Triangle-free reserving

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.