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

Buhlmann Credibility Model: Formula, EPV, VHM and Z

Updated 11 October 2026 · Fact-checked

The Buhlmann model estimates a risk's next claim as a weighted average: Z × (the risk's own sample mean) + (1 − Z) × (the overall mean μ). Z = n/(n+k), where k = EPV ÷ VHM. Work out μ, EPV and VHM, then k, then Z, then the premium.

Understand Buhlmann Credibility Model

Imagine an insurer with many policyholders. Each one has its own true average claim, but you cannot see it. You only see a few years of claims. The question is how much weight to give that policyholder's own experience and how much to give the average of the whole portfolio.

The Buhlmann model answers this. Each risk has a hidden parameter θ. Given θ, the claims X1, X2, ..., Xn are independent and identically distributed. Define the hypothetical mean m(θ) = E[X | θ] and the process variance s²(θ) = Var(X | θ). Both are random variables, because θ is random.

Two numbers summarise them. The expected process variance EPV = E[s²(θ)] measures noise within a risk. The variance of the hypothetical means VHM = Var(m(θ)) measures how different the risks are from each other. The overall mean is μ = E[m(θ)].

The credibility premium is the linear estimate a0 + a1·X̄ that minimises expected squared error against the true m(θ) (or the next claim). The least squares solution gives Z = n/(n+k) with k = EPV ÷ VHM. So the premium is Z·X̄ + (1 − Z)·μ.

The logic is simple. Large EPV means experience is noisy, so k is large and Z is small. Large VHM means risks truly differ, so k is small and Z is large. More years n also raise Z. Z is always between 0 and 1 and never reaches 1 for finite n.

Key rules to remember

Hypothetical mean and process variance
m(θ) = E[X | θ]; s²(θ) = Var(X | θ)
Both are functions of θ, so they are random variables before θ is known.
Overall mean
μ = E[m(θ)]
This is the collective premium, the mean claim of the whole portfolio.
EPV
EPV = E[s²(θ)] (often written v)
Average of the process variances over the distribution of θ.
VHM
VHM = Var(m(θ)) (often written a)
Variance of the hypothetical means.
Parameter k
k = EPV ÷ VHM
Larger k means less weight on own experience.
Credibility factor
Z = n/(n + k) = n·VHM / (n·VHM + EPV)
n is the number of observations (years) for the risk.
Credibility premium
P = Z·X̄ + (1 − Z)·μ
X̄ is the risk's sample mean over n years.
Total variance link
Var(X) = EPV + VHM
Follows from the law of total variance for a single observation.
Variance of sample mean
Var(X̄) = VHM + EPV ÷ n
Useful in the least squares derivation.

How to solve Buhlmann Credibility Model questions

Use this order for any Buhlmann question where the distribution of θ and the claim distribution given θ are known.

  1. 1Write down the model: the distribution of θ (or the list of risk types with probabilities) and the distribution of X given θ.
  2. 2Compute m(θ) and s²(θ) for each value of θ.
  3. 3Compute μ = E[m(θ)] using the probabilities of θ.
  4. 4Compute EPV = E[s²(θ)] and VHM = E[m(θ)²] − μ².
  5. 5Find k = EPV ÷ VHM, then Z = n/(n + k) using the number of years n observed.
  6. 6Find the sample mean X̄ of the risk and compute P = Z·X̄ + (1 − Z)·μ.
  7. 7State the answer with units (for example in rupees) and check that Z lies between 0 and 1 and P lies between X̄ and μ.

Quickest way: Table method for discrete risk types

When to use it: Use when θ takes a few values, such as risk classes A, B and C, each with a stated probability, mean and variance.

  1. Make columns: probability, m, s², then m × probability, m² × probability, s² × probability.
  2. Sum the columns to get μ, E[m²] and EPV.
  3. Compute VHM = E[m²] − μ².
  4. Compute k = EPV ÷ VHM and Z = n/(n + k).
  5. Premium = μ + Z(X̄ − μ). This form needs one subtraction and one multiplication.

Common mistakes in Buhlmann Credibility Model

  • Using the variance of all claims instead of EPV

    Students compute Var(X) across the portfolio and call it the process variance.

    Fix: EPV is the average of within-risk variances. Var(X) = EPV + VHM, so subtract VHM if you only have the total.

  • Computing VHM as the average of m(θ) squared, or forgetting to subtract μ²

    Students stop at E[m²].

    Fix: VHM = E[m²] − μ². Check that it is positive.

