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Insurance ratemaking using the Exponential-Lognormal regression model

Published online by Cambridge University Press:  26 June 2019

George Tzougas ,
Woo Hee Yik  and
Muhammad Waqar Mustaqeem
George Tzougas*
Affiliation:
Department of Statistics, London School of Economics and Political Science, Holborn, London WC2A 2AE, UK
Woo Hee Yik
Affiliation:
Department of Statistics, London School of Economics and Political Science, Holborn, London WC2A 2AE, UK
Muhammad Waqar Mustaqeem
Affiliation:
Department of Statistics, London School of Economics and Political Science, Holborn, London WC2A 2AE, UK
*
*Corresponding author. E-mail: [email protected]
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Abstract

This paper is concerned with presenting the Exponential-Lognormal (ELN) regression model as a competitive alternative to the Pareto, or Exponential-Inverse Gamma, regression model that has been used in a wide range of areas, including insurance ratemaking. This is the first time that the ELN regression model is used in a statistical or actuarial context. The main contribution of the study is that we illustrate how maximum likelihood estimation of the ELN regression model, which does not have a density in closed form, can be accomplished relatively easily via an Expectation-Maximisation type algorithm. A real data application based on motor insurance data is examined in order to emphasise the versatility of the proposed algorithm. Finally, assuming that the number of claims is distributed according to the classic Negative Binomial and Poisson-Inverse Gaussian regression models, both the a priori and a posteriori, or Bonus–Malus, premium rates resulting from the ELN regression model are calculated via the net premium principle and compared to those determined by the Pareto regression model that has been traditionally used for modelling claim sizes.

Keywords

Exponential-Lognormal regression modelEM AlgorithmMotor Third Party Liability Insuranceratemaking
Type
Paper
Information
Annals of Actuarial Science , Volume 14 , Issue 1 , March 2020 , pp. 42 - 71
Copyright
© Institute and Faculty of Actuaries 2019 

