Research article

Optimal reinsurance for both an insurer and a reinsurer under general premium principles

  • Received: 31 January 2020 Accepted: 04 March 2020 Published: 27 March 2020
  • MSC : 62P05, 60E15

  • A reinsurance contract should consider the conflicting interests of the insurer and the reinsurer. An optimal reinsurance contract for one party may not be optimal for another party and it might be unacceptable for another party. Therefore, in this paper, we study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of loss function which is defined by the joint value-at-risk, assuming that the reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. We derive the optimal reinsurance policies over three ceded loss function sets, the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the layer reinsurance is shown to be optimal; the quota-share reinsurance with a limit is always optimal when the ceded loss functions are in the class of increasing concave functions. We further use the expectation premium principle and Dutch premium principle to illustrate the application of our results by deriving the optimal parameters.

    Citation: Ying Fang, Guo Cheng, Zhongfeng Qu. Optimal reinsurance for both an insurer and a reinsurer under general premium principles[J]. AIMS Mathematics, 2020, 5(4): 3231-3255. doi: 10.3934/math.2020208

    Related Papers:

  • A reinsurance contract should consider the conflicting interests of the insurer and the reinsurer. An optimal reinsurance contract for one party may not be optimal for another party and it might be unacceptable for another party. Therefore, in this paper, we study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by minimizing their total costs under the criteria of loss function which is defined by the joint value-at-risk, assuming that the reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. We derive the optimal reinsurance policies over three ceded loss function sets, the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the layer reinsurance is shown to be optimal; the quota-share reinsurance with a limit is always optimal when the ceded loss functions are in the class of increasing concave functions. We further use the expectation premium principle and Dutch premium principle to illustrate the application of our results by deriving the optimal parameters.


