Buhlmann-Straub Model for Heterogeneous Portfolios

Extension to Varying Exposures

The Buhlmann-Straub model generalizes Buhlmann credibility to handle observations with different volumes or exposures. Instead of observing individual claims X1, ..., Xn, we observe averages X_j = (total claims in period j) / m_j, where m_j is the exposure (e.g., number of policies) in period j. The model assumes E[X_j|Theta] = mu(Theta) and Var(X_j|Theta) = v(Theta)/m_j.

The Buhlmann-Straub premium is: P_{BS} = Z * X_w + (1 - Z) * mu, where X_w = sum(m_j * X_j) / sum(m_j) is the exposure-weighted average and Z = m / (m + k), with m = sum(m_j) being the total exposure and k = v/a as in the standard Buhlmann model.

Estimation of Structural Parameters

With multiple risks (policyholders or groups), the structural parameters mu, v, and a can be estimated from data. For r risks, each observed over n_i periods with exposure m_{ij}: the overall mean estimate is the grand weighted mean. The estimated process variance v-hat uses within-risk variability. The estimated between-risk variance a-hat uses between-risk variability, adjusted for within-risk noise. The formulas involve weighted sums and are provided on Exam STAM formula sheets.

Exam STAM Applications

The Buhlmann-Straub model is used when different risk groups have different exposures (e.g., different-sized insurance pools in different years). A common exam setup gives data for several risks across multiple years with varying numbers of policies. You compute X_w for each risk, estimate v and a from the data, compute k and Z for each risk, and then compute the credibility premium for each risk. The computational steps are systematic but require careful organization. Practice with tabular data to build speed and accuracy.