On-Leveling Premiums for a Rate Indication

Why Premiums Must Be On-Leveled

A rate indication compares the premium available at the current rate level with the losses and other costs for the experience being analyzed. Historical earned premium, however, was collected under the rates in force when policies earned that premium. Comparing historical losses with unadjusted historical premium can therefore confuse a change in rate level with a change in underlying risk or adequacy.

On-leveling restates earned premium to a selected common rate level, usually the rate level in effect at the valuation date. The adjustment is not a projection of future premium volume, and it does not change the historical losses. It answers a narrower question: what premium would the earned exposure have produced if the selected rate level had applied?

Definition: A rate-level index represents the relative premium level associated with a set of rates. Set a reference level to 1.00. A rate change of $r$ percent changes the index by the multiplicative factor $1+r$, where $r$ is expressed as a decimal. The on-level factor for premium earned at index $I_h$, restated to current index $I_c$, is $I_c/I_h$. On-level earned premium is historical earned premium multiplied by that factor, with appropriate weighting when the experience spans multiple rate levels.

For example, a 7% increase multiplies the prior index by 1.07. A later 4% decrease multiplies the resulting index by 0.96; it does not subtract four percentage points from the original index. The combined index is $1.07 \times 0.96=1.0272$, or 2.72% above the starting level.

Applying Rate-Level Factors

If all premium in an experience segment was earned at one known rate level, the calculation is direct. Multiply the segment's earned premium by the ratio of the selected current index to its historical index. When an experience period contains more than one rate level, calculate the adjustment for each segment and add the adjusted amounts.

P_{\mathrm{OL}}=\sum_{j=1}^{m} P_j\frac{I_c}{I_j}

Here $P_j$ is reported earned premium for segment $j$, $I_j$ is the rate-level index applicable to that earned premium, and $I_c$ is the selected current index. Segments must reflect the rate level at which premium was actually earned, not merely the date on which a rate change was filed or announced.

The factor is a ratio, not the percentage rate change itself. If the current index is 1.10 and the historical index is 1.00, the on-level factor is 1.10. If the current index is 1.10 and the historical index is 1.05, the factor is $1.10/1.05$, not 1.05 or 1.10.

Worked Example: Restating Premium From One Historical Level

A hypothetical insurer reports $1,260,000 of earned premium for a segment earned entirely at a rate-level index of 0.90. The selected current index is 1.08. Calculate on-level earned premium and state the method and a common mistake to avoid.

Method. Apply the ratio of the current index to the historical index:

\text{On-level factor}=\frac{1.08}{0.90}=1.20

Thus, the current-level equivalent is:

P_{\mathrm{OL}}=\$1{,}260{,}000(1.20)=\$1{,}512{,}000

Check. The current index is 20% above the historical index, since $1.08/0.90=1.20$. The adjusted premium is therefore 20% above $1,260,000, which is $1,512,000.

Common mistake. Multiplying by $0.90/1.08$ would produce approximately $1,050,000. That reverses the adjustment and moves the historical premium farther below the current level. To restate premium upward from index 0.90 to 1.08, the multiplier must exceed 1.

Why the Ratio Works

The rate-level adjustment follows from the assumption that, for the same earned exposure and coverage mix, premium is proportional to the applicable rate level. Let $B$ denote the premium that the segment's exposure would produce at index 1.00. At historical index $I_j$, its earned premium is $P_j=B I_j$. At current index $I_c$, that same exposure would produce $B I_c$. Substituting $B=P_j/I_j$ gives:

B I_c=\frac{P_j}{I_j}I_c=P_j\frac{I_c}{I_j}

This also shows the method's limits. The ratio assumes that the rate change applies proportionally to the relevant premium and that the segment's exposure and coverage mix can be treated consistently. If the rate change affects only some coverages, classifications, or policy groups, an aggregate factor may not represent the actual premium change for the whole segment. Use appropriately separated segments or a more detailed calculation when the available data support it.

