Discussion of ‘A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving

Professor Verrall nicely illustrates how Bayesian models can be applied to claims reserving within the framework of Generalized Linear Models (GLMs) and how they lead to posterior predictive distributions of quantities of interest. In this discussion we apply a Bayesian model in the context of discounted loss reserves. The outcomes of this approach are compared with results on approximations for the distribution of the discounted loss reserve when the run-off triangle is modelled by a GLM. These approximations are based on the concepts of comonotonicity and convex order and are explained in full details in Hoedemakers et al. (2003) (for lognormal claims reserving models) and Hoedemakers et al. (2004) (for claims reserving using GLMs). The comonotonicity approach has been shown to provide elegant solutions to various other actuarial and financial problems involving the distribution of a sum of dependent random variables (check www.kuleuven.ac.be/insurance- research papers for more illustrations). We realize that the Bayesian posterior predictive distribution is a very general workhorse, which takes into account all sources of uncertainty in the model formulation and is applicable to different statistical domains, whereas our approximations originate from a specific actuarial context. We want to illustrate however that the predictive distribution based on the comonotonic bounds provides results that are close to the results obtained via MCMC. The main advantage of the bounds is that several risk measures such as percentiles (VaRs), expected shortfalls (stop-loss premia) and TailVaRs can be calculated easily from it. Generalized linear models for claims reserving: the likelihood-based approach We use the notation from Verrall (2004). An insurer is interested in the aggregated value � n i=2 � n j=n+2−i Cij corresponding with the future part of a classical run-off triangle (as shown in Table 1 of Verrall’s paper). In a (likelihood-based) GLM framework this reserve will be predicted by n � n� reserve