Worked Bühlmann-Straub Example With Unequal Exposures

Why Unequal Exposures Matter

In a Bühlmann-Straub portfolio, each risk contributes experience at a level determined by its exposure. A rate based on four exposure units should generally carry more credibility than one based on one exposure unit, even if both rates come from the same number of observation periods. The calculation therefore uses exposure-weighted means and makes the credibility factor depend on total exposure.

This worked treatment follows the Bühlmann-Straub risk model and the earlier material on weighted experience means, structural parameter estimation, and credibility premiums. The examples use hypothetical portfolios. Observed amounts are stated as rates per exposure unit, so they can be compared across periods with different exposures.

Model and assumptions: For risk $i$ in period $j$, let $X_{ij}$ be the observed rate and $w_{ij}>0$ its exposure. Conditional on the risk’s hypothetical mean $m_i$, assume $E[X_{ij}\mid m_i]=m_i$ and $\operatorname{Var}(X_{ij}\mid m_i)=v/w_{ij}$, where $v$ is the expected process variance (EPV). Conditional observation errors are uncorrelated across periods and risks. Across risks, assume the $m_i$ are independent and identically distributed, with $E[m_i]=\mu$ and $\operatorname{Var}(m_i)=a$, where $a$ is the variance of hypothetical means (VHM). More generally, the standard VHM estimator below requires distinct risk means to be uncorrelated, as well as uncorrelated observation errors across risks. Exposures are treated as fixed. If risk means or observation errors are correlated across risks, the standard between-risk expectation and estimator need not hold.

Portfolio Quantities and Estimators

For risk $i$, let $n_i$ be its number of observations, $w_i=\sum_j w_{ij}$ its total exposure, and $\bar X_i$ its exposure-weighted experience mean. The portfolio mean $\bar X$ weights each risk mean by that risk’s total exposure. Write $I$ for the number of risks, $N=\sum_i n_i$, and $W=\sum_i w_i$.

\bar X_i=\frac{\sum_{j=1}^{n_i}w_{ij}X_{ij}}{w_i}, \qquad \bar X=\frac{\sum_{i=1}^{I}w_i\bar X_i}{W}

The within-risk weighted sum of squares estimates process variation. Provided the observations within a risk are conditionally uncorrelated as specified by the model, the usual moment estimate is

\hat v= \frac{\sum_{i=1}^{I}\sum_{j=1}^{n_i}w_{ij}(X_{ij}-\bar X_i)^2}{N-I}.

For the between-risk calculation, define

B=\sum_{i=1}^{I}w_i(\bar X_i-\bar X)^2, \qquad D=W-\frac{\sum_{i=1}^{I}w_i^2}{W}.

When $D>0$, the moment estimate of VHM is

\hat a=\frac{B-\hat v(I-1)}{D}.

The term $\hat v(I-1)$ removes the expected contribution of process noise from the observed differences between risk means. Omitting it would attribute some within-risk randomness to genuine differences between risks.

Justification: The weighted mean for risk $i$ has conditional error variance $v/w_i$. Expressing $B$ as $\sum_i w_i\bar X_i^2-W\bar X^2$, the independent risk means contribute $aD$ in expectation. The independent mean errors contribute $Iv-v=v(I-1)$: the first term is the sum of $I$ weighted error variances, and subtracting $W$ times the variance of the weighted portfolio mean removes $v$. Therefore $E[B]=aD+v(I-1)$, which gives the stated moment estimate after replacing $a$ and $v$ by estimates. This expectation relies on the cross-risk uncorrelatedness assumptions in the model box.

The credibility constant and risk-specific credibility factor are

\hat K=\frac{\hat v}{\hat a}, \qquad Z_i=\frac{w_i}{w_i+\hat K}.

When $\hat a>0$, the Bühlmann-Straub premium rate for risk $i$ is $Z_i\bar X_i+(1-Z_i)\bar X$. If a future period has exposure $q$, the corresponding expected premium amount is $q$ times this rate. The factor $Z_i$ depends on the risk’s total past exposure, not simply on how many periods were observed.

Worked Portfolio: From Raw Data to Premiums

The table gives hypothetical rates for three risks. Every rate is in dollars per exposure unit. Exposures differ between periods, and the total exposures also differ between risks.

Worked Example: Estimating Parameters and Premiums With Unequal Exposures

Calculate each risk’s weighted mean, $\hat v$, $\hat a$, the credibility constant, and each Bühlmann-Straub premium rate. State the model assumptions that make the between-risk estimator appropriate.

