# American Institute of Mathematical Sciences

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

## Optimal stop-loss reinsurance with joint utility constraints

 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  March 2021 Early access  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.

Citation: Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial and 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. [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. [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. [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. [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. [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. [7] R. E. Beard, T. Pentikainen and E. Pesonen, Risk Theory, second edition, Chapman and Hall, London, 1977. [8] K. Borch, Reciprocal reinsurance treaties, ASTIN Bulletin, 1 (1960), 170-191.  doi: 10.1017/S0515036100009557. [9] K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.  doi: 10.1017/S051503610000814X. [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. [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. [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. [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. [14] Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, ASTIN Bulletin, 41 (2011), 487-509. [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. [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. [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 [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. [19] A. E. van. Heerwarden and R. Kaas, The dutch premium principle, Insurance: Mathematics and Economics, 11 (1992), 129-133. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] R. E. Beard, T. Pentikainen and E. Pesonen, Risk Theory, second edition, Chapman and Hall, London, 1977. [8] K. Borch, Reciprocal reinsurance treaties, ASTIN Bulletin, 1 (1960), 170-191.  doi: 10.1017/S0515036100009557. [9] K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297.  doi: 10.1017/S051503610000814X. [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. [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. [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. [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. [14] Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, ASTIN Bulletin, 41 (2011), 487-509. [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. [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. [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 [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. [19] A. E. van. Heerwarden and R. Kaas, The dutch premium principle, Insurance: Mathematics and Economics, 11 (1992), 129-133. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
The expected utility increment without utility constraint when $\alpha_I<\alpha_R$
The expected utility increment with utility constraints when $\alpha_I<\alpha_R$
The lower bound of total expected utility increments when $\alpha_I<\alpha_R$
The expected utility increment without utility constraint when $\alpha_I>\alpha_R$
The expected utility increment with utility constraints when $\alpha_I>\alpha_R$
The lower bound of total expected utility increment when $\alpha_I>\alpha_R$
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