Why the Adjustment Sequence Matters
On-leveling and trending both change historical data, but they answer different questions. On-leveling restates premium to a specified rate level. Trending projects loss costs from an experience period to a later cost level. Confusing the two can produce plausible arithmetic on the wrong basis: an accurate multiplication is not a valid adjustment unless its factor applies to that component.
The practical sequence is therefore not “multiply every number by every factor.” First identify what each quantity represents; then apply the factor that changes it to the intended basis. In a standard rate-level and loss-trend adjustment, earned premium is on-leveled, while losses are trended. Exposure counts are not multiplied by a loss-dollar trend factor. The period definitions and endpoints used to construct the factors must also agree with the data being adjusted.
What Each Factor Changes
An on-level factor expresses the relationship between the premium at the selected rate level and the premium actually earned, with the relevant exposure and coverage basis held consistent. If the selected rate level is higher than the average level earned, the factor will generally exceed 1; if it is lower, the factor can be below 1. The factor is applied to premium, not to losses merely because those losses arose during the same period.
A loss trend factor expresses the change in the loss-cost level between the chosen experience endpoint and the prospective endpoint. It is applied to losses on the basis for which that trend was selected. For example, a trend factor for total losses should not be substituted for a factor defined for a different measure or endpoint.
An exposure count describes the amount of insured exposure, not its dollar cost. A loss-dollar trend changes the projected cost associated with exposure; it does not convert one exposure unit into another. If the exposure definition itself changes, that issue requires its own treatment rather than an automatic loss trend adjustment.
| Data component | Adjustment in this framework | Purpose |
|---|---|---|
| Earned premium | Multiply by its on-level factor | Restate premium at the selected rate level |
| Losses | Multiply by their loss trend factor | Restate losses at the selected cost level |
| Exposure count | No loss-dollar trend factor | Retain the exposure measure unless a separate exposure adjustment is required |
The Algebra Behind the Sequence
The formulas follow from the meaning of each factor. If an on-level factor is the ratio of premium at the selected rate level to premium at the actual earned rate level, then multiplying actual earned premium by that ratio gives premium at the selected level. Likewise, if a loss trend factor is the ratio of the selected future loss-cost level to the historical loss-cost level, multiplying historical losses by it restates those losses at the future level, assuming the trend applies proportionally to the losses being adjusted.
The multiplication itself is commutative: for a quantity \(x\) that is legitimately subject to both independent factors, \((xO)T=x(OT)=(xT)O\). This is an algebraic identity, not a rule that every quantity should receive both adjustments. In the usual premium-loss comparison, premium and losses are separate bases: premium receives \(O_j\), and losses receive \(T_j\). Applying both factors to both components changes the question being answered.
Separate premium, losses, and exposure, and identify the period represented by each amount.
Use the on-level factor for premium and the loss trend factor for losses. Confirm each factor's target level and endpoint.
Calculate \(P_jO_j\) and \(L_jT_j\) for each period before summing the adjusted amounts.
Verify the arithmetic and confirm that no factor was applied to a component outside its defined basis.
Worked Applications
Worked Example: Adjusting Premium and Losses Separately
A hypothetical insurer has two experience periods. Period A has earned premium of $800,000 and losses of $500,000; its on-level factor is 1.10 and its loss trend factor is 1.08. Period B has earned premium of $600,000 and losses of $420,000; its on-level factor is 1.04 and its loss trend factor is 1.03. Find total on-leveled premium and total trended losses.
Method. Apply each factor to its defined component, period by period.
For Period A, on-leveled premium is \(800{,}000(1.10)=880{,}000\). Trended losses are \(500{,}000(1.08)=540{,}000\).
For Period B, on-leveled premium is \(600{,}000(1.04)=624{,}000\). Trended losses are \(420{,}000(1.03)=432{,}600\).
Add the corresponding adjusted components: total on-leveled premium is \(880{,}000+624{,}000=\$1{,}504{,}000\). Total trended losses are \(540{,}000+432{,}600=\$972{,}600\).
Check. The original premium total is $1,400,000, and the adjusted total is higher, consistent with both on-level factors exceeding 1. The original loss total is $920,000, and its adjusted total is also higher, consistent with both trend factors exceeding 1. Common mistake avoided: this method does not multiply each premium by its loss trend factor or each loss by its on-level factor.
Worked Example: Finding and Correcting a Cross-Application Error
A hypothetical analyst uses the same two-period data and factors as in the previous example. The analyst multiplies both premium and losses by both factors in each period. Calculate the analyst’s totals, then correct them.
