A Worked Bühlmann-Straub Example

Why Unequal Exposures Need Unequal Weights

In a credibility calculation, a class with more exposure generally provides more information about its underlying expected rate. When observations have unequal exposure, the class experience should reflect that difference: an observation based on six exposure units should contribute more to the class average than one based on a single unit. The Bühlmann-Straub model provides a way to combine that weighted experience with the collective mean.

This tutorial applies the Bühlmann-Straub formulas to numerical data. The focus is the calculation and interpretation of class estimates when the exposure weights differ; the model's variance parameters and the collective mean are supplied as inputs.

Definition: In the Bühlmann-Straub model, let $\Theta_i$ denote the unknown expected rate for class $i$, and let $X_{ij}$ be the observed rate for that class in period $j$, with exposure weight $w_{ij}>0$. The moment assumptions are $E[\Theta_i]=m$, $\operatorname{Var}(\Theta_i)=a$, $E[X_{ij}\mid\Theta_i]=\Theta_i$, and $\operatorname{Var}(X_{ij}\mid\Theta_i)=\phi/w_{ij}$. Observations for a class are assumed conditionally uncorrelated given $\Theta_i$. Here $m$ is the collective mean, $a$ is the variance of hypothetical means (VHM), and $\phi$ is the expected process variance (EPV) for one unit of exposure. No particular parametric distribution is required by these moment assumptions.

For a class with observations across periods, its total exposure is $w_i=\sum_j w_{ij}$. Its exposure-weighted average rate is the total observed amount divided by total exposure. If $X_{ij}$ is already a rate per exposure unit, the corresponding observed amount is $w_{ij}X_{ij}$.

\bar{X}_i=\frac{\sum_j w_{ij}X_{ij}}{w_i}, \qquad K=\frac{\phi}{a}, \qquad Z_i=\frac{w_i}{w_i+K}, \qquad \widehat{\Theta}_i=Z_i\bar{X}_i+(1-Z_i)m.

The constant $K$ is the credibility parameter, expressed in exposure units. The factor $Z_i$ is the credibility assigned to the class's own experience; the remaining weight, $1-Z_i$, is assigned to the collective mean. These formulas presume $a>0$, $\phi>0$, and positive exposure. They do not apply as written if the stated model assumptions are unsuitable, such as when the observations have material dependence not represented by the model.

Why the Credibility Factor Has This Form

The credibility factor can be justified by considering the mean squared prediction error of a linear blend. For a single class, write its exposure-weighted average as $\bar{X}$, with total exposure $w$, and consider the estimator $Z\bar{X}+(1-Z)m$. Under the model, conditional independence and the conditional variances imply $\operatorname{Var}(\bar{X}\mid\Theta)=\phi/w$. Its unconditional variance is therefore $a+\phi/w$, while $\operatorname{Cov}(\Theta,\bar{X})=a$.

The estimation error is $Z(\bar{X}-\Theta)+(1-Z)(m-\Theta)$. The first term contains process variation, and the second reflects how far the class's underlying mean is from the collective mean. Using the stated moments, its mean squared error is

\begin{aligned} E\left[\left\{Z\bar{X}+(1-Z)m-\Theta\right\}^2\right] &=Z^2\frac{\phi}{w}+(1-Z)^2a. \end{aligned}

To minimize this expression, differentiate with respect to $Z$ and set the derivative equal to zero:

2Z\frac{\phi}{w}-2(1-Z)a=0 \quad\Longrightarrow\quad Z\left(\frac{\phi}{w}+a\right)=a \quad\Longrightarrow\quad Z=\frac{a}{a+\phi/w} =\frac{w}{w+\phi/a}.

Thus, the credibility factor is $w/(w+K)$. This derivation also explains the direction of the effect: increasing exposure reduces the conditional process variance $\phi/w$, so the experience average receives more weight.

Worked Calculations

Worked Example: Two Classes With Different Exposure Patterns

A hypothetical insurer has two classes, each observed in two periods. The observed rates and exposure weights are shown below. For this illustration, take $m=\$120$ per exposure unit, $a=400$ dollars squared, and $\phi=1{,}600$ dollars squared per exposure unit. The model specifies only the conditional moments above; no parametric distribution, interest rate, or mortality assumption is used.

