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

Empirical Bayes Credibility Theory: Estimating EPV and VHM

Updated 11 October 2026 · Fact-checked

Empirical Bayes credibility theory estimates the Bühlmann parameters from the claims data instead of assuming a prior. You estimate the EPV from variation within each risk, the VHM from variation between risk means, then calculate Z and the premium Z × own mean + (1 − Z) × overall mean. Model 1 has equal exposure; Model 2 has weights.

Understand Empirical Bayes Credibility Theory

In Bühlmann credibility you need three things: the overall mean E[m(θ)], the expected process variance EPV = E[s²(θ)], and the variance of the hypothetical means VHM = Var[m(θ)]. In a Bayesian question the prior gives these. In practice there is often no prior. Empirical Bayes means you estimate them from the data of many similar risks.

The idea is simple. The EPV measures how much one risk's results vary from year to year. So you estimate it from the variation inside each risk and average over risks. The VHM measures how much the true risk levels differ between risks. So you look at the variation between the risks' sample means. But sample means vary partly through pure chance, so you must remove that chance part. That is why the VHM estimate has a subtraction.

Model 1 is the simple case. You have N risks, each observed for the same n years, with equal exposure. Model 2 is the Bühlmann-Straub case. Each year has an exposure (volume) P_ij, and the observations X_ij are claims per unit of exposure. All means become exposure-weighted means. The number of years can differ between risks.

Once you have the estimates, the rest is the usual Bühlmann work. Find k = EPV ÷ VHM, then Z, then the credibility premium for each risk. The estimate of the VHM can come out negative. A variance cannot be negative, so in that case you set it to zero. Then Z = 0 and every risk gets the overall mean.

Key rules to remember

Model 1: overall mean estimate
Ê[m(θ)] = X̄ = (1 ÷ Nn) Σi Σj Xij
N risks, n years each, equal exposure. X̄i is the mean of risk i.
Model 1: EPV estimate
Ê[s²(θ)] = (1 ÷ N) Σi [ (1 ÷ (n − 1)) Σj (Xij − X̄i)² ]
Average of the sample variances of each risk, each with divisor n − 1.
Model 1: VHM estimate
V̂ar[m(θ)] = (1 ÷ (N − 1)) Σi (X̄i − X̄)² − Ê[s²(θ)] ÷ n
If the result is negative, set it to 0.
Model 1: credibility factor and premium
Z = n ÷ (n + k), k = EPV ÷ VHM; premium = Z X̄i + (1 − Z) X̄
Z = n ÷ (n + EPV ÷ VHM) is the same thing.
Model 2: weighted means
Pi = Σj Pij; P = Σi Pi; X̄iP = Σj Pij Xij ÷ Pi; X̄PP = Σi Pi X̄iP ÷ P
Xij is claims per unit exposure. X̄PP is the estimate of E[m(θ)].
Model 2: EPV estimate
ŝ² = Σi Σj Pij (Xij − X̄iP)² ÷ Σi (ni − 1)
With equal years n, the divisor is N(n − 1).
Model 2: VHM estimate
σ̂² = [ Σi Pi (X̄iP − X̄PP)² − (N − 1) ŝ² ] ÷ [ P − Σi Pi² ÷ P ]
Set to 0 if negative.
Model 2: credibility factor and premium
Zi = Pi ÷ (Pi + ŝ² ÷ σ̂²); premium per unit exposure = Zi X̄iP + (1 − Zi) X̄PP
Multiply by next year's exposure to get the total premium.

How to solve Empirical Bayes Credibility Theory questions

Use this order for any empirical Bayes credibility question. First decide which model applies, then work out the pieces one at a time.

