Why Exposure-Weighted Experience Matters
A risk’s experience may be recorded over several periods, but the periods need not represent the same amount of business. One year might cover 40 insured units and another 400. If the observations are loss rates per insured unit, averaging those rates as if each period had equal exposure gives the small period the same influence as the large one. In the Bühlmann-Straub model, the experience mean must reflect how much exposure each observation represents.
This tutorial focuses on two related quantities: the exposure-weighted mean of period-level observations and the total exposure associated with them. The first summarizes the observed rate; the second records how much experience supports that summary. Keep the observation’s units clear: a period’s aggregate loss is not itself a loss rate.
The exposure units must be comparable across periods. Depending on the problem, a unit might be a vehicle-year, property-year, or another specified measure. Do not add unlike quantities, such as policies in one period and policy-years in another, unless the question gives a conversion that puts them on a common basis.
How the Weighting Works
Each period contributes \(w_{ij}X_{ij}\) to the numerator, its implied aggregate loss, and \(w_{ij}\) to the denominator, its exposure. A period with twice the exposure therefore has twice the influence on the mean, all else equal. If all exposures are equal and positive, the weighted mean reduces to the ordinary arithmetic mean.
The weighted mean is not the same as the average of period aggregate losses. Aggregate losses have money units; the weighted mean has loss-per-exposure-unit units. Dividing aggregate loss by exposure is what returns the result to a rate basis.
Determine whether each \(X_{ij}\) is already a rate per exposure unit or is an aggregate amount.
If the data are aggregate losses, divide each amount by its corresponding exposure to obtain a period rate.
Sum the exposures and, for rate observations, sum exposure multiplied by rate.
Divide the weighted loss by total exposure. The answer must have the same units as the period-level rates.
Why Total Exposure Is the Right Denominator
The Bühlmann-Straub model treats observations as having different precision when their exposures differ. In its standard form, conditional on the risk parameter \(\Theta_i\), period observations have a common conditional mean and a variance inversely proportional to exposure:
For the following justification, assume the period observations are conditionally independent given \(\Theta_i\), and the exposures are positive. The weighted mean is conditionally unbiased for the risk-specific mean:
Its conditional variance shows why total exposure summarizes the information in the observations. By conditional independence, the variance of a weighted sum is the sum of its variances:
So the weighted mean preserves the common conditional mean, while its conditional variance decreases as total exposure increases. This justification depends on the stated conditional variance structure and independence; if observations are dependent or their variances do not scale as \(1/w_{ij}\), the variance conclusion need not hold. The arithmetic definition of a weighted mean still applies, but its statistical interpretation may change.
Worked Calculations
Worked Example: Combining Three Period Loss Rates
Hypothetical scenario. A small equipment portfolio has observed loss rates of $80, $100, and $140 per equipment-year in three periods. The respective exposures are 100, 200, and 50 equipment-years. Calculate the total exposure, the exposure-weighted mean, and the unweighted mean. Assume the stated rates are measured on the same basis, with no discounting or other adjustment.
Method. Use the exposure-weighted mean because each period’s observation is a rate and the periods have unequal exposures. The total exposure is \(100+200+50=350\) equipment-years.
The numerator is the sum of each rate multiplied by its exposure: \(100(80)=8{,}000\), \(200(100)=20{,}000\), and \(50(140)=7{,}000\) dollars. Their sum is \(8{,}000+20{,}000+7{,}000=35{,}000\) dollars. Therefore,
The unweighted mean is \((80+100+140)/3=320/3=\$106.666\ldots\), or $106.67 per equipment-year to the nearest cent. The weighted mean is checked by reconstructing the aggregate loss: \(350(100)=\$35{,}000\), matching the sum of the three period amounts. The common mistake is to give each period equal weight despite its different exposure.
Worked Example: Recovering Rates From Aggregate Losses
Hypothetical scenario. A property coverage group reports aggregate losses of $12,000, $18,000, and $8,000 for three periods, with corresponding exposures of 100, 200, and 50 property-years. Find the total exposure and weighted mean loss per property-year. Assume all aggregate amounts and exposures refer to the same periods and basis.
Method. Because the given observations are aggregate losses rather than rates, first divide each aggregate loss by its own exposure. The period rates are \(12{,}000/100=\$120\), \(18{,}000/200=\$90\), and \(8{,}000/50=\$160\) per property-year.
The total exposure is \(100+200+50=350\) property-years. The total aggregate loss is \(12{,}000+18{,}000+8{,}000=\$38{,}000\). Thus the weighted mean rate is
As a cross-check using the rates, \(100(120)+200(90)+50(160)=12{,}000+18{,}000+8{,}000=\$38{,}000\). By contrast, the unweighted rate mean is \((120+90+160)/3=\$123.33\) to the nearest cent. The common mistake is averaging the aggregate losses themselves: that would produce a dollar amount per period, not a rate per property-year.
