American Institute of Mathematical Sciences

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March  2021, 17(2): 841-868. doi: 10.3934/jimo.2020001

Optimal stop-loss reinsurance with joint utility constraints

Nan Zhang 1, , Linyi Qian 1, , Zhuo Jin 2, and  Wei Wang 3,,

1. 

Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China
2. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia
3. 

Department of Financial Engineering, Ningbo University, 818 Fenghua Road, Ningbo 315211, China

* Corresponding author: Wei Wang

Received  May 2018 Revised  August 2019 Published  January 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (11771147, 11571113), the Zhejiang Provincial Natural Science Foundation of China (LY17G010003), "Shuguang Program" supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission(18SG25), the Humanity and Social Science Youth Foundation of the Ministry of Education of China (18YJC910012), the National Social Science Foundation Key Program (17ZDA091), the 111 Project(B14019) and Faculty Research Grant of University of Melbourne

We investigate the optimal reinsurance problems in this paper, specifically, the stop-loss strategies that can bring mutual benefit to both the insurance company and the reinsurance company. The utility improvement constraints are adopted by both contracting parties to guarantee that a reinsurance contract will bring higher expected utilities of wealth to the two participants. We also introduce five risk criteria that reflect the interests of both parties. Under each optimality criterion, we obtain explicit expressions of optimal stop-loss retentions and the corresponding optimised value of objective functions. The upper and lower bounds of expected utility increments under the optimal stop-loss retentions are provided. In the numerical example, we analyse the expected utility improvements under the criterion of minimising total Value-at-Risk. Notable increases in the lower bound of total utility increments are observed after adopting the joint utility improvement constraints.

Keywords: VaR, TVaR, joint survival probability, stop-loss reinsurance, utility improvement.
Mathematics Subject Classification: Primary: 91B30, 91B16; Secondary: 93C95.
Citation: Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001
References:
[1]

K. J. Arrow, Uncertainty and the welfare economics of medical care, Uncertainty in Economics, 1978, Pages 345,347–375. doi: 10.1016/B978-0-12-214850-7.50028-0.  Google Scholar

[2]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[3]

L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance: Mathematics and Economics, 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008.  Google Scholar

[4]

A. Balbás, B. Balbás, R. Balbás and A. Heras, Optimal reinsurance under risk and uncertainty, Insurance: Mathematics and Economics, 60 (2015), 61-74.  doi: 10.1016/j.insmatheco.2014.11.001.  Google Scholar

[5]

I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business & Industry, 28 (2013), 506-528.  doi: 10.1002/asmb.925.  Google Scholar

[6]

A. P. Bazaz and A. T. P. Najafabadi, An optimal reinsurance contract from insurer's and reinsurer's viewpoints, Applications and Applied Mathematics, 10 (2015), 970-982.   Google Scholar

[7]

R. E. Beard, T. Pentikainen and E. Pesonen, Risk Theory, second edition, Chapman and Hall, London, 1977. Google Scholar

[8]

K. Borch, Reciprocal reinsurance treaties, ASTIN Bulletin, 1 (1960), 170-191.  doi: 10.1017/S0515036100009557.  Google Scholar

[9]

K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.  doi: 10.1017/S051503610000814X.  Google Scholar

[10]

J. Cai, Y. Fang, Z. Li and G. E. Willmot, Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, Journal of Risk and Insurance, 80 (2013), 145-168.  doi: 10.1111/j.1539-6975.2012.01462.x.  Google Scholar

[11]

J. Cai and T. Mao, Risk measures derived from a regulator's perspective on the regulatory capital requirements for insurers, SSRN, (2018), 39pp. doi: 10.2139/ssrn.3127285.  Google Scholar

[12]

J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bulletin, 37 (2007), 93-112.  doi: 10.1017/S0515036100014756.  Google Scholar

[13]

Y. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 2014 (2014), 424-438.  doi: 10.1080/03461238.2012.723638.  Google Scholar

[14]

Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, ASTIN Bulletin, 41 (2011), 487-509.   Google Scholar

