Apply Buhlmann and Buhlmann-Straub models and understand their relationship to Bayesian models.

A policyholder files 3 claims in 2 years against a book average of 1.2. Do you believe the policyholder is genuinely riskier, or is 2 years too thin to override the book? Credibility theory is the formal answer.

The structural setup. Each policyholder carries a hidden risk parameter $\Theta$ drawn from a prior. Given $\Theta$, claim observations $X_1, X_2, \ldots, X_n$ are conditionally i.i.d. Three structural quantities drive everything:

• Hypothetical mean: $\mu(\theta) = E[X \mid \Theta = \theta]$. Its prior average is the collective premium $\mu = E[\mu(\Theta)]$.
• Process variance: $v(\theta) = \mathrm{Var}(X \mid \Theta = \theta)$. Its prior average is the expected process variance $v = E[v(\Theta)]$, the EPV. EPV is the within-risk noise actuaries see year over year on a single policy: it represents...
• Variance of hypothetical means: $a = \mathrm{Var}(\mu(\Theta))$, the VHM. VHM is the between-risk spread in true premiums across the book, the part a carrier can extract by classifying policyholders.

Common mistakes

• Plugging in $k = a/v$ instead of $k = v/a$. The right ratio is EPV over VHM so heterogeneous books get lower k and higher Z.
• Computing the sample mean in Buhlmann-Straub as a simple arithmetic mean across years. Use the exposure-weighted mean $\bar{X} = \sum m_{ij} X_{ij} / m$.
• Treating $v(\Theta) = \mathrm{Var}(X \mid \Theta)$ as $\mathrm{Var}(X)$ unconditional. The total variance is $v + a$, not just $v$.

Bottom line

• Buhlmann credibility: $Z = n/(n+k)$, with $k = \text{EPV}/\text{VHM} = v/a$.
• Credibility premium: $P_C = Z\,\bar{X} + (1-Z)\,\mu$, a convex combination of sample and collective.
• Buhlmann-Straub: with exposures $m_i$, $Z_i = m_i/(m_i + k)$, an exposure-weighted $\bar{X}$, and $v$ as EPV per unit exposure.
• Total variance decomposes as $\mathrm{Var}(X) = v + a = \text{EPV} + \text{VHM}$; larger $a$ rewards the data, larger $v$ rewards the prior.

Exam shortcut

Build a small table of $\mu, v, a$ before touching the data. With the three numbers and $n$ (or $m$), $Z$ and $P_C$ drop out in two lines. For Poisson-Gamma, skip to $k = 1/\theta$ and check your Buhlmann answer against the closed-form posterior mean $(\alpha + \sum X)/(1/\theta + n)$. They must match.

The full lesson (about 5,912 words, 39 min read) adds 5 worked examples, all 10 common mistakes, a self-check, free in the app.

Learning objectives

• 5b

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