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Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums

Buhlmann-Straub Model: Credibility with Varying Exposure

Updated 11 October 2026 · Fact-checked

The Buhlmann-Straub model is a credibility model where each observation has a known weight, such as number of policies. The premium is Z × (weighted average of the risk's experience) + (1 − Z) × m. Here Z = P/(P + k), P is total exposure and k = s² ÷ a. More exposure means more credibility.

Understand Buhlmann-Straub Model

The Buhlmann model assumes every year of experience carries equal weight. In practice, a risk may have 200 policies in one year and 500 in the next. A year with more policies is more reliable, so it should count for more.

The Buhlmann-Straub model fixes this. Each observation X_ij (risk i, year j) has a known weight P_ij, such as number of policies or sum insured. X_ij is usually a claim per unit of exposure. Given the risk parameter θ_i, the mean is the same each year, E[X_ij | θ_i] = μ(θ_i). The variance falls as exposure rises: Var(X_ij | θ_i) = σ²(θ_i) ÷ P_ij.

Three parameters drive everything. m = E[μ(θ)] is the overall mean. s² = E[σ²(θ)] is the average process variance (variation within a risk). a = Var[μ(θ)] is the variance of the hypothetical means (variation between risks). Their ratio k = s² ÷ a measures how noisy the data is compared with how different the risks are.

The credibility premium is a weighted average of the risk's own experience and the overall mean. The risk's experience is the exposure-weighted mean of its observations. The credibility factor depends on total exposure P_i, not on the number of years. A risk with few years but huge exposure can get high credibility.

If every P_ij = 1, the model collapses to the Buhlmann model, with P_i = n. That is the key link between the two models.

Key rules to remember

Model assumptions
E[X_ij | θ_i] = μ(θ_i); Var(X_ij | θ_i) = σ²(θ_i) ÷ P_ij
P_ij is the known exposure. Given θ_i, observations are independent.
Parameters
m = E[μ(θ)]; s² = E[σ²(θ)]; a = Var[μ(θ)]; k = s² ÷ a
k is the exposure at which Z = 50%.
Total exposure and weighted mean for risk i
P_i = Σ_j P_ij; X̄_i = Σ_j P_ij X_ij ÷ P_i
Weight each year by its exposure, not equally.
Credibility factor
Z_i = P_i ÷ (P_i + k)
Increases with total exposure P_i and falls as k rises.
Credibility premium (per unit of exposure)
Z_i X̄_i + (1 − Z_i) m
For total claims next year, multiply by next year's exposure.
Empirical Bayes: overall mean
m̂ = X̄ = Σ_i Σ_j P_ij X_ij ÷ P, where P = Σ_i P_i
Weighted mean over all risks and years.
Empirical Bayes: estimate of s²
ŝ² = Σ_i Σ_j P_ij (X_ij − X̄_i)² ÷ Σ_i (n_i − 1)
n_i is the number of years for risk i. Needs at least one risk with n_i ≥ 2.
Empirical Bayes: estimate of a
â = [Σ_i P_i (X̄_i − X̄)² − (N − 1) ŝ²] ÷ [P − Σ_i P_i² ÷ P]
N is the number of risks. If â is negative, set it to 0, which gives Z = 0.

How to solve Buhlmann-Straub Model questions

Use this order for any Buhlmann-Straub question, whether the parameters are given or must be estimated from data.

  1. 1Identify the observations X_ij and the exposures P_ij. Check that X_ij is per unit of exposure. If you are given total claims, divide by exposure first.
  2. 2Compute P_i = Σ P_ij and the weighted mean X̄_i = Σ P_ij X_ij ÷ P_i for each risk.
  3. 3If m, s² and a are given, go to step 5. If not, compute the overall mean X̄ = Σ P_i X̄_i ÷ P.
  4. 4Estimate s² from within-risk variation, using exposure weights and divisor Σ(n_i − 1). Then estimate a using the formula with N − 1 and the denominator P − ΣP_i² ÷ P.
  5. 5Find k = s² ÷ a, then Z_i = P_i ÷ (P_i + k).
  6. 6Compute the premium Z_i X̄_i + (1 − Z_i) m. Use X̄ in place of m if you estimated it.
  7. 7If asked for total expected claims, multiply the per-unit premium by next year's exposure. State any assumption, such as that next year's exposure is known.
  8. 8Sense-check: the premium must lie between X̄_i and m, and Z must lie between 0 and 1.

Quickest way: Shortcut: work with k and totals

When to use it: Use when parameters are given and you only need the premium, or to check an empirical Bayes answer.

  1. Compute k = s² ÷ a once.
  2. Premium = (P_i X̄_i + k m) ÷ (P_i + k). This avoids finding Z separately.
  3. Note P_i X̄_i = Σ P_ij X_ij, the total of exposure × claim per unit. You do not need X̄_i separately.
  4. Check: if P_i = k, the premium is exactly halfway between X̄_i and m.
  5. In empirical Bayes, finish ŝ² and the between-risk sum first, then â, then k. Keep fractions rather than rounding early.

Common mistakes in Buhlmann-Straub Model

  • Using the simple average of yearly claims instead of the exposure-weighted mean.

    Students carry over the Buhlmann model habit of equal weights.

    Fix: Always compute X̄_i = Σ P_ij X_ij ÷ P_i. Equal weights only apply when all P_ij are equal.

  • Using Z = n ÷ (n + k) with n as the number of years.

    The Buhlmann formula is memorised without noting what n stands for.

    Fix: In Buhlmann-Straub, use total exposure: Z = P_i ÷ (P_i + k).

  • Defining k as a ÷ s² instead of s² ÷ a.

    Mixing up which variance is in the numerator.