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References

Barlow, R.E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, USA.Google Scholar
Bhattacharya, S.K. & Kumar, S. (1986). E-IG model in life testing. Calcutta Statistical Association Bulletin, 35(1–2), 85–90.CrossRefGoogle Scholar
Bohning, D. (1999). Computer Assisted Analysis of Mixtures and Applications in Meta-Analysis, Disease Mapping and Others. CRC Press, New York, USA.Google Scholar
Booth, J.G. & Hobert, J.P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(1), 265–285.10.1111/1467-9868.00176CrossRefGoogle Scholar
Booth, J.G., Hobert, J.P. & Jank, W. (2001). A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Statistical Modelling, 1(4), 333–349.10.1177/1471082X0100100407CrossRefGoogle Scholar
Boucher, J.P., Denuit, M. & 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), 110–131.10.1080/10920277.2007.10597487CrossRefGoogle Scholar
Boucher, J.P., Denuit, M. & 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
Brouhns, N., Guillen, M., Denuit, M. & Pinquet, J. (2003). Bonus–Malus scales in segmented tariffs with stochastic migration between segments. Journal of Risk and Insurance, 70(4), 577–599.10.1046/j.0022-4367.2003.00066.xCrossRefGoogle Scholar
Burnham, K.P. & Anderson, D.R. (2003). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer, New York, USA.Google Scholar
Calderín-Ojeda, E., Fergusson, K. & Wu, X. (2017). An EM algorithm for Double-Pareto-Lognormal Generalized Linear Model applied to heavy-tailed insurance claims. Risks, 5(4), 60.10.3390/risks5040060CrossRefGoogle Scholar
De Jong, P. & Heller, G.Z. (2008). Generalized Linear Models for Insurance Data. Cambridge University Press, Cambridge, UK.10.1017/CBO9780511755408CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M. & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 39(1), 1–38.Google Scholar
Denuit, M. & Lang, S. (2004). Nonlife ratemaking with Bayesian GAM’s. Insurance: Mathematics and Economics, 35(3), 627–647.Google Scholar
Denuit, M., Marechal, X., Pitrebois, S. & Walhin, J. F. (2007). Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus–Malus Systems. John Wiley and Sons, Chichester, West Sussex, UK.10.1002/9780470517420CrossRefGoogle Scholar
Dionne, G. & Vanasse, C. (1989). A generalization of actuarial automobile insurance rating models: the negative binomial distribution with a regression component. ASTIN Bulletin, 19(2), 199–212.CrossRefGoogle Scholar
Dionne, G. & Vanasse, C. (1992). Automobile insurance ratemaking in the presence of asymmetrical information. Journal of Applied Econometrics, 7(2), 149–165.10.1002/jae.3950070204CrossRefGoogle Scholar
Dunn, P.K. & Smyth, G.K. (1996). Randomized quantile residuals. Computational and Graphical Statistics, 5(3), 236–245.Google Scholar
Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997). Modelling extremal events. In Applications of Mathematics, vol. 33. Springer-Verlag, Berlin, Heidelberg, DE.Google Scholar
Frangos, N. & 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), 1–22.10.2143/AST.31.1.991CrossRefGoogle Scholar
Frangos, N. & Karlis, D. (2004). Modelling losses using an exponential-inverse Gaussian distribution. Insurance: Mathematics and Economics, 35(1), 53–67.Google Scholar
Frees, E.W. & Valdez, E.A. (2008). Hierarchical insurance claims modeling. Journal of the American Statistical Association, 103(484), 1457–1469.CrossRefGoogle Scholar
Frees, E.W. (2010). Regression Modelling with Actuarial and Financial Applications. Cambridge University Press, Cambridge, UK.Google Scholar
Frees, E.W., Derrig, R.A. & Meyers, G. (2014a). Predictive Modeling Applications in Actuarial Science, vol. 1. Cambridge University Press, New York, USA.10.1017/CBO9781139342674CrossRefGoogle Scholar
Gómez-Déniz, E., Sarabia, J.M. & Calderín-Ojeda, E. (2008). Univariate and multivariate versions of the negative Binomial-Inverse Gaussian distributions with applications. Insurance, Mathematics and Economics, 42(1), 39–49.10.1016/j.insmatheco.2006.12.001CrossRefGoogle Scholar
Gómez-Déniz, E., Hernández-Bastida, A. & Fernández-Sánchez, M.P. (2014). Computing credibility Bonus–Malus premiums using the aggregate claims distribution. Hacettepe Journal of Mathematics and Statistics, 43(6), 1047–1061.Google Scholar
Gómez-Déniz, E. & Calderín-Ojeda, E. (2018). Multivariate credibility in Bonus–Malus systems distinguishing between different types of claims. Risks, 6(2), 34.10.3390/risks6020034CrossRefGoogle Scholar
Haberman, S. & Renshaw, A. E. (1996). Generalized linear models and actuarial science. The Statistician, 45(4), 407–436.10.2307/2988543CrossRefGoogle Scholar
Halliwell, Leigh J. (2013). Classifying the tails of loss distributions. Casualty Actuarial Society E-Forum, 2 (Spring 2013), 1.Google Scholar
Hesselager, O., Wang, S. & Willmot, G. (1998). Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, 1998(2), 125–142.CrossRefGoogle Scholar
Insurance Europe. (2016). European motor insurance markets addendum. https://www.insuranceeurope.eu/sites/default/files/attachments/European%20motor%20insurance%20markets%20addendum.pdf.Google Scholar
Jones, B.L. & Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 44–54.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. (2008). Modern Actuarial Risk Theory: Using R. Springer-Verlag, Berlin, Heidelberg, DE.10.1007/978-3-540-70998-5CrossRefGoogle Scholar
