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MEASURING THE IMPACT OF A BONUS-MALUS SYSTEM IN FINITE AND CONTINUOUS TIME RUIN PROBABILITIES FOR LARGE PORTFOLIOS IN MOTOR INSURANCE

Published online by Cambridge University Press:  21 March 2017

Lourdes B. Afonso
Affiliation:
Departamento de Matemática and CMA, Faculdade Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal, E-Mail: [email protected]
Rui M. R. Cardoso
Affiliation:
Departamento de Matemática and CMA, Faculdade Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal, E-Mail: [email protected]
Alfredo D. Egídio dos Reis*
Affiliation:
Department of Management, ISEG and CEMAPRE, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
Gracinda Rita Guerreiro
Affiliation:
Departamento de Matemática and CMA, Faculdade Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal, E-Mail: [email protected]

Abstract

Motor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

Type
Research Article
Copyright
Copyright © Astin Bulletin 2017 

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References

Afonso, L.B., Egídio dos Reis, A.D. and Waters, H.R. (2009) Calculating continuous time ruin probabilities for a large portfolio with varying premiums. ASTIN Bulletin, 39 (1), 117136.CrossRefGoogle Scholar
Afonso, L.B., Egídio dos Reis, A.D. and Waters, H.R. (2010) Numerical evaluation of continuous time ruin probabilities for a portfolio with credibility updated premiums. ASTIN Bulletin, 40 (1), 399414.CrossRefGoogle Scholar
Andrade e Silva, J. and Centeno, M.L. (2005) A note on bonus scales. Journal of Risk and Insurance, 72 (4), 601607.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities, Advanced Series on Statistical Science & Applied Probability, Vol. 14. World Scientific, New Jersey, London, Singapore.CrossRefGoogle Scholar
Borgan, Ø., Hoem, J. and Norberg, R. (1981) A non asymptotic criterion for the evaluation of automobile bonus system. Scandinavian Actuarial Journal, 1981, 165178.Google Scholar
Constantinescu, C., Dai, S., Ni, W. and Palmowski, Z. (2016) Ruin probabilities with dependence on the number of claims within a fixed time window. Risks, 4 (2), 17. doi:10.3390/risks4020017.CrossRefGoogle Scholar
Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.-F. (2007) Actuarial Modelling of Claim Counts. Willey, Chichester, West Sussex, UK.CrossRefGoogle Scholar
Dubey, A. (1977) Probabilité de ruine lorsque le paramètre de Poisson est ajusté a posteriori. Mitt. Verein. Schweiz. VersicherungsMath., 77, 131141.Google Scholar
Dufresne, F. (1988) Distributions stationnaires d'un système bonus-malus et probabilité de ruine. ASTIN Bulletin, 18 (1), 3146.CrossRefGoogle Scholar
Gilde, V. and Sundt, B. (1989) On bonus systems with credibility scales. Scandinavian Actuarial Journal, 1989, 1322.CrossRefGoogle Scholar
Gómez-Déniz, E. (2016) Bivariate credibility bonus-malus premiums distinguishing between two types of claims. Insurance: Mathematics and Economics, 70, 117124.Google Scholar
Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Springer, New York.CrossRefGoogle Scholar
Li, B., Ni, W. and Constantinescu, C. (2015). Risk models with premiums adjusted to claims number. Insurance: Mathematics and Economics, 65(2015), 94102. https://doi.org/10.1016/j.insmatheco.2015.09.001.Google Scholar
Ni, W., Constantinescu, C. and Pantelous, A.A. (2014). Bonus-Malus systems with Weibull distributed claim severities. Annals of Actuarial Science, 8 (2), 217233. doi:10.1017/S1748499514000062.CrossRefGoogle Scholar
Norberg, R. (1976) A credibility theory for automobile bonus system. Scandianvian Actuarial Journal, 1976, 92107.CrossRefGoogle Scholar
Parzen, E. (1965) Stochastic Processes. Holden-Day series in probability and statistics. Holden-Day, San Francisco, London, Amsterdam.Google Scholar
Tan, C.I. (2016) Varying transition rules in bonus-malus systems: From rules specification to determination of optimal relativities. Insurance: Mathematics and Economics, 68, 134140.Google Scholar
Tan, C.I., Li, J., Li, J.S-H. and Balasooriya, U. (2015) Optimal relativities and transition rules of a bonus-malus system. Insurance: Mathematics and Economics, 61, 255263.Google Scholar
Wagner, C. (2001) A note on ruin in a two state Markov model. ASTIN Bulletin, 31 (2), 349358.CrossRefGoogle Scholar
Wu, X., Chen, M., Guo, J. and Jin, C. (2015) On a discrete-time risk model with claim correlated premiums. Annals of Actuarial Science, 9 (2), 322342.CrossRefGoogle Scholar