Advertisement

European Actuarial Journal

, Volume 7, Issue 1, pp 109–132 | Cite as

A non-linear mixed model approach for excess of loss benchmark rating

Original Research Paper
  • 112 Downloads

Abstract

This paper proposes market conform individual benchmark rates for the excess of loss reinsurance of long tail insurance portfolios, that offer market references for the premium rates taking individual contractual conditions into account. The premium rates are expressed in terms of the percentage of the expected premium income of the covered insurance portfolio. We incorporate the specific reinsurance contractual conditions like stabilisation, interest sharing and deposit clauses as well as payment patterns and ’incurred but not (enough) reported’ information. The parameters of the benchmark model are estimated within the framework of non-linear mixed models. This approach allows to correct for different cedent specific conditions in the model, and so we refine the results from Verlaak et al. (Astin Bull 39, 2009) where only one benchmark was proposed for the whole market. The method is applied to the Belgian Motor Third Party Liability XL rates observed from 2001 till 2004.

Keywords

XL reinsurance Benchmark rates Pareto model Non-linear mixed models XL clauses. 

Notes

Acknowledgements

The authors would like to thank the referees for the many excellent suggestions which led to a significant improvement of the presentation.

References

  1. 1.
    Beirlant J, Goegebeur Y, Teugels J, Segers J (2004) Statistics of extremes: theory and applications. Wiley, UKCrossRefMATHGoogle Scholar
  2. 2.
    Carter RL, Lucas LD, Ralph N (2000) Reinsurance, Reaction Publishing Group (in association with Guy Carpenter & Company)Google Scholar
  3. 3.
    Dobson AJ (2002) An introduction to generalized linear models. Chapman and HallGoogle Scholar
  4. 4.
    Gerathewohl K (1980) Reinsurance principles and practice, vol I. Verlag Versicherungswirtschaft e. V, KarlsruheGoogle Scholar
  5. 5.
    Gerathewohl K (1982) Reinsurance principles and practice, vol II. Verlag Versicherungswirtschaft e. V, KarlsruheGoogle Scholar
  6. 6.
    Hardin J, Hilbe J (2001) Generalized linear models and extensions. Stata Press College Station, TexasMATHGoogle Scholar
  7. 7.
    Kaas R, Goovaerts M, Dhaene J, Denuit M (2008) Modern actuarial risk theory using R. Springer, New YorkCrossRefMATHGoogle Scholar
  8. 8.
    Kiln R (1982) Reinsurance in practice. Witherby & Co., LondonGoogle Scholar
  9. 9.
    Klugman SA, Panjer H, Willmot GE (1988) Loss models, from data to decisions. Wiley, New YorkMATHGoogle Scholar
  10. 10.
    Lei L, Zhangsheng Y (2008) A likelihood reformulation method in non-normal random effects models. Stat Med 27:31053124MathSciNetGoogle Scholar
  11. 11.
    McCullagh P, Nelder JA (1985) Generalized linear models. Chapman and HallGoogle Scholar
  12. 12.
    McCulloch CE, Searle SR (2001) Generalized, linear and mixed models. Wiley, New YorkMATHGoogle Scholar
  13. 13.
    Molenberghs G, Verbeke G (2006) Models for discrete longitudinal data. Springer, New YorkGoogle Scholar
  14. 14.
    Nelson KP, Lipsitz SR, Fitzmaurice GM, Ibrahim J, Parzen M, Strawderman R (2006) Use of the probability integral transformation to fit nonlinear mixed-effects models with nonnormal random effects. J Comput Graph Stat 15:3957MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ohlssonn E, Johansson B (2010) Non-life insurance pricing with generalized linear models. Springer, BerlinCrossRefGoogle Scholar
  16. 16.
    Verbeke G, Molenberghs G (2000) Linear mixed models for longitudinal data. Springer, New YorkGoogle Scholar
  17. 17.
    Verlaak R, Beirlant J (2003) Optimal reinsurance programs: an optimal combination of several reinsurance protections on an heterogeneous insurance portfolio. Insur Math Econ 33:381–403MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Verlaak R, Hürliman W, Beirlant J (2009) Benchmark rates for excess of loss reinsurance programs: a generalised non-linear quasi-likelihood approach. Astin Bull, vol 39Google Scholar
  19. 19.
    Walhin JF, Herfurth L, De Longueville P (2001) The actuarial pricing of excess of loss treaties: actuarial, financial, economic and commercial aspects. Belg Actuar Bull 1:40–57Google Scholar
  20. 20.
    Wang S, Coste CM (2000) From distortion operators to risk loading. SCOR Tech 058, SCOR GroupGoogle Scholar

Copyright information

© EAJ Association 2017

Authors and Affiliations

  1. 1.Aon BenfieldBrusselsBelgium
  2. 2.KU LeuvenLouvainBelgium
  3. 3.University of the Free StateBloemfonteinSouth Africa

Personalised recommendations