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 approximations, it is often applied to conduct statistical inference including the estimation of large quantiles of the reserve and determination of the reserve risk. However, for the Mack model, the literature lacks a rigorous justification of such a normal approximation for the reserve.

In this paper, we propose a suitable stochastic framework which allows to derive meaningful asymptotic theory for the Mack model. For increasing number of accident years, we establish central limit theorems for the commonly used estimators in the Mack model. In particular, these results enable us to derive the limiting distribution of the reserve risk. First, it is split in two random parts that carry the process uncertainty and the estimation uncertainty, respectively. Surprisingly, by deriving their joint limiting distributions, we prove that the limiting distribution of the reserve risk will be usually non-Gaussian. This main result casts the common practice to use a normal approximation for the reserve in the Mack model into doubt. We illustrate our findings by simulations and illustrate that the limiting distributions of the reserve risk might deviate substantially from a Gaussian distribution.