Why Rate Changes Can Be Counted Twice
On-leveling adjusts historical premium to a common rate level so that premium from different periods can be compared consistently. The key diagnostic question is not merely whether the premium is historical; it is what rate level the premium already reflects. A premium base that has already been adjusted for a rate change must not receive that same change again.
The error can occur when an on-level factor is applied to a premium total that was previously recalculated to a newer level, or when a data extract is labeled “earned premium” even though it has already been normalized. Applying an unnecessary factor inflates the denominator of a loss ratio and can understate an indicated rate change. This diagnostic is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Rate-Level Indices and the Correct Factor
Represent rate levels by indices relative to a common reference level. Set the reference index to \(1.00\). If a uniform rate increase of \(r\) has taken effect, the new index is \(1+r\). For example, a 10% increase moves the index from \(1.00\) to \(1.10\). A 5% increase applied after that moves it to \(1.10 \times 1.05=1.155\), not \(1.15\): successive percentage changes multiply.
Suppose premium \(P_b\) is at embedded rate-level index \(I_b\), and the desired target index is \(I_t\). Under the assumption that the same risks and exposure are being repriced proportionally, the equivalent premium at the target level is:
This is a relative adjustment. The factor is not automatically the full change from the reference level to the target; it is the change from the level already in the base to the target. If \(I_b=I_t\), the factor is 1.00. A further adjustment would then be double counting.
Why the Ratio of Indices Works
Assume a fixed set of policies and exposures, with premiums changing in direct proportion to the rate-level index. Let \(B\) be the premium those risks would produce at the reference index \(1.00\). At index \(I_b\), their premium is \(B I_b\). At target index \(I_t\), it is \(B I_t\). Thus the conversion from the base premium to the target premium is:
The reference premium \(B\) cancels. This justification depends on a consistent risk and exposure basis and on premium being proportional to the stated rate-level changes. If the base includes different mixes of risks, coverages, or limits, a single overall index ratio may not represent the required adjustment; the premium may need to be reconstructed on a consistent basis instead.
Find What the Premium Base Already Reflects
Before multiplying premium by an on-level factor, trace how the premium was created. A base might be raw earned premium at the rates in force as policies were written, premium recalculated using current rates, or a prior on-level premium already adjusted to an intermediate or target level. The label alone does not establish which one it is.
For a partial adjustment, identify the embedded level rather than treating the base as either entirely raw or entirely at the target level. For example, if a prior calculation moved premium from index \(1.00\) to \(1.08\), and the current target is \(1.12\), the remaining factor is \(1.12/1.08\). Applying \(1.12\) to that already adjusted premium repeats part of the change.
Confirm the data period, premium measure, and whether values are raw, recalculated, or previously on-leveled.
Use the calculation history or the actual rate changes reflected in the premium. Do not infer it solely from a spreadsheet heading.
State the desired rate level explicitly, using the same reference basis as the embedded level.
Calculate target index divided by embedded index, then verify that no earlier step has already made that adjustment.
Worked Calculations
Worked Example: A Target-Level Premium Is Adjusted Again
A hypothetical insurer has $800,000 of earned premium at an old rate level, index \(1.00\). A 10% rate increase has already been applied to this premium in a data-preparation step, producing a premium base of $880,000 at index \(1.10\). The target level is also \(1.10\), and losses are $600,000. A worksheet then multiplies the $880,000 by 1.10 again. Find the correct premium and loss ratio, the erroneous results, and the effect on a simple indicated rate change using a 65% target loss ratio.
Method. Compare the embedded and target indices before applying any factor. The premium base is already at \(1.10\), so the correct residual factor is \(1.10/1.10=1.00\).
Correct premium. The initial adjustment is \(800{,}000 \times 1.10=880{,}000\). No further adjustment is due, so the correct target-level premium is $880,000.
Correct loss ratio. The loss ratio is \(600{,}000/880{,}000=0.681818\ldots\), or 68.18% to two decimal places.
Erroneous calculation. Applying 1.10 a second time gives \(880{,}000 \times 1.10=968{,}000\). The erroneous loss ratio is \(600{,}000/968{,}000=0.6198347\ldots\), or 61.98% to two decimal places.
Effect on the simple indicated rate change. With losses and other provisions held fixed, divide the loss ratio by the 65% target loss ratio, then subtract 1. Correctly, \(0.681818\ldots/0.65-1=0.048951\ldots\), an indicated increase of 4.90%. Using the erroneous loss ratio gives \(0.6198347\ldots/0.65-1=-0.046408\ldots\), an indicated decrease of 4.64%. The inflated premium reverses the direction of this simplified indication.
Check. The correct factor is 1.00 because the base and target indices match. The incorrect factor adds $88,000 to a base that was already at the target level. The common mistake is to use the full 10% increase without checking whether it is already embedded.
Worked Example: Removing an Intermediate Adjustment
A hypothetical program has $1,000,000 of raw historical premium at index \(1.00\). A prior data step on-leveled it to index \(1.08\), so the premium supplied for analysis is $1,080,000. The current target index is \(1.12\). Losses are $700,000. Calculate the correct target-level premium and loss ratio, then compare them with the results from mistakenly multiplying the supplied premium by 1.12.
