An optimal reinsurance simulation model for non-life insurance in the Solvency II framework

European Actuarial Journal - Tập 12 - Trang 89-123 - 2021
Alberto Zanotto1, Gian Paolo Clemente2
1Quantitative Analyst at Prima Assicurazioni, Milan, Italy
2Department of Mathematics for Economics, Financial and Actuarial Sciences, Università Cattolica del Sacro Cuore, Milan, Italy

Tóm tắt

In this paper, we propose an approach to explore reinsurance optimization for a non-life multi-line insurer through a simulation model that combines alternative reinsurance treaties. Based on the Solvency II framework, the model maximises both solvency ratio and portfolio performance under user-defined constraints. Data visualisation helps understanding the numerical results and, together with the concept of the Pareto frontier, supports the selection of the optimal reinsurance program. We show in the case study that the methodology can be easily restructured to deal with multi-objective optimization, and, finally, the selected programs from each proposed problem are compared.

Tài liệu tham khảo

Aas K, Berg D (2009) Models for construction of multivariate dependence: a comparison study. Eur J Financ 15(7–8):639–659 Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insur Math Econ 44:182–198 Albrecher H, Beirlant J, Teugels JL (2017) Reinsurance: actuarial and statistical aspects. Wiley, New York Asimit AV, Badescu AM, Haberman S, Kim E-S (2016) Efficient risk allocation within a non-life insurance group under Solvency II Regime. Insur Math Econ 66:69–76 Asimit AV, Bignozzi V, Cheung KC, Hu J, Kim E-S (2017) Robust and Pareto Optimality of Insurance Contracts. Eur J Oper Res 4:20 Asimit AV, Boonen TJ (2017) Insurance with Multiple Insurers: A Game-Theoretic Approach. Eur J Oper Res 267(2):778–790 Associazione Nazionale fra le Imprese Assicuratrici. L’assicurazione Italiana 2018-2019. Technical report, ANIA, 2019 Balbás A, Balbás B, Heras A (2009) Optimal reinsurance with general risk measures. Insur Math Econ 44(3):374–384 Beard RE, Pentikäinen T, Pesonen E (1984) Risk theory. Chapman & Hall, London Bernard C, Tian W (2009) Optimal reinsurance arrangements under tail risk measures. J Risk Insur 20:20 Borch K (1960) An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th international congress of actuaries, pp 597–610 Cai J, Liu H, Wang R (2017) Pareto-optimal reinsurance arrangements under general model settings. Insur Math Econ 77:24–37 Cai J, Tan KS (2007) Optimal retention for a stop-loss reinsurance under the VAR and CTE risk measures. ASTIN Bull J IAA 37(1):93–112 Cai J, Tan KS, Weng C, Zhang Y (2008) Optimal reinsurance under VAR and CTE risk measures. Insur Math Econ 43(1):185–196 Cai J, Wang Y (2019) Reinsurance premium principles based on weighted loss functions. Scand Actuarial J 2019(10):903–923 de L. Centeno M, (1995) The effect of the retention limit on risk reserve. Astin Bull 25:1 Cheung KC (2010) Optimal reinsurance revisited—a geometric approach. ASTIN Bull J IAA 40(1):221–239 Clemente GP (2018) The effect of non-proportional reinsurance: a revision of solvency ii standard formula. Risks 6(2):50 Clemente GP, Savelli N (2017) Actuarial improvements of standard formula for non-life underwriting risk. Insurance regulation in the European Union. Springer, Berlin, pp 223–243 Kurowika D, Joe H (2010) Dependence modeling vine copula handbook Coutts SM, Thomas T (1997) Capital and risk and their relationship to reinsurance programmes. In: Proceedings of 5th international conference on insurance solvency and finance Czado C (2010) Copula theory and its applications, chapter pair-copula constructions of multivariate copulas. Springer, Berlin, pp 93–109 Daykin C, Pentikäinen T, Pesonen M (1994) Practical risk theory for actuaries. Chapman & Hall, London Doumpos M, Zopounidis C (2020) Multi-objective optimization models in finance and investments. J Glob Optim 76(2):243–244 Durante F, Fernandex-Sanchez J, Sempi C (2013) A topological proof of Sklar’s theorem. Appl Math Lett 26(9):945–948 Eddelbuettel D, Francois R, Allaire J.J, Ushey K, Kou Q, Russel N, Bates D, Chambers J (2021) Rcpp: seamless r and c++ integration. R package version 1.0.6 Eling M, Gatzert N, Schmeiser H (2008) The swiss solvency test and its market implications. Geneva Pap 33:418–439 European Commission. Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II). Technical report, European Commission, 2009 European Commission. Commission Delegated Regulation (EU) 2015/35 supplementing 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) 10 of October 2014 (published on Official Journal of the EU, vol. 58, 17 January 2015). Technical report, European Commission, 2014 Federal Office of Private Insurance (2004) White Paper of the Swiss Solvency Test. Technical report, FOPI Fischer M, Köck C, Schlüter S, Weigert F (2009) An empirical analysis of multivariate copula models. Quant Financ 9(7):839–854 Gajek L, Zagrodny D (2004) Optimal reinsurance under general risk measures. Insur Math Econ 34(2):227–240 Gajek L, Zagrodny D (2004) Reinsurance arrangements maximizing insurer’s survival probability. J Risk Insur 71(3):421–435 Gisler A (2009) The insurance risk in the SST and in Solvency II: modelling and parameter estimation. In: Proceedings Astin colloquium Guerra M, de L. Centeno M (2008) Optimal reinsurance policy: the adjustment coefficient and the expected utility criteria. Insur Math Econ 42(2):529–539 Guerra M, de L. Centeno M (2012) Are quantile risk measures suitable for risk-transfer decisions? Insur Math Econ 50(3):446–461 Jiang W, Hong H, Ren J (2017) On pareto-optimal reinsurance with constraints under distortion risk measures. Eur Actuarial J 8:2 Joe H (1996) Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. Inst Math Stat Hayward 28:120–141 Joe H (2001) Multivariate models and dependence concepts. Chapman & Hall/CRC, Boca Raton Kaluszka A (2001) Optimal reinsurance under mean-variance premium principles. Insur Math Econ 28(1):61–67 Malamud S, Rui H, Whinston A (2016) Optimal reinsurance with multiple tranches. J Math Econ 65:71–82 Meyers G, Shenker N (1982) Parameter uncertainty in the collective risk model. Technical report, Casualty Actuarial Society Discussion Paper Program Nelsen RB (2010) An introduction to copulas. Springer, Berlin Oesterreicher I, Mitschele A, Schlottmann F, Seese D (2006) Comparison of multi-objective evolutionary algorithms in optimizing combinations of reinsurance contracts. In: GECCO ’06 Román S, Villegas AM, Villegas JG (2018) An evolutionary strategy for multiobjective reinsurance optimization. J Oper Res Soc 69(10):1661–1677 Savelli N, Clemente GP (2011) Hierarchical structures in the aggregation of premium risk for insurance underwriting. Scand Actuarial J 2011(3):193–213 Schepsmeier U, Stoeber J, Brechmann EC, Graeler B, Nagler T, Erhardt T (2015) Vinecopula: statistical inference of vine copulas. R package version 1.6.1 Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231 Sklar A (1996) Random variables, distribution functions, and copulas: a personal look backward and forward. Lect Notes Monograph Ser 20:1–14 Steuer RE, Qi Y, Hirschberger M (2007) Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Ann Oper Res 152(1):297–317 Sundt B (1991) On excess of loss reinsurance with reinstatements. Bull Swiss Assoc Actuaries 20:1