Why Compare Credibility Methods?
An experience-rated premium combines a risk’s own observed experience with a collective complement. Different credibility methods can use the same data and complement but represent the relationship between them differently. A useful comparison asks not only which calculation produces which premium, but also what assumptions make that premium appropriate.
Bühlmann credibility gives the best linear predictor under a squared-error criterion, using the collective mean, expected process variance, and variance of hypothetical means. Bühlmann-Straub extends that framework to unequal exposure. Bayesian rating instead specifies a prior distribution for the risk parameter and a likelihood for the data, then averages the future loss over the posterior distribution. Some models make these approaches coincide; other prior-likelihood combinations produce a genuinely different Bayesian estimate.
The premium in this tutorial is the predicted expected loss cost per unit exposure for a future period. It is not necessarily a final charged premium: expenses, profit provisions, and rating constraints are outside this comparison. In a Bayesian calculation, the target is \(E[Y_{\text{new}}\mid\text{observed data}]\). If \(E[Y_{\text{new}}\mid\Theta]=\Theta\), that target equals the posterior mean of \(\Theta\).
What Each Method Assumes
For the basic Bühlmann model, assume the observations are conditionally independent and identically distributed given \(\Theta\), with finite conditional means and variances. Define \(m=E[\Theta]\), \(\operatorname{EPV}=E[\operatorname{Var}(Y_j\mid\Theta)]\), and \(\operatorname{VHM}=\operatorname{Var}(\Theta)\). Its credibility premium is \(Z\bar Y+(1-Z)m\), where \(Z=n/(n+K)\). The method requires the relevant moments, but it does not require a full prior distribution or a particular likelihood.
Bühlmann-Straub is appropriate when observations have different exposure amounts \(w_j\). In its standard form, conditional means remain \(\Theta\), while conditional variances are \(\operatorname{Var}(Y_j\mid\Theta)=\operatorname{Var}(Y_j\mid\Theta,w_j)=v(\Theta)/w_j\). Thus, a period with more exposure is treated as more informative. Write \(W=\sum_j w_j\) and \(\bar Y_w=\sum_j w_jY_j/W\). With the same definitions of EPV and VHM per unit exposure as in the standard model, the credibility factor is \(Z=W/(W+K)\), and the premium is \(Z\bar Y_w+(1-Z)m\). These formulas do not justify using exposure weights when the data-generating process does not support the corresponding variance relationship.
A Bayesian calculation requires an explicit prior and likelihood. Bayes’ theorem gives the posterior distribution of \(\Theta\) after observing the experience; the premium is the posterior expectation of the future loss. A Bayesian premium may be nonlinear in the observed experience. It should not be assumed to equal a Bühlmann or Bühlmann-Straub premium unless the model and target make that equality hold.
Why the Bühlmann Weight Has This Form
The Bühlmann premium is the best linear predictor of \(\Theta\) from the observed mean \(\bar Y\), under squared-error loss. Conditional independence and the stated equal-period model give \(E[\bar Y]=m\), \(\operatorname{Cov}(\Theta,\bar Y)=\operatorname{VHM}\), and \(\operatorname{Var}(\bar Y)=\operatorname{VHM}+\operatorname{EPV}/n\). The slope of the best linear predictor is covariance divided by variance:
The intercept makes the predictor equal to \(m\) when \(\bar Y=m\), so the predictor is \(Z\bar Y+(1-Z)m\). This derivation explains why credibility increases with the number of equally informative observations and decreases when process variation is large relative to between-risk variation. It also explains the limitation: this is a linear predictor, not generally the full conditional expectation \(E[\Theta\mid\text{data}]\).
Matched Comparisons in Practice
Worked Example: Equal-Exposure Poisson Experience
A hypothetical risk has claim counts of 8, 12, and 14 in three periods, each with exposure 1. Model the counts as conditionally independent Poisson observations with mean \(\Theta\) per unit exposure. The collective prior is \(\Theta\sim\operatorname{Gamma}(20,2)\), using shape and rate parameters. Compare Bühlmann, Bühlmann-Straub, and the exact Bayesian premium. All premiums are expected claims per unit exposure.
