Limited Fluctuation Credibility
Limited fluctuation credibility (also called classical or American credibility) assigns full credibility to an insured’s observed data when the observed claim count $n$ is large enough that the estimated pure premium lies within $\pm k\%$ of the true value with probability $p$.
If claims are Poisson-distributed, the full credibility standard for claim frequency is:
\[n \ge n_0 = \left(\frac{z_{(1+p)/2}}{k}\right)^2\]For $p = 90\%$, $k = 5\%$: $z_{0.95} = 1.645$, so $n_0 = (1.645/0.05)^2 = 1082$.
When $n < n_0$, partial credibility uses:
\[Z = \sqrt{n / n_0}, \quad Z \in [0, 1]\]The credibility estimate is a blend of observed mean $\bar{X}$ and prior mean $\mu_0$:
\[\hat{\mu} = Z\bar{X} + (1-Z)\mu_0\]This approach lacks a firm theoretical basis (the square-root rule is ad hoc), which motivates the Bühlmann approach.
Bühlmann Credibility Model
Let $\theta$ be a risk parameter drawn from a prior distribution $\pi(\theta)$. Given $\theta$, the annual losses $X_1, X_2, \ldots, X_n$ are i.i.d. with:
\(\mu(\theta) = E[X_i \mid \theta] \quad\text{(hypothetical mean)}\) \(\sigma^2(\theta) = \text{Var}(X_i \mid \theta) \quad\text{(process variance)}\)
The three structural parameters are:
\[\mu = E[\mu(\theta)], \quad v = E[\sigma^2(\theta)], \quad a = \text{Var}(\mu(\theta))\]Here $v$ is the expected process variance (within-risk variability) and $a$ is the variance of hypothetical means (between-risk variability).
The Bühlmann premium is the best linear estimator of $\mu(\theta)$ given $X_1, \ldots, X_n$, minimizing the expected squared error among all linear functions of the data:
\[\hat{\mu} = Z\bar{X} + (1-Z)\mu\]where the credibility factor is:
\[Z = \frac{n}{n + k}, \quad k = \frac{v}{a}\]The ratio $k = v/a$ is the Bühlmann $k$: high process variance or low between-risk variance makes $k$ large, reducing the weight given to own experience.
Key properties:
- $Z \to 1$ as $n \to \infty$ (full credibility with infinite data)
- $Z \to 0$ as $a \to 0$ (all risks are homogeneous; prior always dominates)
- $Z \to 1$ as $v \to 0$ (observation is noise-free; one year suffices)
Bühlmann–Straub Model
The Bühlmann–Straub extension handles varying exposure. Let $m_i$ be the exposure (e.g., premium, earned units) in year $i$, and let $\bar{X}_i = X_i / m_i$ be the loss ratio. Assume:
\[E\!\left[\bar{X}_i \mid \theta\right] = \mu(\theta), \quad \text{Var}\!\left[\bar{X}_i \mid \theta\right] = \frac{\sigma^2(\theta)}{m_i}\]The credibility estimate remains:
\[\hat{\mu} = Z\bar{X}_w + (1-Z)\mu\]where $\bar{X}_w = \sum m_i \bar{X}_i / m$ is the exposure-weighted average, $m = \sum m_i$, and:
\[Z = \frac{m}{m + k}\]This is formally identical to Bühlmann but with total exposure $m$ replacing count $n$.
Estimation of structural parameters uses the within-group and between-group sum of squares:
\[\hat{v} = \frac{1}{n-r} \sum_{i=1}^r \sum_{j=1}^{n_i} m_{ij}\!\left(\bar{X}_{ij} - \bar{X}_{i\cdot}\right)^2\] \[\hat{a} = \frac{1}{c}\left[\sum_{i=1}^r m_i(\bar{X}_{i\cdot} - \bar{X})^2 - (r-1)\hat{v}\right]\]where $c$ is a correction factor. This empirical Bayes approach avoids specifying the full prior.
Bayesian Interpretation
The Bühlmann premium is exactly Bayesian when $(X_i \mid \theta)$ is from the exponential family with a conjugate prior. In those cases, the posterior mean is linear in the data.
Example — Poisson-Gamma model: Suppose $(X \mid \lambda) \sim \text{Poisson}(\lambda)$ and $\lambda \sim \text{Gamma}(\alpha, \beta)$ (prior). After observing $n$ years with total claims $s$:
\[(\lambda \mid X_1, \ldots, X_n) \sim \text{Gamma}(\alpha + s, \; \beta + n)\]The posterior mean is:
\[E[\lambda \mid \text{data}] = \frac{\alpha + s}{\beta + n} = \frac{n}{n + \beta}\cdot \frac{s}{n} + \frac{\beta}{n+\beta}\cdot \frac{\alpha}{\beta} = Z\bar{X} + (1-Z)\mu\]with $Z = n/(n+\beta)$ and $k = \beta = v/a$. The Bühlmann formula is exact.
Other conjugate pairs used in credibility:
| Likelihood | Prior | $k$ |
|---|---|---|
| Poisson($\lambda$) | Gamma($\alpha, \beta$) | $\beta$ |
| Normal($\mu, \sigma^2$) | Normal($\mu_0, \tau^2$) | $\sigma^2/\tau^2$ |
| Binomial($m, q$) | Beta($a, b$) | $(a+b+1)/m$ |
Credibility in Practice
Commercial lines insurers apply the Bühlmann-Straub framework to experience rating and schedule rating:
Experience modification factor for workers’ compensation:
\[\text{Mod} = Z \cdot \frac{\text{Actual losses}}{\text{Expected losses}} + (1-Z)\]The credibility factor $Z$ for small accounts uses a simplified formula based on expected losses:
\[Z = \frac{E}{E + K}, \quad K = \frac{v}{a}\]where $K$ is the ballast (NCCI terminology), calibrated periodically from industry data.
Hierarchical credibility (Jewell, 1975) extends the model to multiple levels — e.g., state, industry, individual risk — each with its own credibility weight:
\(\hat{\mu}_{ij} = Z_{ij}\bar{X}_{ij} + (1-Z_{ij})\hat{\mu}_i\) \(\hat{\mu}_i = Z_i \bar{X}_i + (1-Z_i)\mu\)
This recursive structure mirrors empirical Bayes shrinkage estimation and is directly connected to linear mixed models in statistics, where the credibility weights correspond to the optimal BLUP weights.