This paper argues that all reserving methods based on claims triangulations (the “triangle trick”), no matter how sophisticated the subsequent processing of the information contained in the triangle is, are inherently inadequate to accurately model the distribution of reserves, although they may be good enough to produce a point estimate of such reserves. The reason is that the triangle representation involves the compression (and ultimately the loss) of crucial information about the individual losses, which comes back to haunt us when we try to extract detailed information on the distribution of incurred but not reported (IBNR) and reported but not settled (RBNS) losses.

This paper then argues that in order to avoid such loss of information it is necessary to adopt an approach which is similar to that used in pricing, where a separate frequency and severity model are developed and then combined by Monte Carlo simulation or other numerical techniques to produce the aggregate loss distribution.

A specific implementation of this approach is described, whose core feature is a method to produce a frequency model for the incurred but not reported claim count based on the empirical distribution of delays (delay = the time between loss date and reporting date), after adjustments to make up for the bias towards smaller delays. The method also produces a kernel severity model for the individual losses, from which the severity distribution of each year of occurrence can be derived. By combining the frequency and severity model in the usual way (e.g. through Monte Carlo simulation), an aggregate model for IBNR and UPR losses can be produced.

As for RBNS losses, we suggest using one of the many methods to analyse the distribution of IBNER (incurred but not enough reserved) factors to produce a possible distribution of outcomes.

A case study based on real-world liability claims is used to illustrate how the method works in practice.

Also, in a first step towards validating the method for calculating IBNR and comparing it with existing methods, a series of experiments with artificial data sets was undertaken, which show a drastic reduction in the prediction error of both the IBNR claim count and the IBNR total amount with respect to the standard chain ladder method. And what is perhaps most promising, the experiments show that the distribution of IBNR reserves is much closer (in terms of the Kolmogorov-Smirnov distance) to the “true” one than that based on Mack's method in the way it is normally applied. The method promises therefore a more accurate assessment of the uncertainty around reserves.

This work deals with prediction of IBNR reserve under a different data ordering of the non-cumulative runoff triangle. The rows of the triangle are stacked, resulting in a univariate time series with several missing values. Under this ordering, two approaches entirely based on state space models and the Kalman filter are developed, implemented with two real data sets, and compared with two well-established IBNR estimation methods — the chain ladder and an overdispersed Poisson regression model. The remarks from the empirical results are: (i) computational feasibility and efficiency; (ii) accuracy improvement for IBNR prediction; and (iii) flexibility regarding IBNR modeling possibilities.

We consider the problem of claims reserving and estimating run-off triangles. We generalize the gamma cell distributions model which leads to Tweedie's compound Poisson model. Choosing a suitable parametrization, we estimate the parameters of our model within the framework of generalized linear models (see Jørgensen-de Souza [2] and Smyth-Jørgensen [8]). We show that these methods lead to reasonable estimates of the outstanding loss liabilities.

The chain ladder forecast of outstanding losses is known to be unbiased under suitable assumptions. According to these assumptions, claim payments in any cell of a payment triangle are dependent on those from preceding development years of the same accident year. If all cells are assumed stochastically independent, the forecast is no longer unbiased. Section 5 shows that, under rather general assumptions, it is biased upward. This result is linked to earlier work on some stochastic versions of the chain ladder.