Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle.

*(English)*Zbl 1431.91344Summary: This paper studies the optimal dynamic reinsurance policy for an insurance company whose surplus is modeled by the diffusion approximation of the classical Cramér-Lundberg model. We assume the reinsurance premium is calculated according to a proposed mean-CVaR premium principle which generalizes Denneberg’s absolute deviation principle and expected value principle. Moreover, we require that both ceded loss and retention functions are non-decreasing to rule out moral hazard. Under the objective of minimizing the ruin probability, we obtain the optimal reinsurance policy explicitly and we denote the resulting treaty as the dual excess-of-loss reinsurance. This form of the optimal treaty is new to the literature and lends support to the fact that reinsurance contracts in practice often involve layers. It also demonstrates that reinsurance treaties such as the proportional and the standard excess-of-loss, which are typically found to be optimal in the dynamic reinsurance model, need not be optimal when we consider a more general optimization model. We also consider other generalizations including (i) allowing the insurer to manage its business through both reinsurance and investment; and (ii) \(N\)-piecewise mean-CVaR premium principle. In the former case, we not only show that the dual excess-of-loss reinsurance policy remains optimal, but also demonstrate that investing in stock can further enhance insurer’s financial stability with lower ruin probability. For the latter case, we establish that the optimal reinsurance treaty can have at most \(N\) layers, which is also more consistent with practice.

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\textit{K. S. Tan} et al., Eur. J. Oper. Res. 282, No. 1, 345--362 (2020; Zbl 1431.91344)

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##### References:

[1] | Albrecher, H.; Beirlant, J.; Teugels, J. L., Reinsurance: Actuarial and statistical aspects (2017), Wiley Series in Probability and Statistics · Zbl 1376.91004 |

[2] | Asimit, A. V.; Bignozzi, V.; Cheung, K. C.; Hu, J.; Kim, E.-S., Robust and Pareto optimality of insurance contracts, European Journal of Operational Research, 262, 2, 720-732 (2017) · Zbl 1376.91097 |

[3] | Asimit, V.; Boonen, T. J., Insurance with multiple insurers: a game-theoretic approach, European Journal of Operational Research, 267, 2, 778-790 (2018) · Zbl 1403.91187 |

[4] | Asmussen, S.; Højgaard, B.; Taksar, M., Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4, 3, 299-324 (2000) · Zbl 0958.91026 |

[5] | Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15, 2, 261-308 (2005) · Zbl 1136.91016 |

[6] | Babai, M. Z.; Jemai, Z.; Dallery, Y., Analysis of order-up-to-level inventory systems with compound poisson demand, European Journal of Operational Research, 210, 3, 552-558 (2011) · Zbl 1213.90045 |

[7] | Balbás, A.; Balbás, B.; Balbás, R., Good deals and benchmarks in robust portfolio selection, European Journal of Operational Research, 250, 2, 666-678 (2016) · Zbl 1346.91198 |

[8] | Balbás, A.; Balbás, B.; Heras, A., Optimal reinsurance with general risk measures, Insurance: Mathematics and Economics, 44, 3, 374-384 (2009) · Zbl 1162.91394 |

[9] | Balbás, A.; Balbás, B.; Heras, A., Stable solutions for optimal reinsurance problems involving risk measures, European Journal of Operational Research, 214, 3, 796-804 (2011) · Zbl 1219.91064 |

[10] | Bernard, C.; He, X.; Yan, J.-A.; Zhou, X. Y., Optimal insurance design under rank-dependent expected utility, Mathematical Finance, 25, 1, 154-186 (2015) · Zbl 1314.91134 |

[11] | Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, Journal of Risk and Insurance, 76, 3, 709-725 (2009) |

[12] | Bijvank, M.; Johansen, S. G., Periodic review lost-sales inventory models with compound poisson demand and constant lead times of any length, European Journal of Operational Research, 220, 1, 106-114 (2012) · Zbl 1253.90019 |

[13] | Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of operations research, 20, 4, 937-958 (1995) · Zbl 0846.90012 |

[14] | Butler, R.; Durbin, D.; Helvacian, N., Increasing claims for soft tissue injuries in workers’ compensation: cost shifting and moral hazard, Journal of Risk and Uncertainty, 13, 1, 73-87 (1996) |