Class	Period	Exposure $w_{ij}$	Observed rate $X_{ij}$
A	1	1	$100
A	2	3	$140
B	1	2	$90
B	2	6	$150

Step 1: Calculate the credibility parameter. The supplied variance components give

K=\frac{\phi}{a}=\frac{1{,}600}{400}=4.

Step 2: Calculate Class A's weighted average and credibility. Its total exposure is $w_A=1+3=4$. The exposure-weighted amount is $1(100)+3(140)=100+420=520$, so

\bar{X}_A=\frac{520}{4}=130, \qquad Z_A=\frac{4}{4+4}=0.5000.

The estimate is $0.5000(130)+0.5000(120)=65+60=\$125$ per exposure unit.

Step 3: Calculate Class B's weighted average and credibility. Its total exposure is $w_B=2+6=8$, and its weighted amount is $2(90)+6(150)=180+900=1{,}080$. Therefore,

\bar{X}_B=\frac{1{,}080}{8}=135, \qquad Z_B=\frac{8}{8+4}=\frac{2}{3}=0.6667.

The estimate is $\frac{2}{3}(135)+\frac{1}{3}(120)=90+40=\$130$ per exposure unit. Class B receives more credibility because its total exposure is larger. As a check, each estimate lies between its class average and $m=\$120$: Class A's estimate is halfway between $130 and $120, and Class B's is two-thirds of the way from $120 to $135.

Method and mistake avoided. This is a Bühlmann-Straub class estimate using exposure-weighted averages. An ordinary average of the period rates would give the wrong class experience whenever the period exposures differ.

Worked Example: Combining Three Unequal-Exposure Periods

A hypothetical specialty insurer observes one class over three periods. The exposure weights are 2, 5, and 3 units, and the corresponding rates are $70, $100, and $90 per unit. Use $m=\$80$, $a=225$ dollars squared, and $\phi=900$ dollars squared per exposure unit. The conditional moment model is used, with no parametric distribution, interest rate, or mortality assumption.

Step 1: Find $K$ and total exposure.

K=\frac{900}{225}=4, \qquad w=2+5+3=10.

Step 2: Calculate the weighted experience. The weighted amount is $2(70)+5(100)+3(90)=140+500+270=910$. Hence

\bar{X}=\frac{910}{10}=91.

Step 3: Calculate credibility and the estimate.

Z=\frac{10}{10+4}=\frac{5}{7}=0.7143.

Using the exact fraction before rounding, the estimate is

\widehat{\Theta} =\frac{5}{7}(91)+\frac{2}{7}(80) =\frac{455+160}{7} =\frac{615}{7} =87.8571\ldots.

Rounded to the nearest cent, the estimated rate is $87.86 per exposure unit. The result is between the collective mean of $80 and the class weighted average of $91, as it must be for a credibility factor between zero and one. This calculation avoids rounding $Z$ too early: using the exact fraction gives the final amount directly and prevents avoidable intermediate rounding error.

Worked Example: Detecting an Ordinary-Average Error

A hypothetical class has rates of $760 and $940 at exposure weights 2 and 8, respectively. The collective mean is $m=\$800$, the VHM is $a=400$ dollars squared, and the EPV is $\phi=1{,}600$ dollars squared per exposure unit. Use the Bühlmann-Straub moment model; no parametric distribution, interest rate, or mortality assumption is needed.

Step 1: Calculate the class weighted average. Total exposure is $w=2+8=10$, and weighted amount is $2(760)+8(940)=1{,}520+7{,}520=9{,}040$. Thus

\bar{X}=\frac{9{,}040}{10}=904.

Step 2: Calculate the credibility factor.

K=\frac{1{,}600}{400}=4, \qquad Z=\frac{10}{10+4}=\frac{5}{7}=0.7143.

Step 3: Calculate the correct estimate.

\widehat{\Theta} =\frac{5}{7}(904)+\frac{2}{7}(800) =\frac{4{,}520+1{,}600}{7} =\frac{6{,}120}{7} =874.2857\ldots.

