Why the Weighting Basis Matters
A set of class factors does not have one unique “average” unless the question specifies how classes are to count. A small class and a large class contribute equally to an unweighted class average, but they do not contribute equally to the average factor per exposure or to total premium. The weighting basis therefore determines what the summary describes.
This distinction matters when reviewing proposed classification factors. A summary can describe the typical class, the typical exposure, or the factor’s aggregate impact on premium. Those are different quantities. Before calculating an average, identify the population being summarized and the basis on which its members contribute.
The factors must refer to a consistent rating change or comparison, and weights must refer to the same classes and portfolio basis. In particular, the premium-weighted average uses the premiums to which the factors will be applied. If factors are proposed changes from current factors, \(r_j\) represents the proposed multiplier on current class premium for this calculation. If the factors are instead relativities on some other base, the implied premium calculation must use that base consistently.
What Each Average Measures
The equal-weighted average gives every class the same influence, regardless of its size. It answers a class-level question: “What is the average of the listed class factors if each class counts once?” It does not, by itself, describe the average factor for an insured, exposure, or dollar of premium.
The exposure-weighted average gives each exposure unit equal influence. It answers an exposure-level question when exposure units are comparable. If the factors multiply a common rate per exposure, this average also describes the aggregate premium factor. If classes have different current premium per exposure, however, exposure weighting alone does not generally describe the change in total premium.
The premium-weighted average gives each dollar of current premium equal influence. When class factors multiply those current premiums, it is the ratio of indicated total premium to current total premium. This makes it a direct measure of aggregate premium impact under that calculation. It is not the same as the average factor for an exposure, since exposure units can carry different premium amounts.
| Weighting basis | Weight for class \(j\) | Primary interpretation |
|---|---|---|
| Equal | 1 | Average factor across the listed classes |
| Exposure | \(E_j\) | Average factor across exposure units |
| Premium | \(P_j\) | Aggregate factor on current premium, if the factors multiply that premium |
Why Premium Weighting Gives the Aggregate Premium Effect
Suppose a proposed factor \(r_j\) multiplies current premium \(P_j\) in each class. The resulting premium for class \(j\) is \(P_jr_j\). Summing class premiums gives the new total, while summing \(P_j\) gives the current total. Dividing those totals yields:
This is exactly the premium-weighted average of the factors. The result follows because every class contributes its own premium change, \(P_jr_j\), to the aggregate numerator. A class with twice the current premium has twice the influence on the aggregate premium ratio, all else equal.
For comparison, if all classes have the same current premium rate per exposure, say \(b\), then \(P_j=bE_j\). Substituting into the premium-weighted formula cancels \(b\), giving the exposure-weighted average. When class premium rates differ, that cancellation does not occur, so exposure weights and premium weights can produce different results.
Worked Applications
Worked Example: A Few Large Classes Versus Many Exposures
A hypothetical equipment insurer reviews three classes. Class A has 800 exposure units and factor 0.90; Class B has 150 exposure units and factor 1.20; Class C has 50 exposure units and factor 1.80. Calculate the equal-weighted and exposure-weighted averages, and explain why they differ. All amounts and factors are deterministic; no interest rate, mortality assumption, or probability distribution is used.
Method. The equal-weighted average counts the three listed classes once each:
The exposure-weighted average uses the total factor contribution from each class. The numerator is \(800(0.90)+150(1.20)+50(1.80)=720+180+90=990\). Total exposure is \(800+150+50=1{,}000\). Therefore:
The equal-weighted result is 1.30 because the small class with factor 1.80 counts as much as the largest class. The exposure-weighted result is 0.99 because most exposure units are in Class A, whose factor is 0.90. As a check, the weighted contributions sum to \(990\), and dividing by \(1{,}000\) exposure units gives the calculated average factor of 0.99.
Common mistake avoided: Treating 1.30 as the portfolio average would implicitly give the 50-unit class the same weight as the 800-unit class. That is appropriate only if the question is specifically the average across the three classes, not across exposure units.
Worked Example: Exposure Weights and Premium Weights
A hypothetical commercial insurer has the following current earned premium, exposure, and proposed multiplicative factor for three equipment classes: Class A has premium \$120,000, 4,000 exposure units, and factor 0.95; Class B has premium \$300,000, 6,000 exposure units, and factor 1.05; Class C has premium \$80,000, 1,000 exposure units, and factor 1.30. Calculate both weighted averages and the aggregate premium change. All figures are deterministic annual values; no interest rate, mortality assumption, or probability distribution is used.
Exposure-weighted method. The weighted factor contributions are \(4{,}000(0.95)=3{,}800\), \(6{,}000(1.05)=6{,}300\), and \(1{,}000(1.30)=1{,}300\). The total is \(11{,}400\), not 11,530. Total exposures are \(4{,}000+6{,}000+1{,}000=11{,}000\). Thus the exposure-weighted average is \(11{,}400/11{,}000=1.03636\ldots\), or 1.0364 to four decimal places.
