Optimal layer reinsurance on the maximization of the adjustment coefficient

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2016, 6(1): 21-34. doi: 10.3934/naco.2016.6.21

Optimal layer reinsurance on the maximization of the adjustment coefficient

Xuepeng Zhang ^1, and Zhibin Liang ^1,

1.	School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, China, China

Received January 2015 Revised December 2015 Published January 2016

Abstract
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In this paper, we study the optimal retentions for an insurance company, which intends to transfer risk by means of a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained for the Brownian motion risk model as well as the compound Poisson risk model. Moreover, we conclude that under the expected value principle there exists a special layer reinsurance strategy, i.e., excess of loss reinsurance strategy which is better than any other layer reinsurance strategies. Whereas, under the variance premium principle, the pure excess of loss reinsurance is not the optimal layer reinsurance strategy any longer. Some numerical examples are presented to show the impacts of the parameters as well as the premium principles on the optimal results.

Keywords: Brownian motion, compound Poisson process, Ruin probability, layer reinsurance., adjustment coefficient.

Mathematics Subject Classification: Primary: 62P05; Secondary: 91B3.

Citation: Xuepeng Zhang, Zhibin Liang. Optimal layer reinsurance on the maximization of the adjustment coefficient. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 21-34. doi: 10.3934/naco.2016.6.21

References:

[1]	S. Asmussen, Ruin probabilities, World Scientific Press, Singapore, 2000. doi: 10.1142/9789812779311.
[2]	E. Bayraktar and V. Young, Minimizing the probability of lifetime ruin under borrowing constraints, Insurance: Mathematics and Economics, 41 (2007), 196-221. doi: 10.1016/j.insmatheco.2006.10.015.
[3]	C. Bernard and W. Tian, Optimal reinsurance arrangements under tail risk measures, Journal of Risk and Insurance, 76 (2009), 709-725.
[4]	S. Browne, Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937.
[5]	J. Cai and K. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bulletin, 37 (2007), 93-112. doi: 10.2143/AST.37.1.2020800.
[6]	J. Cai, K. Tan, C. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196. doi: 10.1016/j.insmatheco.2008.05.011.
[7]	M. Centeno, Dependent risks and excess of loss reinsurance, Insurance: Mathematics and Economics, 37 (2005), 229-238. doi: 10.1016/j.insmatheco.2004.12.001.
[8]	M. Centeno and O. Simũes, Combining quota-share and excess of loss treaties on the reinsurance of n independent risks, ASTIN Bulletin, 21 (2002), 41-55.
[9]	H. Gerber, An Introduction to Mathematical Risk Theory, In: S. S. Huebner Foundation Monograph, Series No. 8. Irwin, Homewood, Illinois, 1979.
[10]	J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.
[11]	M. Guerra and M. Centeno, Optimal reinsurance for variance related premium calculation principles, ASTIN Bulletin, 40 (2010), 97-121. doi: 10.2143/AST.40.1.2049220.
[12]	M. Hald and H. Schmidli, On the maximization of the adjustment coefficient under proportioal reinsurance, ASTIN Bulletin, 34 (2004), 75-83. doi: 10.2143/AST.34.1.504955.
[13]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51. doi: 10.1016/j.insmatheco.2004.04.004.
[14]	M. Kaluszka, Optimal reinsurance under mean-variance premium principles, Insurance: Mathematics and Economics, 28 (2001), 61-67. doi: 10.1016/S0167-6687(00)00066-4.
[15]	M. Kaluszka, Mean-variance optimal reinsurance arrangements, Scandinavian Actuarial Journal, 1 (2004), 28-41. doi: 10.1080/03461230410019222.
[16]	Z. Liang and E. Bayraktar, Optimal proportional reinsurance and investment with unobservable claim sizes and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166. doi: 10.1016/j.insmatheco.2014.01.011.
[17]	Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350. doi: 10.1080/15326340701300894.
[18]	Z. Liang and J. Guo, Ruin probabilities under optimal combining quota-share and excess of loss reinsurance, Acta Mathematica Sinica, Chinese Series, 9 (2010), 858-870.
[19]	Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.
[20]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.
[21]	D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.
[22]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68. doi: 10.1080/034612301750077338.
[23]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173.
[24]	X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess of loss reinsurance policies in a dynamic setting, Applied Stochastic Model in Business and Industry, 23 (2007), 63-71. doi: 10.1002/asmb.637.

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References:

[1]	S. Asmussen, Ruin probabilities, World Scientific Press, Singapore, 2000. doi: 10.1142/9789812779311.
[2]	E. Bayraktar and V. Young, Minimizing the probability of lifetime ruin under borrowing constraints, Insurance: Mathematics and Economics, 41 (2007), 196-221. doi: 10.1016/j.insmatheco.2006.10.015.
[3]	C. Bernard and W. Tian, Optimal reinsurance arrangements under tail risk measures, Journal of Risk and Insurance, 76 (2009), 709-725.
[4]	S. Browne, Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937.
[5]	J. Cai and K. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bulletin, 37 (2007), 93-112. doi: 10.2143/AST.37.1.2020800.
[6]	J. Cai, K. Tan, C. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196. doi: 10.1016/j.insmatheco.2008.05.011.
[7]	M. Centeno, Dependent risks and excess of loss reinsurance, Insurance: Mathematics and Economics, 37 (2005), 229-238. doi: 10.1016/j.insmatheco.2004.12.001.
[8]	M. Centeno and O. Simũes, Combining quota-share and excess of loss treaties on the reinsurance of n independent risks, ASTIN Bulletin, 21 (2002), 41-55.
[9]	H. Gerber, An Introduction to Mathematical Risk Theory, In: S. S. Huebner Foundation Monograph, Series No. 8. Irwin, Homewood, Illinois, 1979.
[10]	J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.
[11]	M. Guerra and M. Centeno, Optimal reinsurance for variance related premium calculation principles, ASTIN Bulletin, 40 (2010), 97-121. doi: 10.2143/AST.40.1.2049220.
[12]	M. Hald and H. Schmidli, On the maximization of the adjustment coefficient under proportioal reinsurance, ASTIN Bulletin, 34 (2004), 75-83. doi: 10.2143/AST.34.1.504955.
[13]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51. doi: 10.1016/j.insmatheco.2004.04.004.
[14]	M. Kaluszka, Optimal reinsurance under mean-variance premium principles, Insurance: Mathematics and Economics, 28 (2001), 61-67. doi: 10.1016/S0167-6687(00)00066-4.
[15]	M. Kaluszka, Mean-variance optimal reinsurance arrangements, Scandinavian Actuarial Journal, 1 (2004), 28-41. doi: 10.1080/03461230410019222.
[16]	Z. Liang and E. Bayraktar, Optimal proportional reinsurance and investment with unobservable claim sizes and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166. doi: 10.1016/j.insmatheco.2014.01.011.
[17]	Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350. doi: 10.1080/15326340701300894.
[18]	Z. Liang and J. Guo, Ruin probabilities under optimal combining quota-share and excess of loss reinsurance, Acta Mathematica Sinica, Chinese Series, 9 (2010), 858-870.
[19]	Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.
[20]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.
[21]	D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.
[22]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68. doi: 10.1080/034612301750077338.
[23]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173.
[24]	X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess of loss reinsurance policies in a dynamic setting, Applied Stochastic Model in Business and Industry, 23 (2007), 63-71. doi: 10.1002/asmb.637.

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