Why Compare Credibility Methods?
Bühlmann, Bühlmann-Straub, and Bayesian methods all combine a risk’s own experience with an experience complement, such as a portfolio mean. They can produce the same premium when their assumptions and parameters align. They can also differ because they use experience differently or describe the risk with different models. A numerical comparison is meaningful only after those assumptions are made explicit.
The examples use hypothetical claim-frequency experience. A risk’s claim count is modeled relative to its exposure, and the premium is a prospective expected number of claims per unit of exposure. This isolates the credibility calculation; converting frequency into a dollar premium would also require an appropriate claim severity assumption.
Three Methods and Their Assumptions
The Bühlmann-Straub model extends this structure to observations with different exposure weights \(w_j\). It assumes conditional independence, a common conditional mean \(m(\Theta)\), and conditional variance \(\operatorname{Var}(X_j\mid\Theta)=v(\Theta)/w_j\). Let \(W=\sum_j w_j\) and let \(\bar X_w=\sum_j w_jX_j/W\). With \(K=\text{EPV}/\text{VHM}\), its premium is \(Z\bar X_w+(1-Z)\mu\), where \(Z=W/(W+K)\). The exposure weights must represent the relative information in the observations.
For a Bayesian calculation, specify a prior distribution for the risk parameter and a likelihood for the data. The resulting posterior distribution represents updated uncertainty about the parameter. A prospective pure premium is the posterior expected loss cost for the future exposure being rated; it is not automatically the same quantity as a posterior mode or a posterior predictive probability.
Under this model, the Gamma prior supplies a precise experience complement. It also shows why exposure matters: counts from twice the exposure contain twice the Poisson information, whereas treating two observations as equally informative would discard that distinction.
Worked Comparison With Equal Exposure
Worked Example: Three Annual Counts and a Shared Complement
A hypothetical risk has one unit of exposure in each of three years, with claim counts \(1,2,2\). The portfolio’s frequency complement is represented by \(\Theta\sim\operatorname{Gamma}(3,2)\), with rate parameter 2. Calculate the Bühlmann, Bühlmann-Straub, and Bayesian expected frequency for one future unit of exposure. Use the Poisson-Gamma model for the Bayesian calculation and its implied credibility parameters for the two credibility calculations.
Step 1: Find the complement and Bühlmann parameters. The prior mean is \(\mu=3/2=1.5\). For a one-unit exposure, \(X=N\) is the count per unit. Conditional on \(\Theta\), its mean and variance are both \(\Theta\). Therefore \(\text{EPV}=E[\Theta]=1.5\), while \(\text{VHM}=\operatorname{Var}(\Theta)=3/2^2=0.75\). Thus \(K=1.5/0.75=2\).
Step 2: Calculate Bühlmann. There are \(n=3\) equal-exposure observations. Their mean is \(\bar X=(1+2+2)/3=5/3\), and \(Z=3/(3+2)=3/5=0.6\). The premium is \(0.6(5/3)+0.4(1.5)=1+0.6=1.6\) claims per exposure unit.
Step 3: Calculate Bühlmann-Straub. Each weight is 1, so \(W=3\) and \(\bar X_w=5/3\). Consequently \(Z=3/(3+2)=0.6\), giving the same premium, \(0.6(5/3)+0.4(1.5)=1.6\).
Step 4: Calculate the Bayesian premium. The total observed count is \(N=5\) over total exposure \(e=3\). The posterior is \(\operatorname{Gamma}(3+5,2+3)=\operatorname{Gamma}(8,5)\). Its mean is \(8/5=1.6\), which is the expected count for one future exposure unit.
Check and method: All three methods give 1.6. This agreement is expected: equal exposures make the Bühlmann and Bühlmann-Straub calculations identical, and the Poisson-Gamma assumptions align the Bayesian posterior mean with the credibility estimate. The common mistake is to use \(5\) as the future frequency: 5 is the historical total count, not the expected count per future exposure unit.
