Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T19:59:54.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Bonus–Malus systems with Weibull distributed claim severities

Published online by Cambridge University Press:  19 May 2014

Weihong Ni ,
Corina Constantinescu  and
Athanasios A. Pantelous
Weihong Ni
Affiliation:
Department of Mathematical Sciences, Institute for Actuarial and Financial Mathematics, University of Liverpool, UK
Corina Constantinescu
Affiliation:
Department of Mathematical Sciences, Institute for Actuarial and Financial Mathematics, University of Liverpool, UK
Athanasios A. Pantelous*
Affiliation:
Department of Mathematical Sciences, Institute for Actuarial and Financial Mathematics, University of Liverpool, UK; Institute for Risk and Uncertainty, University of Liverpool, UK
*
*Correspondence to: Athanasios A. Pantelous, Department of Mathematical Sciences, Institute for Actuarial and Financial Mathematics, University of Liverpool, UK. E-mail: A.Pantelous@liverpool.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

One of the pricing strategies for Bonus–Malus (BM) systems relies on the decomposition of the claims’ randomness into one part accounting for claims’ frequency and the other part for claims’ severity. By mixing an exponential with a Lévy distribution, we focus on modelling the claim severity component as a Weibull distribution. For a Negative Binomial number of claims, we employ the Bayesian approach to derive the BM premiums for Weibull severities. We then conclude by comparing our explicit formulas and numerical results with those for Pareto severities that were introduced by Frangos & Vrontos.

Keywords

Bonus–Malus SystemsWeibull DistributionBayesian Estimator
Type
Papers
Information
Annals of Actuarial Science , Volume 8 , Issue 2 , September 2014 , pp. 217 - 233
Copyright
© Institute and Faculty of Actuaries 2014 

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

Abramowitz, M. & Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York, NY.Google Scholar
Albrecher, H., Constantinescu, C. & Loisel, S. (2011). Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics, 48(2), 265–270.Google Scholar
Boland, P.J. (2007). Statistical and Probabilistic Methods in Actuarial Science. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Dionne, G. & Vanasse, C. (1989). A generalization of automobile insurance rating models: the negative binomial distribution with a regression component. ASTIN Bulletin, 19(2), 199–212.CrossRefGoogle Scholar
Embrechts, P. (1983). A property of the generalized inverse Gaussian distribution with some applications. Journal of Applied Probability, 3, 537–544.CrossRefGoogle Scholar
Frangos, N.E. & Vrontos, S.D. (2001). Design of optimal Bonus-Malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31, 1–22.Google Scholar
Ismail, M.E.H. & Muldoon, M.E. (1978). Monotonicity of the zeros of a cross-product of Bessel functions. SIAM Journal on Mathematical Analysis, 9(4), 759–767.Google Scholar
Klugman, S.A., Panjer, H.H. & Willmot, G.E. (1998). Loss Models: From D ata to Decisions. John Wiley & Sons, New Jersey.Google Scholar
Lemaire, J. (1995). Bonus-Malus Systems in Automobile Insurance. Kluwer Academic Publishers, Boston, MA.Google Scholar
Lorch, L. (1994). Monotonicity of the zeros of a cross product of Bessel functions. Methods and Applications of Analysis, 1(1), 75–80.Google Scholar
Picard, P. (1976). Généralisation de l’etude sur la survenance des sinistres en assurance automobile. Bulletin Trimestriel de l’Institut des Actuaires Français, 296, 204–268.Google Scholar
Tremblay, L. (1992). Using the poisson inverse Gaussian in Bonus-Malus systems. ASTIN Bulletin, 22(1), 97–106.CrossRefGoogle Scholar
Valdez, E.A. & Frees, E.W. (2005). Longitudinal modeling of Singapore Motor Insurance, working paper.Google Scholar