Asymptotic theory for Mack's model

Abstract

The distribution-free chain ladder reserving model by Mack (1993) belongs to the most popular approaches in non-life insurance mathematics. It was originally proposed to determine the first two moments of the reserve distribution. Together with a normal approximation, it is commonly applied to conduct statistical inference including the estimation of large quantiles of the reserve and determination of the reserve risk. However, for Mack's model, the literature lacks a rigorous justification of such a normal approximation for the reserve.In this paper, we propose a general stochastic framework which allows to derive asymptotic theory for Mack's model. For increasing number of accident years, we establish central limit theorems for the parameter estimators in Mack's model. In particular, these results enable us to derive also unconditional and conditional limiting distributions for the reserve. For this purpose, the reserve risk is split into two random parts that carry the process uncertainty and the estimation uncertainty, respectively. Unconditionally, but also when conditioning on either the whole observed loss triangle or on its diagonal, we show that the limiting distribution of the first part that corresponds to the process uncertainty will be usually non-Gaussian. When properly inflated, the second part corresponding to the estimation uncertainty is measurable with respect to the loss triangle and, unconditionally, turns out to be asymptotically non-Gaussian as well. By contrast, when conditioning only on the diagonal, this results in a Gaussian limit. As the process uncertainty part dominates asymptotically, this leads overall to a non-Gaussian limiting distribution for the reserve in both cases.These findings cast the common practice to use a normal approximation for the reserve in Mack's model into doubt. We illustrate our findings by simulations and show that our setup covers cases, where the limiting distributions of the reserve risk might deviate substantially from a Gaussian distribution.