Why Earned Premium Must Be On-Leveled
A loss ratio compares losses with the premium associated with the exposure that produced them. When earned premium was written at historical rates, it does not necessarily represent the premium that the same exposure would produce at the rate level being evaluated. Before using that premium in a loss ratio indication, restate it to a common rate level.
The adjustment changes the premium denominator, not the historical losses. If the experience contains premium earned at more than one rate level, calculate a factor for each level and adjust each corresponding premium amount. This keeps the calculation tied to the rate actually underlying each part of the earned premium.
A rate-level index is a relative measure of premium rates, with the same coverage and exposure basis assumed across the levels being compared. For example, if the index moves from 1.00 to 1.10, the rate level has increased by 10%. An index of 1.20 is 20% above the 1.00 base; it is not a statement that every premium amount in the experience was 20% higher than the immediately preceding level.
The On-Level Factor
For premium earned at actual rate level \(R_j\), the on-level factor to target level \(R_T\) is the ratio of those levels. Apply it to the premium earned at \(R_j\). The factor can be above, below, or equal to 1, depending on whether the target level is higher than, lower than, or equal to the actual level.
The formula follows from the meaning of a rate-level index. Suppose a fixed quantity of exposure would produce premium \(qR_j\) at level \(R_j\), where \(q\) represents the premium at the index base. If the recorded premium is \(P_j=qR_j\), then the premium for the same exposure at the target level is \(qR_T\). Substituting \(q=P_j/R_j\) gives \(qR_T=P_j(R_T/R_j)\). Thus the ratio converts the recorded premium to the target level without changing the exposure represented.
This reasoning assumes the premium being adjusted is comparable across the rate levels: the exposure and coverage basis is consistent, and the rate-level indices correctly describe the rates underlying the premium. If the experience includes materially different coverage, exposure mix, or premium components, those differences should not be silently treated as rate changes. This calculation also requires premium amounts assigned to their actual rate levels; a calendar-time fraction alone does not establish how much earned premium belongs to each level.
Worked Example: One Historical Level and One Target Level
A hypothetical commercial equipment portfolio has $600,000 of earned premium at rate-level index 1.00 and $440,000 at index 1.10. The loss ratio analysis calls for all earned premium at target index 1.20. The matched losses are $840,000. Calculate the on-level earned premium and actual loss ratio.
Method. Apply the target-to-actual rate-level ratio to each premium amount. The first premium amount has factor \(1.20/1.00=1.20\). The second has factor \(1.20/1.10=1.090909\ldots\).
For premium at index 1.00:
For premium at index 1.10:
The on-level earned premium is \( \$720{,}000+\$480{,}000=\$1{,}200{,}000 \). Using losses of $840,000, the actual loss ratio is:
Check. At the target level, the two premium components total $1,200,000, and 70% of that amount is $840,000, the stated loss amount. The common mistake avoided here is applying the same factor to both components even though they were earned at different rate levels.
Experience With Several Rate Levels
A single experience period can include premium earned under several rate changes. The correct procedure is to identify each rate level, pair it with the premium earned at that level, and adjust each pair to the same target. Rate changes are reflected in the indices themselves: if a level is 1.10 and a later change decreases it by 5%, the new index is \(1.10(0.95)=1.045\), not 0.95.
State the common rate-level index to which the experience premium will be restated.
For each component, identify the index that applies to that earned premium.
For each component, divide the target index by its actual index, then multiply by the component's earned premium.
Add the adjusted premiums and use that total in the loss ratio calculation with matched losses.
Worked Example: A Decrease Before the Target Level
A hypothetical inland marine book has earned premium of $300,000 at index 1.00, $500,000 at index 1.10, and $200,000 at index 1.045. The last level resulted from a 5% decrease applied to index 1.10. The target is 1.045, and matched losses are $690,000. Find the total on-level earned premium and actual loss ratio.
Method. First verify the rate-level sequence: \(1.10(1-0.05)=1.045\). Then adjust each premium to 1.045.
At index 1.00, the factor is \(1.045/1.00=1.045\), giving \( \$300{,}000(1.045)=\$313{,}500 \).