Method. First reproduce the incorrect calculation to identify its effect. In Period A, the combined factor is \(1.10(1.08)=1.188\). In Period B, it is \(1.04(1.03)=1.0712\).
The analyst’s premium total is \(800{,}000(1.188)+600{,}000(1.0712)=950{,}400+642{,}720=\$1{,}593{,}120\). The analyst’s loss total is \(500{,}000(1.188)+420{,}000(1.0712)=594{,}000+449{,}904=\$1{,}043{,}904\).
Those products are arithmetically correct but do not match the requested adjustments. Premium should receive only its on-level factors, giving \(800{,}000(1.10)+600{,}000(1.04)=880{,}000+624{,}000=\$1{,}504{,}000\). Losses should receive only their trend factors, giving \(500{,}000(1.08)+420{,}000(1.03)=540{,}000+432{,}600=\$972{,}600\).
Check. Both erroneous totals exceed their correct counterparts because the analyst applied an additional factor greater than 1 to every component. Common mistake avoided: “apply both adjustments” does not mean apply both factors to every line of data.
Worked Example: Verifying Which Operations May Be Reordered
A hypothetical review uses an amount of $250,000 that has been established as a valid common base for two specified, independent adjustments: an on-level factor of 0.96 and a trend factor of 1.10. Separately, the review has $300,000 of earned premium subject only to the on-level factor and $180,000 of losses subject only to the trend factor. Calculate the adjusted amounts and verify the algebraic ordering for the common-base amount.
Method. For the common-base amount, applying the on-level factor first gives \(250{,}000(0.96)=240{,}000\), followed by \(240{,}000(1.10)=\$264{,}000\). Applying the trend factor first gives \(250{,}000(1.10)=275{,}000\), followed by \(275{,}000(0.96)=\$264{,}000\). The equality follows because \(0.96(1.10)=1.056\), so either order gives \(250{,}000(1.056)=\$264{,}000\).
For the separate premium amount, apply only its on-level factor: \(300{,}000(0.96)=\$288{,}000\). For the losses, apply only their trend factor: \(180{,}000(1.10)=\$198{,}000\).
Check. The common-base amount is unchanged by the order of its two valid multipliers, while the premium and loss amounts receive different adjustments because their bases differ. Common mistake avoided: commutativity does not establish that both factors are appropriate for every amount.
Common Traps and Practical Checks
The most persistent error is to treat on-leveling and trending as interchangeable “uplifts.” They are not interchangeable in interpretation. A rate change affects premium at a given exposure and coverage basis; a loss trend represents change in the cost of losses over time. Even when both factors exceed 1, their numerical direction does not make them substitutes.
- Applying a loss trend to premium. Premium is restated for rate level; loss trend is not a premium adjustment in this framework.
- Applying an on-level factor to losses. A rate change does not automatically change the historical dollar amount of losses.
- Applying the same factor to exposure counts. A loss-dollar trend is not an exposure multiplier.
- Combining factors without checking endpoints. A factor is meaningful only for its stated starting and target levels. Misaligned endpoints cannot be repaired by changing multiplication order.
- Summing first when factors vary by period. In general, \(\sum_j P_jO_j\) is not equal to \((\sum_jP_j)\) times an arbitrary single factor. Use a weighted factor only if it is constructed to preserve the adjusted total.
A useful final audit is to write the adjusted components before doing the arithmetic: \(P_jO_j\) for premium and \(L_jT_j\) for losses. Then check that units are preserved, each factor's base and endpoint match the data, and the adjusted totals move in the direction implied by the factors. This disciplined sequence is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, identify the relevant base before calculating.
- A hypothetical book has $450,000 of earned premium with an on-level factor of 1.06 and $280,000 of losses with a trend factor of 1.05. Find the adjusted premium and losses. (Answer: on-leveled premium $477,000; trended losses $294,000.)
- Two periods have premium of $200,000 and $300,000, with on-level factors 1.10 and 0.98. Find total on-leveled premium. (Answer: $514,000.)
- Two periods have losses of $160,000 and $240,000, with trend factors 1.125 and 1.05. Find total trended losses. (Answer: $432,000.)
- A reviewer applies both an on-level factor of 1.08 and a trend factor of 1.04 to $500,000 of premium, then calls the result on-leveled premium. Find that result and the correct on-leveled premium. (Answer: the reviewer’s result is $561,600; correct on-leveled premium is $540,000.)
- A valid common-base amount of $125,000 is subject to an on-level factor of 0.92 and a trend factor of 1.15. Verify the result in either order. (Answer: $132,250 in either order.)