Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-13T21:34:36.370Z Has data issue: false hasContentIssue false
  • English
  • Français

CALENDAR YEAR EFFECT MODELING FOR CLAIMS RESERVING IN HGLM

Published online by Cambridge University Press:  19 July 2019

Patrizia Gigante ,
Liviana Picech  and
Luciano Sigalotti
Patrizia Gigante
Affiliation:
Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche ’B. de Finetti’University of TriestePiazzale Europa 1, 34127, Trieste, Italy E-Mail: patrizia.gigante@deams.units.it
Liviana Picech*
Affiliation:
Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche ’B. de Finetti’University of TriestePiazzale Europa 1, 34127, Trieste, Italy E-mail: liviana.picech@deams.units.it
Luciano Sigalotti
Affiliation:
Dipartimento di Scienze Economiche e StatisticheUniversity of UdineVia Tomadini 30, 33100, Udine, Italy E-mail: luciano.sigalotti@uniud.it
*
E-mail: liviana.picech@deams.units.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.

Keywords

Calendar year effectsclaims reservingdependence modelinghierarchical generalized linear modelsPoisson-gamma modelmean square error of predictionsimulation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 763 - 786
Copyright
© Astin Bulletin 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BjöRkwall, S., HöSsjer, O. and Ohlsson, E. (2010) Bootstrapping the separation method in claims reserving. ASTIN Bulletin, 40(2), 845–869.Google Scholar
Bohnert, A., Gatzert, N. and Kolb, A. (2016) Assessing inflation risk in non-life insurance. Insurance: Mathematics and Economics, 66, 86–96.Google Scholar
Brydon, D. and Verrall, R.J. (2009) Calendar year effects, claims inflation and the Chain-Ladder technique. Annals of Actuarial Science, 4, 287–301.CrossRefGoogle Scholar
BüHlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and Its Applications, Berlin: Springer.Google Scholar
BüHlmann, H. and Moriconi, F. (2015) Credibility claims reserving with stochastic diagonal effects. ASTIN Bulletin, 45(2), 309–353.CrossRefGoogle Scholar
Gigante, P., Picech, L. and Sigalotti, L. (2013a) Claims reserving in the hierarchical generalized linear models framework. Insurance: Mathematics and Economics, 52, 381–390.Google Scholar
Gigante, P., Picech, L. and Sigalotti, L. (2013b) Prediction error for credible claims reserves: an h-likelihood approach. European Actuarial Journal, 3(2), 453–470.CrossRefGoogle Scholar
Gigante, P., Picech, L. and Sigalotti, L. (2016) A mixture model for payments and payment numbers in claims reserving. ASTIN Bulletin, 48(1), 25–53.CrossRefGoogle Scholar
Kuang, D., Nielsen, B. and Nielsen, J.P. (2011) Forecasting in an extended Chain-Ladder-type model. The Journal of Risk and Insurance, 78(2), 345–359.CrossRefGoogle Scholar
Jessen, A. and Rietdorf, N. (2011) Diagonal effects in claims reserving. Scandinavian Actuarial Journal, 1, 21–37.CrossRefGoogle Scholar
Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8, 77–90.CrossRefGoogle Scholar
Lee, Y. and Ha, I.D. (2010) Orthodox BLUP versus h-likelihood methods for inferences about random effects in Tweedie mixed models. Statistics and Computing, 20(3), 295–303.CrossRefGoogle Scholar
Lee, Y. and Nelder, J.A. (1996) Hierarchical generalized linear models (with Discussion). Journal of the Royal Statistical Society B, 58, 619–678.Google Scholar
Lee, Y. and Nelder, J.A. (2001) Hierarchical generalized linear models: a synthesis of generalized linear models, random-effect models and structured dispersion. Biometrika, 88(4), 987–1006.CrossRefGoogle Scholar
Lee, Y., Nelder, J.A. and Pawitan, Y. (2006) Generalized Linear Models with Random Effects. Unified Analysis via H-likelihood. Boca Raton: Chapman and Hall/CRC.Google Scholar
Nelder, J.A. and Pregibon, D. (1987) An extended quasi-likelihood function. Biometrika, 74, 221–232.CrossRefGoogle Scholar
Ohlsson, E. and Johansson, B. (2006) Exact credibility and Tweedie models. ASTIN Bulletin, 36(1), 121–133.CrossRefGoogle Scholar
Salzmann, R. and WüThrich, M. (2012) Modeling accounting year dependence in runoff triangles. European Actuarial Journal, 2(2), 227–242.CrossRefGoogle Scholar
Taylor, G. (1977) Separation of inflation and other effects from the distribution of non-life insurance claims delays. ASTIN Bulletin, 9(1), 219–230.CrossRefGoogle Scholar
Taylor, G. (2000) Loss Reserving: An Actuarial Perspective, Boston; Kluwer Academic Publishers.CrossRefGoogle Scholar
Verbeek, H. (1972) An approach to the analysis of claims experience in motor liability excess of loss reinsurance. ASTIN Bulletin, 6(3), 195–202.CrossRefGoogle Scholar
Verrall, R.J. and WüThrich, M.V. (2015) Parameter reduction in log-normal chain-ladder models. European Actuarial Journal, 5(2), 355–380.CrossRefGoogle Scholar
WüThrich, M.V. (2010) Accounting year effects modeling in the stochastic Chain Ladder reserving method. North American Actuarial Journal, 14(2), 235–255.CrossRefGoogle Scholar
WüThrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance, Chichester: Wiley.Google Scholar