[15]

W. Cui and J. Yang, Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insurance: Mathematics and Economics, 53 (2013), 74-85.  doi: 10.1016/j.insmatheco.2013.03.007.  Google Scholar

[16]

N. E. D'Ortona and G. Marcarelli, Optimal proportional reinsurance from the point of view of cedent and reinsurer, Scandinavian Actuarial Journal, 2017 (2017), 366-375.  doi: 10.1080/03461238.2016.1148627.  Google Scholar

[17]

European Parliament and the Council, Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency Ⅱ), 2009., http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2009:335:0001:0155:en:PDF Google Scholar

[18]

Y. Fang and Z. Qu, Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability, IMA Journal of Management Mathematics, 25 (2014), 89-103.  doi: 10.1093/imaman/dps029.  Google Scholar

[19]

A. E. van. Heerwarden and R. Kaas, The dutch premium principle, Insurance: Mathematics and Economics, 11 (1992), 129-133.   Google Scholar

[20]

X. Hu, H. Yang and L. Zhang, Optimal retention for a stop-loss reinsurance with incomplete information, Insurance: Mathematics and Economics, 65 (2015), 15-21.  doi: 10.1016/j.insmatheco.2015.08.005.  Google Scholar

[21]

Y. Huang and C. Yin, Optimal reciprocal reinsurance under GlueVaR distortion risk measures, Journal of Mathematical Finance, 9 (2019), 11-24.  doi: 10.4236/jmf.2019.91002.  Google Scholar

[22]

S. Kusuoka, On law invariant coherent risk measures, Advances in Mathematical Economics, Springer, 3 (2001), 83–95. doi: 10.1007/978-4-431-67891-5_4.  Google Scholar

[23]

P. Li, M. Zhou and C. Yin, Optimal reinsurance with both proportional and fixed costs, Statistics and Probability Letters, 106 (2015), 134-141.  doi: 10.1016/j.spl.2015.06.024.  Google Scholar

[24]

Z. Liang and J. Guo, Optimal proportional reinsurance under two criteria: Maximizing the expected utility and minimizing the value at risk, The ANZIAM Journal, 51 (2010), 449-463.  doi: 10.1017/S1446181110000878.  Google Scholar

[25]

Z. Liang and J. Guo, Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, Journal of Applied Mathematics and Computing, 36 (2011), 11-25.  doi: 10.1007/s12190-010-0385-8.  Google Scholar

[26]

H. Meng, T. K. Siu and H. Yang, Optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 306 (2016a), 40-52.  doi: 10.1016/j.cam.2016.04.005.  Google Scholar

[27]

H. Meng, M. Zhou and T. K. Siu, Optimal dividend-reinsurance with two types of premium principles, Probability in the Engineering and Informational Sciences, 30 (2016b), 224-243.  doi: 10.1017/S0269964815000352.  Google Scholar

[28]

H. Meng, M. Zhou and T. K. Siu, Optimal reinsurance policies with two reinsurers in continuous time, Economic Modelling, 59 (2016c), 182-195.  doi: 10.1016/j.econmod.2016.07.009.  Google Scholar

[29]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar

[30]

K. S. Tan, C. Weng and Y. Zhang, Optimality of general reinsurance contracts under CTE risk measure, Insurance: Mathematics and Economics, 49 (2011), 175-187.  doi: 10.1016/j.insmatheco.2011.03.002.  Google Scholar

[31]

P. Vicig, Financial risk measurement with imprecise probabilities, International Journal of Approximate Reasoning, 49 (2008), 159-174.  doi: 10.1016/j.ijar.2007.06.009.  Google Scholar

[32]

D. Yao, H. Yang and R. Wang, Optimal dividend and reinsurance strategies with financing and liquidation value, ASTIN Bulletin, 46 (2016), 365-399.  doi: 10.1017/10.1017/asb.2015.28.  Google Scholar

[33]

K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009.  Google Scholar

[34]