Worked Example: Using the Weighted Mean in a Premium
Hypothetical scenario. A risk has three annual loss rates of $70, $110, and $130 per insured unit-year, based on exposures of 100, 300, and 100 unit-years. For this calculation, take the Bühlmann-Straub credibility weight as \(Z_i=0.40\) and the collective mean as \(m=\$100\) per unit-year, both supplied. Calculate the experience mean and the indicated credibility premium. Assume the supplied credibility weight applies to this risk and that the observations are on a consistent basis.
Method. First calculate the exposure-weighted experience mean. Total exposure is \(100+300+100=500\) unit-years. The weighted numerator is \(100(70)+300(110)+100(130)=7{,}000+33{,}000+13{,}000=\$53{,}000\). Therefore,
Using the Bühlmann-Straub premium form with the supplied \(Z_i\) and \(m\), \(\pi_i=Z_i\bar X_i+(1-Z_i)m\). Substitution gives
If the three period rates were incorrectly averaged without weights, the result would be \((70+110+130)/3=\$103.333\ldots\). That would give \(0.40(103.333\ldots)+0.60(100)=\$101.33\) per unit-year to the nearest cent. The correct calculation is checked by confirming the weighted loss total, \(500(106)=\$53{,}000\). The common mistake is to apply the credibility formula to an unweighted average when the experience periods have different exposures.
Interpreting the Difference From an Unweighted Mean
The ordinary mean answers a different question: what is the average of the listed period rates if every period counts once? The exposure-weighted mean answers what rate results when all observed exposure is combined. When exposures vary, these generally differ. Neither calculation is inherently wrong in every context; the appropriate one depends on the quantity being summarized. For Bühlmann-Straub experience with period rates and exposure weights, use the exposure-weighted mean.
The direction of the difference depends on how the rates align with exposure. If high rates occur in high-exposure periods, the weighted mean tends to exceed the unweighted mean. If high rates occur in low-exposure periods, it tends to be lower. If the rates are all equal, or if the exposures are all equal, the two means coincide.
| Data form | Calculation for the experience mean | Common error |
|---|---|---|
| Rates \(X_{ij}\) and exposures \(w_{ij}\) | \(\sum_j w_{ij}X_{ij}/\sum_j w_{ij}\) | Taking an ordinary mean despite unequal exposures |
| Aggregate losses \(L_{ij}\) and exposures \(w_{ij}\) | \(\sum_j L_{ij}/\sum_j w_{ij}\) | Averaging aggregate losses and calling the result a rate |
| Equal positive exposures and rates | Ordinary mean of the rates | Assuming equal exposures when they are not specified |
Exam Traps and Practical Checks
- Confusing amount and rate. Multiply a rate by exposure to obtain an aggregate amount; divide an amount by exposure to obtain a rate.
- Using the wrong denominator. The denominator is total exposure, not the number of periods, when calculating an exposure-weighted rate.
- Mixing exposure units. Convert all weights to a common basis before adding them. A weight of 0.5 years and a weight of six months are equal only if the units are explicitly reconciled.
- Giving zero exposure a contribution. A period with zero exposure contributes zero to both weighted sums and cannot provide an observed rate through division by exposure. Do not divide its aggregate loss by zero.
- Rounding too early. Keep exact products and sums through the calculation, then round the final monetary rate as requested.
A reliable final check is to multiply the calculated weighted mean by total exposure. The result should equal the sum of the period aggregate losses, subject only to any final rounding. Also check that the final mean has the same units as the original period rates.
Check Your Understanding
For each problem, identify whether the observations are rates or aggregate losses before calculating. Round monetary rates to the nearest cent where needed.
- A hypothetical fleet has annual rates of $75, $90, and $105 per vehicle-year, with exposures of 40, 60, and 100 vehicle-years. Find total exposure and the weighted mean. (Answer: 200 vehicle-years; $94.50 per vehicle-year.)
- A hypothetical benefit group reports aggregate losses of $500, $400, and $1,500 at exposures of 2, 1, and 3 units. Find each period rate, total exposure, and the weighted mean. (Answer: Rates of $250, $400, and $500 per unit; 6 units; $400 per unit.)
- A hypothetical portfolio has period rates of $160, $120, and $100 per exposure unit, with exposures of 0.25, 0.75, and 1.5 units. Calculate the weighted mean and compare it with the unweighted mean. (Answer: Weighted mean $112.00 per unit; unweighted mean $126.67 per unit.)
- A risk’s rates are $70 and $110 per unit-year, with exposures of 100 and 300 unit-years. If the supplied Bühlmann-Straub weight is \(Z_i=0.60\) and the collective mean is $85 per unit-year, find the weighted experience mean and indicated premium. (Answer: Experience mean $100 per unit-year; premium $94.00 per unit-year.)
- A hypothetical coverage group has aggregate losses of $6,500 and $13,500 in periods with exposures of 50 and 150 exposure units. If \(Z_i=0.25\) and the supplied collective mean is $120 per unit, calculate the weighted mean and indicated premium. (Answer: Total exposure 200 units; weighted mean $100 per unit; premium $115.00 per unit.)