[15] | Cheung, E. C.; Rabehasaina, L.; Woo, J.-K.; Xu, R., Asymptotic correlation structure of discounted incurred but not reported claims under fractional poisson arrival process, European Journal of Operational Research, 276, 2, 582-601 (2019) · Zbl 1430.90188 |

[16] | Cheung, E. C.; Wong, J. T., On the dual risk model with parisian implementation delays in dividend payments, European Journal of Operational Research, 257, 1, 159-173 (2017) · Zbl 1394.91204 |

[17] | Chi, Y.; Tan, K. S., Optimal reinsurance with general premium principles, Insurance: Mathematics and Economics, 52, 2, 180-189 (2013) · Zbl 1284.91216 |

[18] | Choulli, T.; Taksar, M.; Zhou, X. Y., Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance, 1, 6, 573-596 (2001) · Zbl 1405.91251 |

[19] | Choulli, T.; Taksar, M.; Zhou, X. Y., A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41, 6, 1946-1979 (2003) · Zbl 1084.91047 |

[20] | Cizek, P.; Härdle, W. K.; Weron, R., Statistical tools for finance and insurance (2005), Springer Science & Business Media · Zbl 1078.62112 |

[21] | Cummins, J.; Tennyson, S., Moral hazard in insurance claiming: Evidence from automobile insurance, Journal of Risk and Uncertainty, 12, 1, 29-50 (1996) |

[22] | Cutler, D. M.; Zeckhauser, R. J., Reinsurance for catastrophes and cataclysms, The financing of catastrophe risk (1999), University of Chicago Press |

[23] | Denneberg, D., Premium calculation: why standard deviation should be replaced by absolute deviation, Astin Bulletin, 20, 02, 181-190 (1990) |

[24] | Diasparra, M.; Romera, R., Inequalities for the ruin probability in a controlled discrete-time risk process, European Journal of Operational Research, 204, 3, 496-504 (2010) · Zbl 1189.91071 |

[25] | Dionne, G.; St-Michel, P., Workers’ compensation and moral hazard, The Review of Economics and Statistics, 73, 2, 236-244 (1991) |

[26] | Drèze, J.; Schokkaert, E., Arrow’s theorem of the deductible: moral hazard and stop-loss in health insurance, Journal of Risk and Uncertainty, 47, 147-163 (2013) |

[27] | Emanuel, D. C.; Michael Harrison, J.; Taylor, A. J., A diffusion approximation for the ruin function of a risk process with compounding assets, Scandinavian Actuarial Journal, 1975, 4, 240-247 (1975) · Zbl 0322.62101 |

[28] | Fleming, W. H.; Soner, H. M., Controlled Markov processes and viscosity solutions (2006), Springer Science & Business Media · Zbl 1105.60005 |

[29] | Froot, K. A., The market for catastrophe risk: a clinical examination, Journal of Financial Economics, 60, 2-3, 529-571 (2001) |

[30] | Gajek, L.; Zagrodny, D., Reinsurance arrangements maximizing insurer’s survival probability, Journal of Risk and Insurance, 71, 3, 421-435 (2004) |

[31] | Grandell, J., Aspects of risk theory (2012), Springer Science & Business Media |

[32] | Gu, J.-W.; Steffensen, M.; Zheng, H., Optimal dividend strategies of two collaborating businesses in the diffusion approximation model, Mathematics of Operations Research, 43, 2, 377-398 (2017) |

[33] | He, L.; Liang, Z., Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44, 1, 88-94 (2009) · Zbl 1156.91395 |

[34] | Hipp, C.; Taksar, M., Optimal non-proportional reinsurance control, Insurance: Mathematics and Economics, 47, 2, 246-254 (2010) · Zbl 1231.91199 |

[35] | Hipp, C.; Vogt, M., Optimal dynamic XL reinsurance, Astin Bulletin, 33, 02, 193-207 (2003) · Zbl 1059.93135 |

[36] | Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998, 2, 166-180 (1998) · Zbl 1075.91559 |

[37] | Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical Finance, 9, 2, 153-182 (1999) · Zbl 0999.91052 |

[38] | Højgaard, B.; Taksar, M., Optimal risk control for a large corporation in the presence of returns on investments, Finance and Stochastics, 5, 4, 527-547 (2001) · Zbl 1049.93090 |