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    [1] K. Borch, An attempt to determine the optimum amount of stop loss reinsurance, In: Transactions of the 16th International Congress of Actuaries, 1 (1960), 597-610.
    [2] P. M. Kahn, Some remarks on a recent paper by Borch. ASTIN Bull., 1 (1961), 265-272.
    [3] K. J. Arrow, Uncertainty and the welfare economics of medical care, Am. Econ. Rev., 53 (1963), 941-973.
    [4] H. U. Gerber, Pareto-optimal risk exchanges and related decision problems, ASTIN Bull., 10 (1978), 25-33. doi: 10.1017/S0515036100006310
    [5] A. Y. Golubin, Pareto-optimal insurance policies in the models with a premium based on the actuarial value, J. Risk Insur., 73 (2006), 469-487. doi: 10.1111/j.1539-6975.2006.00184.x
    [6] N. L. Bowers, H. U. Gerber, J. C. Hickman, et al. Actuarial Mathematics, 2Eds., The Society of Actuaries, Schaumburg, 1997.
    [7] S. Vajda, Minimum variance reinsurance, ASTIN Bull., 2 (1962), 257-260. doi: 10.1017/S0515036100009995
    [8] M. Kaluszka, A. Okolewski, An extension of Arrow's result on optimal reinsurance contract, J. Risk Insur., 75 (2008), 275-288. doi: 10.1111/j.1539-6975.2008.00260.x
    [9] J. Cai, K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, ASTIN Bull., 37 (2007), 93-112. doi: 10.1017/S0515036100014756
    [10] J. Cai, K. S. Tan, C. G. Weng, et al. Optimal reinsurance under VaR and CTE risk measures, Insur. Math. Econ., 43 (2008), 185-196. doi: 10.1016/j.insmatheco.2008.05.011
    [11] K. C. Cheung, Optimal reinsurance revisited-a geometric approach, ASTIN Bull., 40 (2010), 221-239. doi: 10.2143/AST.40.1.2049226
    [12] Y. C. Chi, K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, ASTIN Bull., 41 (2011), 547-574.
    [13] Y. C. Chi, Optimal reinsurance under variance related premium principles, Insur. Math. Econ., 51 (2012), 310-321. doi: 10.1016/j.insmatheco.2012.05.005
    [14] Y. C. Chi, K. S. Tan, Optimal reinsurance with general premium principles, Insur. Math. Econ., 52 (2013), 180-189. doi: 10.1016/j.insmatheco.2012.12.001
    [15] H. H. Huang, Optimal insurance contract under a value-at-risk constraint, Geneva Risk Ins. Rev., 31 (2006), 91-110. doi: 10.1007/s10713-006-0557-5
    [16] Z. Y. Lu, L. P. Liu, S. W. Meng, Optimal reinsurance with concave ceded loss functions under VaR and CTE risk measures, Insur. Math. Econ., 52 (2013), 46-51. doi: 10.1016/j.insmatheco.2012.10.007
    [17] K. S. Tan, C. G. Weng, Y. Zhang, Optimality of general reinsurance contracts under CTE risk measure, Insur. Math. Econ., 49 (2011), 175-187. doi: 10.1016/j.insmatheco.2011.03.002
    [18] K. Borch, The optimal reinsurance treaties, ASTIN Bull., 5 (1969), 293-297. doi: 10.1017/S051503610000814X
    [19] J. Cai, Y. Fang, Z. Li, et al. Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, J. Risk Insur., 80 (2013), 145-168. doi: 10.1111/j.1539-6975.2012.01462.x
    [20] Y. Fang, Z. F. Qu, Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability, IMA J. Manag. Math., 25 (2014), 89-103. doi: 10.1093/imaman/dps029
    [21] J. Cai, C. Lemieux, F. D. Liu, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, ASTIN Bull., 46 (2016), 815-849. doi: 10.1017/asb.2015.23
    [22] A. Lo, A Neyman Pearson perspective on optimal reinsurance with constraints, ASTIN Bull., 47 (2017), 467-499. doi: 10.1017/asb.2016.42
    [23] W. J. Jiang, J. D. Ren, R. Zitikis, Optimal reinsurance policies under the VaR risk measure when the interests of both the cedent and the reinsurer are taken into account, Risks, 5 (2017), 1-22. doi: 10.3390/risks5010011
    [24] J. Cai, H. Y. Liu, R. D. Wang, Pareto-optimal reinsurance arrangements under general model settings, Insur. Math. Econ., 77 (2017), 24-37. doi: 10.1016/j.insmatheco.2017.08.004
    [25] Y. Fang, X. Wang, H. L. Liu, et al. Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles, Commun. Stat-Theory. M., 48 (2019), 6134-6154. doi: 10.1080/03610926.2018.1528364
    [26] A. Lo, Z. F. Tang, Pareto-optimal reinsurance policies in the presence of individual risk constraints, Ann. Oper. Res., 274 (2019), 395-423. doi: 10.1007/s10479-018-2820-4
    [27] Y. X. Huang, C. C. Yin, A unifying approach to constrained and unconstrained optimal reinsurance, J. Comput. Appl. Math., 360 (2019), 1-17. doi: 10.1016/j.cam.2019.03.046
    [28] P. Embrechts, G. Puccetti, Bounds for functions of multivariate risks, J. Multivariate Anal., 97 (2006), 526-547. doi: 10.1016/j.jmva.2005.04.001
    [29] A. J. McNeil, R. Frey, P. Embrechts, Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton University Press, 2005.
    [30] J. Dhaene, M. Denuit, M. J. Goovaerts, et al. The concept of comonotonicity in actuarial science and finance: Theory, Insur. Math. Econ., 31 (2002), 3-33. doi: 10.1016/S0167-6687(02)00134-8
    [31] M. Shaked, J. G. Shanthikumar, Stochastic Orders, Springer, 2007.
    [32] Q. Zhao, Structural learning about directed acyclic graphs from multiple databases, Abstra. Appl. Anala., 2012 (2012), 579543.
    [33] H. Y. Wang, Z. Wu, Eigenvalues of stochastic Hamiltonian systems driven by Poisson process with boundary conditions, Bound Value Probl., 2017 (2017), 1-20. doi: 10.1186/s13661-016-0733-1
    [34] X. L. Wang, F. Chen, L. Lin, Empirical likelihood inference for estimating equation with missing data, Sci. China Math., 56 (2013), 1233-1245. doi: 10.1007/s11425-012-4504-x
    [35] X. L. Wang, Y. Q. Song, L. Lin, Handling estimating equation with nonignorably missing data based on SIR algorithm, J. Comput. Appl. Math., 326 (2017), 62-70. doi: 10.1016/j.cam.2017.05.016
    [36] C. Hu, Strong laws of large numbers for sublinear expectation under controlled 1st moment condition, Chinese Ann. Math. B, 39 (2018), 791-804. doi: 10.1007/s11401-018-0096-2
    [37] C. Hu, Central limit theorems for sub-linear expectation under the Lindeberg condition, J. Inequal. Appl., 2018 (2018), 1-21. doi: 10.1186/s13660-017-1594-6
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