The Role of On-Level Premium in an Indication

Why Historical Premium Needs a Common Rate Level

Historical earned premium reflects the rates charged when the policies were in force. If rates changed during the experience period, the premium total therefore combines two things: the amount and type of business written, and the rate levels applied to that business. To compare historical experience with prospective costs, an actuary needs a premium measure that does not confuse differences in rate level with differences in the underlying experience.

On-level premium restates historical earned premium as though it had been earned at a specified common rate level. It provides a consistent premium denominator for evaluating experience across periods with different rates. The adjustment changes the premium measure; it does not change the historical losses.

Definition: On-level premium is historical earned premium adjusted to the selected common rate level. If the historical rate-level index is $I_h$ and the common rate-level index is $I_c$, the on-level factor is $I_c/I_h$, and the on-level premium is earned premium multiplied by that factor. The indices must use the same reference basis.

In practice, a historical period may contain policies earning premium under multiple rate levels. Its on-level factor must reflect the rate levels applying to the premium earned over the period, including how rate changes affect earning. The earlier material on on-leveling covers how those factors are constructed. Here, the central issue is their role: they restate premium to a common rate level so that differences in the premium denominator do not distort an experience comparison.

How On-Leveling Affects the Comparison

A loss ratio is losses divided by premium. If the rate level rises while the exposures and expected losses remain the same, premium rises and the loss ratio falls. That lower ratio does not, by itself, show that the underlying risks became less costly. It may simply reflect that more premium is now charged for the same exposure.

On-leveling addresses this rate-level effect by restating the historical premium at a common level before comparing it with losses. The losses are not multiplied by the on-level factor: the adjustment is specifically for the premium rate level. Losses may require separate adjustments for such matters as development or trend, but those are distinct from on-leveling.

P_h^{\mathrm{OL}}=P_h^{\mathrm{EP}}\frac{I_c}{I_h}

Here $P_h^{\mathrm{EP}}$ is historical earned premium, $I_h$ is the rate-level index applicable to that premium, $I_c$ is the selected common rate-level index, and $P_h^{\mathrm{OL}}$ is the resulting on-level premium. If a factor is supplied directly, use the equivalent form $P_h^{\mathrm{OL}}=P_h^{\mathrm{EP}}F_h$, where $F_h=I_c/I_h$.

The formula follows from the way premium responds to a rate change. For a fixed set of exposures and coverage, multiplying the rate level by a factor multiplies the corresponding premium by the same factor. Thus, to restate premium from historical level $I_h$ to common level $I_c$, multiply by $I_c/I_h$. If historical losses are held fixed to isolate the rate-level effect, the loss ratio using on-level premium becomes:

\frac{L_h}{P_h^{\mathrm{OL}}} = \frac{L_h}{P_h^{\mathrm{EP}}(I_c/I_h)}

For example, if the historical index is 0.80 and the common index is 1.00, the factor is 1.00/0.80 = 1.25. The historical premium is increased by 25% to express it at the higher common rate level. With losses held constant, the resulting loss ratio is lower than the ratio using unadjusted historical premium. This is a denominator adjustment, not an assertion that the historical losses themselves have changed.

Worked Applications

Worked Example: Restating One Period’s Premium

A hypothetical insurer earned $1,200,000 of premium during a period when the applicable rate-level index was 0.80. The selected common rate-level index is 1.00. Losses for the period are $840,000. Calculate the on-level premium and compare the loss ratios using raw and on-level premium. This is deterministic arithmetic; no interest rate, mortality assumption, or probability distribution is involved.

Method and factor. The common-to-historical index ratio is the on-level factor:

F_h=\frac{1.00}{0.80}=1.25

On-level premium. Multiply the historical earned premium by 1.25:

P_h^{\mathrm{OL}}=\$1{,}200{,}000(1.25)=\$1{,}500{,}000

Raw loss ratio. Using unadjusted earned premium gives:

\frac{\$840{,}000}{\$1{,}200{,}000}=0.70=70.00\%

Loss ratio on the common rate level. Using on-level premium gives:

\frac{\$840{,}000}{\$1{,}500{,}000}=0.56=56.00\%

The arithmetic checks: $1,200,000 multiplied by 1.25 is $1,500,000, and $840,000 divided by $1,500,000 is 0.56. The 14-percentage-point difference between the two ratios comes from changing the premium basis, not from changing the loss amount. The method avoids treating premium earned at a lower rate level as if it were already at the selected common level.

Worked Example: On-Leveling Two Rate-Level Groups

A hypothetical portfolio has two groups of earned premium. Group A earned $900,000 at rate-level index 0.90 and incurred $495,000 of losses. Group B earned $1,260,000 at index 1.05 and incurred $756,000 of losses. Restate each group to a common index of 1.00, then calculate the combined loss ratio using on-level premium. This is deterministic arithmetic; no interest rate, mortality assumption, or probability distribution is involved.

Group A. Its factor is $1.00/0.90=1.111111\ldots$. Its on-level premium is:

\$900{,}000\left(\frac{1.00}{0.90}\right)=\$1{,}000{,}000

Group B. Its factor is $1.00/1.05=0.952380\ldots$. Its on-level premium is:

\$1{,}260{,}000\left(\frac{1.00}{1.05}\right)=\$1{,}200{,}000

Combine the adjusted premiums and losses. Add the amounts after applying each group’s own factor:

P^{\mathrm{OL}}=\$1{,}000{,}000+\$1{,}200{,}000=\$2{,}200{,}000

L=\$495{,}000+\$756{,}000=\$1{,}251{,}000

The combined on-level loss ratio is:

\frac{\$1{,}251{,}000}{\$2{,}200{,}000}=0.568636\ldots\approx56.86\%

As a check, raw premium totals $900,000 + $1,260,000 = $2,160,000, and the raw loss ratio is $1,251,000/$2,160,000 = 57.92% when rounded to two decimals as a percentage. It differs because the groups’ premiums were earned at different rate levels. The method avoids applying one group’s factor to the other group’s premium.

