Stochastic Models for Triangular Tables with Applications to Cohort Data and Claims Reserving

Stochastic models for triangular data are derived and applied to claims reserving data. The standard actuarial technique, the so-called "chain-ladder technique" is given a sound statistical foundation and considered as a linear model. This linear model, the '"Chain Ladder Linear Model" is extended to encompass Bayesian, empirical Bayes and dynamic estimation. The empirical Bayes results are given a credibility theory interpretation, and the advantages and disadvantages of the various approaches are highlighted. Finally, the methods are extended to two-dimensional systems and results based on classical time series and Kalman filtering theory are produced. The empirical Bayes estimation results are very useful, practically, and can be compared to the Kalman filter estimates. They have the advantage that no prior information is required: the Kalman filter method requires the state and observation variances to be specified. For illustration purposes the estimates from the empirical Bayes procedure are used. The empirical Bayes results can also be compared with credibility theory estimators, although they retain the general statistical advantages of the linear modelling approach. For the classical theory, unbiased estimates of outstanding claims, reserves and variances are derived, and prediction intervals for total outstanding claims are produced. Maximum likelihood theory is utilised to derive the distributions of quantities relating to the column parameters which have actuarial interpretations. The row totals are also considered. Bayesian estimates of similar quantities are derived for the methods based on Bayes theory