N. Zhang, Z. Jin, L. Qian and R. Wang, Optimal quota-share reinsurance based on the mutual benefit of insurer and reinsurer, Journal of Computational and Applied Mathematics, 342 (2018), 337-351.  doi: 10.1016/j.cam.2018.04.030.  Google Scholar

[35]

X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling, Insurance: Mathematics and Economics, 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.  Google Scholar

[36]

Y. Zhu, Y. Chi and C. Weng, Multivariate reinsurance designs for minimizing an insurer's capital requirement, Insurance: Mathematics and Economics, 59 (2014), 144-155.  doi: 10.1016/j.insmatheco.2014.09.009.  Google Scholar

show all references

References:
[1]

K. J. Arrow, Uncertainty and the welfare economics of medical care, Uncertainty in Economics, 1978, Pages 345,347–375. doi: 10.1016/B978-0-12-214850-7.50028-0.  Google Scholar

[2]

P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068.  Google Scholar

[3]

L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance: Mathematics and Economics, 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008.  Google Scholar

[4]

A. Balbás, B. Balbás, R. Balbás and A. Heras, Optimal reinsurance under risk and uncertainty, Insurance: Mathematics and Economics, 60 (2015), 61-74.  doi: 10.1016/j.insmatheco.2014.11.001.  Google Scholar

[5]

I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business & Industry, 28 (2013), 506-528.  doi: 10.1002/asmb.925.  Google Scholar

[6]

A. P. Bazaz and A. T. P. Najafabadi, An optimal reinsurance contract from insurer's and reinsurer's viewpoints, Applications and Applied Mathematics, 10 (2015), 970-982.   Google Scholar

[7]

R. E. Beard, T. Pentikainen and E. Pesonen, Risk Theory, second edition, Chapman and Hall, London, 1977. Google Scholar

[8]

K. Borch, Reciprocal reinsurance treaties, ASTIN Bulletin, 1 (1960), 170-191.  doi: 10.1017/S0515036100009557.  Google Scholar

[9]

K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.  doi: 10.1017/S051503610000814X.  Google Scholar

[10]

J. Cai, Y. Fang, Z. Li and G. E. Willmot, Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, Journal of Risk and Insurance, 80 (2013), 145-168.  doi: 10.1111/j.1539-6975.2012.01462.x.  Google Scholar

[11]

J. Cai and T. Mao, Risk measures derived from a regulator's perspective on the regulatory capital requirements for insurers, SSRN, (2018), 39pp. doi: 10.2139/ssrn.3127285.  Google Scholar

[12]

J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bulletin, 37 (2007), 93-112.  doi: 10.1017/S0515036100014756.  Google Scholar

[13]

Y. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 2014 (2014), 424-438.  doi: 10.1080/03461238.2012.723638.  Google Scholar

[14]

Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, ASTIN Bulletin, 41 (2011), 487-509.   Google Scholar

[15]

W. Cui and J. Yang, Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insurance: Mathematics and Economics, 53 (2013), 74-85.  doi: 10.1016/j.insmatheco.2013.03.007.  Google Scholar

[16]

N. E. D'Ortona and G. Marcarelli, Optimal proportional reinsurance from the point of view of cedent and reinsurer, Scandinavian Actuarial Journal, 2017 (2017), 366-375.  doi: 10.1080/03461238.2016.1148627.  Google Scholar

[17]

European Parliament and the Council, Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency Ⅱ), 2009., http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2009:335:0001:0155:en:PDF Google Scholar

[18]

Y. Fang and Z. Qu, Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability, IMA Journal of Management Mathematics, 25 (2014), 89-103.  doi: 10.1093/imaman/dps029.  Google Scholar

[19]

A. E. van. Heerwarden and R. Kaas, The dutch premium principle, Insurance: Mathematics and Economics, 11 (1992), 129-133.   Google Scholar

[20]

X. Hu, H. Yang and L. Zhang, Optimal retention for a stop-loss reinsurance with incomplete information, Insurance: Mathematics and Economics, 65 (2015), 15-21.  doi: 10.1016/j.insmatheco.2015.08.005.  Google Scholar

[21]

