Skip to main content
SpringerLink
Log in

The Design of an Optimal Bonus-Malus System Based on the Sichel Distribution

  • Chapter
  • First Online:
Modern Problems in Insurance Mathematics

Part of the book series: EAA Series ((EAAS))

Abstract

This chapter presents the design of an optimal Bonus-Malus System (BMS) using the Sichel distribution to model the claim frequency distribution. This system is proposed as an alternative to the optimal BMS obtained by the traditional Negative Binomial model [19]. The Sichel distribution has a thicker tail than the Negative Binomial distribution and it is considered as a plausible model for highly dispersed count data. We also consider the optimal BMS provided by the Poisson-Inverse Gaussian distribution (PIG), which is a special case of the Sichel distribution. Furthermore, we develop a generalised BMS that takes into account both the a priori and a posteriori characteristics of each policyholder. For this purpose we consider the generalised additive models for location, scale and shape (GAMLSS) in order to use all available information in the estimation of the claim frequency distribution. Within the framework of the GAMLSS we propose the Sichel GAMLSS for assessing claim frequency as an alternative to the Negative Binomial Type I (NBI) regression model used by Dionne and Vanasse [9, 10]. We also consider the NBI and PIG GAMLSS for assessing claim frequency.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akantziliotou, C., Rigby, R.A., Stasinopoulos, D.M.: The R implementation of generalized additive models for location, scale and shape. In: Stasinopoulos, M., Touloumi, G. (eds.) Statistical Modelling in Society: Proceedings of the 17th International Workshop on Statistical Modelling, pp. 75–83. Chania, Greece (2002)

    Google Scholar 

  2. Bichsel, F.: Erfahrung-Tarifieung in der Motorfahrzeug-haftplichtversiherung. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, pp. 119–129 (1964)

    Google Scholar 

  3. Boucher, J.P., Denuit, M., Guillen, M.: Risk classification for claim counts: a comparative analysis of various zero-inflated mixed Poisson and Hurdle models. North Am. Actuar. J. 11(4), 110–131 (2007)

    Article  MathSciNet  Google Scholar 

  4. Boucher, J.P., Denuit, M., Guillen, M.: Models of insurance claim counts with time dependence based on generalisation of Poisson and negative binomial distributions. Variance 2(1), 135–162 (2008)

    Google Scholar 

  5. Buhlmann, H.: Optimale Pramienstufensysteme. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, pp. 193–213 (1964)

    Google Scholar 

  6. Coene, G., Doray, L.G.: A financially balanced Bonus-Malus system. ASTIN Bull. 26, 107–115 (1996)

    Article  Google Scholar 

  7. Dean, C., Lawless, J.F., Willmot, G.E.: A mixed Poisson-inverse-Gaussian regression model. Can. J. Stat. 17(2), 171–181 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Denuit, M., Marechal, X., Pitrebois, S., Walhin, J.F.: Actuarial modelling of claim counts: risk classification. Credibility and Bonus-Malus systems. Wiley, Chichester (2007)

    Book  Google Scholar 

  9. Dionne, G., Vannasse, C.: A generalization of actuarial automobile insurance rating models: the negative binomial distribution with a regression component. ASTIN Bull. 19, 199–212 (1989)

    Article  Google Scholar 

  10. Dionne, G., Vannasse, C.: Automobile insurance ratemaking in the presence of asymmetrical information. J. Appl. Econom. 7, 149–165 (1992)

    Article  Google Scholar 

  11. Frangos, N., Vrontos, S.: Design of optimal Bonus-Malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bull. 31(1), 1–22 (2001)

    Article  MathSciNet  Google Scholar 

  12. Frangos, N., Tzougas, G., Vrontos, S.: On the design of some optimal Bonus-Malus systems using frequency and severity components. In: Proceedings of the 6th Conference in Actuarial Science and Finance. Samos, Greece (2010)

