Why an On-Level Factor Is Needed
Premium collected or earned during an experience period may reflect rate levels that differ from the level at which a ratemaking analysis is intended to evaluate that premium. Comparing the unadjusted premium with losses can therefore mix two effects: the amount of exposure insured and the rates charged for it. An on-level adjustment restates the experience premium at a specified target rate level, so the premium and the target level are on a consistent basis.
The factor is a ratio: target rate level divided by the average rate level underlying the experience premium. This direction matters. If the target level is higher than the experience level, the factor should increase premium; if it is lower, the factor should decrease premium. The calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
What the Formula Assumes
Rate levels are expressed relative to a common reference level. For example, a level of \(1.00\) can represent the reference, while \(1.10\) represents a level 10% above it. A rate level is not itself a premium amount. It describes how the rates compare with the reference rates.
The formula treats premium as proportional to rate level for a fixed body of exposure. It assumes that the premium being adjusted is attributable to the rates represented by \(L_{\mathrm{exp}}\), and that the target level \(L_T\) is expressed on the same reference basis. It also assumes that the adjustment is intended to change only the rate-level basis—not the exposure, coverage, classification mix, or other components of the premium. If those features change, multiplying by this factor alone does not account for them.
The average level \(L_{\mathrm{exp}}\) represents the rate level embedded in the experience premium. A problem may provide it directly. If experience premium consists of groups with different rate levels, each group can be converted separately; collapsing groups requires a correctly determined average level, not automatically the simple arithmetic mean of their levels. The detailed calculation of average levels across policies is a separate issue.
Why the Ratio Works
The formula follows from the proportional relationship between premium and rate level. Consider a fixed set of exposures and let \(C\) denote the premium those exposures would produce at the common reference level. At rate level \(L\), the premium is \(CL\). Thus the experience premium and target-level premium are respectively \(CL_{\mathrm{exp}}\) and \(CL_T\). Dividing eliminates the common exposure-and-reference amount \(C\):
Therefore, multiplying the experience premium by \(L_T/L_{\mathrm{exp}}\) converts it to the target level. The argument depends on the same exposures being compared and premium scaling proportionally with rate level. It does not imply that losses should be multiplied by the on-level factor: this factor adjusts premium for rate-level differences.
Confirm which experience premium is being converted and identify its average underlying rate level.
Express the target rate level on the same reference basis as the experience level.
Divide target level by experience level, not the reverse.
Multiply experience premium by the factor. A target above the experience level must produce a factor above 1; a target below it must produce a factor below 1.
Applying the Factor
Worked Example: Converting Premium From a Lower Average Level
A hypothetical inland-watercraft program reports \( \$1{,}260{,}000 \) of experience premium. The average rate level underlying that premium is \(0.96\), and the selected target level is \(1.08\). Calculate the on-level factor and the converted premium. State the method and check the direction of the adjustment.
Method. Use the target-to-experience ratio. Because \(1.08\) is greater than \(0.96\), the factor should be greater than 1 and the premium should increase.
Step 1: Calculate the factor. $$ F_{\mathrm{OL}}=\frac{1.08}{0.96}=1.125. $$
Step 2: Apply the factor. $$ P_{\mathrm{OL}}=\$1{,}260{,}000(1.125)=\$1{,}417{,}500. $$
Check. The factor is 12.5% above 1, consistent with moving from a lower level to a higher one. Also, \(1.08/0.96=1.125\), so the converted premium is \(12.5\%\) above the original: \( \$1{,}260{,}000(0.125)=\$157{,}500\), and \( \$1{,}260{,}000+\$157{,}500=\$1{,}417{,}500\). The common mistake avoided is reversing the ratio, which would incorrectly reduce this premium. Money is reported to the cent; here the result is an exact whole-dollar amount.
Worked Example: Using a Target After Sequential Rate Changes
A hypothetical equipment policy program has successive overall rate changes of \(+10\%\), \(-5\%\), and \(+8\%\), each measured against the level immediately before that change. For the experience premium of \( \$840{,}000 \), the average underlying level is \(1.08\). The target is the level after a further \(+4\%\) change from the level following the three stated changes. Find the on-level factor and target-level premium. Use compounding for the sequential changes and explain why the last stated level is not automatically the experience average.
Step 1: Compound the stated changes to find the level after them. $$ 1.00(1.10)(0.95)(1.08)=1.1286. $$ The \(5\%\) decrease is represented by \(0.95\), not by subtracting five percentage points from the accumulated level.
Step 2: Apply the additional target change. $$ L_T=1.1286(1.04)=1.173744. $$
Step 3: Divide the target by the given experience average. $$ F_{\mathrm{OL}}=\frac{1.173744}{1.08}=1.0868. $$
Step 4: Convert the premium. $$ P_{\mathrm{OL}}=\$840{,}000(1.0868)=\$912{,}912. $$
Check. The target is above the average experience level, and the factor \(1.0868\) increases premium by \(8.68\%\). The increase is \( \$840{,}000(0.0868)=\$72{,}912\), giving \( \$840{,}000+\$72{,}912=\$912{,}912\). The common mistake avoided is using \(1.1286\), the level after the three changes, as the experience average. The problem supplies \(1.08\) as the average level underlying the premium; that is the denominator required by the formula.
