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THE RESERVE UNCERTAINTIES IN THE CHAIN LADDER MODEL OF MACK REVISITED

Published online by Cambridge University Press:  01 July 2019

Alois Gisler
Alois Gisler*
Affiliation:
Mathematik, RiskLab ETH Zürich, HG J 58 Rämistrasse 101 CH-8092 ZürichSwitzerland
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Abstract

We revisit the “full picture” of the claims development uncertainty in Mack’s (1993) distribution-free stochastic chain ladder model. We derive the uncertainty estimators in a new and easily understandable way, which is much simpler than the derivation found so far in the literature, and compare them with the well known estimators of Mack and of Merz–Wüthrich.

Our uncertainty estimators of the one-year run-off risks are new and different to the Merz–Wüthrich formulas. But if we approximate our estimators by a first order Taylor expansion, we obtain equivalent but simpler formulas. As regards the ultimate run-off risk, we obtain the same formulas as Mack for single accident years and an equivalent but better interpretable formula for the total over all accident years.

Keywords

claims reservingdistribution free chain-ladder modelBayesian chain-ladder modelconditional mean square error of predictionultimate run-off uncertaintyone-year run-off uncertaintiesMack’s formulaWüthrich–Merz formulascost of capital loadingmarket value margin
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 787 - 821
Copyright
© Astin Bulletin 2019 

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References

Buchwalder, M., Bühlmann, H., Merz, M. and Wüthrich, M.V. (2006) The mean square error of prediction in the chain ladder reserving method (Mack and Murphy Revisited). ASTIN Bulletin, 36(2), 521–542.CrossRefGoogle Scholar
Bühlmann, H., De Felice, M., Gisler, A. and Wüthrich, M.V. (2009) Recursive credibility formulas for chain ladder factors and the claims development result. ASTIN Bulletin, 39(1), 275–306.CrossRefGoogle Scholar
Diers, D., Linde, M. and Hahn, L. (2016) Quantification of multi-year non-life insurance risk in chain ladder reserving models. Insurance: Mathematics and Economics, 67, 187–199.Google Scholar
Gisler, A. (2006) The estimation error in the chain-ladder reserving method: A Bayesian approach. ASTIN Bulletin, 36(2), 554–565.CrossRefGoogle Scholar
Gisler, A. (2016) The chain ladder reserve uncertainties revisited. Conference Paper, ASTIN Colloquium 2016, Lisbon.Google Scholar
Harnau, J., Nielsen, B. (2018). Over-Dispersed Age-Period-Cohort Models. Journal of the American Statistical Association, 113(524), 1722–1732.CrossRefGoogle Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23(2), 213–225.CrossRefGoogle Scholar
Mack, T., Quarg, B. and Braun, C. (2006) The mean square error of prediction in the chain ladder reserving method – a comment. ASTIN Bulletin, 23(2), 523–552.Google Scholar
Merz, M. and Wüthrich, M.V. (2008a) Stochastic Claims Reserving Methods in Insurance, Wiley.Google Scholar
Merz, M. and Wüthrich, M.V. (2008b). Modelling the claims development result for solvency purposes. CAS Forum, Fall 2008, 542–568.Google Scholar
Merz, M. and Wüthrich, M.V. (2014). Claims run-off uncertainty: the full picture. SSRN manuscript 2524352.CrossRefGoogle Scholar
Röhr, A. (2016). Chain ladder and error propagation, ASTIN Bulletin, 46(2), 293–330.CrossRefGoogle Scholar
Wüthrich, M.V., Merz, M. and Lysenko, N. (2009). Uncertainty of the claims development result in the chain ladder method. Scandinavian Actuarial Journal, 2009(1), 63–84.CrossRefGoogle Scholar