Worked Example: Weighting Periods by Earned Premium

A hypothetical insurer is combining two experience periods. Period 1 has $2,000,000 of earned premium at index 0.80 and $1,500,000 of losses. Period 2 has $1,000,000 of earned premium at index 1.20 and $750,000 of losses. The common index is 1.00. Calculate total on-level premium and the combined loss ratio. Then compare with the result from incorrectly applying the simple average of the two factors to total raw premium. This is deterministic arithmetic; no interest rate, mortality assumption, or probability distribution is involved.

Period 1 factor and premium. The factor is $1.00/0.80=1.25$, so:

P_1^{\mathrm{OL}}=\$2{,}000{,}000(1.25)=\$2{,}500{,}000

Period 2 factor and premium. The factor is $1.00/1.20=0.833333\ldots$, so:

P_2^{\mathrm{OL}}=\$1{,}000{,}000\left(\frac{1}{1.20}\right)=\$833{,}333.33

Total on-level premium.

P^{\mathrm{OL}}=\$2{,}500{,}000+\$833{,}333.33=\$3{,}333{,}333.33

Total losses are $1,500,000 + $750,000 = $2,250,000. Therefore, the combined loss ratio on the common rate level is:

\frac{\$2{,}250{,}000}{\$3{,}333{,}333.33}=0.675=67.50\%

The simple average factor is $(1.25+0.833333\ldots)/2=1.041666\ldots$. Applying that average to the $3,000,000 of raw premium produces $3,125,000 of premium and a loss ratio of $2,250,000/$3,125,000 = 72.00%. That is not the correct aggregate result because the two factors should be weighted by their corresponding earned premiums. The correct aggregate factor is $3,333,333.33/$3,000,000 = 1.111111\ldots. The check is that applying each period’s factor separately and then adding gives the same $3,333,333.33 total.

What On-Leveling Does—and Does Not—Do

On-leveling is one part of building a consistent comparison basis. It answers a focused question: what would the historical earned premium have been if the relevant business had been charged at the selected common rate level? It does not automatically make all other aspects of historical and prospective experience comparable.

It adjusts premium for rate-level changes. The factor should correspond to the rate levels that generated the earned premium, not merely to a date label for the period.
It does not adjust losses for loss trend. Trend is a separate adjustment to reflect changes in claim costs over time. Applying an on-level factor to losses would confuse two different adjustments.
It does not correct every change in the portfolio. Shifts in territory, coverage, limits, deductibles, or other risk characteristics may require separate analysis. A common rate level alone does not remove those differences.
It does not make inconsistent data measures consistent. The premium base still needs to match the experience being analyzed, including the earned or written premium choice and the period definition.

A useful conceptual check is to ask what would happen if only the rates changed while the exposures and expected losses stayed fixed. The on-level premium should change to reflect the chosen common rate level, while losses remain on their original basis for this particular adjustment. That separation helps identify which differences are due to rates and which require another treatment.

Common Exam Trap: Reversing the Factor

Suppose historical premium of $800,000 was earned at index 0.80 and must be restated to index 1.00. A common error is to calculate $0.80/1.00=0.80$ and reduce the historical premium to $640,000. That factor goes in the wrong direction: it moves the premium farther from, not toward, the higher common level.

\text{Wrong: }\ \$800{,}000\left(\frac{0.80}{1.00}\right)=\$640{,}000 \qquad \text{Correct: }\ \$800{,}000\left(\frac{1.00}{0.80}\right)=\$1{,}000{,}000

A direction check prevents the error. When the common rate level is higher than the historical level, an unchanged set of exposures would generate more premium at the common level, so the on-level factor must exceed 1. Conversely, if the common level is lower than the historical level, the factor must be less than 1. This check does not replace verifying the underlying rate-level calculation, but it quickly catches an inverted ratio.

Key takeaway: On-level premium puts historical earned premium on a selected common rate level so rate changes do not masquerade as changes in experience. Apply the common-to-historical factor to premium, aggregate adjusted amounts rather than unweighted factors, and keep loss trend and other adjustments separate. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the common rate-level basis before calculating.

A hypothetical book earned $640,000 at rate-level index 0.80. The common index is 1.00. What is the on-level premium? (Answer: $800,000.)
A hypothetical segment earned $900,000 at index 1.125. Restate it to common index 1.00. (Answer: $800,000.)
Two hypothetical groups have earned premiums of $500,000 at index 0.80 and $700,000 at index 1.00. Their losses total $500,000, and the common index is 1.00. Find total on-level premium and the combined loss ratio. (Answer: $1,325,000 and 37.74%.)
Two hypothetical periods have earned premiums of $1,200,000 and $800,000, with on-level factors 1.10 and 0.95. Find total on-level premium and the earned-premium-weighted aggregate factor. (Answer: $2,080,000 and 1.040.)
A hypothetical period has $1,000,000 of earned premium at index 1.25 and $600,000 of losses. Restate premium to index 1.00 and calculate the loss ratio on that basis. (Answer: On-level premium is $800,000; the loss ratio is 75.00%.)

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