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A continuous-time theory of reinsurance chains. (English) Zbl 1452.91266
Summary: A chain of reinsurance is a hierarchical system formed by the subsequent interactions among multiple (re)insurance agents, which is quite often encountered in practice. This paper proposes a novel continuous-time framework for studying the optimal reinsurance strategies within a chain of reinsurance. The transactions between reinsurance buyers and sellers are formulated by means of Stackelberg games, in order to reflect the conflicting interests and unequal negotiation powers in the bargaining process. Assuming the variance premium principle and the mean-variance criterion on the surplus processes, we solve the time-consistent optimal reinsurance demands and pricing strategies in explicit forms, which are surprisingly plain.
Based on the proposed reinsurance chain models, our in-depth theoretical analysis shows that: (a.) it is optimal to situate more (resp. less) risk averse reinsurers to the latter (resp. former) positions in a chain of reinsurance; (b.) adding new reinsurers will lower the reinsurance prices at all levels in a chain of reinsurance, promoting the existing agents to rationally control their respective risk exposures; and essentially (c.) alleviate the systemic risk in the chain structure.
##### MSC:
 91G05 Actuarial mathematics 91A65 Hierarchical games (including Stackelberg games) 91A80 Applications of game theory 91G45 Financial networks (including contagion, systemic risk, regulation)
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##### References:
 [1] Aase, K. K., The Nash bargaining solution vs. equilibrium in a reinsurance syndicate, Scand. Actuar. J., 2009, 3, 219-238 (2009) · Zbl 1224.91035 [2] Bäuerle, N., Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62, 1, 159-165 (2005) · Zbl 1101.93081 [3] Björk, T.; Murgoci, A.; Zhou, X. Y., Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24, 1, 1-24 (2014) · Zbl 1285.91116 [4] Boonen, T. J.; Tan, K. S.; Zhuang, S. C., Pricing in reinsurance bargaining with comonotonic additive utility functions, Astin Bull., 46, 2, 507-530 (2016) · Zbl 1390.91164 [5] Boonen, T. J.; Tan, K. S.; Zhuang, S. C., The role of a representative reinsurer in optimal reinsurance, Insurance Math. Econom., 70, 196-204 (2016) · Zbl 1371.91082 [6] Borch, K. H., The Mathematical Theory of Insurance (1974), Lexington Books: Lexington Books Lexington [7] Chen, S.; Li, Z.; Zeng, Y., Optimal dividend strategies with time-inconsistent preferences, J. Econom. Dynam. Control, 46, 150-172 (2014) · Zbl 1402.91671 [8] Chen, L.; Shen, Y., On a new paradigm of optimal reinsurance: A stochastic stackelberg differential game between an insurer and a reinsurer, Astin Bull., 48, 02, 905-960 (2018) · Zbl 1390.91170 [9] Chen, L.; Shen, Y., Stochastic Stackelberg differential reinsurance games under time-inconsistent mean-variance framework, Insurance Math. Econom., 88, 120-137 (2019) · Zbl 1425.91217 [10] Chi, Y.; Meng, H., Optimal reinsurance arrangements in the presence of two reinsurers, Scand. Actuar. J., 2014, 5, 424-438 (2014) · Zbl 1401.91113 [11] Chi, Y.; Zhou, M., Optimal reinsurance design: A mean-variance approach, N. Am. Actuar. J., 21, 1, 1-14 (2017) · Zbl 1414.91175 [12] Chiu, M. C.; Wong, H. Y., Mean-variance portfolio selection of cointegrated assets, J. Econom. Dynam. Control, 35, 8, 1369-1385 (2011) · Zbl 1217.91166 [13] Cournot, A., Recherches sur les principes mathematiques de la theorie des richesses, (Recherches sur Les Principes Mathématique de la Théorie des Richesses (1838), Hachette: Hachette Paris) · Zbl 0174.51801 [14] Cui, X.; Li, D.; Shi, Y., Self-coordination in time inconsistent stochastic decision problems: A planner-doer game framework, J. Econom. Dynam. Control, 75, 91-113 (2017) · Zbl 1401.91043 [15] Cui, W.; Yang, J.; Wu, L., Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insurance Math. Econom., 53, 1, 74-85 (2013) · Zbl 1284.91222 [16] Dai, M.; Jin, H.; Kou, S.; Xu, Y., A dynamic mean-variance analysis for log returns, Manage. Sci. (2020), (in press) [17] Doherty, N.; Smetters, K., Moral hazard in reinsurance markets, J. Risk Insurance, 72, 3, 375-391 (2005) [18] d’Ursel, L.; Lauwers, M., Chains