Asymptotic consistency and inconsistency of the chain ladder

The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators f̂j of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of f̂j. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well. ► Consistency of the distribution-free chain ladder method is examined. ► Unbiasedness of the development factors can be of less importance than consistency. ► A necessary and sufficient condition for the conditional consistency is derived. ► Mean square error of the development factors’ estimates is calculated. ► Real data example and numerical simulations are shown.