  1. 1Decide the model. Equal exposure in every cell means Model 1. Different exposures P_ij mean Model 2.
  2. 2Calculate each risk's mean (X̄i for Model 1, X̄iP weighted for Model 2) and the overall mean.
  3. 3Estimate the EPV from within-risk variation. Use divisor n − 1 per risk in Model 1, or Σ(ni − 1) in Model 2.
  4. 4Estimate the VHM from between-risk variation of the means, and subtract the correction term (EPV ÷ n in Model 1, or (N − 1) ŝ² in Model 2 before dividing).
  5. 5If the VHM estimate is zero or negative, set it to 0. Then Z = 0 for all risks.
  6. 6Calculate k = EPV ÷ VHM and Z (n ÷ (n + k) or Pi ÷ (Pi + k)).
  7. 7Calculate the credibility premium: Z × own mean + (1 − Z) × overall mean. In Model 2, multiply by the exposure if the question asks for a total.
  8. 8State your assumptions and the final answer with sensible rounding.

Quickest way: Table-then-plug method

When to use it: Use it when you must do the full calculation by hand with a small data table, in Paper A or on a calculator.

  1. Write a small table with one row per risk: mean, sum of squared deviations (weighted in Model 2), and total exposure.
  2. Get the EPV by adding the sums of squared deviations and dividing once by the total degrees of freedom: N(n − 1), or Σ(ni − 1) in Model 2.
  3. Compute the between-risk sum Σ (X̄i − X̄)² (weighted by Pi in Model 2) in one pass.
  4. Plug into the VHM formula and check the sign before going further.
  5. Compute k once, then each Z, then each premium. Check that each premium lies between the risk's own mean and the overall mean.

Common mistakes in Empirical Bayes Credibility Theory

  • Dividing by n instead of n − 1 when estimating the EPV in Model 1.

    Students treat each risk's variance like a population variance.

    Fix: Use the sample variance with divisor n − 1 for each risk, then average over the N risks. Equivalent: total squared deviations ÷ N(n − 1).

  • Forgetting to subtract EPV ÷ n from the between-means variance in Model 1.

    The sample variance of the means looks like the VHM already.

    Fix: Sample means also vary through random noise. That noise has variance EPV ÷ n. Subtract it to get the VHM estimate.

  • Using a negative VHM estimate to calculate Z.

    Students follow the formula blindly.

    Fix: A negative result means no evidence of difference between risks. Set the VHM estimate to 0, so Z = 0, and give every risk the overall mean.

  • Using unweighted means in Model 2.

    Students copy the Model 1 steps.

    Fix: In Model 2 use exposure-weighted means X̄iP and X̄PP, and weight the squared deviations by Pij or Pi.

  • Using the number of years in Z in Model 2 instead of the total exposure Pi.

    Z = n ÷ (n + k) is remembered from Model 1.

    Fix: In Model 2, Zi = Pi ÷ (Pi + ŝ² ÷ σ̂²), with Pi the total exposure of risk i.

  • Using the wrong denominator for the VHM in Model 2.

    The term P − ΣPi² ÷ P is easy to misremember.

    Fix: Compute P and ΣPi² first. The denominator is P − ΣPi² ÷ P. With equal Pi this reduces to P(N − 1) ÷ N.

Worked examples

Example 1

Three policyholders each have claim amounts (₹ thousands) for 3 years with equal exposure. Policy 1: 2, 4, 6. Policy 2: 5, 5, 5. Policy 3: 8, 10, 12. Using empirical Bayes credibility Model 1, estimate the credibility premium for Policy 1.

Show the solution
  1. N = 3, n = 3. Means: X̄1 = 4, X̄2 = 5, X̄3 = 10. Overall mean X̄ = 19 ÷ 3 = 6.3333.
  2. Sample variances (divisor 2): Policy 1: (4 + 0 + 4) ÷ 2 = 4. Policy 2: 0. Policy 3: (4 + 0 + 4) ÷ 2 = 4.
  3. EPV estimate = (4 + 0 + 4) ÷ 3 = 8/3 = 2.6667.
  4. Between-means: Σ(X̄i − X̄)² = 5.4444 + 1.7778 + 13.4444 = 20.6667. Divide by N − 1 = 2: 10.3333.
  5. VHM estimate = 10.3333 − (2.6667 ÷ 3) = 10.3333 − 0.8889 = 9.4444 (positive, so use it).
  6. k = 2.6667 ÷ 9.4444 = 24/85 = 0.2824. Z = 3 ÷ (3 + 0.2824) = 0.9140.
  7. Premium for Policy 1 = 0.9140 × 4 + 0.0860 × 6.3333 = 3.6559 + 0.5448 = 4.2007.

