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Mack's estimator motivated by large exposure asymptotics in a compound Poisson setting

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  • Nils Engler
  • Filip Lindskog

Abstract

The distribution-free chain ladder of Mack justified the use of the chain ladder predictor and enabled Mack to derive an estimator of conditional mean squared error of prediction for the chain ladder predictor. Classical insurance loss models, i.e. of compound Poisson type, are not consistent with Mack's distribution-free chain ladder. However, for a sequence of compound Poisson loss models indexed by exposure (e.g. number of contracts), we show that the chain ladder predictor and Mack's estimator of conditional mean squared error of prediction can be derived by considering large exposure asymptotics. Hence, quantifying chain ladder prediction uncertainty can be done with Mack's estimator without relying on the validity of the model assumptions of the distribution-free chain ladder.

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  • Nils Engler & Filip Lindskog, 2023. "Mack's estimator motivated by large exposure asymptotics in a compound Poisson setting," Papers 2310.12056, arXiv.org.
    • Handle: RePEc:arx:papers:2310.12056
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    References listed on IDEAS

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    1. Buchwalder, Markus & Bühlmann, Hans & Merz, Michael & Wüthrich, Mario V., 2006. "The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mack and Murphy Revisited)," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 521-542, November.
    2. Mack, Thomas, 1993. "Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 213-225, November.
    3. Gisler, Alois, 2019. "The Reserve Uncertainties In The Chain Ladder Model Of Mack Revisited," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 787-821, September.
    4. Gisler, Alois, 2006. "The Estimation Error in the Chain-Ladder Reserving Method: A Bayesian Approach," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 554-565, November.
    5. Renshaw, A.E. & Verrall, R.J., 1998. "A Stochastic Model Underlying the Chain-Ladder Technique," British Actuarial Journal, Cambridge University Press, vol. 4(4), pages 903-923, October.
    6. Mack, Thomas & Quarg, Gerhard & Braun, Christian, 2006. "The Mean Square Error of Prediction in the Chain Ladder Reserving Method – A Comment," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 543-552, November.
    7. Venter, Gary G., 2006. "Discussion of the Mean Square Error of Prediction in the Chain Ladder Reserving Method," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 566-571, November.
    8. Mack, Thomas & Venter, Gary, 2000. "A comparison of stochastic models that reproduce chain ladder reserve estimates," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 101-107, February.
    9. Buchwalder, Markus & Bühlmann, Hans & Merz, Michael & Wüthrich, Mario V., 2006. "The Mean Square Error of Prediction in the Chain Ladder Reserving Method – Final Remark," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 553-553, November.
    10. Taylor, G. C. & Ashe, F. R., 1983. "Second moments of estimates of outstanding claims," Journal of Econometrics, Elsevier, vol. 23(1), pages 37-61, September.
    11. Lindholm, Mathias & Lindskog, Filip & Wahl, Felix, 2020. "Estimation of conditional mean squared error of prediction for claims reserving," Annals of Actuarial Science, Cambridge University Press, vol. 14(1), pages 93-128, March.
    12. Verrall, R. J. & England, P. D., 2000. "Comments on: "A comparison of stochastic models that reproduce chain ladder reserve estimates", by Mack and Venter," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 109-111, February.
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