Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:08:38.698Z Has data issue: false hasContentIssue false
  • English
  • Français

JOINT OPTIMIZATION OF TRANSITION RULES AND THE PREMIUM SCALE IN A BONUS-MALUS SYSTEM

Published online by Cambridge University Press:  11 September 2020

Kolos Csaba Ágoston  and
Márton Gyetvai
Kolos Csaba Ágoston
Affiliation:
Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest, H-1093, Fővám tér 8., Budapest, Hungary, E-Mail: kolos.agoston@uni-corvinus.hu
Márton Gyetvai*
Affiliation:
Department of Operations Research and Actuarial Sciences, Corvinus University of Budapest, H-1093, Fővám tér 8., Budapest, Hungary, Institute of Economics, Centre for Economic and Regional Studies, H-1097, Tóth Kálmán u. 4., Budapest, Hungary, E-Mail: gyetvai.marton@krtk.mta.hu
*
E-Mail: gyetvai.marton@krtk.mta.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

Bonus-malus systems (BMSs) are widely used in actuarial sciences. These systems are applied by insurance companies to distinguish the policyholders by their risks. The most known application of BMS is in automobile third-party liability insurance. In BMS, there are several classes, and the premium of a policyholder depends on the class he/she is assigned to. The classification of policyholders over the periods of the insurance depends on the transition rules. In general, optimization of these systems involves the calculation of an appropriate premium scale considering the number of classes and transition rules as external parameters. Usually, the stationary distribution is used in the optimization process. In this article, we present a mixed integer linear programming (MILP) formulation for determining the premium scale and the transition rules. We present two versions of the model, one with the calculation of stationary probabilities and another with the consideration of multiple periods of the insurance. Furthermore, numerical examples will also be given to demonstrate that the MILP technique is suitable for handling existing BMSs.

