Buhlmann-Straub model

Plain Buhlmann credibility quietly assumes every period or every risk you observe is the same size — one driver, one year, one unit. Real data are lumpier than that: one employer's group plan covers 5,000 lives, another covers 50; one year you wrote twice as much business as the next. The Buhlmann-Straub model is the natural extension that lets each observation carry a different exposure (number of lives, car-years, policies) and weights the data accordingly.

The idea is intuitive: an observation built on more exposure is less noisy, so it should count for more. Buhlmann-Straub replaces the simple claim counts with exposure-weighted averages and uses total exposure m in place of the count n, giving Z = m / (m + k) with the same k = EPV / VHM. A risk averaged over 5,000 lives lands near Z = 1; a tiny 50-life group gets a small Z and is pulled hard toward the collective rate. Concretely, if k = 800 and a group has m = 3,200 life-years of exposure, Z = 3,200/(3,200 + 800) = 0.8.

Buhlmann-Straub is the workhorse actually used in experience rating of group insurance, reinsurance treaties and any setting where exposures vary widely from risk to risk and year to year. It is the bridge from textbook theory to practice. The honest caveat is that it still assumes the within-risk variance scales cleanly with exposure (like a Poisson or compound-Poisson process) and that exposures are measured correctly — if a 'large' account is large only on paper, its high Z will mislead.