Title: Asymptotic theory and bootstrap inference for Mack's model
Authors: Julia Steinmetz - TU Dortmund University (Germany) [presenting]
Carsten Jentsch - TU Dortmund University (Germany)
Abstract: The distribution-free chain ladder reserving model by Mack belongs to the most popular approaches in non-life insurance mathematics. As originally proposed, it serves well to determine the first two moments of the reserve distribution, but it does not allow to identify its whole distribution. To estimate also quantiles of the reserve, e.g. .~to determine the value-at-risk and tail value-at-risk, Mack's model is usually equipped with a tailor-made bootstrap procedure. For this purpose, the resulting Mack bootstrap proposal requires additional parametric assumptions and postulates a normal distribution for the individual development factors. Although the Mack bootstrap is widely used in applications, no bootstrap consistency results exist that justify this approach. We establish a rigorous model framework that allows, for an increasing number of accident years, to derive asymptotic theory for the estimators in the Mack model. We investigate parametric and non-parametric implementations of the Mack bootstrap and prove bootstrap consistency results based on a suitable set of assumptions. We illustrate our findings in simulations and discuss the validity of our approaches in situations where wrong distributional assumptions are.