Unbiased estimator for the ultimate prediction uncertainty in the chain-ladder model of Mack

We derive in a straightforward manner and through a unified framework the main known results related to the estimation of the ultimate prediction uncertainty within the famous Mack's distribution-free chain-ladder model, specifically, the Mack and BBMW formulas, and explain the deviation between the two formulas in terms of the applied estimation principles.We demonstrate that by applying a well-defined estimation principle to estimate the quadratic difference between the estimated and true development factors, the BBMW formula can be derived without using the conditional resampling approach. Moreover, we highlight that none of the estimators are unbiased for estimating the true uncertainty. Furthermore, even if the Mack estimator can be proved to be (slightly) less overestimated (on average, given the first triangle column) than the BBMW estimator, for each and any claims development triangle, the Mack formula does not provide the most accurate estimate.By applying an enhanced estimation principle, we derive a novel formula to quantify the ultimate prediction uncertainty, which can be proved to be unbiased (conditionally given the first triangle column) under some harmless additional conditions. However, this novel estimator, as well as the Mack and BBMW formulas, can materially fail to estimate the true uncertainty