The correct estimate, rounded to the nearest cent, is $874.29 per exposure unit.

Common mistake and correction. A trainee might take the ordinary average $(760+940)/2=\$850$ and blend it with the collective mean: $\frac{5}{7}(850)+\frac{2}{7}(800)=\frac{4{,}250+1{,}600}{7}=\$835.71$. That is not the Bühlmann-Straub estimate because the ordinary average gives the low-exposure observation the same weight as the high-exposure observation. It is $\$874.29-\$835.71=\$38.58$ too low after rounding to cents. The weighted average, not the ordinary average, is required by the unequal-exposure model.

Interpreting and Checking the Result

A class estimate is a blend, not a replacement for either input. With positive $a$, $\phi$, and $w$, the factor satisfies $0<Z<1$. Therefore, $\widehat{\Theta}_i$ lies between $\bar{X}_i$ and $m$. If the class experience is above the collective mean, the estimate is pulled downward toward $m$; if it is below the collective mean, the estimate is pulled upward toward $m$.

The factor depends on total exposure and the ratio of the variance components. For fixed $a$ and $\phi$, a larger total exposure makes $Z_i$ larger. For fixed exposure, a larger process variance $\phi$ lowers credibility, while a larger VHM $a$ raises it. These statements describe the implications of the supplied model inputs; they do not determine whether those inputs have been estimated appropriately.

Compute total exposure.
Add the exposure weights for all observations belonging to the class.

Compute weighted experience.
Multiply each observed rate by its exposure, add the products, and divide by total exposure.

Compute the credibility parameter and factor.
Use $K=\phi/a$, then $Z_i=w_i/(w_i+K)$.

Blend and check.
Calculate $Z_i\bar{X}_i+(1-Z_i)m$, then verify the estimate lies between the class average and the collective mean.

A useful arithmetic check is to confirm that the weighted-average numerator uses every rate multiplied by its own exposure. A second check is that the two credibility weights sum to one: $Z_i+(1-Z_i)=1$. Finally, preserve unrounded values through the calculation and round the monetary estimate only at the final step.

Key takeaway: In Bühlmann-Straub credibility, unequal exposures enter twice: they determine the class's weighted average experience and its total-exposure credibility factor. Use both pieces consistently, then blend the class average with the collective mean.

Check Your Understanding

For each problem, use the Bühlmann-Straub moment model, calculate with unrounded values, and report monetary estimates to the nearest cent when needed.

A hypothetical class has one observation with exposure 6 and rate $58. Given $m=\$50$, $a=100$ dollars squared, and $\phi=400$ dollars squared per exposure unit, calculate $K$, $Z$, and the class estimate. (Answer: $K=4$, $Z=0.6000$, estimate $54.80)
A class has rates of $30 and $45 at exposures 1 and 4. Given $m=\$40$, $a=64$ dollars squared, and $\phi=256$ dollars squared per exposure unit, find the weighted average, credibility factor, and estimate. (Answer: weighted average $42, $Z=5/9=0.5556$, estimate $41.11)
A class has two period rates, $180 at exposure 3 and $240 at exposure 5. Given $m=\$200$, $a=900$ dollars squared, and $\phi=1{,}800$ dollars squared per exposure unit, calculate the class estimate. (Answer: weighted average $217.50, $Z=0.8000$, estimate $214.00)
A hypothetical class has rates of $80 and $120 at exposures 1 and 3. Given $m=\$100$, $a=225$ dollars squared, and $\phi=900$ dollars squared per exposure unit, calculate $K$, the weighted average, $Z$, and the estimate. (Answer: $K=4$, weighted average $110, $Z=0.5000$, estimate $105.00)
A class has rates of $60 and $100 at exposures 2 and 6. Given $m=\$70$, $a=100$ dollars squared, and $\phi=400$ dollars squared per exposure unit, calculate the correct estimate and the estimate obtained if the ordinary average is incorrectly used for experience. (Answer: weighted average $90, $Z=0.6667$, correct estimate $83.33; ordinary-average estimate $76.67, which is $6.66 too low to the nearest cent)

What You'll Learn