Why the Poisson-Gamma Estimates Match
The equivalence in the first example is not a coincidence. For the Gamma prior with shape \(\alpha\) and rate \(\beta\), the prior mean frequency is \(\mu=\alpha/\beta\), and its variance is \(\alpha/\beta^2\). At exposure \(e\), the observed frequency \(X=N/e\) has conditional mean \(\Theta\) and conditional variance \(\Theta/e\). Hence \(\text{EPV}=\alpha/(\beta e)\), \(\text{VHM}=\alpha/\beta^2\), and \(K=\beta/e\) for observations all having exposure \(e\).
With \(n\) equal exposures, total exposure is \(ne\). The Bühlmann credibility is therefore
The posterior mean can be written as a weighted average of the observed frequency \(N/(ne)\) and prior mean \(\alpha/\beta\):
The first weight is \(Z\); the second is \(1-Z\). This establishes why the Bayesian posterior mean and the credibility estimate match under these assumptions. If the likelihood, prior, exposure treatment, or target quantity changes, the equality need not hold.
Unequal Exposure: Weight the Information
Worked Example: Comparing Three Unequal-Exposure Periods
A hypothetical account has claim counts \(1,4,2\) over exposures \(1,2,1\), respectively. Use the same \(\operatorname{Gamma}(3,2)\) portfolio complement and conditional Poisson model. Calculate the Bühlmann-Straub and Bayesian premiums. Then calculate what ordinary Bühlmann would produce if the three annualized frequencies were incorrectly treated as equally informative.
Step 1: Calculate the Bühlmann-Straub weighted frequency. The annualized observations are \(1/1=1\), \(4/2=2\), and \(2/1=2\). Total exposure is \(W=1+2+1=4\), and total count is \(1+4+2=7\). Therefore \(\bar X_w=7/4=1.75\).
Step 2: Determine the parameters and premium. For this Poisson-Gamma model, the unit-exposure \(\text{EPV}=1.5\), \(\text{VHM}=0.75\), and \(K=2\), as before. The Bühlmann-Straub credibility is \(Z=4/(4+2)=2/3\). The premium is \((2/3)(1.75)+(1/3)(1.5)=7/6+1/2=5/3\), or about 1.667 claims per exposure unit.
Step 3: Verify using Bayes. The posterior is \(\operatorname{Gamma}(3+7,2+4)=\operatorname{Gamma}(10,6)\). Its mean is \(10/6=5/3\), matching the Bühlmann-Straub result.
Step 4: Diagnose the unweighted calculation. If the three annualized frequencies are incorrectly given equal weight, their mean is \((1+2+2)/3=5/3\). Applying ordinary Bühlmann with \(n=3\) and \(K=2\) gives \(Z=3/5\) and premium \((3/5)(5/3)+(2/5)(1.5)=1+0.6=1.6\). That is not a valid equal-information treatment of this Poisson data: the observation based on exposure 2 has twice the exposure of either other observation.
Check and method: Bühlmann-Straub and Bayes both give \(5/3\); the unweighted calculation gives 1.6 because it ignores exposure. The trap is not an arithmetic error but a model mismatch. Ordinary Bühlmann is appropriate only when observations satisfy its equal-information assumptions, or when data have been appropriately transformed and the model supports that treatment.
In practice, an exposure weight should reflect the information represented by the observation, not merely its calendar duration. For a Poisson count model, exposure is the natural weight. In other settings, the appropriate variance relationship may differ. Bühlmann-Straub is not a license to attach arbitrary weights; the assumed conditional variance structure must justify them.
Different Risks, Different Amounts of Experience
Worked Example: Two Risks With Separate Exposure Totals
A hypothetical portfolio uses the prior \(\operatorname{Gamma}(2,1)\). Risk A has 3 claims over 2 exposure units; Risk B has 8 claims over 3 exposure units. Calculate each risk’s prospective frequency for one future exposure unit using the Bayesian model and Bühlmann-Straub. Also calculate the expected combined claims if each risk receives one future exposure unit.
Risk A, Bayesian calculation. The posterior is \(\operatorname{Gamma}(2+3,1+2)=\operatorname{Gamma}(5,3)\), with mean \(5/3\). Its expected future count over one unit is therefore \(5/3\).