Rate-Change Timing and Premium Earning

A rate change's effective date does not necessarily mean that all earned premium immediately moves to the new level. For annual policies, a rate change commonly applies as policies renew. If renewals occur throughout the year, policies already in force may continue earning at the old level while renewing policies begin earning at the new level. The share of premium earned at the new level then builds over time.

A useful timing calculation is possible under a specific simplifying assumption: policies have 12-month terms, earn premium uniformly through the term, renew uniformly throughout the year, and the rate change applies at renewal. Under these conditions, the proportion of earned premium at the new level rises approximately linearly from zero to one during the 12 months after the effective date. This is often represented by a parallelogram earning pattern. The resulting average level depends on the experience-period dates; it is not automatically the full-year average of old and new rates.

Worked Example: Midyear Increase With Uniform Renewal Earning

A hypothetical insurer's 12% increase takes effect July 1, 2025, at renewal. Consider calendar-year 2025. Assume annual policies earn premium uniformly over 12 months and renew uniformly throughout the year. Assume the underlying premium at the old level is constant by month. The current level for the calculation is the new level. Find the average rate level earned during 2025 and the factor to on-level $2,030,000 of 2025 earned premium.

Step 1: Establish the rate indices. Set the old level to 1.00. The new level is $1.00(1.12)=1.12$.

Step 2: Determine the new-level earning share. The increase takes effect at the start of July. During January through June, the new-level share is zero. For July through December, the monthly average shares are $0.5/12,\ 1.5/12,\ 2.5/12,\ 3.5/12,\ 4.5/12,$ and $5.5/12$. Their sum is $18/12=1.5$ month-equivalents of fully new-level premium. Across the full 12-month experience period, the average new-level share is $1.5/12=0.125$.

Step 3: Calculate the average historical level. A share of 0.875 is at index 1.00 and a share of 0.125 is at index 1.12:

I_{\mathrm{avg}}=(0.875)(1.00)+(0.125)(1.12)=0.875+0.140=1.015

Step 4: Restate premium to the new level. The average on-level factor is $1.12/1.015=1.103448\ldots$. Applying it to reported earned premium gives:

\$2{,}030{,}000\left(\frac{1.12}{1.015}\right)=\$2{,}240{,}000

The on-level earned premium is $2,240,000. The check is that $1.015 \times \$2,240,000=\$2,273,600$, which also equals $1.12 \times \$2,030,000$. Round money to the nearest cent only at the final step; the result here is exact.

Common mistake. Applying 1.12 to all 2025 premium would assume the full year earned at the new level, ignoring the renewal transition. Conversely, treating all 2025 premium as old-level would ignore the premium that did earn at the increased level. The timing pattern makes the average historical index 1.015 under the stated assumptions.

The linear earning approximation is not a universal rule. Actual renewal patterns, policy term lengths, seasonality, endorsements, and the precise effective-date convention can change the earning pattern. If policy-level effective dates and earned premium are available, direct segmentation by the rate level actually earned is generally preferable to an aggregate approximation.

Multiple Changes and the Indicated Loss Ratio

With successive changes, compound the changes to establish each index. Then apply the current-to-historical ratio to each premium segment. A loss ratio based on on-level earned premium is more comparable across experience periods than one using unadjusted premium, provided the losses and premium are otherwise on consistent bases. This adjustment isolates rate-level differences; it does not correct for loss development, trend, exposure changes, or other adjustments.

Worked Example: Two Changes and Their Effect on a Loss Ratio

A hypothetical insurer has an old rate index of 1.00, followed by a 10% increase and then a 5% decrease. The current index is the level after both changes. Reported earned premium consists of $320,000 earned at the old level, $280,000 earned after the increase but before the decrease, and $200,000 earned after both changes. For illustration, assume the associated losses have already been adjusted for any loss-side considerations and total $1,440,720. Calculate on-level premium and compare the raw and on-level loss ratios. Round loss ratios to four decimal places.

