Claim Reserving Using Distance-Based Generalized Linear Models

  • Eva BojEmail author
  • Teresa Costa
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 175)


Generalized linear models (GLM) can be considered a stochastic version of the classical Chain-Ladder (CL) method of claim reserving in nonlife insurance. In particular, the deterministic CL model is reproduced when a GLM is fitted assuming over-dispersed Poisson error distribution and logarithmic link. Our aim is to propose the use of distance-based generalized linear models (DB-GLM) in the claim reserving problem. DB-GLM can be considered a generalization of the classical GLM to the distance-based analysis, because DB-GLM contains as a particular instance ordinary GLM when the Euclidean, \(l^2\), metric is applied. Then, DB-GLM can be considered too a stochastic version of the CL claim reserving method. In DB-GLM, the only information required is a predictor distance matrix. DB-GLM can be fitted using the dbstats package for R. To estimate reserve distributions and standard errors, we propose a nonparametric bootstrap technique adequate to the distance-based regression models. We illustrate the method with a well-known actuarial dataset.


Reserving Chain-Ladder Generalized linear models Distance-based prediction dbstats. 



Work supported by the Spanish Ministerio de Educación y Ciencia, grant MTM2014-56535-R.


  1. 1.
    Albarrán, I., Alonso, P.: Métodos estocásticos de estimación de las provisiones tenicas en el marco de Solvencia II. Cuadernos de la Fundación MAPFRE 158. Fundación MAPFRE Estudios, Madrid (2010)Google Scholar
  2. 2.
    Boj, E., Costa, T.: Modelo lineal generalizado y cálculo de la provisión técnica. Depósito digital de la Universidad de Barcelona. Colección de objetos y materiales docentes (OMADO) (2014).
  3. 3.
    Boj, E., Costa, T.: Provisions for claims outstanding, incurred but not reported, with generalized linear models: prediction error formulation by calendar years. Cuad. Gestión (2015). (to appear)Google Scholar
  4. 4.
    Boj, E., Claramunt, M.M., Fortiana, J.: Selection of predictors in distance-based regression. Commun. Stat. A Theory Methods 36, 87–98 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Boj, E., Costa, T., Espejo, J.: Provisiones técnicas por años de calendario mediante modelo lineal generalizado. Una aplicación con RExcel. An. Inst. Actuar. Esp. 20, 83–116 (2014)Google Scholar
  6. 6.
    Boj, E., Caballé, A., Delicado, P., Fortiana, J.: dbstats: distance-based statistics (dbstats). R package version 1.4 (2014).
  7. 7.
    Boj, E., Delicado, P., Fortiana, J., Esteve, A., Caballé, A.: Global and local distance-based generalized linear models. TEST (2015). doi: 10.1007/s11749-015-0447-1
  8. 8.
    Boj, E., Costa, T., Fortiana, J., Esteve, A.: Assessing the importance of risk factors in distance-based generalized linear models. Methodol. Comput. Appl. 17, 951–962 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Efron, B., Tibshirani, J.: An Introduction to the Bootstrap. Chapman and Hall, New York (1998)zbMATHGoogle Scholar
  10. 10.
    England, P.D.: Addendum to ‘Analytic and bootstrap estimates of prediction errors in claim reserving’. Insur. Math. Econ. 31, 461–466 (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    England, P.D., Verrall, R.J.: Analytic and bootstrap estimates of prediction errors in claims reserving. Insur. Math. Econ. 25, 281–293 (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    England, P.D., Verrall, R.J.: Predictive distributions of outstanding liabilities in general insurance. Ann. Actuar. Sci. 1:II, 221–270 (2006)Google Scholar
  13. 13.
    Gower, J.C.: A general coeficient of similarity and some of its properties. Biometrics 27, 857–874 (1971)CrossRefGoogle Scholar
  14. 14.
    Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern Actuarial Risk Theory: Using R, 2nd edn. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman and Hall, London (1989)CrossRefzbMATHGoogle Scholar
  16. 16.
    Renshaw, A.E.: Chain ladder and interactive modelling (claims reserving and GLIM). J. Inst. Actuar. 116:III, 559–587 (1989)Google Scholar
  17. 17.
    Renshaw, A. E.: On the second moment properties and the implementation of certain GLIM based stochastic claims reserving models. Actuarial Research Paper 65. Department of Actuarial Science and Statistics, City University, London (1994)Google Scholar
  18. 18.
    Renshaw, A.E., Verrall, R.J.: A stochastic model underlying the Chain-Ladder technique. Br. Actuar. J. 4, 903–923 (1998)CrossRefGoogle Scholar
  19. 19.
    Taylor, G., Ashe, F.R.: Second moments of estimates of outstanding claims. J. Econom. 23, 37–61 (1983)CrossRefGoogle Scholar
  20. 20.
    van Eeghen, J., Greup, E.K., Nijssen, J.A.: Loss reserving methods. Surveys of Actuarial Studies 1, National Nederlanden (1981)Google Scholar
  21. 21.
    Verrall, R.J.: An investigation into stochastic claims reserving models and the chain-ladder technique. Insur. Math. Econ. 26, 91–99 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Verrall, R.J., England, P.D.: Comments on: ‘A comparison of stochastic models that reproduce chain ladder reserve estimates’, by Mack and Venter. Insur. Math. Econ. 26, 109–111 (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Facultat d’Economia i EmpresaUniversitat de BarcelonaBarcelonaSpain

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