Skip to main content

Statistical concepts of a priori and a posteriori risk classification in insurance

Abstract

Everyday we face all kinds of risks, and insurance is in the business of providing us a means to transfer or share these risks, usually to eliminate or reduce the resulting financial burden, in exchange for a predetermined price or tariff. Actuaries are considered professional experts in the economic assessment of uncertain events, and equipped with many statistical tools for analytics, they help formulate a fair and reasonable tariff associated with these risks. An important part of the process of establishing fair insurance tariffs is risk classification, which involves the grouping of risks into various classes that share a homogeneous set of characteristics allowing the actuary to reasonably price discriminate. This article is a survey paper on the statistical tools for risk classification used in insurance. Because of recent availability of more complex data in the industry together with the technology to analyze these data, we additionally discuss modern techniques that have recently emerged in the statistics discipline and can be used for risk classification. While several of the illustrations discussed in the paper focus on general, or non-life, insurance, several of the principles we examine can be similarly applied to life insurance. Furthermore, we also distinguish between a priori and a posteriori ratemaking. The former is a process which forms the basis for ratemaking when a policyholder is new and insufficient information may be available. The latter process uses additional historical information about policyholder claims when this becomes available. In effect, the resulting a posteriori premium allows one to correct and adjust the previous a priori premium making the price discrimination even more fair and reasonable.

References

  1. Antonio, K., Beirlant, J.: Actuarial statistics with generalized linear mixed models. Insur. Math. Econ. 40(1), 58–76 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  2. Antonio, K., Frees, E.W., Valdez, E.A.: A multilevel analysis of intercompany claim counts. ASTIN Bull. 40(1), 151–177 (2010)

    MathSciNet  Article  Google Scholar 

  3. Beirlant, J., Goegebeur, Y., Verlaak, R., Vynckier, P.: Burr regression and portfolio segmentation. Insur. Math. Econ. 23(3), 231–250 (1998)

    MATH  Article  Google Scholar 

  4. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)

    MATH  Book  Google Scholar 

  5. Bolancé, C., Guillén, M., Pinquet, J.: Time-varying credibility for frequency risk models: estimation and tests for autoregressive specifications on random effects. Insur. Math. Econ. 33(2), 273–282 (2003)

    MATH  Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  7. Boucher, J.-P., Denuit, M., Guillén, M.: Modelling of insurance claim count with hurdle distribution for panel data. In: Arnold, B.C., Balakrishnan, N., Sarabia, J.M., Mínquez, R. (eds.) Advances in Mathematical and Statistical Modeling: Statistics for Industry and Technology, pp. 45–60. Birkhäuser, Boston (2008). Chap. 4

    Chapter  Google Scholar 

  8. Boucher, J.-P., Denuit, M., Guillén, M.: Number of accidents or number of claims? J. Risk Insur. 76(4), 821–846 (2009)

    Article  Google Scholar 

  9. Bühlmann, H.: Experience rating and credibility I. ASTIN Bull. 4(3), 199–207 (1967)

    Google Scholar 

  10. Bühlmann, H.: Experience rating and credibility II. ASTIN Bull. 5(2), 157–165 (1969)

    Google Scholar 

  11. Bühlmann, H., Gisler, A.: A Course in Credibility Theory and Its Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  12. Cameron, A.C., Trivedi, P.K.: Regression Analysis of Count Data. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  13. de Jong, P., Heller, G.Z.: Generalized Linear Models for Insurance Data. Cambridge University Press, Cambridge (2008)

    MATH  Book  Google Scholar 

  14. Denuit, M., Lang, S.: Non-life ratemaking with Bayesian GAMs. Insur. Math. Econ. 35(3), 627–647 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  15. Denuit, M., Maréchal, X., Pitrebois, S., Walhin, J.-F.: Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus–Malus Systems. Wiley, Chichester (2007)

    MATH  Book  Google Scholar 

  16. Dickson, D.C.M., Hardy, M.R., Waters, H.R.: Actuarial Mathematics for Life Contingent Risks. Cambridge University, Cambridge (2009)

    MATH  Google Scholar 

  17. Finger, R.J.: Risk classification. In: Foundations of Casualty Actuarial Science, 4th edn., pp. 75–148. Casualty Actuarial Society, Arlington (2001). Chap. 6