  • Inverting k, using Z = n/(n + VHM/EPV)

    The ratio EPV ÷ VHM is easy to flip.

    Fix: Remember: noisy data (big EPV) should lower Z. If Z rises when EPV rises, you inverted k.

  • Using the wrong n

    Students use the number of policies or total claims instead of years of experience for the risk.

    Fix: n is the number of observations per risk. Use Buhlmann-Straub if exposures differ.

  • Applying Z to the wrong mean in the premium

    Students write Z·μ + (1 − Z)·X̄.

    Fix: Z multiplies the risk's own sample mean X̄. (1 − Z) multiplies the collective mean μ.

  • Treating the Buhlmann premium as the Bayesian posterior mean

    Both are credibility-type answers.

    Fix: Buhlmann is the best linear approximation. It equals the Bayes premium only in special cases such as conjugate exponential-family models.

Worked examples

Example 1

Risks fall into two classes. Class 1 (probability 0.6): claim amount per year has mean 100 and variance 400. Class 2 (probability 0.4): mean 200 and variance 900. A policyholder, whose class is unknown, claimed a total of 540 over 3 years. Find the Buhlmann credibility premium for next year.

Show the solution
  1. μ = 0.6 × 100 + 0.4 × 200 = 60 + 80 = 140.
  2. EPV = 0.6 × 400 + 0.4 × 900 = 240 + 360 = 600.
  3. E[m²] = 0.6 × 10,000 + 0.4 × 40,000 = 6,000 + 16,000 = 22,000.
  4. VHM = 22,000 − 140² = 22,000 − 19,600 = 2,400.
  5. k = 600 ÷ 2,400 = 0.25. Z = 3 ÷ (3 + 0.25) = 3 ÷ 3.25 = 0.923077.
  6. X̄ = 540 ÷ 3 = 180.
  7. P = 0.923077 × 180 + 0.076923 × 140 = 166.154 + 10.769 = 176.92.

Answer: Credibility premium ≈ 176.92 per year (Z ≈ 0.923, k = 0.25).

Example 2

For a portfolio, EPV = 50 and VHM = 10, and the collective mean is μ = 80. A risk has a sample mean of 100 over n years. Find the number of years needed for Z to be at least 0.5, and the premium when n = 5.

Show the solution
  1. k = EPV ÷ VHM = 50 ÷ 10 = 5.
  2. Z = n/(n + 5) ≥ 0.5 means n ≥ 0.5n + 2.5, so 0.5n ≥ 2.5 and n ≥ 5.
  3. So at least 5 years are needed. At n = 5, Z = 5 ÷ 10 = 0.5.
  4. P = 0.5 × 100 + 0.5 × 80 = 50 + 40 = 90.

Answer: n must be at least 5 years. For n = 5, Z = 0.5 and the premium is 90.

Exam tips

  • Show EPV, VHM, k and Z as separate labelled lines. Written papers give marks for each step even if a later number is wrong.
  • In discrete problems, always build the probability table. It prevents arithmetic slips with E[m²].
  • Sanity-check: the premium must lie between X̄ and μ, and Z must be between 0 and 1.
  • For the derivation question, set up minimising E[(m(θ) − a0 − a1X̄)²], differentiate with respect to a0 and a1, and use Var(X̄) = VHM + EPV ÷ n and Cov(m(θ), X̄) = VHM. Then state a1 = Z.
  • Paper B (computer-based) may ask you to compute EPV and VHM from a data table in R or Excel. Know the empirical Bayes estimators for that.

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

Buhlmann Credibility Model: frequently asked questions

What is the difference between EPV and VHM?

EPV is the average variability of claims within a single risk. VHM is how much the true average claims differ between risks. EPV measures noise, VHM measures real differences.

Why does Z increase with n?

More years of data make the sample mean more reliable, because its variance from process noise is EPV ÷ n. So you trust the risk's own experience more. Z approaches 1 as n grows but never equals 1.

How is the Buhlmann premium derived?

You look for the linear function a0 + a1X̄ that minimises the expected squared error against the true mean m(θ). Setting derivatives to zero gives a1 = n/(n + k) = Z and a0 = (1 − Z)μ.

When does the Buhlmann premium equal the Bayesian premium?

It happens when the Bayesian posterior mean is itself linear in the data. This holds for standard conjugate pairs such as Poisson-gamma and normal-normal. In other cases the Buhlmann premium is only the best linear approximation.