Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-11T07:16:09.323Z Has data issue: false hasContentIssue false
  • English
  • Français

MODELING DEPENDENCE BETWEEN LOSS TRIANGLES WITH HIERARCHICAL ARCHIMEDEAN COPULAS

Published online by Cambridge University Press:  19 June 2015

Anas Abdallah ,
Jean-Philippe Boucher  and
Hélène Cossette
Anas Abdallah
Affiliation:
École d'Actuariat, Université Laval, Quebec City (Quebec) E-Mail: anas.abdallah.1@ulaval.ca
Jean-Philippe Boucher*
Affiliation:
Département de mathématiques, UQAM Montreal (Quebec)
Hélène Cossette
Affiliation:
École d'Actuariat, Université Laval, Quebec City (Quebec) E-Mail: helene.cossette@act.ulaval.ca
*
E-Mail: boucher.jean-philippe@uqam.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

One of the most critical problems in property/casualty insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. Recently, Shi and Frees (2011) proposed to put dependence between lines of business with a copula that captures dependence between two cells of two different runoff triangles. In this paper, we propose to generalize this model in two steps. First, by using an idea proposed by Barnett and Zehnwirth (1998), we will suppose a dependence between all the observations that belong to the same calendar year (CY) for each line of business. Thereafter, we will then suppose another dependence structure that links the CYs of different lines of business. This model is done by using hierarchical Archimedean copulas. We show that the model provides more flexibility than existing models, and offers a better, more realistic and more intuitive interpretation of the dependence between the lines of business. For illustration, the model is applied to a dataset from a major US property-casualty insurer, where a bootstrap method is proposed to estimate the distribution of the reserve.

Keywords

Runoff trianglescopulahierarchical Archimedean copulamaximum likelihood estimationbootstrap

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 577 - 599
Copyright
Copyright © Astin Bulletin 2015 

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.)

Article purchase

Temporarily unavailable

References

Barnett, G. and Zehnwirth, B. (1998) Best estimates for reserves. Casualty Actuarial Society Forum (Fall), 1–54.Google Scholar
Braun, C. (2004) The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin, 34 (2), 399–434.CrossRefGoogle Scholar
Brehm, P. (2002) Correlation and the aggregation of unpaid loss distributions. Casualty Actuarial Society Forum (Fall), 1–23.Google Scholar
De Jong, P. (2006) Forecasting runoff triangles. North American Actuarial Journal, 10 (2), 28–38.CrossRefGoogle Scholar
De Jong, P. (2012) Modeling dependence between loss triangles. North American Actuarial Journal, 16 (1), 74–86.CrossRefGoogle Scholar
England, P. and Verrall, R. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8 (3), 443–518.CrossRefGoogle Scholar
Hofert, M., Mächler, M. and McNeil, A. J. (2012) Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis, 110, 133–150.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. London: Chapman and Hall.Google Scholar
Johnson, P.H.J., Qi, Y. and Chueh, Y. (2011) Bias-corrected maximum likelihood estimation in actuarial science. Working paper.Google Scholar
Kirschner, G., Kerley, C. and Isaacs, B. (2008) Two approaches to calculating correlated reserve indications across multiple lines of business. Variance, 2 (1), 15–38.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J. (2008) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika, 95 (4), 987–991.CrossRefGoogle Scholar
Kuang, D., Nielsen, B. and Nielsen, J. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance, 78 (2), 345–359.CrossRefGoogle Scholar
Lowe, J. (1994) A practical guide to measuring reserve variability using bootstrapping, operational times and a distribution-free approach. General Insurance Convention, Institute of Actuaries and Faculty of Actuaries.Google Scholar
Marshall, A.W. and Olkin, I. (1988) Families of multivariate distributions. Journal of the American Statistical Association, 83 (403), 834–841.CrossRefGoogle Scholar
McNeil, A. and Nešlehová, J. (2009) Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions. The Annals of Statistics 37 (5B), 3059–3097.CrossRefGoogle Scholar
Meyers, G. (2013) Stochastic loss reserving using Bayesian MCMC models. ASTIN Colloquium May 23, 2013.Google Scholar
Okhrin, O., Okhrin, Y. and Schmid, W. (2013) On the structure and estimation of hierarchical Archimedean copulas. Journal of Econometrics 173 (2), 189–204.CrossRefGoogle Scholar
Savu, C. and Trede, M. (2010) Hierarchies of Archimedean copulas. Quantitative Finance, 10 (3), 295–304.CrossRefGoogle Scholar
Shi, P. (2014) A copula regression for modeling multivariate loss triangles and quantifying reserving variability. ASTIN Bulletin 44 (1), 85–102.CrossRefGoogle Scholar
Shi, P., Basu, S. and Meyers, G. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal 16 (1), 29–51.CrossRefGoogle Scholar
Shi, P. and Frees, E. (2011) Dependent loss reserving using copulas. ASTIN Bulletin, 41 (2), 449–486.Google Scholar
Taylor, G. and McGuire, G. (2007) A synchronous bootstrap to account for dependencies between lines of business in the estimation of loss reserve prediction error. North American Actuarial Journal, 11 (3), 70–88.CrossRefGoogle Scholar
Wüthrich, M. (2010) Accounting year effects modeling in the stochastic chain ladder reserving method. North American Actuarial Journal, 14 (2), 235–255.CrossRefGoogle Scholar
Wüthrich, M. (2012) Discussion of “A Bayesian log-normal model for multivariate loss reserving” by Shi-Basu-Meyers. North American Actuarial Journal, 16 (3), 398–401.Google Scholar
Wüthrich, M. (2013) Calendar year dependence modeling in run-off triangles. ASTIN Colloquium, May 21–24, The Hague.Google Scholar
Wüthrich, M., Merz, M. and Hashorva, E. (2013) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7 (1), 3–25.Google Scholar
Wüthrich, M. and Salzmann, R. (2012) Modeling accounting year dependence in runoff triangles. European Actuarial Journal, 2 (2), 227–242.Google Scholar