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Bayesian multivariate Poisson models for insurance ratemaking

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  • Bermúdez, Lluís
  • Karlis, Dimitris

Abstract

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner's insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.

Suggested Citation

  • Bermúdez, Lluís & Karlis, Dimitris, 2011. "Bayesian multivariate Poisson models for insurance ratemaking," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 226-236, March.
  • Handle: RePEc:eee:insuma:v:48:y:2011:i:2:p:226-236
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    Cited by:

    1. Pechon, Florian & Denuit, Michel & Trufin, Julien, 2019. "Home and Motor insurance joined at a household level using multivariate credibility," LIDAM Discussion Papers ISBA 2019013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Tsyganov, Aleksander & Baskakov, Valery & Yazykov, Andrey & Sheparnev, Nikolay & Yanenko, Evgeny & Grysenkova, Yulia, 2019. "The impact of the bonus-malus system on the insurance ratemaking in the system of compulsory insurance of the responsibility of transport owners in Russia," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 56, pages 123-141.
    3. Payandeh Najafabadi Amir T. & MohammadPour Saeed, 2018. "A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 12(2), pages 1-31, July.
    4. Lluís Bermúdez & Dimitris Karlis, 2021. "Multivariate INAR(1) Regression Models Based on the Sarmanov Distribution," Mathematics, MDPI, vol. 9(5), pages 1-13, March.
    5. Shi, Peng & Valdez, Emiliano A., 2014. "Multivariate negative binomial models for insurance claim counts," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 18-29.
    6. Ramon Alemany & Catalina Bolance & Montserrat Guillen, 2014. "Accounting for severity of risk when pricing insurance products," Working Papers 2014-05, Universitat de Barcelona, UB Riskcenter.
    7. Tzougas, George & Hoon, W. L. & Lim, J. M., 2019. "The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking," LSE Research Online Documents on Economics 101728, London School of Economics and Political Science, LSE Library.
    8. Bermúdez, Lluís & Karlis, Dimitris, 2012. "A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3988-3999.
    9. Fuzi, Mohd Fadzli Mohd & Jemain, Abdul Aziz & Ismail, Noriszura, 2016. "Bayesian quantile regression model for claim count data," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 124-137.
    10. Tzougas, George & Pignatelli di Cerchiara, Alice, 2021. "The multivariate mixed Negative Binomial regression model with an application to insurance a posteriori ratemaking," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 602-625.
    11. Zhao, Xiaobing & Zhou, Xian, 2012. "Copula models for insurance claim numbers with excess zeros and time-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 191-199.
    12. Fung, Tsz Chai & Badescu, Andrei L. & Lin, X. Sheldon, 2019. "A class of mixture of experts models for general insurance: Theoretical developments," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 111-127.
    13. Bolancé, Catalina & Vernic, Raluca, 2019. "Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 89-103.
    14. Jacek Osiewalski & Jerzy Marzec, 2019. "Joint modelling of two count variables when one of them can be degenerate," Computational Statistics, Springer, vol. 34(1), pages 153-171, March.
    15. Pechon, Florian & Denuit, Michel & Trufin, Julien, 2018. "Multivariate Modelling of Multiple Guarantees in Motor Insurance of a Household," LIDAM Discussion Papers ISBA 2018019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Ramon Alemany & Catalina Bolancé & Roberto Rodrigo & Raluca Vernic, 2020. "Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity," Mathematics, MDPI, vol. 9(1), pages 1-18, December.
    17. Bermúdez, Lluís & Guillén, Montserrat & Karlis, Dimitris, 2018. "Allowing for time and cross dependence assumptions between claim counts in ratemaking models," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 161-169.
    18. Tzougas, George & Makariou, Despoina, 2022. "The multivariate Poisson-Generalized Inverse Gaussian claim count regression model with varying dispersion and shape parameters," LSE Research Online Documents on Economics 117197, London School of Economics and Political Science, LSE Library.
    19. Catalina Bolancé & Raluca Vernic, 2017. "“Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution”," IREA Working Papers 201718, University of Barcelona, Research Institute of Applied Economics, revised Oct 2017.

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