The Balance Property: From Insurance Pricing to Risk-Sharing

The balance property is an in-sample global calibration constraint requiring the total fitted actuarial price to be equal to the total observed loss. For generalized linear models (GLMs) fitted by maximum likelihood estimation (MLE), this property holds automatically under the canonical link choice, but it generally fails under non-canonical link choices. Existing balance correction methods include quasi-MLE, simple post-processing of the intercept, and constrained MLE. This note proposes a fourth approach, motivated by the linear regression approximation of the conditional mean risk-sharing (CMRS) rule in large heterogeneous pools. Starting from the correctly specified MLE, the proposed correction allocates the aggregate balance defect according to the fitted conditional variances. We compare the four methods through two complementary asymptotic criteria. First, a delta method expansion gives the first-order mean squared error (MSE) of the fitted pure premium. Under this criterion, the proposed correction shares the same MLE component as the simple post-processing and the constrained MLE corrections, but differs in the way it allocates the balance defect. Second, we derive its average Kullback–Leibler (KL) divergence and show that the variance-weighted correction is locally optimal among additive first-order balance corrections acting directly on the fitted mean function. Thus, the proposed method complements constrained MLE: the latter is optimal within the parametric GLM class, whereas the former is optimal in a larger class of mean function corrections and has a direct risk-sharing interpretation. The paper finally highlights the link between insurance pricing models possessing the balance property and risk-sharing rules. This view reconciles two seemingly disconnected core actuarial topics, ex-ante premium calculation and ex-post loss allocation to participants in a peer-to-peer insurance pool.