Bühlmann–Straub Credibility for a Four-Class Commercial Auto Book
You are the pricing actuary for a regional commercial auto carrier whose four rating classes were last repriced three years ago. Your analyst has assembled four accident years of experience and fitted a Bühlmann–Straub credibility model with exposure measured in vehicle-years. The variance decomposition returned a pooled within-class process variance of v̂ = 1,764 (pure-premium squared, per vehicle-year) and an estimated variance of hypothetical means of â = 446. Observed experience:
| Class | Exposure (vehicle-years) | Observed pure premium |
|---|---|---|
| A | 12.0 | $210 |
| B | 28.0 | $185 |
| C | 4.0 | $272 |
| D | 36.0 | $196 |
The exposure-weighted book mean is $198.05, and your analyst has already reported Z_A = 0.752. Class C was carved out of Class B only last year to isolate long-haul tractors; it contains 22 risks, one of which produced a single $1.4M bodily injury claim in the latest accident year. The chief underwriter wants the Class C indication by Friday.
1. Complete the credibility calculation for Classes B, C and D and give the indicated pure premium for each class. Show the intermediate quantities.
Read the worked answer
Step 1 - the Bühlmann–Straub constant. k = v̂/â = 1,764/446 ≈ 3.96 vehicle-years. This is the exposure at which a class earns exactly 50% credibility, so roughly four vehicle-years buys half weight.
Step 2 - credibility factors, Z = m/(m + k):
| Class | m | Z = m/(m+3.96) |
|---|---|---|
| A | 12.0 | 0.752 (given) |
| B | 28.0 | 0.876 |
| C | 4.0 | 0.503 |
| D | 36.0 | 0.901 |
Step 3 - credibility estimates, Z·(class mean) + (1−Z)·$198.05:
- B: 0.876(185) + 0.124(198.05) = 162.1 + 24.6 = $186.6
- C: 0.503(272) + 0.497(198.05) = 136.8 + 98.4 = $235.2
- D: 0.901(196) + 0.099(198.05) = 176.6 + 19.6 = $196.2
- (A, for completeness: 0.752(210) + 0.248(198.05) = $207.0)
Reading the result. The estimator is a weighted average of own experience and the book mean, which plays the role of the hypothetical mean μ. Two things a reviewer should flag. First, Class C keeps only half its own experience even though it also has the highest raw pure premium, so the indication ($235) sits far below the observed $272 - the model is asserting that the $272 is largely noise. Second, shrinkage compresses the range of indications from $87 (raw) to roughly $49; every indication moves toward the book, so the ranking survives but the relativities are heavily dampened. If the book mean itself is wrong (for example because C is still contaminated by B's experience), every indication inherits that error.
2. Should the $1.4M claim stay in the calculation? Discuss its effect on v̂, on the credibility factors, and on the indications for the other classes, and say what you would do before quoting the Class C indication.
Read the worked answer
Scale of the problem. Class C has 4.0 vehicle-years and a mean pure premium of $272, so its total observed losses are on the order of $1,100. A single $1.4M claim is roughly 1,000 times the class's expected losses - the class mean of $272 is essentially just that claim divided by exposure, with a slightly negative remainder. The credibility model is being asked to shrink a number that carries almost no information.
Effect on v̂. The claim enters the pooled within-class sum of squares, inflating v̂. Because k = v̂/â, a larger v̂ means a larger k and lower credibility for every class - B and D are pulled toward the book mean more than their own data justifies. Note the partial offset: â is computed as [Σmᵢ(x̄ᵢ − x̄)² − (k−1)v̂]/(m.. − Σmᵢ²/m..), so an inflated v̂ is subtracted there too, which can push â down and destabilise k further. A single large loss therefore contaminates the whole structure, not just Class C.
Before quoting C:
- Verify the claim's reserving status - is $1.4M incurred or paid, is it closed, is subrogation expected?
- Check whether it belongs in C at all (a long-haul tractor risk may have been misclassified).
- Cap losses at a working layer (e.g., $250k) for the credibility calculation and load the excess separately as a large-loss provision - this is the standard treatment and keeps v̂ interpretable.
- Alternatively place this risk in a large-account/schedule-rated tier outside the class plan.
Until then, quote C at the credibility estimate with an explicit caveat, not as a filed rate.
3. The analyst wants to apply the same k to every class. What assumption does that embed, and how would you test or relax it?
Read the worked answer
The assumption. A single k = v̂/â implies that the process variance per vehicle-year is identical across classes (homoscedasticity): Zᵢ = mᵢ/(mᵢ + v/â) uses the same v for all i. Equivalently, it assumes the within-class variability of pure premium is the same for local delivery vans and for long-haul tractors.
Why that is doubtful here. Commercial auto classes differ enormously in severity volatility. Long-haul tractor-trailer experience is dominated by a few catastrophic bodily injury claims, so its process variance per exposure is far higher than a local artisan-contractor class. If v_C ≫ v_A, the correct credibility is Zᵢ = mᵢ/(mᵢ + vᵢ/â), and C should get less weight than the pooled k implies at the same exposure (a Hachemeister-type, heteroscedastic credibility).
How to test. Compute class-specific variance estimates v̂ᵢ from each class's within-class sum of squares, and compare ratios (a variance-ratio or Bartlett-type check). Recognise the power problem: with four accident years and 22 risks in C, v̂_C has a huge standard error, so failing to reject equality is not evidence of equality.
How to relax. The pragmatic answer is a hierarchical model - a Poisson or normal GLMM with a random class effect - which estimates class-specific process variance and lets the data set the shrinkage rather than imposing a single k. With four classes and four years, a full Bayes model would need a strongly regularising hyperprior; a cheaper alternative is to estimate v̂ᵢ by credibility-weighting the class-specific and pooled variance estimates, or to segment the book into two volatility tiers (local vs long-haul) and estimate k within tier.