    Fix: k = s² ÷ a: process variance over variance of hypothetical means. Check: a larger, so risks differ more, means k smaller and Z larger.

  • Dividing ŝ² by the total number of observations instead of Σ(n_i − 1).

    Forgetting that each risk loses one degree of freedom for its own mean.

    Fix: Divide the exposure-weighted sum of squares by Σ(n_i − 1).

  • Leaving out the (N − 1)ŝ² term, or using the wrong denominator, when estimating a.

    The formula is long and parts get dropped.

    Fix: Write it in two pieces: numerator Σ P_i(X̄_i − X̄)² − (N − 1)ŝ², denominator P − ΣP_i² ÷ P. If â < 0, take a = 0.

  • Giving a per-unit premium when total claims were asked, or the reverse.

    X_ij is a rate per unit of exposure, and this is lost in the working.

    Fix: State the unit of your answer. Multiply by next year's exposure for total expected claims.

Worked examples

Example 1

A risk has exposures of 20, 30 and 50 policies in three years, with average claims per policy of ₹10,000, ₹12,000 and ₹9,000. Assume the Buhlmann-Straub model with m = ₹11,000, s² = 40 and a = 2, where amounts are in ₹ thousand. Find the credibility premium per policy for next year.

Show the solution
  1. Total exposure: P = 20 + 30 + 50 = 100.
  2. Weighted mean (₹ thousand): (20 × 10 + 30 × 12 + 50 × 9) ÷ 100 = (200 + 360 + 450) ÷ 100 = 10.1.
  3. k = s² ÷ a = 40 ÷ 2 = 20.
  4. Z = 100 ÷ (100 + 20) = 5/6.
  5. Premium = (5/6) × 10.1 + (1/6) × 11 = 8.4167 + 1.8333 = 10.25 (₹ thousand).

Answer: The credibility premium is ₹10,250 per policy. The credibility factor is 5/6.

Example 2

Two risks have these observations (claim per policy, exposure in brackets). Risk 1: year 1: 5 (10), year 2: 7 (30). Risk 2: year 1: 9 (20), year 2: 11 (20). Use empirical Bayes Buhlmann-Straub estimation to find the credibility premium per policy for each risk.

Show the solution
  1. Total exposures: P₁ = 40, P₂ = 40, P = 80.
  2. Weighted means: X̄₁ = (10 × 5 + 30 × 7) ÷ 40 = 260 ÷ 40 = 6.5. X̄₂ = (20 × 9 + 20 × 11) ÷ 40 = 400 ÷ 40 = 10.
  3. Overall mean: X̄ = (260 + 400) ÷ 80 = 8.25. So m̂ = 8.25.
  4. Within-risk sum of squares. Risk 1: 10 × (5 − 6.5)² + 30 × (7 − 6.5)² = 22.5 + 7.5 = 30. Risk 2: 20 × 1² + 20 × 1² = 40. Total = 70.
  5. Σ(n_i − 1) = 1 + 1 = 2, so ŝ² = 70 ÷ 2 = 35.
  6. Between-risk sum: Σ P_i (X̄_i − X̄)² = 40 × 1.75² + 40 × 1.75² = 80 × 3.0625 = 245.
  7. N = 2, so subtract (N − 1) ŝ² = 35, giving 210.
  8. Denominator: P − Σ P_i² ÷ P = 80 − (1600 + 1600) ÷ 80 = 80 − 40 = 40. So â = 210 ÷ 40 = 5.25.
  9. k = 35 ÷ 5.25 = 20/3. Z = 40 ÷ (40 + 20/3) = 120 ÷ 140 = 6/7.
  10. Risk 1 premium = (6/7) × 6.5 + (1/7) × 8.25 = (39 + 8.25) ÷ 7 = 6.75.
  11. Risk 2 premium = (6/7) × 10 + (1/7) × 8.25 = (60 + 8.25) ÷ 7 = 9.75.

Answer: Z = 6/7 for both risks (their exposures are equal). Premium per policy: Risk 1 = 6.75, Risk 2 = 9.75.

Exam tips

  • Write the model assumptions in standard notation first: E[X_ij | θ_i] = μ(θ_i) and Var(X_ij | θ_i) = σ²(θ_i) ÷ P_ij. Written questions give marks for this.
  • Tabulate P_i, X̄_i and the within-risk sums for each risk before using any formula. It makes errors easy to spot and earns method marks.
  • Paper B style questions in R or Excel ask for the same estimators on a dataset. Compute them by formula, then show the result for ŝ², â, k and Z.
  • Comment on the answer. Say how Z changes with exposure, or that a negative â means no credibility. Examiners reward interpretation.
  • In MCQs, check whether the question gives claims per unit or total claims. Check whether Z uses P_i or n.

Practice questions from Credibility theory: Bayesian and empirical Bayes credibility premiums

Buhlmann-Straub Model: frequently asked questions

What is the difference between the Buhlmann and Buhlmann-Straub models?

The Buhlmann model treats all years as equally reliable, so Z = n ÷ (n + k). The Buhlmann-Straub model allows each year a different exposure P_ij and uses weighted means. Its credibility factor is Z = P_i ÷ (P_i + k). With all P_ij = 1 it reduces to the Buhlmann model.

What does the credibility factor depend on in the Buhlmann-Straub model?

It depends on the risk's total exposure P_i and on k = s² ÷ a. More exposure, or a larger a relative to s², raises Z. It does not depend on the number of years directly.

Why is the variance given as σ²(θ) ÷ P_ij?

An observation with exposure P_ij is like an average over P_ij independent units. Averaging reduces variance in proportion to the number of units, so more exposure means less process variance.

What if the estimate of a comes out negative?

A variance cannot be negative. Set â = 0, which makes k infinite and Z = 0. The premium is then just the overall mean m̂. Mention this in your answer.