Comments on: “A comparison of stochastic models that reproduce chain ladder reserve estimates”, by Mack and Venter

This note contains a discussion of the paper of Mack and Venter (2000), and also refers to the paper of Verrall (2000) (both in this issue). The issue is the relationship between the over-dispersed Poisson (ODP) model of Renshaw and Verrall (1998), the distribution-free stochastic model (DFCL) of Mack (1993) and the chain-ladder technique. For a full understanding of the chain-ladder technique, it is necessary to have read first Verrall (2000), and many of the points made in this discussion are taken from that paper. It was shown in Verrall (2000) that the ODP model can be re-expressed in a number of different ways, one of which is closely related to the DFCL model. The DFCL model is a reasonable approach, particularly when the incremental data contain a significant number of negative values, but it is not necessarily the best approach to use, nor it is sensible to argue that it is the only approach which should be used. This note is constructed as follows. Firstly, we consider the five arguments which Mack and Venter make in order to attempt to show that the ODP model should not be used in claims reserving and that the DFCL model is the only model which “can qualify to be referred to as the model underlying the chain-ladder algorithm”. Secondly, we clarify the relationship between the chain-ladder technique and the various stochastic models which have been put forward which give the same reserve estimates. This provides clear guidance about which approach is likely to be most suitable for each application considered by practicing actuaries. Mack and Venter make five points which they claim as “evidence that the models are different” (the models are referred to as ODP and DFCL). We list these and refute them below: 1. “ODP has more parameters than DFCL.” Firstly, Mack and Venter try to argue that the chain-ladder technique conditions on the row totals, so that there are no row parameters to estimate. This argument cannot be settled by appealing to the chain-ladder technique since the same estimates are obtained in both cases. The practitioner must decide whether to treat the row parameter as fixed or as an estimate. In other words, to choose between using the conditional likelihood (LC) or the unconditional likelihood (L), as discussed in Section 4 of Verrall (2000). If the row totals are really conditioned upon, then this must be taken into account in the variance of the estimates, and it implies that the row totals obtained (i.e. the values of {Di ,n − i + 1: i=1,2,. . . ,n}) are the only ones which could have been