    Google Scholar 

  13. Gourieroux, C., Montfort, A., Trognon, A.: Pseudo maximum likelihood methods: theory. Econometrica 52, 681–700 (1984a)

    Google Scholar 

  14. Gourieroux, C., Montfort, A., Trognon, A.: Pseudo maximum likelihood methods: applications to Poisson models. Econometrica 52, 701–720 (1984b)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hilbe, J.M.: Negative Binomial Regression Extensions. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  16. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions. Wiley, Chichester (1994)

    MATH  Google Scholar 

  17. Jørgensen, B.: Statistical properties of the generalized inverse Gaussian distribution. In: Lecture Notes in Statistics, vol. 9. Springer, New York (1982)

    Google Scholar 

  18. Lawless, J.F.: Negative binomial distribution and mixed Poisson regression. Can. J. Stat. 15(3), 209–225 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lemaire, J.: Bonus-Malus Systems in Automobile Insurance. Kluwer, Boston (1995)

    Book  Google Scholar 

  20. Mahmoudvand, R., Hassani, H.: Generalized Bonus-Malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bull. 39, 307–315 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mahmoudvand, R., Aziznasiri, S.: Bonus-Malus systems in open and closed portfolios. In: International Cramér Symposium on Insurance Mathematics. Stockholm, Sweden (2013)

    Google Scholar 

  22. Pinquet, J., et al.: Long-range contagion in automobile insurance data: estimation and implications for experience rating. ASTIN Bull. 31(2), 337–348 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rigby, R.A., Stasinopoulos, D.M.: The GAMLSS project—a flexible approach to statistical modelling. In: Klein, B., Korsholm, L. (eds.) New Trends in Statistical Modelling: Proceedings of the 16th International Workshop on Statistical Modelling, pp. 249–256. Odense, Denmark (2001)

    Google Scholar 

  24. Rigby, R.A., Stasinopoulos, D.M.: Generalized additive models for location, scale and shape, (with discussion). Appl. Stat. 54, 507–554 (2005)

    MATH  MathSciNet  Google Scholar 

  25. Rigby, R.A., Stasinopoulos, D.M., Akantziliotou, C.: A framework for modeling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution. Comput. Stat. Data Anal. 53, 381–393 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rigby, R.A., Stasinopoulos, D.M.: A flexible regression approach using GAMLSS in R. Lancaster (2009) Available via DIALOG. http://www.gamlss.org/wp-content/uploads/2013/01/Lancaster-booklet.pdf. (Cited 13 Nov 2009)

  27. Sichel, H.S.: A bibliometric distribution which really works. J. Am. Soc. Inf. Sci. 36(5), 314–321 (1985)

    Article  Google Scholar 

  28. Tzougas, G., Frangos, N.: Design of an optimal Bonus-Malus system using the Sichel distribution as a model of claim counts. In: International Cramér Symposium on Insurance Mathematics. Stockholm, Sweden (2013)

    Google Scholar 

  29. Vuong, Q.: Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57, 307–333 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  30. Walhin, J.F., Paris, J.: Using mixed Poisson distributions in connection with Bonus-Malus systems. ASTIN Bull. 29, 81–99 (1999)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Athens University of Economics and Business, Athens, Greece

    George Tzougas & Nicholas Frangos

Authors
  1. George Tzougas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Frangos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Tzougas .

Editor information

Editors and Affiliations

  1. Department of Mathematics Professor of Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Dmitrii Silvestrov

  2. Department of Mathematics Professor in Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Anders Martin-Löf

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Tzougas, G., Frangos, N. (2014). The Design of an Optimal Bonus-Malus System Based on the Sichel Distribution. In: Silvestrov, D., Martin-Löf, A. (eds) Modern Problems in Insurance Mathematics. EAA Series. Springer, Cham. https://doi.org/10.1007/978-3-319-06653-0_15

Download citation

Publish with us

Policies and ethics

Search

Navigation