Introduction
Historical earned premium reflects both the exposure that earned coverage and the rates applicable to that coverage. For ratemaking, the objective is often to compare historical losses with premium expressed at a selected rate level. On-leveling makes that premium comparison on a consistent price basis: it restates premium for the rate-level difference without rewriting the underlying exposure or loss experience.
This distinction matters because a premium change can arise from a different price per unit of exposure, a different amount or mix of exposure, or both. On-leveling is intended to isolate the price-level part. It is not a way to create additional exposure, revise historical claims, or change how much coverage was provided.
What Is Being Adjusted?
For a group of policies with a consistent rate level and otherwise comparable coverage, earned premium can be viewed conceptually as the product of exposure and the rate applicable to that exposure. If the exposure is unchanged but the selected rate level differs, premium is adjusted to reflect that price difference. The historical rate level is the level already reflected in the earned premium; the target rate level is the level to which that premium is restated.
Let \(P_h\) be earned premium at historical rate level \(L_h\), and let \(L_t\) be the selected target level. When the group is homogeneous enough for one rate-level index to represent it, and the premium varies proportionally with that index, the restatement is:
Here \(P_t\) is the premium restated at the target level. The ratio compares prices, not exposures. This relationship assumes the rate-level indices use the same reference basis and that the premium being adjusted is proportional to the indexed rates. It does not automatically apply to fees, taxes, or other premium components that do not change proportionally with rates. Such components must be treated consistently with the purpose of the analysis.
Why the Rate-Level Ratio Works
The formula follows by holding the exposure fixed. Suppose the exposure amount is \(E\), and the applicable rate per exposure unit is proportional to the rate-level index \(L\). Write the premium at a level \(L\) as \(P(L)=E r(L)\), where \(r(L)\) is the rate for that exposure. If the rate changes in proportion to the index, then \(r(L_t)/r(L_h)=L_t/L_h\). Therefore:
The exposure \(E\) cancels because the same exposure is being priced at each level. Multiplying historical premium by \(L_t/L_h\) therefore changes the price basis while retaining the amount of historical exposure. If exposure changes between the groups being compared, or if their coverage or rating mix differs, that cancellation does not justify applying one common ratio to the entire aggregate.
Establish which earned premium is being restated and the historical level that it represents.
Use the stated selected level on the same index basis as the historical level.
For a suitable homogeneous group, multiply by the target-to-historical level ratio.
Do not alter exposure or loss amounts merely because the premium has been restated.
Worked Applications
Worked Example: Restating Two Groups to One Target Level
A hypothetical insurer has two groups of earned premium. Group A has $180,000 earned at rate level 0.90, and Group B has $320,000 earned at level 1.20. Restate both groups to a target level of 1.08 and find the combined on-level premium. Assume the indices use a common reference and each group’s premium is proportional to its stated level.
Method. Restate each group separately using \(P_t=P_h(L_t/L_h)\). This preserves each group’s earned exposure while changing its rate-level basis.
Group A. Its target-to-historical ratio is \(1.08/0.90=1.20\). Its restated premium is \( \$180{,}000 \times 1.20 = \$216{,}000\).
Group B. Its ratio is \(1.08/1.20=0.90\). Its restated premium is \( \$320{,}000 \times 0.90 = \$288{,}000\).
Combined result. Add the two restated amounts: \( \$216{,}000+\$288{,}000=\$504{,}000\). The original combined earned premium was \( \$180{,}000+\$320{,}000=\$500{,}000\); the on-level total is higher because the target lies above Group A’s level, and the increase for A exceeds the decrease for B.
Check and trap avoided. The separate adjustments are \( \$36{,}000\) up for A and \( \$32{,}000\) down for B, for a net increase of \( \$4{,}000\). Thus \( \$500{,}000+\$4{,}000=\$504{,}000\), consistent with the direct sum. A common error is to calculate one group correctly and then mis-add the group totals; the total must be the sum of both adjusted premiums.
Worked Example: Changing Premium Does Not Change Losses
For a hypothetical coverage group, earned premium is $264,000 at rate level 1.10. The selected target level is 1.21. The group’s recorded losses are $198,000. Find the on-level premium and compare the loss ratio using the original and restated premiums. Assume the losses and exposure are held fixed. Report loss ratios to two decimal places.
Step 1: Restate premium. The level ratio is \(1.21/1.10=1.10\). Therefore the on-level premium is \( \$264{,}000 \times 1.10=\$290{,}400\).
Step 2: Calculate the historical-premium ratio. The loss ratio using the original earned premium is \( \$198{,}000/\$264{,}000=0.75\), or 75.00%.
Step 3: Calculate the ratio on the selected level. The losses remain $198,000, so the ratio is \( \$198{,}000/\$290{,}400=0.681818\ldots\), or 68.18% rounded to two decimal places.