Worked Example: Converting Two Premium Segments Separately
A hypothetical specialty insurer has two experience-premium segments. Segment A has \( \$500{,}000 \) of premium at an average level of \(0.90\); Segment B has \( \$700{,}000 \) at an average level of \(1.05\). Both are to be restated at target level \(1.08\). Find the converted premium for each segment and the total. Explain why applying a factor based on the simple average of the two levels is not valid here.
Segment A. Its factor is \(1.08/0.90=1.20\). Therefore, $$ P_{\mathrm{OL},A}=\$500{,}000(1.20)=\$600{,}000. $$
Segment B. Its factor is \(1.08/1.05=1.028571\ldots\). Therefore, $$ P_{\mathrm{OL},B}=\$700{,}000\left(\frac{1.08}{1.05}\right) =\$700{,}000\left(\frac{36}{35}\right)=\$720{,}000. $$
Total converted premium. $$ P_{\mathrm{OL}}=\$600{,}000+\$720{,}000=\$1{,}320{,}000. $$
Check and trap. Segment A increases because its target exceeds \(0.90\); Segment B also increases, but by a smaller percentage because its average is already \(1.05\). A simple mean of the levels, \((0.90+1.05)/2=0.975\), ignores that the segments have different premium amounts and does not supply the correct combined experience level. Using it would produce the incorrect result \(\$1{,}200{,}000(1.08/0.975)=\$1{,}329{,}230.77\), rather than the segment-by-segment total of \(\$1{,}320{,}000\). The safe method is to convert each segment at its own stated average level and then add the converted amounts. Money is rounded to the nearest cent only at the final step.
Interpretation and Common Exam Traps
The on-level factor is a conversion multiplier, not a new rate level and not a percentage change to be added to premium. For example, a factor of \(1.06\) means multiply the experience premium by \(1.06\); it does not mean add \(0.06\) dollars or divide by \(1.06\). Likewise, if the target is \(0.94\) and the experience average is \(1.00\), the factor is \(0.94\): the converted premium is 6% lower.
A useful check is to compare the factor with 1 before doing the final multiplication. If \(L_T>L_{\mathrm{exp}}\), then \(F_{\mathrm{OL}}>1\). If \(L_T<L_{\mathrm{exp}}\), then \(F_{\mathrm{OL}}<1\). If the two levels are equal, the factor is 1 and the premium is unchanged. A factor moving in the opposite direction usually signals that the ratio has been reversed.
- Reversing numerator and denominator. Use target level divided by experience level. The opposite ratio converts in the wrong direction.
- Using a current or final level without checking the data basis. The denominator is the average level underlying the experience premium, which may differ from the most recent level.
- Adding sequential percentage changes. When sequential changes are supplied to establish a level, compound their multipliers. For example, \(+10\%\) followed by \(-5\%\) gives \(1.10(0.95)=1.045\), not \(1.05\).
- Applying the factor to losses. The formula restates premium for rate-level differences. It is not a loss trend or loss adjustment.
- Combining unlike segments prematurely. If segments have different average levels, convert them separately unless a valid combined average level has been established.
The result is useful because it places experience premium on a consistent rate-level basis for later ratemaking calculations. The factor itself does not select the experience period or target level; those are inputs to this calculation. Its role is precise: given the right premium basis and the two comparable levels, it converts the premium by their ratio.
Check Your Understanding
For each problem, calculate the factor and converted premium unless the question asks otherwise. Treat all scenarios and data as hypothetical.
- A hypothetical forestry-equipment portfolio has \( \$320{,}000 \) of experience premium at average level \(0.80\). The target is \(0.92\). Calculate the factor and target-level premium. (Answer: \(F_{\mathrm{OL}}=1.15\); premium \(=\$368{,}000\).)
- A hypothetical marine account has \( \$2{,}400{,}000 \) of experience premium at level \(1.04\). The target is 5% above that level. Find the target level, factor, and converted premium. (Answer: target level \(1.092\); factor \(1.05\); premium \(=\$2{,}520{,}000\).)
- A hypothetical insurer reports two segments: \( \$180{,}000 \) at average level \(0.90\), and \( \$270{,}000 \) at average level \(1.08\). Convert both to target level \(1.17\) and add them. (Answer: segment premiums \(=\$234{,}000\) and \(\$292{,}500\); total \(=\$526{,}500\).)
- A hypothetical program’s sequential rate changes are \(+12\%\) and then \(-10\%\). The experience premium is \( \$510{,}000 \), its average underlying level is \(1.02\), and the target level is \(1.08\). Calculate the cumulative level after the two changes and the converted premium. (Answer: cumulative level \(=1.008\); factor \(=18/17\approx1.0588\); premium \(=\$540{,}000\).)
- A hypothetical coverage has \( \$640{,}000 \) of experience premium at average level \(1.12\), to be converted to target level \(0.98\). Calculate the factor and premium, and state whether premium increases or decreases. (Answer: factor \(=0.875\); premium \(=\$560{,}000\); it decreases.)