square-root rule

When your data fall short of the full-credibility bar, you still want to give them some weight — but how much? The square-root rule is the simple recipe classical credibility uses: the credibility weight Z equals the square root of the ratio of your data to the data needed for full credibility. If you have a quarter of what you need, Z is the square root of one quarter, which is one half — not one quarter. Weight grows with the square root, not in a straight line.

Written out, Z = sqrt(n / N) (capped at 1), where n is your observed number of claims (or exposure) and N is the full-credibility standard. The square root is not arbitrary: the random error of an average shrinks in proportion to the square root of the sample size, so to halve the noise you need four times the data. Setting Z this way makes the fluctuation of the partial-credibility estimate just meet the same safety target that full credibility meets exactly. Example: with n = 120 and N = 1,080, Z = sqrt(120/1080) = sqrt(1/9) = 1/3.

The square-root rule is prized for being transparent and easy to defend to a regulator. But it inherits the blind spot of classical credibility: it depends only on how much data you have, never on how different this risk truly is from the benchmark. The Buhlmann approach replaces it with Z = n/(n + k), a different shape that does account for genuine between-risk variation — so do not assume the square-root rule is the 'right' formula; it is one principled choice among several.