Repeated Periods in Bühlmann-Straub Credibility

Why Repeated Periods Need Careful Combining

A risk class may have experience recorded over several policy periods, with a different amount of exposure in each. The key is not to give every period equal influence automatically. A period based on many exposures generally provides more information about the class risk level than a period based on few exposures.

The Bühlmann-Straub model handles this by assigning each period an exposure weight. For a class observed over multiple periods, first combine its observations using those weights, then calculate credibility from the class’s total exposure. This tutorial focuses on that repeated-period calculation; it does not address differences in risk mix between classes or adjustments for changing rates over time.

Definition: For one risk class, let $X_j$ be its observed rate or loss cost in period $j$, and let $w_j$ be the exposure weight for that observation. Let $\Theta$ be the class’s underlying risk level. The Bühlmann-Straub moment assumptions are $$ E[X_j\mid \Theta]=\Theta,\qquad \operatorname{Var}(X_j\mid \Theta)=\frac{\sigma^2}{w_j}, $$ with observations conditionally independent given $\Theta$. Across classes, $E[\Theta]=\mu$ and $\operatorname{Var}(\Theta)=\tau^2$. The weights are known and positive for observed periods.

The model says that, conditional on the class risk level, each period’s experience is centered on that same level. Larger $w_j$ means lower conditional variance. In applications, $w_j$ must be a measure of exposure appropriate to the experience being modeled, such as earned exposure units for a rate per exposure unit. Period observations must also be comparable: they should represent the same measure on a consistent basis.

The model does not require a particular probability distribution; it specifies moments and conditional independence. The standard credibility estimate is the best linear estimate under these assumptions. It is also the conditional expectation when suitable normality assumptions are added. In the examples below, no interest rate or mortality assumption applies, and no parametric distribution is assumed.

Weighted Class Experience Across Periods

For $J$ periods with positive exposure, define total exposure $W$ and the exposure-weighted class mean $\bar X_w$ as follows:

W=\sum_{j=1}^{J}w_j,\qquad \bar X_w=\frac{\sum_{j=1}^{J}w_jX_j}{W}.

The numerator is the sum of exposure multiplied by the observed rate in each period. If $X_j$ is a rate per exposure unit, $w_jX_j$ is the corresponding total amount of experience on that basis. Dividing the combined amount by combined exposure gives the overall rate. An observation with zero exposure supplies no information and is omitted; it does not receive a zero-valued rate or count as a period with observed experience.

The credibility factor is based on $W$, not on the number of periods. Define $K=\sigma^2/\tau^2$, the ratio of process variance to variance between risk levels. Then the credibility estimate of the class risk level is:

Z=\frac{W}{W+K},\qquad \widehat{\Theta}=Z\bar X_w+(1-Z)\mu.

Here $\mu$ is the collective mean, or prior mean of class risk levels, and $\widehat{\Theta}$ is the credibility-weighted estimate. The estimate approaches the class’s own weighted mean as total exposure grows. With little exposure, it remains closer to the collective mean.

Why the Total-Exposure Formula Works

The model assumptions explain both the weighted average and the use of total exposure. Conditional independence gives the variance of the weighted mean:

\operatorname{Var}(\bar X_w\mid\Theta) =\frac{1}{W^2}\sum_{j=1}^{J}w_j^2\operatorname{Var}(X_j\mid\Theta) =\frac{1}{W^2}\sum_{j=1}^{J}w_j^2\frac{\sigma^2}{w_j} =\frac{\sigma^2}{W}.

Also, $E[\bar X_w\mid\Theta]=\Theta$, so the weighted mean is an unbiased observation of the class risk level, with error variance $\sigma^2/W$. For a linear credibility estimate $a\bar X_w+(1-a)\mu$, the best linear credibility coefficient is the covariance of $\Theta$ with $\bar X_w$ divided by the variance of $\bar X_w$. Since $E[\bar X_w\mid\Theta]=\Theta$, the covariance is $\tau^2$, while the total variance is $\tau^2+\sigma^2/W$. Therefore:

a=\frac{\tau^2}{\tau^2+\sigma^2/W} =\frac{W}{W+\sigma^2/\tau^2} =\frac{W}{W+K}.

Thus the weighted mean summarizes the repeated experience, and its precision depends on total exposure. This derivation requires the stated conditional-independence and variance assumptions. If period observations are dependent given $\Theta$, the variance calculation and resulting credibility factor need not hold.