At index 1.10, the factor is \(1.045/1.10=0.95\), giving \( \$500{,}000(0.95)=\$475{,}000 \).
At index 1.045, the factor is \(1.045/1.045=1.00\), giving \( \$200{,}000 \).
Therefore:
The actual loss ratio is \( \$690{,}000/\$988{,}500=0.698027\ldots \), or 69.80% to two decimal places.
Check. Premium at the target level remains unchanged, while premium at 1.10 is reduced by 5% to the target index of 1.045. The mistake avoided is using 0.95 as the new rate-level index: 0.95 is the change factor, not the resulting index.
Using the Adjusted Premium in a Loss Ratio
Once premium has been put at a common rate level, calculate the loss ratio using losses matched to the same experience period and the total on-level earned premium. Do not calculate separate period ratios and take an unweighted average when the goal is the aggregate loss ratio. The aggregate is total losses divided by total adjusted premium.
Worked Example: Combining Periods After On-Leveling
A hypothetical specialty property insurer is comparing two experience periods at target index 1.20. In Period A, earned premium was $500,000 at index 1.00 and $220,000 at index 1.10; losses were $588,000. In Period B, earned premium was $200,000 at index 1.10 and $300,000 at index 1.20; losses were $311,000. Calculate the combined actual loss ratio at index 1.20.
Method. On-level each premium component first, then combine the periods' losses and adjusted premiums.
For Period A, the factors are \(1.20/1.00=1.20\) and \(1.20/1.10=12/11\). Its adjusted premium is:
For Period B, the factors are \(12/11\) for index 1.10 and 1.00 for index 1.20. Its adjusted premium is:
The combined on-level premium is \( \$840{,}000+\$518{,}181.82=\$1{,}358{,}181.82 \). Combined losses are \( \$588{,}000+\$311{,}000=\$899{,}000 \). Therefore:
Check. Both periods' premiums now represent index 1.20, and the numerator and denominator each include both periods. The common mistake avoided is averaging period loss ratios without regard to the adjusted premium weights; the combined ratio is formed from totals.
Common Mistakes and Practical Checks
A stated rate change and an on-level factor answer different questions. A 10% increase describes the movement from one rate level to another. The factor needed to move premium from an actual level to a target is the ratio of those two levels. For example, moving from index 1.10 to index 1.20 requires \(1.20/1.10=1.090909\ldots\), not a factor of 1.10.
- Reversing the ratio. To move premium from actual level \(R_j\) to target \(R_T\), use \(R_T/R_j\). The reverse ratio moves in the opposite direction.
- Applying one factor to unlike premium components. Premium at different actual levels needs different factors unless the components happen to share the same level.
- Adjusting losses as though they were premium. On-leveling earned premium does not, by itself, change the historical loss amount.
- Using calendar time as an unsupported premium split. A fraction of the year is not automatically the same fraction of earned premium at a particular level. Use a supported allocation of premium by actual rate level.
- Dividing by unadjusted premium. If the indication calls for a common target level, the loss ratio denominator must be the on-level premium, not the original earned premium.
Check Your Understanding
For each problem, calculate the requested result using the stated target rate level. Round loss ratios to two decimal places.
- A hypothetical repair-liability portfolio has $240,000 earned at index 1.00. Restate it to index 1.08. (Answer: $259,200)
- A hypothetical crop book has $180,000 earned at index 1.20. Its target is index 1.26. Calculate the adjusted premium. (Answer: $189,000)
- A hypothetical small-business portfolio has $100,000 earned at index 1.00 and $220,000 at index 1.10. The target is 1.21. Calculate total on-level earned premium. (Answer: $363,000)
- A hypothetical equipment insurer has $300,000 earned at index 1.00 and $330,000 at index 1.10, with matched losses of $525,000. Restate premium to target index 1.21 and calculate the loss ratio. (Answer: $726,000 on-level premium; 72.31%)
- A hypothetical marine portfolio has $400,000 earned at index 1.00 and $200,000 at index 1.10, with matched losses of $489,000. Calculate the loss ratio at target index 1.20. (Answer: $698,181.82 on-level premium; 70.04%)