of reinsurance: Non-cooperative equilibria and Pareto optimality, Insurance Math. Econom., 4, 4, 279-285 (1985) · Zbl 0596.62109 [19] d’Ursel, L.; Lauwers, M., Pareto optimal chains of reinsurance, Econom. Lett., 20, 4, 307-310 (1986) · Zbl 1328.91140 [20] Dutang, C.; Albrecher, H.; Loisel, S., Competition among non-life insurers under solvency constraints: A game-theoretic approach, European J. Oper. Res., 231, 3, 702-711 (2013) · Zbl 1317.91042 [21] Gerber, H. U., Chains of reinsurance, Insurance Math. Econom., 3, 1, 43-48 (1984) · Zbl 0532.62083 [22] Guerra, M.; Centeno, M.d. L., Optimal reinsurance for variance related premium calculation principles, Astin Bull., 40, 1, 97-121 (2010) · Zbl 1230.91073 [23] Horst, U.; Moreno-Bromberg, S., Risk minimization and optimal derivative design in a principal agent game, Math. Financ. Econ., 2, 1, 1-27 (2008) · Zbl 1177.91082 [24] Jørgensen, S.; Kort, P. M.; van Schijndel, G. J.C., Optimal investment, financing, and dividends: A stackelberg differential game, J. Econom. Dynam. Control, 13, 3, 339-377 (1989) · Zbl 0674.90008 [25] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern Actuarial Risk Theory (2008), Springer: Springer Heidelberg · Zbl 1148.91027 [26] Kaluszka, M., Mean-variance optimal reinsurance arrangements, Scand. Actuar. J., 2004, 1, 28-41 (2004) · Zbl 1117.62115 [27] Kanno, M., The network structure and systemic risk in the global non-life insurance market, Insurance Math. Econom., 67, 38-53 (2016) · Zbl 1348.91278 [28] Laffont, J.-J.; Martimort, D., The theory of incentives: The principal-agent model, (The Theory of Incentives (2002), Princeton University Press: Princeton University Press New Jersey), 421 [29] Lemaire, J.; Quairiere, J.-P., Chains of reinsurance revisited, Astin Bull., 16, 2, 77-88 (1986) [30] Li, B.; Li, D.; Xiong, D., Alpha-robust mean-variance reinsurance-investment strategy, J. Econom. Dynam. Control, 70, 101-123 (2016) · Zbl 1401.91521 [31] Liang, Z.; Yuen, K. C., Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016, 1, 18-36 (2016) · Zbl 1401.91167 [32] Lin, X.; Zhang, C.; Siu, T. K., Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Math. Methods Oper. Res., 75, 1, 83-100 (2012) · Zbl 1276.91095 [33] Massey, R.; Hart, D.; Widdows, J.; Law, D.; Bhattacharya, K.; Hawes, W.; Shaw, R., Insurance Company FailureTechnical report (2003), Institute and Faculty of Actuaries: Institute and Faculty of Actuaries London [34] Meng, H.; Siu, T. K.; Yang, H., Optimal insurance risk control with multiple reinsurers, J. Comput. Appl. Math., 306, 40-52 (2016) · Zbl 1339.93124 [35] Meng, H.; Zhou, M.; Siu, T. K., Optimal reinsurance policies with two reinsurers in continuous time, Econ. Model., 59, 182-195 (2016) [36] Øksendal, B.; Sandal, L.; Ubøe, J., Stochastic stackelberg equilibria with applications to time-dependent newsvendor models, J. Econom. Dynam. Control, 37, 7, 1284-1299 (2013) · Zbl 1402.90010 [37] Plantin, G., Does reinsurance need reinsurers?, J. Risk Insurance, 73, 1, 153-168 (2006) [38] Simaan, M.; Cruz, J. B., On the Stackelberg strategy in nonzero-sum games, J. Optim. Theory Appl., 11, 5, 533-555 (1973) · Zbl 0243.90056 [39] Strotz, R. H., Myopia and inconsistency in dynamic utility maximization, Rev. Econom. Stud., 23, 3, 165 (1955) [40] Tan, K. S.; Wei, P.; Wei, W.; Zhuang, S. C., Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle, European J. Oper. Res., 282, 1, 345-362 (2020) · Zbl 1431.91344 [41] von Stackelberg, H., Marktform und gleichgewicht, (Economica (1934), Springer: Springer Vienna) · Zbl 1405.91003 [42] Wang, J.; Forsyth, P., Continuous time mean variance asset allocation: A time-consistent strategy, European J. Oper. Res., 209, 2, 184-201 (2011) · Zbl 1208.91139 [43] Wang, S. S.; Young, V. R.; Panjer, H. H., Axiomatic characterization of insurance prices, Insurance Math. Econom., 21, 2, 173-183 (1997) · Zbl 0959.62099 [44] Young, V. R., Optimal insurance under Wang’s premium principle, Insurance Math. Econom., 25, 2, 109-122 (1999) · Zbl 1156.62364 [45] Zhuang, S. C.; Weng, C.; Tan, K. S.; Assa, H., Marginal indemnification function formulation for optimal reinsurance, Insurance Math. Econom., 67, 65-76 (2016) · Zbl 1348.91196
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