Putting the Calculation Into Practice

Use a consistent sequence: check that the observations and weights are defined on the same basis, combine experience with exposure weights, calculate total exposure, and then apply credibility. The steps below keep the class mean and credibility calculation distinct.

Check the model basis.
Confirm that each $X_j$ measures the same quantity and that the $w_j$ values are valid exposure weights. Omit periods with no exposure.

Combine period experience.
Calculate $\sum_j w_jX_j$ and divide by $W=\sum_jw_j$ to obtain $\bar X_w$.

Calculate credibility.
Find $K=\sigma^2/\tau^2$, then use $Z=W/(W+K)$.

Blend class and collective experience.
Calculate $\widehat{\Theta}=Z\bar X_w+(1-Z)\mu$, and check that it lies between $\bar X_w$ and $\mu$.

Worked Example: Three Periods With Different Exposures

A hypothetical specialty equipment class has observed rates of $72, $68, and $80 per exposure unit in three periods, with exposures $100$, $200$, and $100$. The collective mean is $\mu=\$70$, the process variance parameter is $\sigma^2=900$, and the between-class variance is $\tau^2=4.5$, in squared rate units. Use the Bühlmann-Straub moment model; no interest, mortality, or parametric distribution assumption applies.

1. Total exposure. Add the period weights: $$ W=100+200+100=400. $$

2. Exposure-weighted class mean. Multiply each rate by its period exposure, add, and divide by total exposure: $$ \bar X_w=\frac{100(72)+200(68)+100(80)}{400} =\frac{7{,}200+13{,}600+8{,}000}{400} =\frac{28{,}800}{400} =72. $$

3. Credibility. First calculate $K=900/4.5=200$. Therefore: $$ Z=\frac{400}{400+200}=\frac{2}{3}. $$

4. Credibility estimate. Blend the weighted class mean with the collective mean: $$ \widehat{\Theta}=\frac{2}{3}(72)+\frac{1}{3}(70)=48+23.3333\ldots=71.3333\ldots. $$

The estimated class rate is $71.33 per exposure unit, rounded to the nearest cent. It lies between the class mean of $72 and the collective mean of $70, as expected. The method avoids treating the three periods equally despite their different exposure amounts.

Worked Example: Handling a Period With No Exposure

A hypothetical roadside-assistance portfolio has experience of $60 per exposure unit at exposure $50$, no exposure in the next period, and $74 per exposure unit at exposure $150$. The collective mean is $\mu=\$66$, with $\sigma^2=500$ and $\tau^2=5$ in squared rate units. Use the Bühlmann-Straub moment model with conditionally independent periods; no interest, mortality, or parametric distribution assumption applies.

1. Exclude the period without experience. Its weight is zero, so it contributes neither to the numerator nor to the total exposure: $$ W=50+150=200. $$

2. Calculate the weighted mean. $$ \bar X_w=\frac{50(60)+150(74)}{200} =\frac{3{,}000+11{,}100}{200} =\frac{14{,}100}{200} =70.5. $$

3. Calculate the credibility factor. Here $K=500/5=100$, so: $$ Z=\frac{200}{200+100}=\frac{2}{3}. $$

4. Blend with the collective mean. $$ \widehat{\Theta}=\frac{2}{3}(70.5)+\frac{1}{3}(66)=47+22=69. $$

The estimate is $69.00 per exposure unit. A tempting but incorrect calculation would count the empty period as a third observation and average $60, an undefined rate, and $74. There is no observed rate for the zero-exposure period; it contributes no information.

Worked Example: Why Averaging Period Estimates Is Not the Same

A hypothetical commercial vehicle class has rates of $90, $75, and $70 per exposure unit with exposures $80$, $120$, and $200$. Let $\mu=\$72$, $\sigma^2=1{,}600$, and $\tau^2=8$ in squared rate units. Use the Bühlmann-Straub moment model, with conditional independence and no interest, mortality, or parametric distribution assumption.

