Model computing the optimal weight to give an individual risk's own experience rather than the collective mean, as the ratio of between-risk variance to within-risk variance.
A garage has had zero claims in three years: should it be priced at zero. Obviously not, but then what weight should its experience carry. Buhlmann credibility answers with a formula rather than a convention: the premium is Z times individual experience plus one minus Z times the collective premium, where Z equals n over n plus k, n being the number of observations and k the ratio of within-risk process variance to between-risk variance of the mean. The reading is intuitive and drives the whole rating: if risks closely resemble each other, between-risk variance is small, k is large, Z stays small and individual experience counts for little; if the book is very heterogeneous, k is small and individual experience quickly takes over. Buhlmann-Straub extends the model to unequal exposure volumes, which is the version used in practice, notably in reinsurance to weight treaty experience against market statistics.
Rating a 40 vehicle commercial fleet, 2026 policy year, five years of observation. Collective frequency 12%, observed fleet frequency 7%. Process variance estimated at 0.11, between-fleet variance at 0.0018, so k equals 61 in exposure units. With n equal to 200 vehicle-years, Z equals 200 over 261, that is 0.77. The retained frequency is 0.77 times 7% plus 0.23 times 12%, that is 8.2%.
modèle de Bühlmann, Bühlmann-Straub, greatest accuracy credibility, facteur de crédibilité Z