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Predicting Ibnyr Events and Delays: I. Continuous Time

Published online by Cambridge University Press:  29 August 2014

William S. Jewell
William S. Jewell*
Affiliation:
Engineering Systems Research Center, University of California, Berkely
*
Engineering Systems Research Center, University of California, Berkeley, CA 94720.
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Abstract

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An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. Both the rate at which such events occur and the parameters of the delay distribution are unknown random quantities. Given the number of events that have been reported during some observation interval, plus various secondary data on the dates of the events, the problem is to estimate the true values of the unknown parameters and to predict the number of events that are still unreported. A full-distributional Bayesian model is used, and it is shown that the amount of secondary data is critical. A recursive procedure calculates the predictive density; however, an explicit formula for the predictive mode can be obtained. The main computational work is the evaluation of an integral involving the prior density of the delay parameters, but this can be simplified in the exponential case using Gammoid approximations.

Keywords

Observations delaysInccurred But Not Reported (IBNR) modelsBayesian estimation and prediction
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 19 , Issue 1 , April 1989 , pp. 25 - 55
Copyright
Copyright © International Actuarial Association 1989

References

Bühlmann, H., Schnieper, R. and Straub, E. (1980). Claims reserves in casualty insurance based on a probabilistic model. B. Assoc. Swiss Actuaries 80, 21–45.Google Scholar
De Vylder, F. (1982). Estimation of IBNYR claims by credibility theory. Insurance: Math. & Econ. 1, 35–40.Google Scholar
Hesselager, O. and Witting, T. (1988). A credibility model with random fluctuations in delay probabilities for the prediction of IBNR claims. ASTIN Bulletin to appear.Google Scholar
Jewell, W.S. (1980). Models in insurance: paradigms, puzzles, communications, and revolutions. Transactions, 21st International Congress of Actuaries, Zurich and Lausanne, 19-26 June 1980, S, 87–141.Google Scholar
Jewell, W. S. (1985a). Bayesian estimation of undetected errors. Bayesian Statistics 2, Bernardo, J. M.et al. (Eds.), Elsevier Science Publishers (North-Holland), 1985, 663–672.Google Scholar
Jewell, W. S. (1985b). Bayesian extension to a basic model of software reliability. IEEE Trans. Software Eng. SE-11, no. 12, 1465–1471.CrossRefGoogle Scholar
Hachemeister, C. (1980). A stochastic model for loss reserving. Transactions, 21st International Congress of Actuaries, Zurich and Lausanne, 19-26 June 1980, 1, 185–194.Google Scholar
Karlsson, J.-E. (1974). A stochastic model for time lag in reporting of claims, J. Appl. Prob. II, 382–387.CrossRefGoogle Scholar
Karlsson, J.-E. (1976). The expected value of IBNR-claims, Scand. Actuarial J. 1976, 108–110.CrossRefGoogle Scholar
Kaminsky, K. S. (1987). Prediction of IBNR claim counts by modelling the distribution of report lags. Insurance: Math. & Econ. 6, 161–159.Google Scholar
Kramreiter, H. and Straub, E. (1973). On the calculation of IBNYR reserves II. Mitt. Ver. Schweiz. Vers. Math. 73. 177–190.Google Scholar
Norberg, R. (1986). A contribution to modelling of IBNR claims. Scand. Actuarial J., 1986, 155–203.CrossRefGoogle Scholar
Straub, E. (1972). On the calculation of IBNR reserves. Prizewinning papers in the Boleslaw-Monk-Fund Competition 1971. Nederlandse Reassurantie Groep N.V., Amsterdam, 1972, 123–131.Google Scholar
Tierney, L. and Kadane, J.B. (1986). Accurate approximations for posterior moments and marginal densities. Jour. Amer. Stat. Assoc., 81, 82–86.CrossRefGoogle Scholar
Van Eeghen, J. (1981). Loss reserving methods. Surveys of actuarial studies no. 1. Nationale-Nederlanden N.B., Rotterdam, 1981.Google Scholar