Method. The supplied base is at \(1.08\), not \(1.00\). The residual on-level factor is \(1.12/1.08=1.037037\ldots\).
Correct target premium. \(1{,}080{,}000 \times (1.12/1.08)=1{,}120{,}000\). This agrees with adjusting the original $1,000,000 directly from \(1.00\) to \(1.12\).
Correct loss ratio. \(700{,}000/1{,}120{,}000=0.625\), or 62.50%.
Erroneous target premium and loss ratio. Applying the full target index to the supplied premium gives \(1{,}080{,}000 \times 1.12=1{,}209{,}600\). The resulting loss ratio is \(700{,}000/1{,}209{,}600=0.5787037\ldots\), or 57.87%.
Check. The erroneous premium exceeds the correct premium by \(1{,}209{,}600-1{,}120{,}000=\)$89,600. The correct computation moves only from \(1.08\) to \(1.12\); applying 1.12 as a multiplier treats the already adjusted base as if it were at \(1.00\).
Worked Example: A Base Contains Premium at Two Rate Levels
A hypothetical book has $300,000 of reference-level premium from one group of exposures and $400,000 of reference-level premium from another. The second group received a 5% rate increase before the premium was recorded; the first did not. The combined recorded premium is the base for an indication. The target index is \(1.12\), and losses are $470,400. Find the correct target premium and loss ratio, and quantify the error from applying 1.12 to the combined recorded premium.
Method. Reconstruct the recorded premium from the underlying reference-level amounts. The first group contributes \(300{,}000 \times 1.00=300{,}000\); the second contributes \(400{,}000 \times 1.05=420{,}000\). Thus the recorded base is \(300{,}000+420{,}000=720{,}000\).
The total reference-level premium is \(300{,}000+400{,}000=700{,}000\). Its weighted embedded index is \(720{,}000/700{,}000=1.028571\ldots\). Therefore the residual factor to target is \(1.12/1.028571\ldots=1.088888\ldots\), equivalently \(784{,}000/720{,}000\).
Correct target premium. \(700{,}000 \times 1.12=784{,}000\), or, from the recorded base, \(720{,}000 \times 1.088888\ldots=784{,}000\).
Correct loss ratio. \(470{,}400/784{,}000=0.60\), or 60.00%.
Erroneous calculation. Multiplying the recorded base by 1.12 gives \(720{,}000 \times 1.12=806{,}400\). The erroneous loss ratio is \(470{,}400/806{,}400=0.583333\ldots\), or 58.33%.
Check. The correct adjustment recognizes that some of the recorded premium already reflects the 5% increase. The wrong calculation overstates target premium by \(806{,}400-784{,}000=\)$22,400. The common trap is assuming that a mixed-level premium base is wholly at the reference level or wholly at the target level; here its embedded level is weighted by the underlying reference-level premium.
Common Diagnostic Traps
An extra on-level factor does not change losses; it changes the premium denominator. If losses remain fixed, an overstated premium produces a lower loss ratio. In a simplified loss-ratio indication, that can make the indicated rate change too small or even change an indicated increase into an apparent decrease. The examples hold all other provisions constant to isolate this effect; a complete indication may include additional adjustments and provisions.
- Using the target index as the multiplier. The multiplier is target index divided by embedded index, not necessarily the target index itself.
- Treating “historical” as “unadjusted.” A historical period can be represented by premium that has already been restated to another level.
- Ignoring intermediate adjustments. If a prior step moved a base to an intermediate level, calculate the remaining factor from that level.
- Applying one factor to a heterogeneous base without checking its composition. If groups have different embedded levels, determine the appropriate weighted embedded level or adjust the groups separately.
- Assuming a premium label documents the calculation. Verify the data definition and workpaper sequence. A label such as “on-level premium” is not a substitute for tracing the factor actually applied.
Check Your Understanding
For each problem, identify the embedded level before calculating the adjustment.
- A hypothetical premium base of $500,000 has already been adjusted from index \(1.00\) to \(1.04\). The target index is \(1.10\). Find the residual factor and target-level premium. (Answer: Factor \(1.10/1.04=1.057692\ldots\); target premium $528,846.15.)
- A hypothetical insurer has $750,000 of raw premium at index \(1.00\), losses of $510,000, and a target index of \(1.08\). Find the correct target premium and loss ratio, then the loss ratio if the premium is mistakenly adjusted twice. (Answer: Correct premium $810,000 and loss ratio 62.96%; double-adjusted premium $874,800 and loss ratio 58.30%.)
- A recorded premium base consists of $300,000 of reference-level premium at index \(1.00\) and $400,000 at index \(1.05\). The target index is \(1.12\). Find the recorded base, the residual factor, and the target premium. (Answer: Recorded base $720,000; residual factor \(784{,}000/720{,}000=1.088889\); target premium $784,000.)
- A hypothetical base of $1,200,000 has already been adjusted to index \(1.06\). The target index is \(1.09\), and losses are $720,000. Calculate the correct target premium and loss ratio. Also find the loss ratio if someone multiplies the base by 1.09. (Answer: Correct factor \(1.09/1.06=1.028302\); target premium $1,233,962.26 and loss ratio 58.35%; erroneous loss ratio 55.05%.)