Method and model moments. The prior mean is \(m=20/2=10\), and the prior variance is \(20/2^2=5\), so \(\operatorname{VHM}=5\). Conditional Poisson variance equals its mean, giving \(\operatorname{EPV}=E[\Theta]=10\). Therefore \(K=10/5=2\).
Bühlmann. The observed mean is \(\bar Y=(8+12+14)/3=34/3\), and \(Z=3/(3+2)=3/5\). The premium is \( (3/5)(34/3)+(2/5)(10)=34/5+4=10.8\).
Bühlmann-Straub. All three exposures are 1, so \(W=3\) and \(\bar Y_w=34/3\). Its credibility is \(3/(3+2)=3/5\), giving the same premium, 10.8.
Bayesian. For a Gamma prior with shape \(\alpha\) and rate \(\beta\), and Poisson counts totaling \(N\) over total exposure \(W\), the posterior is \(\operatorname{Gamma}(\alpha+N,\beta+W)\). Here \(N=34\) and \(W=3\), so the posterior mean is \((20+34)/(2+3)=54/5=10.8\).
Check and trap. All three methods give 10.80, and the result lies between the collective mean 10 and the observed mean \(11.3333\). A common mistake is to use the Gamma rate parameter 2 as the prior mean; the mean is shape divided by rate, or 10 here. The equality of the three premiums follows from this conjugate Poisson-Gamma model, not from a universal equivalence of the methods.
Worked Example: Unequal Exposure Changes the Weight
A hypothetical portfolio risk has claim counts 8, 18, and 36 over exposures 1, 2, and 4 units, respectively. Use the same conditional Poisson model and collective prior \(\operatorname{Gamma}(20,2)\) as in the preceding comparison. Calculate the Bühlmann-Straub and Bayesian premiums, then show what an ordinary Bühlmann calculation would produce if it incorrectly treated the three period rates as equally informative.
Exposure-weighted experience. Total exposure is \(W=1+2+4=7\), and total claims are \(N=8+18+36=62\). Thus \(\bar Y_w=N/W=62/7\), approximately 8.8571. The model moments remain \(\operatorname{EPV}=10\), \(\operatorname{VHM}=5\), and \(K=2\).
Bühlmann-Straub. The credibility is \(Z=7/(7+2)=7/9\). Therefore the premium is \((7/9)(62/7)+(2/9)(10)=62/9+20/9=82/9\), or 9.1111.
Bayesian. The posterior is \(\operatorname{Gamma}(20+62,2+7)=\operatorname{Gamma}(82,9)\), so its mean is \(82/9=9.1111\). This agrees with Bühlmann-Straub because the same Poisson-Gamma model and future expected-count target apply.
Unequal-weight trap. The three period rates are \(8/1=8\), \(18/2=9\), and \(36/4=9\). An ordinary Bühlmann calculation that averages these equally gets \(26/3=8.6667\). It then uses \(Z=3/(3+2)=3/5\), producing \((3/5)(26/3)+(2/5)(10)=9.2\). This is not the exposure-adjusted result: it gives the one-unit period as much influence as the four-unit period. The check is that the valid weighted premium, 9.1111, lies between the complement 10 and weighted rate 8.8571.
Worked Example: A Discrete Bayesian Prior Can Give a Different Premium
A hypothetical risk’s annual loss cost \(Y\), measured in thousands of dollars, is conditionally normal: \(Y\mid\Theta\sim N(\Theta,4)\), where the second argument is the variance. The prior assigns probability \(1/2\) to each of \(\Theta=6\) and \(\Theta=14\). Two independent annual observations are both 12. Compare the Bühlmann and exact Bayesian predictions for the next annual loss cost.
Bühlmann moments. The complement is \(m=(6+14)/2=10\). The within-risk variance is 4, so \(\operatorname{EPV}=4\). The prior variance is \(((6-10)^2+(14-10)^2)/2=16\), so \(\operatorname{VHM}=16\), \(K=4/16=1/4\), and \(Z=2/(2+1/4)=8/9\). The sample mean is 12, giving premium \((8/9)(12)+(1/9)(10)=106/9=11.7778\) thousand dollars.