Risk	Period	Exposure $w_{ij}$	Rate $X_{ij}$
A	1	1	$40
A	2	2	$50
A	3	1	$60
B	1	1	$55
B	2	1	$75
B	3	1	$65
C	1	1	$70
C	2	2	$80
C	3	1	$90

Method and assumptions. Use the unequal-exposure Bühlmann-Straub moment estimators. Assume fixed exposures, conditionally uncorrelated observation errors with variance $v/w_{ij}$, and independent, identically distributed risk means. These assumptions justify both the within-risk estimate and the between-risk expectation.

Step 1: Find the risk means and total exposures. For A, $w_A=1+2+1=4$, and its weighted rate total is $1(40)+2(50)+1(60)=200$, so $\bar X_A=200/4=50$. For B, $w_B=3$ and its weighted total is $55+75+65=195$, so $\bar X_B=195/3=65$. For C, $w_C=4$, its weighted total is $70+2(80)+90=320$, and $\bar X_C=320/4=80$. Thus $W=4+3+4=11$, and $\bar X=[4(50)+3(65)+4(80)]/11=715/11=65$.

Step 2: Estimate EPV. For A the weighted squared deviations from 50 are $1(40-50)^2+2(50-50)^2+1(60-50)^2=100+0+100=200$. For B they are $1(55-65)^2+1(75-65)^2+1(65-65)^2=100+100+0=200$. For C they are $1(70-80)^2+2(80-80)^2+1(90-80)^2=100+0+100=200$. The total is $600$. There are $N=9$ observations and $I=3$ risks, so the degrees of freedom are $9-3=6$, giving $\hat v=600/6=100$.

Step 3: Estimate VHM. The between-risk sum of squares is $B=4(50-65)^2+3(65-65)^2+4(80-65)^2=4(225)+0+4(225)=1800$. Also, $D=11-(4^2+3^2+4^2)/11=11-41/11=80/11$. Subtract the expected process contribution, $100(3-1)=200$, before dividing: $\hat a=(1800-200)/(80/11)=1600(11/80)=220$.

Step 4: Calculate credibility and premiums. The credibility constant is $\hat K=100/220=5/11$ exposure units. For A and C, with exposure 4, $Z=4/(4+5/11)=44/49$. For B, with exposure 3, $Z=3/(3+5/11)=33/38$. The premium rates are $A:(44/49)(50)+(5/49)(65)=2525/49=\$51.53$; $B:(33/38)(65)+(5/38)(65)=\$65.00$; and $C:(44/49)(80)+(5/49)(65)=3845/49=\$78.47$. Dollar results are rounded to the nearest cent.

Check and common mistake avoided. Each $Z_i$ lies between 0 and 1, and each premium lies between that risk’s experience mean and the portfolio mean of $65. The common mistake is to use the unweighted average of the three risk means as the collective mean. Here it happens to equal $65, but that coincidence should not be relied on: the correct portfolio mean is exposure-weighted. Another mistake is to omit the process correction in $\hat a$, which would incorrectly give $1800/(80/11)=247.5$, not 220.

Using Summary Statistics for a New Risk

A portfolio calculation may provide within-risk and between-risk summaries rather than every observation. The same estimators apply, provided the summaries were formed with the stated exposure weights and the model assumptions hold.

Worked Example: Applying Estimated Credibility to a Future Exposure

A hypothetical portfolio has three risks with total exposures $1,2,3$, respectively, and observed weighted means $10,20,30$ dollars per exposure unit. The within-risk weighted sum of squares is 60 across three within-risk degrees of freedom. Calculate the structural estimates and the expected premium amount for a fourth risk whose observed rate is $26, experience exposure is 2.5 units, and next-period exposure is 1.5 units.

Method and assumptions. Apply the same Bühlmann-Straub moment method, assuming the exposure-weighted observations satisfy the model and distinct risk means and observation errors are uncorrelated.

Step 1: Estimate EPV. The within-risk sum of squares is 60 and the degrees of freedom are 3, so $\hat v=60/3=20$.

Step 2: Calculate the portfolio mean and between-risk terms. Total exposure is $W=1+2+3=6$, and $\bar X=[1(10)+2(20)+3(30)]/6=140/6=70/3$. Then $B=1(10-70/3)^2+2(20-70/3)^2+3(30-70/3)^2$ $=1600/9+200/9+1200/9=3000/9=1000/3$. Also $D=6-(1^2+2^2+3^2)/6=6-14/6=11/3$. Thus $\hat a=[1000/3-20(3-1)]/(11/3)=(1000/3-40)/(11/3)=80$.

