Statistical inference for the reserve risk

Abstract

The major part of the liability of an insurance company's balance belongs to the reserves. Reserves are built to pay for all future, known or unknown, claims that happened so far. Hence an accurate prediction of the outstanding claims to determine the reserve is important. For non-life insurance companies, Mack (1993) proposed a distribution-free approach to calculate the first two moments of the reserve. In this cumulative dissertation, we derive first asymptotic theory for the unconditional and conditional limit distribution of the reserve risk. Therefore, we enhance the assumptions from Mack's model and derive a fully stochastic framework. The distribution of the reserve risk can be split up into two additive random parts covering the process and parameter uncertainty. The process uncertainty part dominates asymptotically and is in general non-Gaussian distributed unconditional and conditional on the whole observed loss triangle or the last observed diagonal of the loss triangle. In contrast, the parameter uncertainty part is measurable with respect to the whole observed upper loss triangle. Properly inflated, the parameter uncertainty part is Gaussian distributed conditional on the last observed diagonal of the loss triangle, and unconditional, it leads to a non-Gaussian distribution. Hence, the parameter uncertainty part is asymptotically negligible. In total, the reserve risk has asymptotically the same distribution as the process uncertainty part since this part dominates asymptotically leading to a non-Gaussian distribution conditional and unconditional. Using the theoretical asymptotic distribution results regarding the distribution of the reserve risk, we can now establish bootstrap consistency results, where the derived distribution of the reserve risk serves as a benchmark. Splitting the reserve risk into two additive parts enables a rigorous investigation of the validity of the Mack bootstrap. If the parametric family of distributions of the individual development factors is correctly specified, we prove that the (conditional) distribution of the asymptotically dominating process uncertainty part is correctly mimicked by the proposed Mack bootstrap approach. On the contrary, the corresponding (conditional) distribution of the estimation uncertainty part is generally not correctly captured by the Mack bootstrap. To address this issue, we propose an alternative Mack bootstrap, which uses a different centering and is designed to capture also the distribution of the estimation uncertainty part correctly.