Premium-weighted method. The proposed class premiums are \(120{,}000(0.95)=\$114{,}000\), \(300{,}000(1.05)=\$315{,}000\), and \(80{,}000(1.30)=\$104{,}000\). The current total premium is \(\$120{,}000+\$300{,}000+\$80{,}000=\$500{,}000\); the proposed total is \(\$114{,}000+\$315{,}000+\$104{,}000=\$533{,}000\). Therefore:
The premium-weighted average is 1.0660 to four decimal places, corresponding to a 6.6% aggregate premium increase. The exposure-weighted average differs because premium per exposure is not constant: the three classes have current premiums per exposure of \(\$30\), \(\$50\), and \(\$80\), respectively. Check: \(\$533{,}000-\$500{,}000=\$33{,}000\), and \(\$33{,}000/\$500{,}000=6.6\%\).
Common mistake avoided: Using exposure weights to claim a 3.64% premium increase would be incorrect here. The factors multiply current premium, so the aggregate premium effect is the premium-weighted result.
Worked Example: Unequal Base Premiums per Exposure
A hypothetical specialty insurer applies proposed factors to current class premiums. Class X has 1,000 exposures, a current premium of \$200 per exposure, and factor 1.10; Class Y has 300 exposures, a current premium of \$400 per exposure, and factor 0.95; Class Z has 100 exposures, a current premium of \$1,000 per exposure, and factor 0.80. Calculate the exposure-weighted average, the premium-weighted average, and the actual aggregate premium change. All figures are deterministic; no interest rate, mortality assumption, or probability distribution is used.
Exposure-weighted average. The class contributions are \(1{,}000(1.10)=1{,}100\), \(300(0.95)=285\), and \(100(0.80)=80\). Total contribution is \(1{,}465\), and total exposures are \(1{,}400\). The exposure-weighted average is \(1{,}465/1{,}400=1.046428\ldots\), or 1.0464 to four decimal places.
Premium-weighted average and portfolio check. Current class premiums are \(1{,}000(\$200)=\$200{,}000\), \(300(\$400)=\$120{,}000\), and \(100(\$1{,}000)=\$100{,}000\), totaling \(\$420{,}000\). After applying the factors, the class premiums are \(\$220{,}000\), \(\$114{,}000\), and \(\$80{,}000\), totaling \(\$414{,}000\). Therefore, the premium-weighted average is \(\$414{,}000/\$420{,}000=0.985714\ldots\), or 0.9857 to four decimal places. The change is \(-\$6{,}000/\$420{,}000=-0.0142857\ldots\), a decrease of about 1.43%.
The premium-weighted factor is below 1 even though the exposure-weighted factor is above 1. The higher-premium-per-exposure classes have relatively low factors, and their premium dollars materially affect the portfolio result. The premium calculation checks: \(\$220{,}000+\$114{,}000+\$80{,}000=\$414{,}000\).
Common mistake avoided: Multiplying the exposure-weighted average by total premium would imply a premium increase of about 4.64%, despite the direct class calculation showing a decrease. The exposure-weighted average is not the premium ratio when class premium per exposure varies.
Selecting and Reporting the Weight
A defensible summary states both the calculation and its interpretation. Use the following sequence to avoid answering a different question from the one intended.
Decide whether the requested summary is across classes, across exposure units, or across premium dollars.
Use equal weights for the mean listed class factor, exposure weights for the average per exposure, and current premium weights for the aggregate premium ratio when the factors multiply those premiums.
Confirm that factors, exposures, and premiums refer to the same classes and consistent populations and periods. Check whether current premium per exposure is constant before equating exposure-weighted and premium-weighted results.
Report, for example, “the premium-weighted factor implies a 2% aggregate increase,” rather than simply calling a number “the average relativity.”
These averages are summaries for specified weights; they are not interchangeable. In particular, a higher equal-weighted average than exposure-weighted average may simply reflect that high-factor classes are small. It does not alone establish that the portfolio premium will rise. Likewise, an exposure-weighted average can be useful for describing the factor per exposure while failing to represent premium impact.
A common exam trap is to see exposure counts and factors and immediately calculate a weighted mean, without checking what result the question asks for. Another is to use current premiums as weights when the factors do not multiply those premiums. In either case, write the intended numerator and denominator in words first: factor contributions per exposure divided by exposures, or proposed premium divided by current premium. This simple check clarifies the calculation and its meaning.
Check Your Understanding
For each problem, select the weighting basis that answers the question and calculate the requested result.
- A hypothetical insurer has two classes with factors 0.80 and 1.40. What is the equal-weighted average? (Answer: 1.10)
- A hypothetical fleet program has 600 exposures at factor 0.95 and 400 exposures at factor 1.15. Calculate the exposure-weighted average factor. (Answer: 1.03)
- Current premiums are \$90,000 for Class A and \$210,000 for Class B. Proposed factors are 1.08 and 0.96, respectively. What is the premium-weighted average and aggregate percentage change? (Answer: 0.996, or a 0.4% decrease)
- Class M has 2,000 exposures, \$150 current premium per exposure, and factor 1.10. Class N has 500 exposures, \$600 current premium per exposure, and factor 0.90. Calculate the exposure-weighted average and the premium-weighted average. (Answer: exposure-weighted \(2{,}650/2{,}500=1.06\); premium-weighted \(\$600{,}000/\$600{,}000=1.00\))
- Three classes have equal current premiums of \$100,000 each and factors 0.92, 1.00, and 1.14. Calculate the aggregate premium ratio, then explain why it equals the equal-weighted factor in this case. (Answer: 1.02; each class has the same premium weight, so premium weighting and equal weighting coincide)