Logistic Credibility with Temporal Decay: Extending B\"uhlmann--Straub for Commercial Lines

Abstract

B\"uhlmann--Straub (B-S) credibility assigns each account a weight $Z_i = E_i/(E_i+K)$, where $K$ is a single portfolio-wide ratio. The formula assumes $K$ is the same for every account regardless of size, history length, or volatility, and that recent and older years carry equal weight. On a held-out US commercial auto dataset these assumptions fail: standard B-S applied to 96 companies produces a calibration slope of 29 for small accounts, a signature of severe under-crediting. We propose a joint framework that retains B-S interpretability while addressing these limitations. The credibility weight $Z_i$ is modelled as a logistic function of account characteristics; historical experience is discounted by an EWMA decay parameter $\lambda$ estimated from the data; and $Z$, $\lambda$, and the complement are optimised in a single likelihood pass. The framework formally nests B\"uhlmann--Straub as a special case, admitting a likelihood-ratio test for any proposed extension. On a two-year held-out test set the proposed model restores calibration (slope = 1.00) and reduces exposure-weighted prediction error by 38% (90% bootstrap interval: 26%--50%). A size gradient in the decay rate emerges ($\hat\lambda \approx 0.6$, $0.84$, $0.13$ for Small, Mid, Large) and replicates qualitatively on Other Liability. A simulation study confirms the mechanisms. The model requires only account-year summaries and delivers three transparent outputs: credibility weight, complement, and recommended renewal rate.