1. Combine the periods first. Total exposure is: $$ W=80+120+200=400. $$ The weighted mean is: $$ \bar X_w=\frac{80(90)+120(75)+200(70)}{400} =\frac{7{,}200+9{,}000+14{,}000}{400} =\frac{30{,}200}{400} =75.5. $$

2. Calculate the combined credibility estimate. Since $K=1{,}600/8=200$: $$ Z=\frac{400}{400+200}=\frac{2}{3},\qquad \widehat{\Theta}=\frac{2}{3}(75.5)+\frac{1}{3}(72) =50.3333\ldots+24 =74.3333\ldots. $$

The combined estimate is $74.33 per exposure unit, rounded to the nearest cent. To see why the periods should not be treated as equally informative, calculate their separate estimates using the same collective mean and $K$. The period credibility factors are $80/(80+200)=2/7$, $120/(120+200)=3/8$, and $200/(200+200)=1/2$. The corresponding estimates are: $$ 72+\frac{2}{7}(90-72)=77.1429,\qquad 72+\frac{3}{8}(75-72)=73.125,\qquad 72+\frac{1}{2}(70-72)=71. $$

Their unweighted average is $(77.1429+73.125+71)/3=73.7560$, not $74.33. Averaging those period estimates gives each period equal weight after credibility has been applied; it does not reproduce the estimate from all available experience. For one stable class under the stated model, aggregate the exposure-weighted experience and total exposure before applying credibility.

Common Errors and Practical Checks

Repeated-period calculations are easy to misread because the number of observations, the amount of exposure, and the credibility factor all appear in the same problem. The following checks help keep the calculation aligned with the model.

Using a simple average when exposures differ. The class mean is $\sum_j w_jX_j/W$, not $\sum_jX_j/J$, unless all positive period weights are equal.
Using the number of periods in the credibility factor. Use total exposure $W$ in $Z=W/(W+K)$. Two periods with exposure $100$ each provide total exposure $200$, not exposure $2$.
Counting an empty period as a zero rate. Zero exposure is not a zero observed rate. Omit that period from both the weighted numerator and total exposure.
Reversing the two variances. The standard parameter is $K=\sigma^2/\tau^2$. Larger process variance relative to between-class variance means more exposure is needed to achieve the same credibility.
Forgetting the collective complement. The estimate is $Z\bar X_w+(1-Z)\mu$, not $Z\bar X_w+\mu$. The two weights must sum to one.
Combining incomparable experience without adjustment. If period rates are on different bases or no longer represent the same underlying risk level, the common-$\Theta$ model may not apply as stated.

A useful final check is that $0\leq Z\leq1$ for nonnegative exposure and positive $K$. Thus the estimate must lie between the weighted class mean and the collective mean. If it does not, check the complement, the weighted mean, and the inputs to $K$.

Key takeaway: For repeated Bühlmann-Straub experience from one stable class, exposure-weight the period observations, sum the period exposures, and apply credibility to that combined experience. Unequal periods should not automatically receive equal influence, and a period with no exposure contributes no observation.

Check Your Understanding

For each problem, use the Bühlmann-Straub moment model with conditionally independent periods. No interest, mortality, or parametric distribution assumption is needed.

A hypothetical class has rates of $48 and $56 with exposures $40$ and $60$. Its collective mean is $50, and $K=100$. Calculate the exposure-weighted mean and credibility estimate. (Answer: weighted mean $52.80; estimate $51.40 per exposure unit.)
A hypothetical class has rates of $30 and $25 at exposures $20$ and $80$. The collective mean is $28, $\sigma^2=400$, and $\tau^2=4$. Calculate $W$, $Z$, and the credibility estimate. (Answer: $W=100$, $Z=0.5000$, estimate $27.00 per exposure unit.)
A hypothetical class records rates of $40, $34, and $31 at exposures $30$, $50$, and $120$. Its collective mean is $32 and $K=100$. Find the weighted mean and credibility estimate. (Answer: weighted mean $33.10; estimate $32.73 per exposure unit, rounded to the nearest cent.)
A hypothetical class has rates of $55, $65, and $60 at exposures $60$, $140$, and $200$. Its collective mean is $58, $\sigma^2=800$, and $\tau^2=4$. Calculate the credibility estimate. (Answer: $W=400$, weighted mean $61.00, $Z=2/3$, estimate $60.00 per exposure unit.)
A hypothetical class has observed rates of $80 and $68 at exposures $25$ and $75$; a third period has zero exposure. The collective mean is $70, $\sigma^2=600$, and $\tau^2=6$. Determine the weighted mean, credibility factor, and estimate, making clear how the zero-exposure period is treated. (Answer: omit the zero-exposure period; $W=100$, weighted mean $71.00, $Z=0.5000$, estimate $70.50 per exposure unit.)

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