Bayesian posterior. For one observation of 12, the likelihood ratio of \(\Theta=14\) to \(\Theta=6\) is \(\exp(((12-6)^2-(12-14)^2)/(2\cdot4))=\exp(4)\). The two conditionally independent observations therefore give likelihood ratio \(\exp(8)\). With equal prior probabilities, the posterior probability of \(\Theta=14\) is \(\exp(8)/(1+\exp(8))\), approximately 0.9996646. The posterior mean is \(6+8(0.9996646)=13.9973\) thousand dollars.
Check and trap. The Bayesian prediction is approximately $13,997.30, while the Bühlmann prediction is approximately $11,777.78. Both use the same complement, 10 thousand dollars, and the same observed data. The discrete prior places probability only on two possible risk levels, and the observations make the higher level overwhelmingly likely. Bühlmann uses a linear predictor and does not make that full posterior update. Treating the Bayesian result as the posterior mean of the risk parameter is valid here because \(E[Y_{\text{new}}\mid\Theta]=\Theta\). The posterior probabilities sum to 1, and the Bayesian mean lies between the two possible risk levels.
Reading the Comparison Correctly
| Method | Information used | What to check |
|---|---|---|
| Bühlmann | Equal-information observations, collective mean, EPV, and VHM | Are the observations appropriately treated as equally informative? |
| Bühlmann-Straub | Exposure-weighted experience, collective mean, EPV, and VHM | Does conditional variance decrease in proportion to exposure? |
| Bayesian | Prior distribution, likelihood, observed data, and future-loss target | Does the chosen prior and likelihood represent the risk and data? |
Agreement is informative only when the models and targets have been matched. In a Poisson-Gamma frequency model, the posterior mean and Bühlmann-Straub premium agree because the posterior mean is linear in total claims and exposure. With another prior, likelihood, or prediction target, that algebra may not hold. Conversely, a difference does not by itself show that one method is wrong: it can reflect a different model or the fact that Bayesian updating is not restricted to a linear estimate.
A reliable exam comparison states the risk parameter, complement, exposure basis, variance assumptions, and target premium before substituting values. Then check that the credibility factor is between 0 and 1 and that a credibility-weighted premium lies between the collective complement and the applicable experience estimate. That range check is a useful error detector, but it does not replace checking whether the model assumptions fit the data.
Check Your Understanding
For each problem, identify the relevant weighting or model assumption before calculating.
- A hypothetical risk has two equally exposed annual loss costs of 1,500 and 900 dollars. The collective mean is $1,200, EPV is 90,000 dollars squared, and VHM is 30,000 dollars squared. Calculate the Bühlmann premium. (Answer: $1,320.00)
- Under a Bühlmann-Straub Poisson model, the collective rate is 5 claims per exposure unit, EPV is 5, and VHM is 10. A risk has 4 claims over exposure 1 and 18 claims over exposure 3. Calculate the credibility-weighted claim rate. (Answer: \(49/9=5.4444\) claims per exposure unit)
- A hypothetical claim probability has a \(\operatorname{Beta}(3,9)\) prior. The risk has 4 claims in 8 independent exposure units, each with a binary claim outcome. Find the Bayesian posterior mean claim probability. (Answer: \(7/20=0.3500\))
- A risk parameter is equally likely to be 4 or 10. An observation has distribution \(Y\mid\Theta\sim N(\Theta,9)\), and the observed value is 8. Calculate the exact Bayesian posterior mean and the one-observation Bühlmann premium, using the prior moments and the conditional variance. (Answer: Bayesian \(4+6e^{2/3}/(1+e^{2/3})\approx7.9648\); Bühlmann \(7.5000\))
- Claim counts follow a Poisson model with a Gamma prior. The collective mean is 12 claims per exposure unit and the VHM is 6; a risk records 20 claims over exposure 2 and 42 claims over exposure 3. Calculate the Bühlmann-Straub premium and the matching Bayesian posterior mean. (Answer: \(86/7=12.2857\) claims per exposure unit)