Answer: The estimated credibility premium for Policy 1 is about ₹4.20 thousand, that is ₹4,200 (Z ≈ 0.914).

Example 2

Two risks are observed for two years. Risk 1 has exposures 10 and 20 with claims per unit exposure 5 and 8. Risk 2 has exposures 20 and 20 with claims per unit exposure 9 and 11. Using empirical Bayes credibility Model 2, find the credibility premium per unit exposure for Risk 1.

Show the solution
  1. P1 = 30, P2 = 40, P = 70. N = 2.
  2. X̄1P = (10×5 + 20×8) ÷ 30 = 210 ÷ 30 = 7. X̄2P = (20×9 + 20×11) ÷ 40 = 400 ÷ 40 = 10.
  3. X̄PP = (30×7 + 40×10) ÷ 70 = 610 ÷ 70 = 8.7143.
  4. Within-risk sums: Risk 1: 10(5 − 7)² + 20(8 − 7)² = 40 + 20 = 60. Risk 2: 20(9 − 10)² + 20(11 − 10)² = 40. Total = 100.
  5. EPV estimate ŝ² = 100 ÷ [(2 − 1) + (2 − 1)] = 50.
  6. Between-risk sum: 30(7 − 8.7143)² + 40(10 − 8.7143)² = 88.163 + 66.122 = 154.286.
  7. Numerator = 154.286 − (N − 1) × 50 = 104.286. Denominator = 70 − (30² + 40²) ÷ 70 = 70 − 35.714 = 34.286. σ̂² = 104.286 ÷ 34.286 = 3.0417.
  8. k = 50 ÷ 3.0417 = 16.438. Z1 = 30 ÷ (30 + 16.438) = 0.6460.
  9. Premium = 0.6460 × 7 + 0.3540 × 8.7143 = 4.522 + 3.085 = 7.607.

Answer: The credibility premium for Risk 1 is about 7.61 per unit of exposure (Z1 ≈ 0.646).

Exam tips

  • Write the model name and the notation (X̄iP, X̄PP, Pi, P) at the start. Marks are given for method even if arithmetic slips.
  • Always check the sign of the VHM estimate before computing Z. Examiners like to set data where it comes out negative or zero.
  • Keep the divisors straight: N(n − 1) for the EPV, N − 1 for the spread of means. Write them out.
  • Sense-check the answer. Each premium must lie between the risk's own mean and the overall mean, and a larger exposure should give a larger Z.
  • In computer-based papers, compute the means, sums of squares and Z in separate lines or cells so you can show and check each stage.

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

Empirical Bayes Credibility Theory: frequently asked questions

What is the difference between Empirical Bayes Model 1 and Model 2?

Model 1 assumes equal exposure in every year for every risk, so ordinary means are used. Model 2 is the Bühlmann-Straub model, where exposures differ, so you use exposure-weighted means and weighted sums of squares. Model 2 also allows different numbers of years per risk.

How do I estimate the EPV and VHM from claims data?

Estimate the EPV from the variation inside each risk, averaged over all risks. Estimate the VHM from the variation between the risk means, then subtract the part caused by random noise. The formulas differ slightly between Model 1 and Model 2.

What if the VHM estimate is negative?

Set it to zero. This means the data show no real difference between the risks. Then Z = 0 and every risk's premium is the overall mean.

Is the empirical Bayes premium different from the Bühlmann premium?

The premium formula is the same: Z × own mean + (1 − Z) × overall mean. The only difference is that the EPV, VHM and overall mean are estimated from the data rather than known from a prior distribution.