[39] | Huberman, G.; Mayers, D.; Smith Jr, C. W., Optimal insurance policy indemnity schedules, The Bell Journal of Economics, 14, 2, 415-426 (1983) |

[40] | Iglehart, L. D., Diffusion approximations in collective risk theory, Journal of Applied Probability, 6, 2, 285-292 (1969) · Zbl 0191.51202 |

[41] | Kaluszka, M.; Okolewski, A., An extension of Arrow’s result on optimal reinsurance contract, Journal of Risk and Insurance, 75, 2, 275-288 (2008) |

[42] | Larsen, L. S.; Munk, C., The costs of suboptimal dynamic asset allocation: general results and applications to interest rate risk, stock volatility risk, and growth/value tilts, Journal of Economic Dynamics and Control, 36, 2, 266-293 (2012) · Zbl 1238.91127 |

[43] | Liu, J.; Pan, J., Dynamic derivative strategies, Journal of Financial Economics, 69, 3, 401-430 (2003) |

[44] | Meng, H.; Zhang, X., Optimal risk control for the excess of loss reinsurance policies, Astin Bulletin, 40, 1, 179-197 (2010) · Zbl 1230.91079 |

[45] | Müller, H.; Brammertz, R., Moral hazard, The Geneva Papers on Risk and Insurance, 11, 39, 130-144 (1986) |

[46] | Picard, P., On the design of optimal insurance policies under manipulation of audit cost, International Economic Review, 41, 4, 1049-1071 (2000) |

[47] | Rowell, D.; Connelly, L., A history of the term “moral hazard”, Journal of Risk and Insurance, 79, 4, 1051-1075 (2012) |

[48] | Schmidli, H., Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001, 1, 55-68 (2001) · Zbl 0971.91039 |

[49] | Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12, 3, 890-907 (2002) · Zbl 1021.60061 |

[50] | Schmidli, H., Stochastic control in insurance (2007), Springer Science & Business Media |

[51] | Shavell, S., On moral hazard and insurance, The Quarterly Journal of Economics, 93, 4, 541-562 (1979) · Zbl 1254.91280 |

[52] | Taksar, M. I.; Markussen, C., Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7, 1, 97-121 (2003) · Zbl 1066.91052 |

[53] | Wang, S. S., A class of distortion operators for pricing financial and insurance risks, Journal of Risk and Insurance, 67, 1, 15-36 (2000) |

[54] | Wei, P., Essays on risk management (2017), University of Oxford, Ph.D. thesis. |

[55] | Winter, R., Optimal insurance contracts under moral hazard, (Dionne, G., Handbook of insurance (2013), Springer: Springer New York) |

[56] | Xu, Z. Q.; Zhou, X. Y.; Zhuang, S. C., Optimal insurance under rank-dependent utility and incentive compatibility, Mathematical Finance, 29, 2, 659-692 (2019) · Zbl 1411.91325 |

[57] | Xue, X.; Wei, P.; Weng, C., Derivatives trading for insurers, Insurance: Mathematics and Economics, 84, 40-53 (2019) · Zbl 1419.91387 |

[58] | Yang, H.; Zhang, L., Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37, 3, 615-634 (2005) · Zbl 1129.91020 |

[59] | Yong, J.; Zhou, X. Y., Stochastic controls: Hamiltonian systems and HJB equations (1999), Springer Science & Business Media · Zbl 0943.93002 |

[60] | Young, V. R., Premium principles, Encyclopedia of Actuarial Science (2004) |

[61] | Zhang, J.; Liu, S.; Kannan, D., Optimal investment and proportional reinsurance under no short-selling and no borrowing, Dynamic Systems and Applications, 20, 2, 205 (2011) · Zbl 1243.93139 |

[62] | Zhao, Y., Analysis and evaluation of an assemble-to-order system with batch ordering policy and compound poisson demand, European Journal of Operational Research, 198, 3, 800-809 (2009) · Zbl 1176.90189 |

[63] | Zhuang, S. C.; Weng, C.; Tan, K. S.; Assa, H., Marginal indemnification function formulation for optimal reinsurance, Insurance: Mathematics and Economics, 67, 65-76 (2016) · Zbl 1348.91196 |

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