Step 3: Price the risk and future period. The credibility constant is $\hat K=20/80=0.25$. For experience exposure 2.5, $Z=2.5/(2.5+0.25)=10/11$. The indicated premium rate is $(10/11)(26)+(1/11)(70/3)=280/11=\$25.45$ per exposure unit, rounded to the nearest cent. For next-period exposure 1.5, the expected premium amount is $1.5(280/11)=420/11=\$38.18$, rounded to the nearest cent.

Check and common mistake avoided. The premium rate is between the risk’s observed rate of $26 and the collective mean of $70/3$, as required by a credibility-weighted average. Multiplying by the future exposure gives the premium amount; using 2.5 instead would price the historical exposure, not the stated future period.

Why Cross-Risk Assumptions Are Essential

The formula $E[B]=aD+v(I-1)$ is not implied just by specifying the marginal mean and variance of each risk mean. Correlation between distinct $m_i$ changes the expected between-risk variation. Consequently, the standard VHM estimator is not justified unless distinct risk means are uncorrelated; the standard derivation also requires uncorrelated observation errors across risks.

Worked Example: Detecting a Cross-Risk Correlation Trap

Two hypothetical risks each have exposure $w=2$. Their risk means have common variance $a=25$ and covariance $c=10$. Their aggregated observed means have independent process errors, each with variance $v/w$, where $v=4$. Calculate the expected between-risk statistic and compare it with the value obtained by wrongly assuming independent risk means.

Method and assumptions. For two equal exposures, the statistic is $B=2(\bar X_1-\bar X)^2+2(\bar X_2-\bar X)^2=(w/2)(\bar X_1-\bar X_2)^2$. The process errors are independent, but the risk means are correlated as specified.

Step 1: Find the difference variance. The true-mean difference has variance $\operatorname{Var}(m_1-m_2)=a+a-2c=25+25-20=30$. Each observed mean has process-error variance $v/w=4/2=2$; the two independent errors therefore contribute $2+2=4$ to the variance of the observed difference. Hence $E[(\bar X_1-\bar X_2)^2]=30+4=34$.

Step 2: Calculate the expected statistic. Since $w/2=1$, $E[B]=1(34)=34$. Equivalently, the true-means contribution is $w(a-c)=2(25-10)=30$, and the process contribution for two risks is $v=4$, totaling 34.

Step 3: Compare with the incorrect calculation. If independence were wrongly assumed, the formula would use $wa+v=2(25)+4=54$. That is 20 too high because it fails to subtract the covariance contribution $wc=2(10)=20$. The standard estimator must not be used as though the risk means were independent in this situation.

Check and common mistake avoided. Positive covariance makes the two risk means move together, reducing their expected difference; the result 34 is therefore less than the incorrect independent-means result 54. Marginal assumptions $E[m_i]=\mu$ and $\operatorname{Var}(m_i)=a$ alone do not settle this issue.

Interpreting the Calculation

Unequal exposures affect the calculation in two distinct ways. First, each risk’s mean and the collective mean use exposure weights. Second, the credibility factor uses the risk’s total exposure $w_i$. A risk can therefore have a different credibility factor from another risk even when both have the same number of observed periods.

The VHM estimate is a moment estimate, so a particular data set can produce a negative $\hat a$. That result signals that the observed between-risk variation, after process variation is removed, is not positive in the sample; it does not change the derivation or justify silently replacing the estimate with zero. Any prescribed practical remedy must be distinguished from the estimator itself.

A reliable exam calculation keeps the exposure and variance roles separate: use $w_{ij}$ in weighted means and within-risk sums, use $w_i$ in between-risk sums and credibility, and subtract $\hat v(I-1)$ before estimating VHM. Then check that parameter estimates, credibility factors, and premium rates are consistent with their definitions.

Check Your Understanding

Use the stated Bühlmann-Straub assumptions for each question, and retain full precision until the final reported amount.

A risk has rates 12 and 18 with exposures 1 and 3. Find its weighted mean. (Answer: 16.5)
Three risks have total exposures 1, 2, and 1, with weighted means 10, 20, and 30. Find the exposure-weighted portfolio mean. (Answer: 20)
A portfolio has $I=4$ risks and a total within-risk weighted sum of squares of 450 over 9 degrees of freedom. Find $\hat v$. (Answer: 50)
A two-risk portfolio has exposures 2 and 2, $\bar X_1=10$, and $\bar X_2=20$. If $\hat v=4$, calculate $B$, $D$, and $\hat a$. (Answer: $B=100$, $D=2$, $\hat a=46$)
A risk has total experience exposure 3 and experience mean 40. The collective mean is 32, $\hat v=18$, and $\hat a=72$. Find its credibility factor and premium rate. (Answer: $Z=12/13=0.9231$, premium rate $=\$39.38$ per exposure unit)

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