Advanced Credibility Theory for Exam MAS-II

Hierarchical Credibility

Hierarchical credibility (Buhlmann-Straub with multiple levels) handles data organized in nested groups, such as policies within territories within states. At each level, credibility weights blend the group estimate with the next higher level's estimate. For a two-level hierarchy, the territory estimate is Z_territory * (territory mean) + (1-Z_territory) * (state estimate), and the state estimate is Z_state * (state mean) + (1-Z_state) * (countrywide mean). The credibility factors at each level use the within-group and between-group variance components appropriate to that level.

Advanced Extensions

Multidimensional credibility handles multiple correlated observations per risk (e.g., frequency and severity, or multiple coverage types). The credibility matrix Z replaces the scalar credibility factor. Regression credibility (Hachemeister model) allows the mean to depend on covariates, with credibility applied to the regression coefficients. The credibility premium is a weighted average of the individual regression prediction and the collective regression prediction. These advanced methods connect credibility theory to modern mixed effects models. Exam MAS-II tests hierarchical models, multidimensional credibility, and the Hachemeister regression approach.