Skip to main content


  • Lourdes B. Afonso (a1), Rui M. R. Cardoso (a2), Alfredo D. Egídio dos Reis (a3) and Gracinda Rita Guerreiro (a4)

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.

Corresponding author
Hide All
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.
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.
Andrade e Silva, J. and Centeno, M.L. (2005) A note on bonus scales. Journal of Risk and Insurance, 72 (4), 601607.
Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities, Advanced Series on Statistical Science & Applied Probability, Vol. 14. World Scientific, New Jersey, London, Singapore.
Borgan, Ø., Hoem, J. and Norberg, R. (1981) A non asymptotic criterion for the evaluation of automobile bonus system. Scandinavian Actuarial Journal, 1981, 165178.
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.
Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.-F. (2007) Actuarial Modelling of Claim Counts. Willey, Chichester, West Sussex, UK.
Dubey, A. (1977) Probabilité de ruine lorsque le paramètre de Poisson est ajusté a posteriori. Mitt. Verein. Schweiz. VersicherungsMath., 77, 131141.
Dufresne, F. (1988) Distributions stationnaires d'un système bonus-malus et probabilité de ruine. ASTIN Bulletin, 18 (1), 3146.
Gilde, V. and Sundt, B. (1989) On bonus systems with credibility scales. Scandinavian Actuarial Journal, 1989, 1322.
Gómez-Déniz, E. (2016) Bivariate credibility bonus-malus premiums distinguishing between two types of claims. Insurance: Mathematics and Economics, 70, 117124.
Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Springer, New York.
Li, B., Ni, W. and Constantinescu, C. (2015). Risk models with premiums adjusted to claims number. Insurance: Mathematics and Economics, 65(2015), 94102.
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.
Norberg, R. (1976) A credibility theory for automobile bonus system. Scandianvian Actuarial Journal, 1976, 92107.
Parzen, E. (1965) Stochastic Processes. Holden-Day series in probability and statistics. Holden-Day, San Francisco, London, Amsterdam.
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.
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.
Wagner, C. (2001) A note on ruin in a two state Markov model. ASTIN Bulletin, 31 (2), 349358.
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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
Please enter your name
Please enter a valid email address
Who would you like to send this to? *