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OPTIMAL BONUS-MALUS SYSTEMS USING FINITE MIXTURE MODELS

Published online by Cambridge University Press:  10 January 2014

George Tzougas ,
Spyridon Vrontos  and
Nicholas Frangos
George Tzougas
Affiliation:
Department of Statistics, Athens University of Economics and Business, Athens, Attica, Greece
Spyridon Vrontos
Affiliation:
Department of Accounting, Finance and Governance, University of Westminster, London, UK, Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Attica, Greece
Nicholas Frangos*
Affiliation:
Department of Statistics, Athens University of Economics and Business, 76 Patission Street, Athens 10434, Attica, Greece, Tel: +302108203579
*
E-Mail: nef@aueb.gr
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Abstract

This paper presents the design of optimal Bonus-Malus Systems using finite mixture models, extending the work of Lemaire (1995; Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Norwell, MA: Kluwer) and Frangos and Vrontos (2001; Frangos, N. and Vrontos, S. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31(1), 1–22). Specifically, for the frequency component we employ finite Poisson, Delaporte and Negative Binomial mixtures, while for the severity component we employ finite Exponential, Gamma, Weibull and Generalized Beta Type II mixtures, updating the posterior probability. We also consider the case of a finite Negative Binomial mixture and a finite Pareto mixture updating the posterior mean. The generalized Bonus-Malus Systems we propose, integrate risk classification and experience rating by taking into account both the a priori and a posteriori characteristics of each policyholder.

Keywords

Optimal BMSclaim frequencyclaim severitymixtures of distributionsa priori classification criteriaa posteriori classification criteria
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 44 , Issue 2 , May 2014 , pp. 417 - 444
Copyright
Copyright © ASTIN Bulletin 2014 

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References

[1]Boucher, J.P., Denuit, M. and Guillen, M. (2007) Risk classification for claim counts: A comparative analysis of various zero-inflated mixed Poisson and hurdle models. North American Actuarial Journal, 11 (4), 110–131.CrossRefGoogle Scholar
[2]Boucher, J.P., Denuit, M. and Guillen, M. (2008) Models of insurance claim counts with time dependence based on generalisation of Poisson and negative binomial distributions. Variance, 2 (1), 135–162.Google Scholar
[3]Brouhns, N., Guillen, M., Denuit, M. and Pinquet, J. (2003) Bonus-malus scales in segmented tariffs with stochastic migration between segments. Journal of Risk and Insurance, 70, 577–599.Google Scholar
[4]Denuit, M., Marechal, X., Pitrebois, S. and Walhin, J.F. (2007) Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems. Chichester, England: Wiley.CrossRefGoogle Scholar
[5]Dionne, G. and Vanasse, C. (1989) A generalization of actuarial automobile insurance rating models: The negative binomial distribution with a regression component. ASTIN Bulletin, 19, 199–212.Google Scholar
[6]Dionne, G. and Vanasse, C. (1992) Automobile insurance ratemaking in the presence of asymmetrical information. Journal of Applied Econometrics, 7, 149–165.Google Scholar
[7]Frangos, N. and Vrontos, S. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31 (1), 1–22.Google Scholar
[8]Gourieroux, C., Montfort, A. and Trognon, A. (1984a) Pseudo maximum likelihood methods: Theory. Econometrica, 52, 681–700.Google Scholar
[9]Gourieroux, C., Montfort, A. and Trognon, A. (1984b) Pseudo maximum likelihood methods: Applications to Poisson models. Econometrica, 52, 701–720.Google Scholar
[10]Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. New York: Wiley.Google Scholar
[11]Johnson, N.L., Kotz, S. and Kemp, A.W. (2005) Univariate Discrete Distributions. New Jersey: Wiley.Google Scholar
[12]Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer Academic Publishers.Google Scholar
[13]Mahmoudvand, R. and Hassani, H. (2009) Generalized bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 39, 307–315.Google Scholar
[14]McDonald, J.B. (1996) Probability distributions for financial models. In Handbook of Statistics (eds. Maddala, G.S. and Rao, C.R.), pp. 427–460. Amsterdam: Elsevier.Google Scholar
[15]McDonald, J.B. and Xu, Y.J. (1995) A generalisation of the beta distribution with applications. Journal of Econometrics, 66, 133–152.CrossRefGoogle Scholar
[16]McLachlan, G. and Peel, D. (2000) Finite Mixture Models. New York: John Wiley & Sons.CrossRefGoogle Scholar
[17]Picech, L. (1994) The merit-rating factor in a multiplicating rate-making model. Conference Paper, ASTIN Colloquium 1994, Cannes.Google Scholar
[18]Pinquet, J. (1997) Allowance for cost of claims in bonus-malus systems. ASTIN Bulletin, 27, 33–57.CrossRefGoogle Scholar
[19]Pinquet, J. (1998) Designing optimal bonus-malus systems from different types of claims. ASTIN Bulletin, 28, 205–220.Google Scholar
[20]Rigby, R.A. and Stasinopoulos, D.M. (2005) Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54, 507–554.Google Scholar
[21]Rigby, R.A. and Stasinopoulos, D.M. (2009) A Flexible Regression Approach Using GAMLSS in R. Lancaster, England.Google Scholar