European Actuarial Journal

, Volume 5, Issue 2, pp 355–380 | Cite as

Parameter reduction in log-normal chain-ladder models

  • Richard J. VerrallEmail author
  • Mario V. Wüthrich
Original Research Paper


Multiplicative chain-ladder (CL) models are characterized by CL factors that explain the development of claims from one period to the next. In classical CL models every development period has its own CL factor. In the present paper we give a method describing how some of these CL factors can be modeled by a joint functional dependence. This joint functional form reduces the number of model parameters needed.


Development Period Prior Model Confidence Bound Parameter Reduction Reversible Jump Markov Chain Monte Carlo 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boor J (2006) Estimating tail development factors: what to do when the triangle runs out. CAS Forum Winter 2006:345–390Google Scholar
  3. 3.
    Bühlmann H, Gisler A (2005) A course in credibility theory and its applications. Springer, BerlinzbMATHGoogle Scholar
  4. 4.
    De Jong P, Zehnwirth B (1983) Claims reserving, state-space models and the Kalman filter. J Inst Actuar 110:157–182CrossRefGoogle Scholar
  5. 5.
    England PD, Verrall RJ (2001) A flexible framework for stochastic claims reserving. In: Proc. CAS, vol LXXXVIII, pp 1–38Google Scholar
  6. 6.
    Gigante P, Picech L, Sigalotti L (2013) Prediction error for credible claims reserves: an \(h\)-likelihood approach. Eur Actuar J 3(2):453–470MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hertig J (1985) A statistical approach to the IBNR-reserves in marine insurance. Ast Bull 15(2):171–183CrossRefGoogle Scholar
  8. 8.
    Johnson RA, Wichern DW (1988) Applied multivariate statistical analysis, 2nd edn. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  9. 9.
    Lee Y, Nelder JA (1996) Hierarchical generalized linear models. J R Stat Soc B 58(4):619–678MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee Y, Nelder JA (2001) Hierarchical generalised linear models: a synthesis of generalised linear models, random-effects models and structured dispersions. Biometrika 88(4):987–1006MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Merz M, Wüthrich MV, Hashorva E (2013) Dependence modelling in multivariate claims run-off triangles. Ann Actuar Sci 7(1):3–25CrossRefGoogle Scholar
  12. 12.
    Shi P, Basu S, Meyers GG (2012) A Bayesian log-normal model for multivariate loss reserving. North Am Actuar J 16(1):29–51MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Verrall RJ (1996) Claims reserving and generalised additive models. Insur Math Econ 19(1):31–43MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Verrall RJ, Wüthrich MV (2012) Reversible jump Markov chain Monte Carlo method for parameter reduction in claims reserving. North Am Actuar J 16(2):240–259CrossRefzbMATHGoogle Scholar
  15. 15.
    Wüthrich MV (2013) Non-life insurance: mathematics & Statistics. SSRN manuscript ID 2319328. doi: 10.2139/ssrn.2319328
  16. 16.
    Wüthrich MV (2012) Discussion of “A Bayesian log-normal model for multivariate loss reserving” by Shi–Basu–Meyers. North Am Actuar J 16(3):398–401CrossRefzbMATHGoogle Scholar
  17. 17.
    Wüthrich MV, Merz M (2008) Stochastic claims reserving methods in insurance. Wiley, New YorkzbMATHGoogle Scholar

Copyright information

© DAV / DGVFM 2015

Authors and Affiliations

  1. 1.Cass Business SchoolCity University LondonLondonUK
  2. 2.RiskLab Switzerland, Department of MathematicsETH ZurichZurichSwitzerland

Personalised recommendations