Why the Extension Method Works Policy by Policy
Premium in an experience period may reflect more than one rate level. If rate changes took effect at different times, policies may have been written or earned under different levels. To compare premiums on a common basis, the extension method recalculates each policy’s applicable premium at the chosen target rate level, then sums those recalculated amounts.
The method is most transparent when policy-level premium and rate-level information are available. It preserves the actual mix of premiums across policies rather than applying one assumed average adjustment to the entire total. The resulting total is an estimate of what the included premium would have been at the target level, under the method’s assumptions.
A rate level may be represented by an index, with a chosen reference level set to 100, or by a multiplicative factor. Either convention works if the same convention is used for the historical and target levels. The extension calculation uses their ratio, not the difference between them. For instance, moving from level 100 to level 110 multiplies a rate-sensitive premium by \(110/100=1.10\), a 10% increase.
The calculation is deterministic premium arithmetic. It does not require an interest rate, mortality assumption, or probability distribution. The relevant assumptions are that the listed rate level applies to the premium being extended, that the premium changes proportionally with that level, and that the selected target level is the common basis for comparison.
The Three Steps
For each policy or rate-sensitive premium component, determine the rate level that produced the recorded premium. Do not assume all premiums in the experience period are at the same level.
Multiply the applicable premium by the ratio of target rate level to historical rate level. Perform this calculation for every included policy or component.
Add the recalculated premiums. Compare that extended total with the corresponding actual premium total to identify the dollar adjustment and the aggregate on-level factor.
Let \(P_j\) be the actual rate-sensitive premium for item \(j\), \(L_j\) its applicable historical rate level, and \(T\) the target rate level. The item’s extension factor is \(T/L_j\), so its extended premium is \(P_jT/L_j\). The extended total and aggregate factor are:
The dollar adjustment is \(P_{\text{ext}}-\sum_jP_j\). A positive result means the premium at the target level exceeds actual premium; a negative result means it is lower. The aggregate factor is a premium-weighted result of the individual extensions, not generally the simple average of their factors.
Why the Extension Formula Holds
Suppose a policy’s rate-sensitive premium is exposure multiplied by a base rate and a rate-level multiplier. Let \(B_j\) denote the product of the exposure and the base rate for item \(j\). At historical level \(L_j\), the premium is \(P_j=B_jL_j\). At target level \(T\), the corresponding premium is \(P_{j,T}=B_jT\). Dividing the target premium by the historical premium eliminates the common exposure and base rate:
This justification depends on proportionality: the historical and target premiums must use the same underlying exposure and base-rate basis, with the rate level acting as a multiplicative change. The formula does not automatically apply to flat fees, premiums subject to a different adjustment, or amounts whose relationship to the rate level is not proportional. Such amounts should be treated according to their own basis rather than indiscriminately multiplied by a rate-level ratio.
Worked Example: Extending Two Policy Premiums
A hypothetical insurer has two policy premiums in an experience period. Policy A has actual premium of $1,000 at rate level 100; Policy B has actual premium of $2,200 at rate level 110. The target level is 120. Find the extended total, the change from actual premium, and the aggregate factor. All amounts are rate-sensitive premium. This is deterministic arithmetic; no interest, mortality, or probability distribution is assumed.
Method. Apply the extension method separately to each policy. The extension factors are the target level divided by each policy’s own historical level.
For Policy A, the factor is \(120/100=1.20\), so its extended premium is \(1{,}000(1.20)=\$1{,}200\).
For Policy B, the factor is \(120/110=12/11\), so its extended premium is \(2{,}200(12/11)=\$2{,}400\).
The actual total is \(\$1{,}000+\$2{,}200=\$3{,}200\). The extended total is \(\$1{,}200+\$2{,}400=\$3{,}600\). Therefore the dollar adjustment is \(\$3{,}600-\$3{,}200=\$400\), and the aggregate factor is \(\$3{,}600/\$3{,}200=1.125\), or 1.125 to three decimal places.
Check and common mistake avoided. The extended total exceeds actual premium, as expected because both policies are moved to a higher level. Applying the Policy A factor of 1.20 to the entire actual total would give $3,840, which is wrong: Policy B starts at level 110, not 100.
Adding Policy-Level Results Correctly
The order of operations matters. First extend each premium using its own applicable level, then add the extended amounts. The arithmetic is equivalent to multiplying the actual total by one factor only when that factor is the premium-weighted aggregate factor calculated from those policy-level results.
In particular, an unweighted average of policy factors generally gives the wrong total. A factor on a small premium should not have the same dollar influence as a factor on a much larger premium. Keep the premium amounts attached to their corresponding historical levels throughout the calculation.
Worked Example: Several Historical Rate Levels
A hypothetical book has three policy premiums: $800 at level 100, $1,250 at level 105, and $1,800 at level 120. The target level is 126. Calculate the extended total, the dollar adjustment, and the aggregate factor. All listed premiums are rate-sensitive. No interest, mortality, or probability distribution is assumed.
