and Mikosch, T., Modelling extremal events (Springer, Berlin, 1997).CrossRefGoogle Scholar
[9]Emmer, S., Klüppelberg, C. and Korn, R., “Optimal portfolios with bounded capital at risk”, Math. Finance 11 (2001) 365–384.CrossRefGoogle Scholar
[10]Gerber, H., An introduction to mathematical risk theory (S. S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Philadelphia, 1979).Google Scholar
[11]Goovaerts, M., De Vylder, F. and Haezendonck, J., Insurance premiums (North-Holland, Amsterdam, 1984).Google Scholar
[12]Grandell, J., Aspects of risk theory (Springer, New York, 1991).CrossRefGoogle Scholar
[13]Hald, M. and Schmidli, H., “On the maximization of the adjustment coefficient under proportional reinsurance”, Astin Bull. 34 (2004) 75–83.CrossRefGoogle Scholar
[14]Huang, H. H., “Optimal insurance contract under a value-at-risk constraint”, Geneva Risk Insurance Rev. 31 (2006) 91–110.CrossRefGoogle Scholar
[15]Irgens, C. and Paulsen, J., “Optimal control of risk exposure, reinsurance and investments for insurance portfolios”, Insurance Math. Econom. 35 (2004) 21–51.CrossRefGoogle Scholar
[16]Kaluszka, M., “Optimal reinsurance under mean-variance premium principles”, Insurance Math. Econom. 28 (2001) 61–67.CrossRefGoogle Scholar
[17]Kaluszka, M., “Mean-variance optimal reinsurance arrangements”, Scand. Actuar. J. 1 (2004) 28–41.CrossRefGoogle Scholar
[18]Kaluszka, M. and Okolewski, A., “An extension of Arrow’s result on optimal reinsurance contract”, J. Risk Insurance 75 (2008) 275–288.CrossRefGoogle Scholar
[19]Liang, Z., “Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion”, Acta Math. Appl. Sin. Engl. Ser. 23 (2007) 477–488.CrossRefGoogle Scholar
[20]Liang, Z. and Guo, J., “Optimal proportional reinsurance and ruin probability”, Stoch. Models 23 (2007) 333–350.CrossRefGoogle Scholar
[21]Liang, Z. and Guo, J., “Upper bound for ruin probabilities under optimal investment and proportional reinsurance”, Appl. Stoch. Models Bus. Ind. 24 (2008) 109–128.CrossRefGoogle Scholar
[22]Luo, S., Taksar, M. and Tsoi, A., “On reinsurance and investment for large insurance portfolios”, Insurance Math. Econom. 42 (2008) 434–444.CrossRefGoogle Scholar
[23]Promislow, S. D. and Young, V. R., “Minimizing the probability of ruin when claims follow Brownian motion with drift”, N. Am. Actuar. J. 9 (2005) 109–128.Google Scholar
[24]Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J., Stochastic processes for insurance and finance (Wiley, Chichester, 1999).CrossRefGoogle Scholar
[25]Schmidli, H., “Optimal proportional reinsurance policies in a dynamic setting”, Scand. Actuar. J. 1 (2001) 55–68.CrossRefGoogle Scholar
[26]Schmidli, H., “On minimizing the ruin probability by investment and reinsurance”, Ann. Appl. Probab. 12 (2002) 890–907.CrossRefGoogle Scholar
[27]Straub, E., Non-life insurance mathematics (Springer, Berlin, 1988).CrossRefGoogle Scholar
[28]Wang, C. P., Shyu, D. and Huang, H. H., “Optimal insurance design under value-at-risk framework”, Geneva Risk Insurance Rev. 30 (2005) 161–179.CrossRefGoogle Scholar
[29]Waters, H., “Excess of loss reinsurance limits”, Scand. Actuar. J. 1 (1979) 37–43.CrossRefGoogle Scholar
[30]Zhang, X., Zhang, K. and Yu, X., “Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth”, Insurance Math. Econom. 44 (2009) 473–478.CrossRefGoogle Scholar