Risk A, Bühlmann-Straub check. The complement is \(\mu=2\). Here \(\text{EPV}=2\), \(\text{VHM}=2\), and \(K=1\). The observed frequency is \(3/2=1.5\); \(W=2\), so \(Z=2/(2+1)=2/3\). The premium is \((2/3)(1.5)+(1/3)(2)=1+2/3=5/3\).
Risk B, Bayesian calculation. The posterior is \(\operatorname{Gamma}(2+8,1+3)=\operatorname{Gamma}(10,4)\), with mean \(10/4=2.5\). This is the expected count for one future exposure unit.
Risk B, Bühlmann-Straub check. Its observed frequency is \(8/3\), and \(W=3\). Thus \(Z=3/(3+1)=3/4\). The premium is \((3/4)(8/3)+(1/4)(2)=2+0.5=2.5\).
Combined check: With one future exposure unit for each risk, expected combined claims are \(5/3+2.5=5/3+15/6=25/6\), approximately 4.167. Risk B receives more credibility because it has greater exposure. Its premium is also higher, reflecting its higher observed frequency. The common mistake is to combine the two risks’ experience first and assign the resulting portfolio frequency to both; that would erase the distinct information about each risk.
How to Interpret a Difference
A difference between estimates is informative only after checking that they target the same quantity and use compatible assumptions. For a disciplined comparison, identify the response variable, the exposure basis, the experience complement, and the variance model. Then verify whether the methods use the same data and whether the proposed weights are justified.
| Method | How experience enters | Key condition to check |
|---|---|---|
| Bühlmann | Arithmetic mean of equal-information observations | Conditional observations have the specified common distribution and variance |
| Bühlmann-Straub | Exposure-weighted mean and total weight | Conditional variance scales inversely with the chosen weight |
| Bayesian | Likelihood updates a specified prior | The prior and likelihood reasonably describe risk uncertainty and data generation |
For the Poisson-Gamma model, Bühlmann-Straub and Bayes agree on the posterior mean frequency because both encode the same prior complement and exposure information. A different Bayesian prior can change the complement and the resulting premium. A different likelihood can also change how observed experience is interpreted. Such differences are model-driven, not evidence that one method has made an arithmetic error.
These comparisons are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. In an exam calculation, state the method, write the assumptions that make its formula applicable, and keep frequencies, counts, and exposure units distinct throughout.
Check Your Understanding
For each problem, identify the relevant model and keep exposure units explicit.
- A hypothetical risk has 6 claims over 3 exposure units. Its prior is \(\operatorname{Gamma}(4,2)\), with rate parameter 2. Find the Bayesian posterior mean frequency and the expected claims for 2 future exposure units. (Answer: posterior mean \(10/5=2\) claims per unit; 4 expected claims over 2 units.)
- Under a Bühlmann-Straub model, \(\mu=2\), \(K=3\), and a risk has total weight \(W=6\) and weighted observed frequency \(2.5\). Find its premium. (Answer: \(Z=6/(6+3)=2/3\), so premium \(=(2/3)(2.5)+(1/3)(2)=7/3\), approximately 2.333 per unit.)
- Under ordinary Bühlmann, a risk has \(n=4\) equal-information observations with mean 1.8; the complement is 1.2 and \(K=2\). Find the credibility premium. (Answer: \(Z=4/(4+2)=2/3\), so premium \(=(2/3)(1.8)+(1/3)(1.2)=1.6\).)
- A Poisson-Gamma model has prior \(\operatorname{Gamma}(3,1.5)\). The risk has 5 claims over 2.5 exposure units. Find the posterior mean frequency. (Answer: \((3+5)/(1.5+2.5)=8/4=2\) claims per unit.)
- For a Poisson-Gamma model with prior \(\operatorname{Gamma}(2,2)\), a risk has 0 claims over exposure 1 and 6 claims over exposure 3. Find its Bühlmann-Straub premium and check it using Bayes. (Answer: \(\mu=1\), \(K=2\), \(W=4\), \(\bar X_w=6/4=1.5\), so premium \(=(4/6)(1.5)+(2/6)(1)=4/3\); Bayes gives \((2+6)/(2+4)=4/3\).)