Step 1: Calculate each rate index. The old index is 1.00. After the increase it is $1.00(1.10)=1.10$. After the decrease it is $1.10(0.95)=1.045$. The current index is therefore 1.045, not 1.05.

Step 2: Adjust each premium segment. The three factors are:

\frac{1.045}{1.00}=1.045,\qquad \frac{1.045}{1.10}=0.95,\qquad \frac{1.045}{1.045}=1.00

The corresponding on-level amounts are $ \$320{,}000(1.045)=\$334{,}400$, $ \$280{,}000(0.95)=\$266{,}000$, and $ \$200{,}000(1.00)=\$200{,}000$. Their sum is:

P_{\mathrm{OL}}=\$334{,}400+\$266{,}000+\$200{,}000=\$800{,}400

Step 3: Compare loss ratios. Reported premium totals $ \$320{,}000+\$280{,}000+\$200{,}000=\$800{,}000$. Thus the raw loss ratio is $ \$1{,}440{,}720/\$800{,}000=1.8009$, or 180.0900%. The on-level loss ratio is $ \$1{,}440{,}720/\$800{,}400=1.8000$, or 180.0000%.

Check. Each segment's premium has been moved to index 1.045. Premium already at that index remains unchanged; premium at lower indices increases, while the segment at 1.10 decreases when restated to 1.045. The on-level total exceeds reported premium by $400, consistent with these adjustments.

Common mistake. Treating the 10% increase and 5% decrease as a net 5% increase gives an incorrect current index of 1.05. The changes are multiplicative: $1.10(0.95)=1.045$. Using 1.05 would apply the wrong factor to every historical segment and distort the resulting loss ratio.

Practical Checks Before Using an Aggregate Factor

A single factor is convenient, but it is defensible only when it represents the mix of premium actually earned. When the experience spans multiple levels, one can either adjust each segment explicitly or calculate an appropriate weighted average. Be careful about what supplies the weights. Rate indices describe premium at a given level for comparable exposure; weights should reflect the relevant exposure or base-premium mix, not be chosen mechanically from the reported premium shares when those shares themselves differ because of the rate levels.

Useful checks include:

Confirm the target. State the current or selected rate level to which premium is being restated.
Compound changes. Multiply each successive percentage-change factor rather than adding percentages.
Use earning dates. Identify when premium was earned at each level, not only when rates were approved or written.
Check factor direction. If the target index exceeds a segment's index, its factor must exceed 1; if it is lower, the factor must be below 1.
Keep the comparison consistent. On-leveling premium alone does not make losses current or otherwise adjust the loss side of an indication.

On-level earned premium matters because the premium denominator influences measured loss ratios and, in turn, the apparent adequacy of rates. An overstated on-level premium depresses the measured loss ratio; an understated on-level premium elevates it. Rate-change timing is therefore not a cosmetic detail. A filing analysis should make its effective-date and earning assumptions explicit and apply them consistently. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, identify the relevant rate levels and timing assumptions before calculating.

A hypothetical segment has $540,000 of earned premium at index 0.96. The selected current index is 1.056. Calculate its on-level earned premium. (Answer: $594,000)
A rate level of 1.00 is increased by 8%, then decreased by 3%. What is the resulting index? (Answer: 1.0476)
A hypothetical book has $150,000 earned at index 1.00 and $250,000 earned at index 1.05. The current index is 1.05. Calculate total on-level premium. (Answer: $407,500)
A 10% increase applies at renewal on July 1. Under the 12-month, uniform-renewal and uniform-earning assumptions used in this tutorial, what is the average rate index for the calendar year that begins January 1, with the old index set to 1.00? (Answer: 1.0125)
A hypothetical experience segment has $400,000 earned at index 1.00 and $600,000 earned at index 1.10. The current index is 1.10, and the losses are $900,000. Calculate on-level premium and the on-level loss ratio to four decimal places. (Answer: $1,040,000; 86.5385%)

What You'll Learn