European Actuarial Journal

, Volume 9, Issue 2, pp 575–602 | Cite as

Multivariate modelling of multiple guarantees in motor insurance of a household

  • Florian PechonEmail author
  • Michel Denuit
  • Julien Trufin
Original Research Paper


Actuarial risk classification is usually performed at a guarantee and policyholder level: for each policyholder, the claim frequencies corresponding to each guarantee are modelled in isolation, without accounting for the correlation between the different guarantees and the different policyholders from the same household. However, sometimes, a common event will trigger both guarantees at the same time. Moreover, the claim frequencies for policyholders from the same household appear to be correlated. This paper aims to supplement the standard actuarial approach by combining two guarantees and the policyholders from the household, which allows to refine the prediction on the claim frequencies and account for the common shocks on multiple guarantees. Some possible cross-selling opportunities can also be identified.


Multivariate Poisson mixture model Poisson-lognormal A posteriori risk revaluation Credibility 



The financial support of the AXA Research Fund through the JRI project “Actuarial dynamic approach of customer in P&C” is gratefully acknowledged. We thank our colleagues from AXA Belgium, especially Arnaud Deltour, Mathieu Lambert, Alexis Platteau, Stanislas Roth and Louise Tilmant for interesting discussions that greatly contributed to the success of this research project. Also, we thank our colleagues from the SMCS, the UCLouvain platform for statistical computing, for setting us up a comfortable and efficient working environment.


  1. 1.
    Antonio K, Frees EW, Valdez EA (2010) A multilevel analysis of intercompany claim counts. ASTIN Bull J IAA 40(1):151–177MathSciNetCrossRefGoogle Scholar
  2. 2.
    Antonio K, Guillén M, Pérez Martín AM et al (2010) Multidimensional credibility: a bayesian analysis of policyholders holding multiple policies. Technical report, Amsterdam School of Economics Research InstituteGoogle Scholar
  3. 3.
    Antonio K, Zhang Y (2014) Nonlinear mixed models. In: Edward WF, Richard AD, Glenn M (eds) Predictive modeling applications in actuarial science of International Series on Actuarial Science, vol 1. Cambridge University Press, Cambridge, pp 398–424CrossRefGoogle Scholar
  4. 4.
    Bermúdez L, Guillén M, Karlis D (2018) Allowing for time and cross dependence assumptions between claim counts in ratemaking models. Insur Math Econ 83:161–169MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bermúdez L (2009) A priori ratemaking using bivariate poisson regression models. Insur Math Econ 44(1):135–141MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bermúdez L, Karlis D (2011) Bayesian multivariate poisson models for insurance ratemaking. Insur Math Econ 48(2):226–236MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bermúdez L, Karlis D (2012) A finite mixture of bivariate poisson regression models with an application to insurance ratemaking. Comput Stat Data Anal 56(12):3988–3999MathSciNetCrossRefGoogle Scholar
  8. 8.
    Denuit M, Maréchal X, Pitrebois S, Walhin J-F (2007) Risk classification, credibility and bonus-malus systems. Actuarial modelling of claim counts. Wiley, HobokenCrossRefGoogle Scholar
  9. 9.
    Eddelbuettel D, François R (2011) Rcpp: seamless R and C++ integration. J Stat Softw 40(8):1–18CrossRefGoogle Scholar
  10. 10.
    Englund M, Guillén M, Gustafsson J, Nielsen LH, Nielsen JP (2008) Multivariate latent risk: a credibility approach. ASTIN Bull 38(1):137–146MathSciNetCrossRefGoogle Scholar
  11. 11.
    Englund M, Gustafsson J, Nielsen JP, Thuring F (2009) Multidimensional credibility with time effects: An application to commercial business lines. J Risk Insur 76(2):443–453CrossRefGoogle Scholar
  12. 12.
    Frees EW, Jin X, Lin X (2013) Actuarial applications of multivariate two-part regression models. Ann Actuar Sci 7(2):258–287CrossRefGoogle Scholar
  13. 13.
    Frees EW, Lee G, Yang L (2016) Multivariate frequency-severity regression models in insurance. Risks. CrossRefGoogle Scholar
  14. 14.
    Frees EW, Wang P (2006) Copula credibility for aggregate loss models. Insur Math Econ 38(2):360–373MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pechon F, Trufin J, Denuit M (2018) Multivariate modelling of household claim frequencies in motor third-party liability insurance. ASTIN Bull 48(3):969–993MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pinquet J (1998) Designing optimal bonus-malus systems from different types of claims. ASTIN Bull 28(2):205–220MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shi P, Feng X, Boucher J-P (2016) Multilevel modeling of insurance claims using copulas. Ann Appl Stat 10(2):834–863MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shi P, Valdez EA (2014) Multivariate negative binomial models for insurance claim counts. Insur Math Econ 55(1):18–29MathSciNetCrossRefGoogle Scholar
  19. 19.
    Thuring F (2012) A credibility method for profitable cross-selling of insurance products. Ann Actuar Sci 6(1):65–75CrossRefGoogle Scholar
  20. 20.
    Tuerlinckx F, Rijmen F, Verbeke G, De Boeck P (2006) Statistical inference in generalized linear mixed models: a review. Br J Math Stat Psychol 59(2):225–255MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wood SN (2017) Generalized additive models: an introduction with R, 2nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar

Copyright information

© EAJ Association 2019

Authors and Affiliations

  1. 1.Institute of Multidisciplinary Research for Quantitative Modelling and AnalysisUniversite catholique de LouvainOttignies-Louvain-la-NeuveBelgium
  2. 2.Department of MathematicsUniversité Libre de Bruxelles (ULB)BruxellesBelgium

Personalised recommendations