Loss Reserving: Stochastic Methods and Bootstrap

Stochastic Chain Ladder

The Mack model provides a distribution-free stochastic framework for the chain ladder. It assumes E[C_{i,j+1}|C_{i,j}] = f_j * C_{i,j} and Var(C_{i,j+1}|C_{i,j}) = sigma_j^2 * C_{i,j}. The development factors f_j are estimated as in the standard chain ladder. The mean squared error of prediction (MSEP) for reserves combines process variance and estimation variance. Mack's formulas give analytical estimates of prediction uncertainty for each accident year and in total, accounting for correlation between accident year reserve estimates.

Bootstrap Reserving

The bootstrap approach to reserving generates a distribution of reserves. The procedure: (1) fit the chain ladder model and compute Pearson residuals, (2) resample residuals with replacement to create pseudo-triangles, (3) re-estimate development factors and project reserves for each pseudo-triangle, (4) add process variance by simulating from an appropriate distribution. Repeating steps 2 to 4 many times gives a predictive distribution of reserves. Percentiles provide reserve ranges, and the standard deviation estimates prediction error. The over-dispersed Poisson (ODP) model provides a GLM framework for the chain ladder, facilitating bootstrap implementation. Exam MAS-II tests both Mack's analytical formulas and bootstrap procedures.