Keywords

Adverse selectionbonus-malus systeminteger programmingC44C61D82D86G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 743 - 776
Copyright
© 2020 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbring, J.H., Chiappori, P-A. and Zavadil, T. (2008) Better Safe than Sorry? Ex Ante and Ex Post Moral Hazard in Dynamic InsuranceData. Tinbergen Institute Discussion Paper No. 08-075/3 CentER Discussion Paper No. 2008-77.Google Scholar
Arató, M. and Martinek, L. (2014) Estimation of claim numbers in automobile insurance. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, 42, 19–35.Google Scholar
Bolton, P. and Dewatripont, M. (2005) Contract Theory. Cambridge: The MIT Press.Google Scholar
Bonsdorff, H. (1992) On the convergence rate of Bonus-Malus systems. ASTIN Bulletin: The Journal of the IAA, 22(2), 217–223.CrossRefGoogle Scholar
Brouhns, N., Guillél’n, M., Denuit, M. and Pinquet, J. (2003) Bonus-Malus scales in segmented Tariffs with stochastic migration between segments. The Journal of Risk and Insurance, 70(4), 577–599.CrossRefGoogle Scholar
Chiappori, P. and Salanié, B. (2000) Testing for asymmetric information in insurance markets. Journal of Political Economy, 108(1), 56–78.CrossRefGoogle Scholar
Coene, G. and Doray, L. (1996) A financially balanced Bonus-Malus system. ASTIN Bulletin, 26(1), 107–116.CrossRefGoogle Scholar
Cooper, R. and Hayes, B. (1987) Multi-period insurance contracts. International Journal of Industrial Organization, 5, 211–231.CrossRefGoogle Scholar
Crocker, K.J. and Snow, A. (1986) The efficiency effects of categorical discrimination in the insurance industry. Journal of Political Economy, 94(2), 321–344.CrossRefGoogle Scholar
Crocker, K.J. and Snow, A. (2000) The theory of risk classification. In Handbook of Insurance (ed. Dionne, G.). Boston, Dordrecht, London: Kluwer Academic Publishers.Google Scholar
Dahlby, B.G. (1983) Adverse selection and statistical discrimination: An analysis of Canadian automobile insurance. Journal of Public Economics, 20(1), 121–130.Google Scholar
Denuit, D. and Dhaene, J. (2001) Bonus-Malus scales using exponential loss functions. Blätter der DGVFM, 25(1), 13–27.CrossRefGoogle Scholar
Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.-F. (2007) Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems. New York: Wiley.CrossRefGoogle Scholar
de Ghellinck, G.T. and Eppen, G.D. (1967) Linear programming solutions for separable Markovian decision. Management Science, Series A, Sciences, 13(5), 371–394.CrossRefGoogle Scholar
De Prill, N. (1978) The efficiency of a Bonus-Malus system. ASTIN Bulletin: The Journal of the IAA, 10(1), 59–72.CrossRefGoogle Scholar
De Prill, N. (1979) Optimal claim decisions for a Bonus-Malus system: A continuous approach. ASTIN Bulletin: The Journal of the IAA, 10(2), 215–222.CrossRefGoogle Scholar
Dionne, G., Michaud, P.-C. and Dahchour, M. (2013) Separating moral hazard from adverse selection and learning in automobile insurance: Longitudinal evidence from France. Journal of the European Economic Association, 11(4), 897–917.CrossRefGoogle Scholar
Eppen, G. and Fama, E. (1968) Solutions for cash-balance and simple dynamic-portfolio problems. The Journal of Business, 41(1), 94–112.CrossRefGoogle Scholar
Gyetvai, M. and Ágoston, K.Cs. (2018) Optimization of transition rules in a Bonus-Malus system. Electronic Notes in Discrete Mathematics, 69, 5–12.CrossRefGoogle Scholar
Heras, A.T., Gil, J.A, Garca-Pineda, P. and Vilar, J.L. (2004) An application of linear programming to Bonus Malus system design. ASTIN Bulletin: The Journal of the IAA, 34(2), 435–456.CrossRefGoogle Scholar
Heras, A.T., Vilar, J.L. and Gil, J.A. (2002) Asymptotic fairness of Bonus-Malus systems and optimal scales of premiums. The Geneva Papers on Risk and Insurance Theory, 27(1), 61–82.CrossRefGoogle Scholar
Holton, J. (2001) Optimal insurance coverage under Bonus-Malus contracts. ASTIN Bulletin: The Journal of the IAA, 31(1), 175–186.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory Using R. Heidelberg: Springer.Google Scholar
Lee, B.J. and Kim, D.H. (2016) Moral hazard in insurance claiming from a Korean natural experiment. The Geneva Papers on Risk and Insurance - Issues and Practice, 41(3), 455–467.Google Scholar
Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer Academic Publisher.CrossRefGoogle Scholar
Lodi, A. (2010) Mixed integer programming computation. In 50 Years of Integer Programming 1958–2008 (eds. Jönger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G. and Wolsey, L.A.), pp. 619–645. Berlin: Springer.CrossRefGoogle Scholar
Loimaranta, K. (1972) Some asymptotic properties of bonus systems. ASTIN Bulletin: The Journal of the IAA, 6(3), 233–245.CrossRefGoogle Scholar
Lourenço, H.R., Martin, O.C. and Stützle, T. (2010) Iterated local search: Framework and applications. In Handbook of Metaheuristics. International Series in Operations Research & Management Science (eds. Gendreau, M. and Potvin, J.Y.), pp. 363–397. Boston: Springer.Google Scholar
Mert, M. and Saykan, Y. (2005) On a bonus-malus system where the claim frequency distribution is geometric and the claim severity distribution is Pareto. Hacettepe Journal of Mathematics and Statistics, 34, 75–81.Google Scholar
Molnar, D.E. and Rockwell, T.H. (1966) Analysis of policy movement in a Merit-Rating program: An application of Markov processes. The Journal of Risk and Insurance, 33(2), 265–276.CrossRefGoogle Scholar
Norberg, R. (1976) A credibility theory for automobile bonus systems. Scandinavian Actuarial Journal, 1976(2), 92–10.CrossRefGoogle Scholar
Payandeh, A.T. and Sakizadeh, M. (2017) Designing an Optimal Bonus-Malus System Using the Number of Reported Claims, Steady-State Distribution, and Mixture Claim Size Distribution. Working Paper. https://arXiv.org/abs/1701.05441arXiv:1701.05441, 1–29.Google Scholar
Puelz, R. and Snow, A. (1994) Evidence on adverse selection: Equilibrium signaling and cross subsidization in the insurance market. Journal of Public Economics, 102(2), 236–257.Google Scholar
Rothschild, M. and Stiglitz, J. (1976) Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. The Quarterly Journal of Economics, 90(4), 629–649.CrossRefGoogle Scholar
Shavell, S. (1979) On moral hazard and insurance. The Quarterly Journal of Economics, 93, 541–562.CrossRefGoogle Scholar
Sundt, B. (1989) Bonus hunger and credibility estimators with geometric weights. Insurance: Mathematics and Economics, 8(2), 119–126.Google Scholar
Tan, C.I., Li, J., Li, J.S. and Balasooriya, U. (2015) Optimal relativities and transition rules of a bonus–malus system. Insurance: Mathematics and Economics, 61, 255–263.Google Scholar
Vanderbroek, M. (1993) Bonus-malus system or partial coverage to oppose moral hazard problems? Insurance: Mathematics and Economics, 13, l–5.CrossRefGoogle Scholar
Vukina, T. and Nestić, D. (2015) Do people drive safer when accidents are more expensive: Testing for moral hazard in experience rating schemes. Transportation Research Part A, 71, 46–58.Google Scholar