Insurance: Mathematics and Economics 26 (2000) 109–111
Discussion
Comments on: “A comparison of stochastic models that reproduce chain
ladder reserve estimates”, by Mack and Venter
R.J. Verrall
∗, P.D. England
Department of Actuarial Science and Statistics, City University, Northampton Square, London EC1V 0HB, UK
Received April 1999; received in revised form May 1999
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
∗Corresponding author.
E-mail address:[email protected] (R.J. Verrall).
110 R.J. Verrall, P.D. England / Insurance: Mathematics and Economics 26 (2000) 109–111
obtained. If the same portfolio were observed again, you would still get the same values for the row totals, because these are the only possible values. It is shown in Verrall (2000), for example (3.13), that the ODP model can be written in such a way that the row parameters are not estimated explicitly, as long as this is taken into account in the variance assumption. We also believe that this is the case with DFCL, which allows for the fact that the row parameters have been estimated in the variance assumption. It is misleading to say that “the parametersxi of ODP have an equivalent neither in the chain ladder algorithm nor in the DFCL model”. It is simply the case that they have been integrated out, so that they do appear explicitly in the model, but it is taken into account in the variance that they have been estimated. In other words, there are two approaches which can be taken when using the unconditional likelihood. You can include the row parameters in the model, in which case the fact that they have been estimated must be taken into account in the calculation of the prediction errors. Alternatively, you can integrate the row parameters out of the model, which will increase the model variance. In both cases you end up with the same result. The only alternative which gives different prediction mean square error is to use the conditional likelihood, but we do not believe this to be the correct approach to use. Secondly, Mack and Venter have overlooked the fact that they estimate the parametersσj2. The recursive model, (3.11), in which the row parameters have been integrated out, has the same number of explicit parameters as the chain-ladder technique. DFCL has a distinct disadvantage that it has the extra parameters in the variance that must be estimated. This makes it possible to deal with negative incremental claims, but at a cost: there are more parameters to estimate. Apart from this point, which shows that DFCL has more parameters than the recursive formulation of the Poisson model, counting parameters is not helpful. What really matters is the prediction mean square error, and this is the same for both models based on the full likelihood: the ODP model and the recursive model. It is only different when you use the conditional likelihood.
2. “ODP and DFCL have diverging independence assumptions.”
Certainly the ODP is formulated with Cij independent. However, once you look more carefully at the conditional model, (2.6) of Verrall (2000), or the recursive model in section 3 it is clear that the data are not independent. Once the estimate of the row parameter is taken into account, the data are not independent. There is no conflict between the independence assumption of ODP and the assumptions of DFCL or other recursive formulations of the model.
3. “The fitted values and therefore also the residuals for ODP are different from those of DFCL.”
The fitted values are in fact the same whichever model is used, if you use the correct method to get them. Unfortunately, an incorrect method for obtaining fitted values is sometimes used, and this is the one which Mack and Venter refer to. When a recursive model is used, the fitted values should be obtained by “back-casting”. Thus, the fitted values for the chain-ladder technique are:
ˆ
When this method is used, therearefitted values for the first column, and the chain-ladder technique, ODP and DFCL give the same fitted values.
4. “The true expected reserves described by the models are different.” 5. “The simulated future emergence is different.”
Both these relate to simulating the claims run-off and so will be dealt with together.
R.J. Verrall, P.D. England / Insurance: Mathematics and Economics 26 (2000) 109–111 111
reserve estimator will have a greater estimated variance and the variability of the simulated values will be lower. The simulation of future outcomes, and bootstrapping of reserve estimates can be approached in a number of different ways as long as care is taken to include all sources of variability. Again there is no conflict between the overall approaches of DFCL, ODP and the recursive models in Verrall (2000).
Mack and Venter also list “other differences between the ODP and the chain ladder algorithm”. 1. ODP does not work in all situations where the chain ladder algorithm works.
This does not make the underlying model different. The recursive model (3.14) in Verrall (2000) will work for the data described by Mack and Venter. It cannot be argued that these are different models since it is merely a case of formulating the model in a more convenient way.
2. ODP cannot be adapted if the weights infˆj are changed.
Renshaw and Verrall (1998), Verrall (2000) and others make it clear that it is the intention that the data should be examined more carefully using the statistical framework of generalised linear models, non-linear models or other methods. It has never been argued that the various variants of the deterministic chain-ladder technique can be reproduced as statistical models, and our approach takes a completely different attitude to how the data should be analysed. The point made by Mack and Venter is irrelevant to their overall argument since it is the standard version of the chain ladder technique that is at issue.
