Mack's distribution-free chain ladder reserving model belongs to the most popular approaches in non-life insurance mathematics. Proposed to determine the first two moments of the reserve, it does not allow to identify the whole distribution of the reserve. For this purpose, Mack's model is usually equipped with a tailor-made bootstrap procedure. Although widely used in practice to estimate the reserve risk, no theoretical bootstrap consistency results exist that justify this approach.

To fill this gap in the literature, we adopt the framework proposed by Steinmetz and Jentsch (2022) to derive asymptotic theory in Mack's model. By splitting the reserve into two parts corresponding to process and estimation uncertainty, this enables - for the first time - a rigorous investigation also of the validity of the Mack bootstrap. We prove that the (conditional) distribution of the asymptotically dominating process uncertainty part is correctly mimicked by Mack's bootstrap if the parametric family of distributions of the individual development factors is correctly specified. Otherwise, this is not the case. In contrast, the (conditional) distribution of the estimation uncertainty part is generally not correctly captured by Mack's bootstrap. To tackle this, we propose an alternative Mack-type bootstrap, which is designed to capture also the distribution of the estimation uncertainty part.

We illustrate our findings by simulations and show that the newly proposed alternative Mack bootstrap performs superior to the Mack bootstrap.

麦克的无分配链梯准备金模型属于非寿险数学中最流行的方法。建议确定储备的前两个矩，但不允许确定储备的整个分布。为此，麦克模型通常配备定制的引导程序。尽管在实践中广泛用于估计准备金风险，但不存在理论引导一致性结果来证明这种方法的合理性。

为了填补文献中的这一空白，我们采用Steinmetz 和 Jentsch (2022)提出的框架来推导 Mack 模型中的渐近理论。通过将储备分为与过程和估计不确定性相对应的两部分，这首次使得能够对麦克引导程序的有效性进行严格的调查。我们证明，如果正确指定了各个发展因素的分布参数族，则麦克引导程序可以正确模拟渐进主导过程不确定性部分的（条件）分布。否则，情况并非如此。相反，麦克的引导程序通常无法正确捕获估计不确定性部分的（条件）分布。为了解决这个问题，我们提出了一种替代的 Mack 型引导程序，其旨在捕获估计不确定性部分的分布。

我们通过模拟来说明我们的发现，并表明新提出的替代 Mack bootstrap 的性能优于 Mack bootstrap。