Worked Example With a Changed Current Rate Level

Recorded Premium and Current Rate Level

Earned premium in an experience period reflects the rates that were actually in force as policies earned. If rates changed during or after that period, this recorded premium may not represent the premium that the same business would produce at the current rate level. A rate-level adjustment restates the premium for comparison on that current basis; it does not alter the amount actually earned.

The distinction matters in a rate indication: the premium base should be expressed at the intended rate level before it is compared with projected losses and other indication inputs. This tutorial focuses on the premium restatement when historical premium was earned at one or more rate levels.

Definition: A rate-level index is a relative measure of the premium rate level, with a chosen reference level assigned an index such as 1.000. The recorded earned premium is the premium actually earned under rates in force during the experience period. The current-rate-level earned premium is the hypothetical premium those same exposures would have earned if rated at the current rate level, holding the relevant coverage and business mix constant.

Let $R_j$ denote the rate-level index applicable to a segment of recorded earned premium, and let $R_c$ denote the current rate-level index. Under the proportional rate-level model, the adjustment factor for that segment is $R_c/R_j$. If the recorded premium for the segment is $EP_j^{\mathrm{rec}}$, its current-level equivalent is:

EP_j^{\mathrm{cur}} = EP_j^{\mathrm{rec}}\frac{R_c}{R_j}.

The index can be normalized in different ways. What matters is that the segment index and current index use the same scale. For example, indexes of 1.00 and 1.10 mean that the second rate level is 10% higher than the first. A 10% increase changes the index by multiplication: $1.00(1.10)=1.10$, not by adding 0.10 to a dollar premium.

Why the Adjustment Formula Works

The formula follows from the proportionality assumption. For the same exposure and coverage, suppose premium at rate level $R$ is proportional to $R$. If the exposure would produce premium $K R_j$ at the segment's rate level, where $K$ represents the fixed exposure and rating basis, then its recorded premium is $EP_j^{\mathrm{rec}}=K R_j$. At the current level, that same business would produce $K R_c$. Therefore:

EP_j^{\mathrm{cur}} = K R_c = (K R_j)\frac{R_c}{R_j} = EP_j^{\mathrm{rec}}\frac{R_c}{R_j}.

This is a restatement, not a prediction that the insurer collected more premium. It applies when the index accurately represents the rate change and the business being restated is comparable on the relevant coverage and mix basis. If exposures, coverage, or business mix differ materially, a single rate-level factor may not isolate the effect of rate changes. The formula also does not substitute for loss trending: it adjusts the premium rate basis, not the cost of losses over time.

One Historical Rate Change

When all recorded premium is at one rate level, the calculation is direct. The key is to use the current index divided by the index that applies to the recorded premium. The premium is multiplied by this ratio; it is not increased by the percentage-point difference between the two indexes.

Worked Example: Restating Premium After a Rate Increase

A hypothetical specialty equipment insurer recorded $480,000 of earned premium entirely at rate-level index 1.00. A subsequent 12% rate increase establishes the current index at 1.12. Assume the exposure, coverage, and business mix are unchanged, and the rate change applies proportionally. Find the current-level equivalent and the change from recorded premium.

Method and factor. The recorded segment is at $R_j=1.00$, and the current level is $R_c=1.12$. Thus the adjustment factor is:

\frac{R_c}{R_j}=\frac{1.12}{1.00}=1.12.

Restate the premium.

EP^{\mathrm{cur}} = \$480{,}000(1.12) = \$537{,}600.

Compare with recorded premium. The difference is $\$537{,}600-\$480{,}000=\$57{,}600$, which is 12% of the recorded premium. The current-level equivalent is therefore $537,600, while the actual amount recorded remains $480,000.

Check and common mistake. Dividing the current index by the old index gives a factor greater than 1, as it should for a rate increase. The mistake to avoid is adding 0.12 dollars to the recorded premium, or treating the $57,600 restatement as additional cash actually earned.

Experience Earned at Several Rate Levels

If an experience period includes multiple rate levels, adjust each premium segment using its own applicable index and then sum the adjusted amounts. The current-level equivalent is not generally obtained by multiplying total recorded premium by an unweighted average of segment factors. Segments with more recorded premium contribute more to the total.

Key result: For segments $j=1,\ldots,n$, the total current-level equivalent is the sum of the segment equivalents: $$ EP^{\mathrm{cur}} = \sum_{j=1}^{n}EP_j^{\mathrm{rec}}\frac{R_c}{R_j}. $$ The corresponding aggregate adjustment factor is $EP^{\mathrm{cur}}/\sum_j EP_j^{\mathrm{rec}}$, when total recorded premium is positive. It is a premium-weighted result, not a simple average of the segment factors.

Worked Example: Combining Premium at Three Rate Levels

A hypothetical insurer has recorded earned premium from three groups with unchanged coverage and mix. The groups' recorded premium and rate-level indexes are $200,000 at 1.000, $324,000 at 1.080, and $237,600 at 1.188. The current rate level is 1.188. The indexes reflect successive increases of 8% and 10%, since $1.000(1.08)=1.080$ and $1.080(1.10)=1.188$. Calculate the current-level equivalent for all three groups and the overall adjustment factor.

