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

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

References:

[1]	S. Asmussen, Ruin probabilities,, World Scientific Press, (2000). doi: 10.1142/9789812779311. Google Scholar
[2]	E. Bayraktar and V. Young, Minimizing the probability of lifetime ruin under borrowing constraints,, Insurance: Mathematics and Economics, 41 (2007), 196. doi: 10.1016/j.insmatheco.2006.10.015. Google Scholar
[3]	C. Bernard and W. Tian, Optimal reinsurance arrangements under tail risk measures,, Journal of Risk and Insurance, 76 (2009), 709. Google Scholar
[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. doi: 10.1287/moor.20.4.937. Google Scholar
[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. doi: 10.2143/AST.37.1.2020800. Google Scholar
[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. doi: 10.1016/j.insmatheco.2008.05.011. Google Scholar
[7]	M. Centeno, Dependent risks and excess of loss reinsurance,, Insurance: Mathematics and Economics, 37 (2005), 229. doi: 10.1016/j.insmatheco.2004.12.001. Google Scholar
[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. Google Scholar
[9]	H. Gerber, An Introduction to Mathematical Risk Theory,, In: S. S. Huebner Foundation Monograph, (1979). Google Scholar
[10]	J. Grandell, Aspects of Risk Theory,, Springer-Verlag, (1991). doi: 10.1007/978-1-4613-9058-9. Google Scholar
[11]	M. Guerra and M. Centeno, Optimal reinsurance for variance related premium calculation principles,, ASTIN Bulletin, 40 (2010), 97. doi: 10.2143/AST.40.1.2049220. Google Scholar
[12]	M. Hald and H. Schmidli, On the maximization of the adjustment coefficient under proportioal reinsurance,, ASTIN Bulletin, 34 (2004), 75. doi: 10.2143/AST.34.1.504955. Google Scholar
[13]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios,, Insurance: Mathematics and Economics, 35 (2004), 21. doi: 10.1016/j.insmatheco.2004.04.004. Google Scholar
[14]	M. Kaluszka, Optimal reinsurance under mean-variance premium principles,, Insurance: Mathematics and Economics, 28 (2001), 61. doi: 10.1016/S0167-6687(00)00066-4. Google Scholar
[15]	M. Kaluszka, Mean-variance optimal reinsurance arrangements,, Scandinavian Actuarial Journal, 1 (2004), 28. doi: 10.1080/03461230410019222. Google Scholar
[16]	Z. Liang and E. Bayraktar, Optimal proportional reinsurance and investment with unobservable claim sizes and intensity,, Insurance: Mathematics and Economics, 55 (2014), 156. doi: 10.1016/j.insmatheco.2014.01.011. Google Scholar
[17]	Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability,, Stochastic Models, 23 (2007), 333. doi: 10.1080/15326340701300894. Google Scholar
[18]	Z. Liang and J. Guo, Ruin probabilities under optimal combining quota-share and excess of loss reinsurance,, Acta Mathematica Sinica, 9 (2010), 858. Google Scholar
[19]	Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin,, Insurance: Mathematics and Economics, 50 (2012), 437. Google Scholar
[20]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance: Mathematics and Economics, 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002. Google Scholar
[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. Google Scholar
[22]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scandinavian Actuarial Journal, 1 (2001), 55. doi: 10.1080/034612301750077338. Google Scholar
[23]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, Annals of Applied Probability, 12 (2002), 890. doi: 10.1214/aoap/1031863173. Google Scholar
[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. doi: 10.1002/asmb.637. Google Scholar

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References:

[1]	S. Asmussen, Ruin probabilities,, World Scientific Press, (2000). doi: 10.1142/9789812779311. Google Scholar
[2]	E. Bayraktar and V. Young, Minimizing the probability of lifetime ruin under borrowing constraints,, Insurance: Mathematics and Economics, 41 (2007), 196. doi: 10.1016/j.insmatheco.2006.10.015. Google Scholar
[3]	C. Bernard and W. Tian, Optimal reinsurance arrangements under tail risk measures,, Journal of Risk and Insurance, 76 (2009), 709. Google Scholar
[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. doi: 10.1287/moor.20.4.937. Google Scholar
[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. doi: 10.2143/AST.37.1.2020800. Google Scholar
[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. doi: 10.1016/j.insmatheco.2008.05.011. Google Scholar
[7]	M. Centeno, Dependent risks and excess of loss reinsurance,, Insurance: Mathematics and Economics, 37 (2005), 229. doi: 10.1016/j.insmatheco.2004.12.001. Google Scholar
[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. Google Scholar
[9]	H. Gerber, An Introduction to Mathematical Risk Theory,, In: S. S. Huebner Foundation Monograph, (1979). Google Scholar
[10]	J. Grandell, Aspects of Risk Theory,, Springer-Verlag, (1991). doi: 10.1007/978-1-4613-9058-9. Google Scholar
[11]	M. Guerra and M. Centeno, Optimal reinsurance for variance related premium calculation principles,, ASTIN Bulletin, 40 (2010), 97. doi: 10.2143/AST.40.1.2049220. Google Scholar
[12]	M. Hald and H. Schmidli, On the maximization of the adjustment coefficient under proportioal reinsurance,, ASTIN Bulletin, 34 (2004), 75. doi: 10.2143/AST.34.1.504955. Google Scholar
[13]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios,, Insurance: Mathematics and Economics, 35 (2004), 21. doi: 10.1016/j.insmatheco.2004.04.004. Google Scholar
[14]	M. Kaluszka, Optimal reinsurance under mean-variance premium principles,, Insurance: Mathematics and Economics, 28 (2001), 61. doi: 10.1016/S0167-6687(00)00066-4. Google Scholar
[15]	M. Kaluszka, Mean-variance optimal reinsurance arrangements,, Scandinavian Actuarial Journal, 1 (2004), 28. doi: 10.1080/03461230410019222. Google Scholar
[16]	Z. Liang and E. Bayraktar, Optimal proportional reinsurance and investment with unobservable claim sizes and intensity,, Insurance: Mathematics and Economics, 55 (2014), 156. doi: 10.1016/j.insmatheco.2014.01.011. Google Scholar
[17]	Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability,, Stochastic Models, 23 (2007), 333. doi: 10.1080/15326340701300894. Google Scholar
[18]	Z. Liang and J. Guo, Ruin probabilities under optimal combining quota-share and excess of loss reinsurance,, Acta Mathematica Sinica, 9 (2010), 858. Google Scholar
[19]	Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin,, Insurance: Mathematics and Economics, 50 (2012), 437. Google Scholar
[20]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance: Mathematics and Economics, 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002. Google Scholar
[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. Google Scholar
[22]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scandinavian Actuarial Journal, 1 (2001), 55. doi: 10.1080/034612301750077338. Google Scholar
[23]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, Annals of Applied Probability, 12 (2002), 890. doi: 10.1214/aoap/1031863173. Google Scholar
[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. doi: 10.1002/asmb.637. Google Scholar

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