    Google Scholar 

  18. Frees, E.W.: Regression Modeling with Actuarial and Financial Applications. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  19. Frees, E.W., Young, V.R., Luo, Y.: A longitudinal data analysis interpretation of credibility models. Insur. Math. Econ. 24(3), 229–247 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  20. Haberman, S., Renshaw, A.E.: Generalized linear models and actuarial science. Statistician 45(4), 407–436 (1996)

    Article  Google Scholar 

  21. Hachemeister, C.A.: Credibility for regression models with application to trend. In: Kahn, P.M. (ed.) Credibility: Theory and Applications, pp. 129–163. Academic Press, New York (1975)

    Google Scholar 

  22. Hausman, J.A., Hall, B.H., Griliches, Z.: Econometric models for count data with application to patents-R&D relationship. Econometrica 52(4), 909–938 (1984)

    Article  Google Scholar 

  23. Jewell, W.S.: The use of collateral data in credibility theory: a hierarchical model. G. Ist. Ital. Attuari 38, 1–6 (1975)

    MATH  Google Scholar 

  24. Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern Actuarial Risk Theory: Using R, 2nd edn. Springer, Berlin (2008)

    MATH  Book  Google Scholar 

  25. Klugman, S.A., Panjer, H.H., Willmot, G.E.: Loss Models: From Data to Decisions, 3rd edn. Wiley, Hoboken (2008)

    MATH  Book  Google Scholar 

  26. Lee, A.H., Wang, K., Scott, J.A., Yau, K.K.W., McLachlan, G.J.: Multi-level zero-inflated Poisson regression modelling of correlated count data with excess zeros. Stat. Methods Med. Res. 15(1), 47–61 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  27. Lemaire, J.: Automobile Insurance: Actuarial Models. Kluwer Academic, Dordrecht (1985)

    Google Scholar 

  28. McClenahan, C.L.: Ratemaking. In: Foundations of Casualty Actuarial Science, 4th edn., pp. 75–148. Casualty Actuarial Society, Arlington (2001). Chap. 3

    Google Scholar 

  29. McDonald, J.B.: Some generalized functions for the size distribution of income. Econometrica 52(3), 647–663 (1984)

    MATH  Article  Google Scholar 

  30. Mullahy, J.: Specification and testing of some modified count data models. J. Econom. 33(3), 341–365 (1986)

    MathSciNet  Article  Google Scholar 

  31. Nelder, J.A., Wedderburn, R.W.M.: Generalized linear models. J. R. Stat. Soc. A 135(3), 370–384 (1972)

    Article  Google Scholar 

  32. Norberg, R.: A credibility theory for automobile bonus systems. Scand. Actuar. J., 92–107 (1976)

  33. Panjer, H.H., Willmot, G.E.: Insurance Risk Models. Society of Actuaries, Schaumburg (1992)

    Google Scholar 

  34. Pinquet, J.: Allowance for cost of claims in Bonus–Malus systems. ASTIN Bull. 27(1), 33–57 (1997)

    MathSciNet  Article  Google Scholar 

  35. Pinquet, J.: Designing optimal Bonus–Malus systems from different types of claims. ASTIN Bull. 28(2), 205–229 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  36. Pinquet, J., Guillén, M., Bolancé, C.: Allowance for age of claims in Bonus–Malus systems. ASTIN Bull. 31(2), 337–348 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  37. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1999)

    MATH  Book  Google Scholar 

  38. Sun, J., Frees, E.W., Rosenberg, M.A.: Heavy-tailed longitudinal data modeling using copulas. Insur. Math. Econ. 42(2), 817–830 (2008)

    MATH  Article  Google Scholar 

  39. Winkelmann, R.: Econometric Analysis of Count Data. Springer, Berlin (2003)

    MATH  Google Scholar 

  40. Yau, K.K.W., Wang, K., Lee, A.H.: Zero-inflated negative binomial mixed regression modeling of over-dispersed count data with extra zeros. Biom. J. 45(4), 437–452 (2003)

    MathSciNet  Article  Google Scholar 

  41. Yip, K.C.H., Yau, K.K.W.: On modeling claim frequency data in general insurance with extra zeros. Insur. Math. Econ. 36(2), 153–163 (2005)

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Katrien Antonio.

Additional information

Katrien Antonio acknowledges financial support from NWO through a Veni 2009 grant.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Antonio, K., Valdez, E.A. Statistical concepts of a priori and a posteriori risk classification in insurance. AStA Adv Stat Anal 96, 187–224 (2012). https://doi.org/10.1007/s10182-011-0152-7

Download citation

Keywords

  • Actuarial science
  • Regression and credibility models
  • Bonus–Malus systems