Method. Restate each segment to index 1.188, using its own starting index.

Group 1. The factor is $1.188/1.000=1.188$, so the adjusted premium is $\$200{,}000(1.188)=\$237{,}600$.

Group 2. The factor is $1.188/1.080=1.10$, so the adjusted premium is $\$324{,}000(1.10)=\$356{,}400$.

Group 3. It is already at the current level, so the factor is $1.188/1.188=1.00$, and the adjusted premium remains $237,600.

Aggregate. Recorded premium totals $\$200{,}000+\$324{,}000+\$237{,}600=\$761{,}600$. Current-level premium totals $\$237{,}600+\$356{,}400+\$237{,}600=\$831{,}600$. The overall factor is $\$831{,}600/\$761{,}600\approx1.0919$, or about a 9.19% increase.

Check and common mistake. The group already at index 1.188 does not change. An unweighted average of the three segment factors, $(1.188+1.10+1.00)/3$, would ignore the different premium amounts and would not produce the aggregate factor. Weighting each factor by its recorded premium gives the same result as summing adjusted dollars.

When the Current Level Changes After the Experience Period

The current level may include a rate change effective after the experience period ended. That change still belongs in $R_c$ when the task is to express historical premium at the current rate level. Do not stop at the rate level that was most recent during the experience period.

Worked Example: Including a Later Rate Change

A hypothetical commercial product has recorded $400,000 at index 1.000 and $600,000 at index 1.100. After the experience period, a further 5% increase takes effect. Assume no other changes to exposure, coverage, or business mix. Find the current index, the premium at the current level, and the overall adjustment factor.

Find the current index. The additional 5% change is multiplicative:

$$ R_c=1.100(1.05)=1.155. $$

Restate the first segment. Its factor is $1.155/1.000=1.155$, so its equivalent is $\$400{,}000(1.155)=\$462{,}000$.

Restate the second segment. Its factor is $1.155/1.100=1.05$, so its equivalent is $\$600{,}000(1.05)=\$630{,}000$.

Aggregate and check. Recorded premium is $\$400{,}000+\$600{,}000=\$1{,}000{,}000$. Current-level premium is $\$462{,}000+\$630{,}000=\$1{,}092{,}000$, giving an overall factor of 1.092. The result is 9.2% above recorded premium. Applying only the earlier index of 1.100 to all premium would miss the later 5% increase for the current-level comparison.

Interpretation and Common Traps

A current-level premium is a comparison basis, not a replacement for recorded premium in source records. Keep both amounts labelled clearly in calculations. A useful reconciliation is to show, for each segment, its recorded amount, its rate-level factor, and its adjusted amount; the segment amounts should sum to the aggregate restatement.

Using the wrong direction. To restate old premium upward to a higher current level, use current index divided by the segment index. Reversing the ratio would reduce premium when the current level is higher.
Applying one factor to mixed-level premium. If segments were earned at different indexes, adjust them separately unless a valid aggregate factor has been calculated from their premium-weighted totals.
Adding rate changes instead of compounding them. Sequential increases of 8% and 10% produce $1.08(1.10)=1.188$, not 1.18.
Confusing a restatement with collected premium. The adjusted amount is what the same business would represent at the current level under the stated assumptions. It is not additional historical revenue.
Ignoring the basis. Rate-level adjustment alone does not correct differences in coverage, exposure, or business mix. Those need to be addressed separately before interpreting the comparison.

Key takeaway: Identify the rate-level index for each recorded premium segment and the target current index. Multiply each recorded amount by current index divided by segment index, then sum the adjusted dollars. The result is a current-level equivalent, not a change to what was actually earned.

Check Your Understanding

For each problem, assume the stated rate-level indexes accurately reflect proportional rate changes and that coverage and business mix are unchanged.

A hypothetical policy group has $150,000 of recorded earned premium at index 1.00. The current index is 1.08. What is its current-level equivalent? (Answer: $162,000)
A hypothetical insurer recorded $250,000 at index 1.00 and $210,000 at index 1.05. The current index is 1.155. Restate each segment and find the total current-level equivalent. (Answer: $519,750)
A hypothetical product has $320,000 at index 1.00 and $480,000 at index 1.12. A further 5% increase is now effective. Find the current index, current-level premium, and aggregate factor. (Answer: current index 1.176; current-level premium $880,320; factor 1.1004)
Recorded premium is $100,000 at index 1.00, $270,000 at index 1.08, and $237,600 at index 1.188. The current index is 1.188. Find the adjusted total and aggregate factor. (Answer: $653,400; factor approximately 1.0754)
A book's recorded premium was earned under one coverage mix, while the proposed current-level comparison uses materially broader coverage. Can a rate-level index alone establish the current-level equivalent? (Answer: No; the coverage-basis difference must also be addressed.)

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