Y. Huang and C. Yin, Optimal reciprocal reinsurance under GlueVaR distortion risk measures, Journal of Mathematical Finance, 9 (2019), 11-24.  doi: 10.4236/jmf.2019.91002.  Google Scholar

[22]

S. Kusuoka, On law invariant coherent risk measures, Advances in Mathematical Economics, Springer, 3 (2001), 83–95. doi: 10.1007/978-4-431-67891-5_4.  Google Scholar

[23]

P. Li, M. Zhou and C. Yin, Optimal reinsurance with both proportional and fixed costs, Statistics and Probability Letters, 106 (2015), 134-141.  doi: 10.1016/j.spl.2015.06.024.  Google Scholar

[24]

Z. Liang and J. Guo, Optimal proportional reinsurance under two criteria: Maximizing the expected utility and minimizing the value at risk, The ANZIAM Journal, 51 (2010), 449-463.  doi: 10.1017/S1446181110000878.  Google Scholar

[25]

Z. Liang and J. Guo, Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, Journal of Applied Mathematics and Computing, 36 (2011), 11-25.  doi: 10.1007/s12190-010-0385-8.  Google Scholar

[26]

H. Meng, T. K. Siu and H. Yang, Optimal insurance risk control with multiple reinsurers, Journal of Computational and Applied Mathematics, 306 (2016a), 40-52.  doi: 10.1016/j.cam.2016.04.005.  Google Scholar

[27]

H. Meng, M. Zhou and T. K. Siu, Optimal dividend-reinsurance with two types of premium principles, Probability in the Engineering and Informational Sciences, 30 (2016b), 224-243.  doi: 10.1017/S0269964815000352.  Google Scholar

[28]

H. Meng, M. Zhou and T. K. Siu, Optimal reinsurance policies with two reinsurers in continuous time, Economic Modelling, 59 (2016c), 182-195.  doi: 10.1016/j.econmod.2016.07.009.  Google Scholar

[29]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar

[30]

K. S. Tan, C. Weng and Y. Zhang, Optimality of general reinsurance contracts under CTE risk measure, Insurance: Mathematics and Economics, 49 (2011), 175-187.  doi: 10.1016/j.insmatheco.2011.03.002.  Google Scholar

[31]

P. Vicig, Financial risk measurement with imprecise probabilities, International Journal of Approximate Reasoning, 49 (2008), 159-174.  doi: 10.1016/j.ijar.2007.06.009.  Google Scholar

[32]

D. Yao, H. Yang and R. Wang, Optimal dividend and reinsurance strategies with financing and liquidation value, ASTIN Bulletin, 46 (2016), 365-399.  doi: 10.1017/10.1017/asb.2015.28.  Google Scholar

[33]

K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009.  Google Scholar

[34]

N. Zhang, Z. Jin, L. Qian and R. Wang, Optimal quota-share reinsurance based on the mutual benefit of insurer and reinsurer, Journal of Computational and Applied Mathematics, 342 (2018), 337-351.  doi: 10.1016/j.cam.2018.04.030.  Google Scholar

[35]

X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling, Insurance: Mathematics and Economics, 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.  Google Scholar

[36]

Y. Zhu, Y. Chi and C. Weng, Multivariate reinsurance designs for minimizing an insurer's capital requirement, Insurance: Mathematics and Economics, 59 (2014), 144-155.  doi: 10.1016/j.insmatheco.2014.09.009.  Google Scholar

Figure 1.  The expected utility increment without utility constraint when $ \alpha_I<\alpha_R $
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Figure 2.  The expected utility increment with utility constraints when $ \alpha_I<\alpha_R $
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Figure 3.  The lower bound of total expected utility increments when $ \alpha_I<\alpha_R $
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Figure 4.  The expected utility increment without utility constraint when $ \alpha_I>\alpha_R $
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Figure 5.  The expected utility increment with utility constraints when $ \alpha_I>\alpha_R $
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Figure 6.  The lower bound of total expected utility increment when $ \alpha_I>\alpha_R $
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