Check and trap avoided. The premium increased by 10%, while the loss amount did not change. It would be incorrect to multiply losses by 1.10 and use \( \$217{,}800\) as the on-level loss amount: that would change the recorded loss experience rather than restate premium. The change in the calculated ratio here results solely from using a different premium denominator.
Worked Example: Why an Aggregate May Need Group-by-Group Treatment
A hypothetical book consists of two distinct groups with different historical rate levels. Group C has $100,000 of earned premium at level 0.80; Group D has $300,000 at level 1.20. Restate both to target level 1.00. A colleague proposes using the unweighted average historical level as the level for all $400,000. Determine the correct total and assess the proposal. Assume each group’s premium is proportional to its stated level.
Step 1: Restate Group C. Its ratio is \(1.00/0.80=1.25\), so its on-level premium is \( \$100{,}000 \times 1.25=\$125{,}000\).
Step 2: Restate Group D. Its ratio is \(1.00/1.20=0.833333\ldots\), so its on-level premium is \( \$300{,}000 \times 0.833333\ldots=\$250{,}000\).
Step 3: Add the results. The correct total is \( \$125{,}000+\$250{,}000=\$375{,}000\). Equivalently, the group amounts at reference level 1.00 are \( \$100{,}000/0.80=\$125{,}000\) and \( \$300{,}000/1.20=\$250{,}000\).
Step 4: Check the proposed shortcut. The unweighted average of the historical indices is \((0.80+1.20)/2=1.00\). Applying a ratio of \(1.00/1.00=1.00\) to the $400,000 aggregate would leave it at $400,000, which differs from the correct $375,000. The shortcut fails because the groups do not contribute equal amounts of premium at the reference level. A single aggregate level must represent the underlying composition appropriately; an unweighted average of group indices does not establish that.
Conclusion. When group levels differ, restate the groups separately if their premiums and levels are available, then sum. The result preserves the actual mix of exposure represented by those group premiums.
What On-Leveling Does Not Do
On-leveling changes the price basis used for premium; it does not remake the historical record. In particular, it does not:
- increase or decrease the amount of exposure that earned the historical premium;
- change claim counts, loss amounts, or the period to which losses are assigned;
- convert losses to the target rate level; or
- correct a mismatch between the premium and loss experience being compared.
The earlier material on matching losses and expenses to experience periods addresses whether the observations belong together. That is a separate issue from whether premium has been restated to a consistent rate level. On-leveling cannot repair a data mismatch. It also cannot make an inappropriate comparison appropriate merely by changing the premium amount.
Assumptions, Scope, and Common Traps
A single ratio is suitable only when the premium being adjusted has a defined historical level, the target uses the same index basis, and the premium responds proportionally to the rate-level change. It also assumes that the group’s exposure, coverage basis, and relevant mix are held fixed for the comparison. If multiple groups have different levels, the group-level restatements should be retained until the information needed to combine them is available. If a component of premium is not rate-sensitive in the same way, applying the full ratio to that component may overstate or understate the restatement.
- Reversing the ratio. To move from historical level \(L_h\) to target \(L_t\), use target divided by historical. If the target is higher, the rate-sensitive premium should increase under the proportionality assumption.
- Changing losses along with premium. The historical loss amount is not part of this adjustment. Restating premium does not mean inflating or deflating losses.
- Treating exposure as price. More earned exposure can produce more premium even when the rate level is unchanged. On-leveling isolates a rate-level difference; it does not standardize exposure volume.
- Using an unweighted average for unlike groups. A simple average of level indices need not represent the level embedded in an aggregate premium. Preserve group detail where possible.
- Applying a ratio to every dollar without checking the component. Confirm that the premium amount being adjusted is actually proportional to the rate level in question.
On-level premium is useful because rate adequacy comparisons require premium on a consistent price basis. But the result is interpretable only if the underlying exposure and loss measures are appropriately matched and the rate-level adjustment is applied to the correct premium component. The central discipline is to change the price basis and leave the experience itself intact.
Check Your Understanding
For each hypothetical situation, identify the appropriate restatement and keep exposure and losses conceptually separate from premium.
- A homogeneous group has $144,000 of earned premium at level 0.96. Restate it to level 1.08. (Answer: $162,000)
- A group has $210,000 of earned premium at level 1.05 and is to be restated to level 1.155. What is its on-level premium? (Answer: $231,000)
- Two groups have $80,000 at level 0.80 and $180,000 at level 1.20. Restate both to level 1.00 and add the results. (Answer: $250,000)
- A group has $375,000 of earned premium at level 1.25 and $300,000 of recorded losses. Restate premium to level 1.00 and calculate the loss ratio using the restated premium, to two decimal places. (Answer: $300,000 on-level premium; losses remain $300,000; loss ratio 100.00%)
- A hypothetical group has 1,200 exposure units and $96,000 of earned premium at level 0.80. Assuming the exposure is unchanged and premium is proportional to the level, find the premium at level 1.00 and state the exposure used after restatement. (Answer: $120,000 premium; 1,200 exposure units)