Karlis, D., Tzougas, G. & Frangos, N. (2018). Confidence intervals of the premiums of optimal Bonus Malus systems. Scandinavian Actuarial Journal, 2018(2), 129–144.CrossRefGoogle Scholar
Klein, N., Denuit, M., Lang, S. & Kneib, T. (2014). Nonlife ratemaking and risk management with Bayesian generalized additive models for location, scale, and shape. Insurance: Mathematics and Economics, 55, 225–249.Google Scholar
Klugman, S., Panjer, H. & Willmot, G. (2012). Loss Models: From Data to Decisions, vol. 715. John Wiley and Sons, New York, USA.Google Scholar
Lemaire, J. (1995). Bonus–Malus Systems in Automobile Insurance. Kluwer Academic Publishers, Dordrecht, NE.10.1007/978-94-011-0631-3CrossRefGoogle Scholar
Louis, T.A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 44 (2), 226–233.Google Scholar
Mahmoudvand, R. & Hassani, H. (2009). Generalized Bonus–Malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 39(1), 307–315.10.2143/AST.39.1.2038066CrossRefGoogle Scholar
McDonald, J.B. & Xu, Y.J. (1995). A generalization of the beta distribution with applications. Journal of Econometrics 66(1–2), 133–152.10.1016/0304-4076(94)01612-4CrossRefGoogle Scholar
McDonald, J.B. (1996). Probability distributions for financial models, vol. 14. In: Maddala, G. S. & Rao, C. R. (Eds.), Handbook of Statistics. Elsevier Science. Amsterdam, NL.Google Scholar
McLachlan, G. & Krishnan, T. (2007). The EM Algorithm and Extensions, vol. 382. John Wiley and Sons, Hoboken, NJ, USA.Google Scholar
Mert, M. & Saykan, Y. (2005). On a Bonus–Malus system where the claim frequency distribution is geometric and the claim severity distribution is Pareto. Hacettepe Journal of Mathematics and Statistics, 34, 75–81.Google Scholar
Munich, Re (2017). Group Annual Report 2017. https://www.munichre.com/site/corporate/get/documents_E-1569124979/mr/assetpool.shared/Documents/0_Corporate_Website/Financial_Reports/2018/annual-report-2017/302-09072_en.pdf.Google Scholar
Ni, W., Constantinescu, C. & Pantelous, A.A. (2014). Bonus–Malus systems with Weibull distributed claim severities. Annals of Actuarial Science, 8(2), 217–233.CrossRefGoogle Scholar
Ni, W., Li, B., Constantinescu, C. & Pantelous, A.A. (2014). Bonus-Malus systems with hybrid claim severity distributions. In Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of the 2nd International Conference on Vulnerability and Risk Analysis and Management, ICVRAM 2014 and the 6th International Symposium on Uncertainty Modeling and Analysis, ISUMA 2014 (pp. 1234–1244). https://doi.org/10.1061/9780784413609.124.CrossRefGoogle Scholar
Pierce, D.A. & Schafer, D.W. (1986). Residuals in generalized linear models. Journal of the American Statistical Association, 81(396), 977–986.CrossRefGoogle Scholar
Picech, L. (1994). The Merit-Rating Factor in a Multiplicating Rate-Making Model. ASTIN Colloquium, Cannes, France.Google Scholar
Pinquet, J. (1997). Allowance for cost of claims in Bonus-Malus systems. ASTIN Bulletin, 27(1), 33–57.10.2143/AST.27.1.542066CrossRefGoogle Scholar
Pinquet, J. (1998). Designing optimal Bonus-Malus systems from different types of claims. ASTIN Bulletin, 28(2), 205–220.10.2143/AST.28.2.519066CrossRefGoogle Scholar
Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 5(3), 375–383.10.1080/00401706.1963.10490105CrossRefGoogle Scholar
Raftery, A.E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–163.CrossRefGoogle Scholar
Reed, W.J. & Jorgensen, M. (2004). The Double Pareto-Lognormal distribution – a new parametric model for size distributions. Communications in Statistics-Theory and Methods 33(8), 1733–1753.CrossRefGoogle Scholar
Rigby, R.A. & Stasinopoulos, D.M. (2009). A flexible regression approach using GAMLSS in R. http://www.gamlss.org.Google Scholar
Santi, D.N., Purnaba, I.G.P & Mangku, I.W. (2016). Bonus–Malus system with the claim frequency distribution is geometric and the severity distribution is truncated Weibull. In Conference Series: Earth and Environmental Science, vol. 31(1), pp. 1755-1315, IOP Publishing.Google Scholar
Stasinopoulos, D.M., Rigby, B. & Akantziliotou, C. (2008). Instructions on how to use the gamlss package in R. Second Edition. http://www.gamlss.org.Google Scholar
Trembley, L. (1992). Using the Poisson Inverse Gaussian in Bonus–Malus systems. ASTIN Bulletin, 22(1), 97–106.10.2143/AST.22.1.2005129CrossRefGoogle Scholar
Tzougas, G. & Frangos, N. (2014). The design of an optimal Bonus–Malus system based on the Sichel distribution. In Modern Problems in Insurance Mathematics (pp. 239–260). Cham, Switzerland, Springer.Google Scholar
Tzougas, G., Vrontos, S. & Frangos, N. (2014). Optimal Bonus–Malus systems using finite mixture models. ASTIN Bulletin, 44(2), 417–444.10.1017/asb.2013.31CrossRefGoogle Scholar
Tzougas, G., Vrontos, S. & Frangos, N. (2015). Risk classification for claim counts and losses using regression models for location, scale and shape. Variance, 9(1), 140–157.Google Scholar
Tzougas, G., Vrontos, S. & Frangos, N. (2018). Bonus–Malus systems with two component mixture models arising from different parametric families. North American Actuarial Journal, 22(1), 55–91.Google Scholar
Tzougas, G., Hoon, W.L. & Lim, J.M. (2018). The Negative Binomial-Inverse Gaussian Regression Model with an application to insurance ratemaking. European Actuarial Journal, (2018), 1–22. https://doi.org/10.1007/s13385-018-0186-2.Google Scholar
Wang, S. (1998). An actuarial index of the right-tail risk. North American Actuarial Journal, 2(2), 88–101.10.1080/10920277.1998.10595708CrossRefGoogle Scholar