How much weight should a risk's own experience receive relative to that of the collective? Credibility theory gives the optimal answer.
Last year a policyholder's loss experience was far above average. Should we charge them based on their own experience (sparse and volatile) or on the group premium (stable but impersonal)? Credibility theory resolves this dilemma with a weighted average:
where $\bar{X}$ is the observed individual experience, $\mu$ the collective (or a priori) premium, and $Z\in[0,1]$ the credibility factor: how much confidence the risk's own experience deserves. The more information (exposure) and the less heterogeneity, the larger $Z$. The various schools of credibility differ in how $Z$ is computed.
The "American" approach (Mowbray, 1914): require enough volume for the experience to be statistically stable.
Expected number of claims required to assign $Z=1$, so that the observation falls within $\pm r$ of the mean with probability $1-\alpha$. Typical standard: $\pm5\%$ at $90\%$ ⇒ 1 082 claims.
When the volume $n$ falls short of the full credibility standard $n_F$, partial credibility is granted in proportion to the square root of the ratio of exposures.
For the pure premium (frequency × severity), the full credibility standard incorporates the coefficient of variation of severity $Y$, requiring more data when claim amounts are dispersed.
The classical approach is simple but arbitrary: neither the full credibility standard nor the form of the partial credibility adjustment is derived from an optimality criterion. Greatest accuracy credibility (Bühlmann) addresses this.
Hans Bühlmann (1967) derived the factor $Z$ as the best linear approximation to the Bayesian premium, minimizing expected squared error. It is the foundation of modern ratemaking.
$\mu$: collective mean. $v$ = Expected Process Variance (variability within a risk). $a$ = Variance of Hypothetical Means (heterogeneity across risks).
$Z$ increases with exposure $n$ and with heterogeneity across risks $a$; it decreases with internal noise $v$. If all risks were identical ($a=0$), $Z=0$.
It minimizes $E[(P_c-\mu(\Theta))^2]$ among all estimators that are linear in the observations. Hence the name "greatest accuracy credibility".
It generalizes Bühlmann to risks with different exposure in each period (portfolio size, policy years). This is the model used in practice in property and casualty insurance.
The ideal credibility premium is the posterior mean $E[\mu(\Theta)\mid \mathbf{X}]$: we update the a priori belief about the risk parameter $\Theta$ with the observed experience through Bayes' theorem. In general this posterior mean is not linear, but for certain conjugate families it coincides exactly with the Bühlmann formula; this is known as exact credibility.
| Model (likelihood / prior) | Posterior | Credibility |
|---|---|---|
| Poisson / Gamma | Gamma | Exact (linear) |
| Binomial / Beta | Beta | Exact (linear) |
| Normal / Normal | Normal | Exact (linear) |
| Exponential / Gamma | Gamma | Exact (linear) |
In practice the structural parameters $\mu$, $v$ and $a$ are unknown: they are estimated from the data of the whole portfolio (nonparametric or empirical Bayes credibility). The collective experience thus calibrates the model that is then applied to each individual risk.
Credibility is the conceptual precursor of linear mixed models and of the modern hierarchical Bayesian models used in pricing, where the factor $Z$ reappears as the shrinkage of random effects toward the overall mean.