Cumulative On-Level Factors Across Rate Changes

Why Cumulative On-Level Factors Are Needed

A loss ratio indication compares losses with premium on a consistent rate level. If rates changed during or before the experience period, the earned premium may reflect several different levels. To compare it with losses, the premium is adjusted to the selected current rate level. When there have been multiple changes, each historical level needs a cumulative factor that accounts for all subsequent changes up to the current level.

The key is to treat each percentage change as applying to the rate level that exists immediately before it. Therefore, successive percentage changes compound. They are not added. If premiums were earned at different levels, apply the factor for each level to its corresponding premium amount, then sum the adjusted amounts.

Definition: A rate-level index assigns a numerical index to each rate level in a sequence, with a chosen base level often set to 1.000. A cumulative on-level factor for a historical level is the ratio of the current rate-level index to the index for that historical level. Multiplying earned premium at the historical level by this factor restates that premium at the current level, assuming the rate changes apply to the premium being adjusted.

Building the Rate-Level Sequence

Suppose a hypothetical insurer starts at rate level $L_0$, assigned index 1.000. A rate change of $r_1$ takes the book from $L_0$ to $L_1$, so the index at $L_1$ is the previous index multiplied by $1+r_1$. Continue in the same way for each change. A decrease is represented by a negative percentage; for example, a 4% decrease multiplies the preceding index by 0.96.

I_j=I_{j-1}(1+r_j) \qquad\text{and}\qquad F_{k\to n}=\frac{I_n}{I_k}

Here $I_j$ is the index at level $L_j$, $r_j$ is the percentage change from $L_{j-1}$ to $L_j$, and $F_{k\to n}$ is the factor that adjusts premium from level $L_k$ to current level $L_n$. When the sequence starts at $I_0=1$, the factor from $L_0$ to $L_n$ is the product of all intervening change factors:

F_{0\to n}=\prod_{j=1}^{n}(1+r_j)

The ratio-of-indices form is useful when premium is earned at more than one historical level. A premium already at $L_k$ should be adjusted only for the changes after $L_k$, not for changes that occurred before that premium reached its existing level.

Why the Factors Multiply

A percentage change is defined relative to the rate level immediately before the change. If a rate level is represented by $R$, an increase of $r$ produces the new level $R(1+r)$. Applying a second change $s$ to that new level gives $R(1+r)(1+s)$, not $R(1+r+s)$. Repeating this argument establishes the cumulative product.

R_1=R_0(1+r_1),\quad R_2=R_1(1+r_2)=R_0(1+r_1)(1+r_2)

Dividing the final level $R_n$ by the historical level $R_k$ gives the adjustment required to restate a premium from $L_k$ to $L_n$. Since both levels are proportional to their indices, the adjustment is $I_n/I_k$. This also explains why a premium earned at a later level does not receive the full factor from the original base: its starting rate level is already higher.

Worked Example: Three Successive Increases

A hypothetical coverage has $100,000 of earned premium entirely at its base rate level. The rates then change successively by +8%, +5%, and +3%. Calculate the cumulative factor and the premium at the final level. A reviewer adds the percentages and reports $116,000; assess that calculation.

Method: build the rate-level index and multiply the premium by the cumulative factor. Starting with $I_0=1.000$, the first change gives:

$$ I_1=1.000(1.08)=1.08 $$

The second change applies to level $L_1$, and the third applies to level $L_2$:

$$ I_2=1.08(1.05)=1.134 $$ $$ I_3=1.134(1.03)=1.16802 $$

Thus the cumulative factor from the base level to the final level is $1.16802$. The on-level premium is:

\$100{,}000(1.16802)=\$116{,}802.00

Check. The index has increased by 16.802%, which is consistent with three compounded increases. Adding the stated percentages gives 16% and a premium of $116,000, which is $802.00 too low. The common mistake is treating each percentage as though it were calculated on the original base rather than on the preceding rate level.

Applying Factors to Premium at Different Levels

The total earned premium for an experience period may include amounts earned at multiple levels. In that case, first identify the rate level associated with each amount. Then calculate each amount's factor to the current level and adjust the amounts separately. The adjusted premiums are additive; the factors generally are not averaged or applied indiscriminately to the entire total.

List the rate levels in order.
Record each percentage change and construct the corresponding indices, starting from a consistent base index.

Match each premium amount to its level.
Use earned premium already assigned to a rate level. If a change took effect partway through a period, the applicable extension or parallelogram procedure determines the relevant premium amounts or weights.

Calculate a factor for each level.
Divide the current-level index by the index for the level at which that premium was earned.

Adjust and sum.
Multiply each premium amount by its own factor, then add the adjusted amounts to obtain total on-level earned premium.

Worked Example: Premium Earned at Three Rate Levels

A hypothetical insurer makes a 10% increase, then a 4% decrease, then a 6% increase. Its current level is $L_3$. Earned premium is $100,000 at $L_0$, $200,000 at $L_1$, and $100,000 at $L_2$. Calculate total earned premium on the current level.

