Bayesian Estimation of Outstanding Claims Reserves

When a given process is seasonal and the seasonal pattern is stable the proportion of events that occur in a given fraction of the total period of the seasonality is constant over time. In this paper we present an application of Bayesian forecasting methods with stable seasonal patterns to insurance claims; specifically to claims incurred but not yet reported, IBNR. We assume that the time it takes for the claims to be completely paid is fixed and known, that payments are made annually and that the development of partial payments follow a stable pay-off pattern. We present a Bayesian approach to forecasting either the total number claims or the total amount, given complete information for one or two past years and partial information on some development years for several occurrence years. Essentially the data corresponds to a run-off triangle in IBNR. Bayesian analysis of these problems allows the computation of forecasts for outstanding claims when there is not enough data to carry out time series analysis. Two different exact models are presented: one for discrete processes and one for continuous, which can be applied to forecast the number and amount of claims, respectively. The advantage of using the Bayesian procedure presented here is that we can derive the complete predictive distribution of the reserve requirements, from which, in turn, we can obtain a point estimate as well as probability intervals. So the possible range of values can be formally quantified. We compare our method with others, using previously published data.