Optimal premium allocation under stop-loss insurance using exposure curves

Year 2022, Volume: 51 Issue: 1, 288 - 307, 14.02.2022

Özenç Murat Mert , Sevtap Selcuk-kestel

https://doi.org/10.15672/hujms.889619

Abstract

Determining the retention level in the stop-loss insurance risk premium for both insurer and reinsurer is an important factor in pricing. This paper aims to set optimal reinsurance with respect to the joint behavior of the insurer and the reinsurer under stop-loss contracts. The dependence between the costs of insurer and reinsurer is expressed as a function of retention ($d$) and maximum-cap ($m$) levels. Based on the maximum degree of correlation, the optimal levels for $d$ and $m$ are derived under certain claim distributions (Pareto, Gamma and Inverse Gamma). Accordingly, the risk premium and exposure curves for both parties are based on the selected distributions. Quantification of the premium share over derived exposure curves based on the optimized retention and maximum levels and the maximum loss risk is obtained using VaR and CVaR as risk measures.

Keywords

Stop-Loss reinsurance, Optimal Retention, Optimal Cap, Premium Share, Exposure Curves, Risk Measures

References

• [1] G. Akarsu, Reinsurance pricing using exposure curve of two dependent risks, MSc thesis, Middle East Technical University, 2018.
• [2] A. Balbás, B. Balbás and A. Heras, Optimal reinsurance with general risk measures, Insurance Math. Econom. 44 (3), 374-384, 2009.
• [3] S. Bernegger, The Swiss Re exposure curves and the MBBEFD distribution class 1, Astin Bull. 27 (1), 99-111, 1997.
• [4] P.J. Boland, Statistical methods in general insurance, in: International Conference on Teaching Statistics, ICOTS-7, 1-6, 2006.
• [5] K. Borch, The optimal reinsurance treaty, Astin Bull. 5 (2), 293-297, 1969.
• [6] J. Cai, Y. Fang, Z. Li and G.E. Willmot, Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability, J. Risk Insur. 80 (1), 145-168, 2013.
• [7] J. Cai, C. Lemieux and F. Liu, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, Astin Bull. 46 (3), 815-849, 2016.
• [8] J. Cai and K.S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, Astin Bull. 37 (1), 93-112, 2007.
• [9] A. Castañer and M.M. Claramunt Bielsa, Optimal stop-loss reinsurance: a dependence analysis, Hacet. J. Math. Stat. 45 (2), 497-519, 2014.
• [10] D.S. Dimitrova and V.K. Kaishev, Optimal joint survival reinsurance: An efficient frontier approach, Insurance Math. Econom. 47 (1), 27-35, 2010.
• [11] L. Gajek and D. Zagrodny, Insurers optimal reinsurance strategies, Insurance Math. Econom. 27 (1), 105-112, 2000.
• [12] M. Guerra and M.D.L. Centeno, Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria, Insurance Math. Econom. 42 (2), 529-539, 2008.
• [13] M. Kaluszka, Mean-variance optimal reinsurance arrangements, Scand. Actuar. J. 1, 28-41, 2004.
• [14] G. Kara, A. Özmen and G.W. Weber, Stability advances in robust portfolio optimization under parallelepiped uncertainty, CEJOR Cent. Eur. J. Oper. Res. 27 (1), 241-261, 2019.
• [15] T. Mack and M. Fackler, Exposure-rating in liability reinsurance, Blätter der DGVFM 26 (2), 229-238, 2003.
• [16] S. Salcedo-Sanz, L. Carro-Calvo, M. Claramunt, A. Castañer and M. Mármol, Effectively tackling reinsurance problems by using evolutionary and swarm intelligence algorithms, Risks 2 (2), 132-145, 2014.
• [17] R.E. Salzmann, Rating by layer of insurance, in: Proceedings of the Casualty Actuarial Society L, 15-26, 1963.
• [18] Y.K. Tse, Nonlife Actuarial Models: Theory, Methods and Evaluation, Cambridge University Press, 2009.
• [19] X. Zhou, H. Zhang and Q. Fan, Optimal limited stop-loss reinsurance under VaR, TVaR, and CTE risk measures, Math. Probl. Eng., Doi:10.1155/2015/143739, 2015.