Method: calculate the level indices, then use a separate current-to-historical index ratio for each premium amount. Set $I_0=1.000$. The successive indices are:

$$ I_1=1.000(1.10)=1.10 $$ $$ I_2=1.10(0.96)=1.056 $$ $$ I_3=1.056(1.06)=1.11936 $$

The factors to current level $L_3$ are $I_3/I_k$ for the relevant historical level. From $L_0$, the factor is $1.11936/1.000=1.11936$. From $L_1$, it is $1.11936/1.10=1.0176$. From $L_2$, it is $1.11936/1.056=1.06$. Apply each:

\$100{,}000(1.11936)=\$111{,}936.00 $$ $$ \$200{,}000(1.0176)=\$203{,}520.00 $$ $$ \$100{,}000(1.06)=\$106{,}000.00 $$ $$ \text{Total on-level premium} =\$111{,}936+\$203{,}520+\$106{,}000 =\$421{,}456.00

Check. The original premium totals $400,000. The final level is 11.936% above $L_0$, but the total increase to the mixed-level premium is not 11.936%, because some premium was already earned after one or more changes. The level-specific adjustments produce $421,456.00. A common mistake is to apply the full base-to-current factor of 1.11936 to all $400,000, which would overstate the adjusted premium as $447,744.00.

Using the Factors in a Loss Ratio Indication

On-level premium is the premium denominator restated to a common rate level. For a loss ratio indication, the losses and premium must correspond to the same experience period, and the premium adjustment must reflect the rate changes relevant to that period. The cumulative factor changes the premium basis; it does not itself change the losses. Development, trend, or other adjustments to losses are separate considerations.

If the experience premium consists of amounts $P_k$ at levels $L_k$, its total on-level premium is:

P_{\text{on-level}} =\sum_{k=0}^{n} P_kF_{k\to n} =\sum_{k=0}^{n}P_k\frac{I_n}{I_k}

This sum is equivalent to applying a premium-weighted average of the factors to total premium, but only if the weights are calculated from the amounts of premium at each level. Directly adjusting each amount is usually clearer and reduces the risk of using an unweighted average. Once the adjusted premium is established, it can be used consistently in the loss ratio calculation.

Worked Example: On-Leveling After a Midyear Change

A hypothetical policy portfolio has earned premium amounts already allocated by an earlier parallelogram calculation: $50,000 at level $L_0$, $75,000 at $L_1$, and $40,000 at $L_2$. The rate changes were +12% from $L_0$ to $L_1$, followed by +5% from $L_1$ to current level $L_2$. Calculate current-level earned premium. Assume the stated amounts are the applicable earned premium at each level, not unadjusted calendar-year totals.

Method: use the allocated level amounts and cumulative ratios to current level. Set $I_0=1.000$. Then:

$$ I_1=1.000(1.12)=1.12 $$ $$ I_2=1.12(1.05)=1.176 $$

The factors to $L_2$ are 1.176 from $L_0$, $1.176/1.12=1.05$ from $L_1$, and 1.000 from $L_2$. The adjusted components are:

\$50{,}000(1.176)=\$58{,}800.00 $$ $$ \$75{,}000(1.05)=\$78{,}750.00 $$ $$ \$40{,}000(1.000)=\$40{,}000.00 $$ $$ P_{\text{on-level}} =\$58{,}800+\$78{,}750+\$40{,}000 =\$177{,}550.00

Check. The unadjusted amounts total $165,000. The $40,000 earned at $L_2$ requires no further change; the $L_1$ amount receives only the 5% increase, while the $L_0$ amount receives both increases. A common mistake is applying 1.176 to all $165,000, which incorrectly treats every dollar as if it had been earned at the base level.

Common Exam Traps and Practical Checks

Several checks help catch errors before an on-level premium is used in an indication:

Do not add percentage changes. Build each new level from the preceding level. For example, +10% followed by −4% gives $1.10(0.96)=1.056$, a net increase of 5.6%, not 6%.
Do not give every premium dollar the same factor. A premium amount earned at a later level should receive only the changes after that level.
Do not confuse a premium level with a partial-year earning pattern. A change effective during the earning period can affect only part of the earned premium. Use the applicable extension or parallelogram result to assign or weight the premium before applying cumulative factors.
Keep the current level consistent. Every factor used in a total must lead to the same selected current level. A factor ending at an intermediate level cannot be mixed into a total intended to reflect a later level.
Check the direction of the result. For a sequence consisting only of increases, premium moved forward to the later level should not decrease. A decrease within the sequence can reduce the cumulative factor, so assess the full product rather than the sign of just one change.

The purpose of cumulative on-leveling is comparability, not merely arithmetic. A sound calculation makes clear what the base and current levels are, what changes connect them, and which historical premium belongs at each starting level. Those details make the result reproducible and help prevent a plausible-looking but inconsistent loss ratio denominator.

Key takeaway: Construct the rate-level indices by compounding each successive percentage change. For each earned-premium component, multiply by the current index divided by its own historical-level index, then sum the adjusted components.

Check Your Understanding

For each problem, identify the applicable rate levels before calculating. Answers are rounded to cents where relevant.

A hypothetical book has $75,000 of earned premium at its base level. Rates then increase by 8% and 5%, in that order. What is the cumulative factor and the premium at the latest level? (Answer: factor 1.134; premium $85,050.00)
A portfolio has $60,000 of premium at $L_0$ and $90,000 at $L_1$. Rates change by +10% from $L_0$ to $L_1$, then −5% from $L_1$ to current level $L_2$. Calculate total on-level premium at $L_2$. (Answer: $148,200.00)
A hypothetical coverage has $125,000 at its base level and experiences successive rate changes of +7%, −3%, and +4%. Calculate the cumulative factor and restated premium. (Answer: factor 1.079416; premium $134,927.00)
Earned premium is $35,000 at $L_0$ and $65,000 at $L_1$. The rate changes are +10% from $L_0$ to $L_1$, then +4% to current level $L_2$. What is total earned premium on level $L_2$? (Answer: $107,640.00)

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