Method. Extend each premium to level 126 using its own ratio. The respective factors are \(126/100=1.26\), \(126/105=1.20\), and \(126/120=1.05\).
The first extended premium is \(\$800(1.26)=\$1{,}008\). The second is \(\$1{,}250(1.20)=\$1{,}500\). The third is \(\$1{,}800(1.05)=\$1{,}890\).
The actual total is \(\$800+\$1{,}250+\$1{,}800=\$3{,}850\). The extended total is \(\$1{,}008+\$1{,}500+\$1{,}890=\$4{,}398\). The adjustment is \(\$4{,}398-\$3{,}850=\$548\). The aggregate factor is \(4{,}398/3{,}850=1.142337662\ldots\), or 1.142 to three decimal places.
Check and common mistake avoided. The factors have simple average \((1.26+1.20+1.05)/3=1.17\). Multiplying the actual total by 1.17 would give \(\$4{,}504.50\), not \(\$4{,}398\). That average gives each policy equal weight even though their premiums differ. The extension method weights each factor by the premium to which it applies.
Extending Multiple Components and Handling Non-Rate-Sensitive Amounts
A policy can contain more than one premium component. If components have different applicable historical levels, extend them separately and then add them. A policy label alone does not guarantee that all premium on the policy shares one rate level. The data must identify the historical basis for each amount being adjusted.
Likewise, define the comparison consistently. If the task concerns rate-sensitive premium only, compare extended rate-sensitive premium with actual rate-sensitive premium. If a total includes a flat charge that does not change with rate level, keep that charge outside the extension calculation and add it back only when the requested total includes it. Extending a fixed charge would incorrectly treat it as proportional to the rate-level change.
Worked Example: Extending Policy Components Separately
A hypothetical policy has liability premium of $600 at level 100 and property premium of $400 at level 110. A second policy has liability premium of $900 at level 105 and property premium of $700 at level 115. The target is level 115. Find the extended total and adjustment from actual premium. All four components are rate-sensitive. This calculation uses no interest, mortality, or probability distribution.
Method. Treat each coverage component as a separate item because the supplied historical levels differ. The four extension factors are \(115/100=1.15\), \(115/110=23/22\), \(115/105=23/21\), and \(115/115=1\).
The first liability component extends to \(\$600(1.15)=\$690\). The first property component extends to \(\$400(23/22)=\$9{,}200/22=\$418.181818\ldots\). The second liability component extends to \(\$900(23/21)=\$20{,}700/21=\$985.714285\ldots\). The second property component remains \(\$700(1)=\$700\).
The extended total, using the exact component calculations before rounding, is \(690+9{,}200/22+20{,}700/21+700=\$2{,}793.896103896\ldots\). The actual total is \(\$600+\$400+\$900+\$700=\$2{,}600\). The adjustment is \(\$193.896103896\ldots\). Rounded only at the final step to cents, the extended total is \(\$2{,}793.90\) and the adjustment is \(\$193.90\).
Check and common mistake avoided. The component at level 115 is unchanged, and all other components increase because their historical levels are below 115. Rounding the component results before summing can slightly change the reported total. Retaining the exact fractions until the final rounding gives the stated cents.
Interpreting the Comparison and Avoiding Scope Errors
The difference between extended and actual premium is a rate-level adjustment for the included premium under the stated assumptions. It is not, by itself, a change in exposure, a forecast of future premium, or evidence that the underlying risk changed. The aggregate factor summarizes the extension for this particular premium mix and target level.
Before calculating, verify three items: that the actual premiums and historical levels refer to the same policies or components; that the target is stated on the same rate-level scale; and that the premium amounts included in the sum are the amounts intended for on-leveling. A correct formula applied to mismatched inputs still produces an invalid result.
Check Your Understanding
For each problem, use the stated target level and compare the extended amount with the corresponding actual premium.
- A policy has $500 of rate-sensitive premium at level 100 and $900 at level 125. The target is 110. Find the extended total and the dollar adjustment. (Answer: Extended total $1,342; adjustment \(-\$58\).)
- Premiums of $720, $1,500, and $800 apply at levels 90, 100, and 120, respectively. The target is 108. Find the extended total and aggregate factor, rounding the factor to three decimal places. (Answer: Extended total $3,204; aggregate factor 1.061.)
- A book has $450 at level 105, $600 at level 112, and $1,000 at level 125. The target is 112. Find the extended total and adjustment from actual premium. (Answer: Extended total $1,976; adjustment \(-\$74\).)
- A policy’s actual premium consists of $1,400 rate-sensitive premium at level 100 and a fixed $75 charge. The target is 110. Find the extended total including the unchanged charge and the change from actual total. (Answer: Extended total $1,615; change $140.)
- Two premiums are $400 at level 100 and $1,800 at level 115. The target is 120. Calculate the extended total and dollar adjustment, rounding final amounts to cents. (Answer: Extended total $2,358.26; adjustment $158.26.)