Compounding Sequential Rate Changes

Why Sequential Rate Changes Compound

A percentage rate change is measured relative to the rate level immediately before that change. Therefore, after one change, the base for the next percentage change is different from the original base. To find the final rate level, calculate each change from the level then in force. This is why successive percentage changes must be compounded rather than simply added.

For instance, a 10% increase followed by a 10% decrease does not return a rate to its starting level. The decrease applies to the already increased rate. This tutorial focuses on calculating that cumulative effect and on moving correctly in the reverse direction. It does not determine which historical premium should be adjusted or how policies are weighted across rate levels.

Definition: If a rate changes by $r$ as a decimal, its change factor is $1+r$. An increase of $r$ has $r>0$, while a decrease of $q$ has $r=-q$ and factor $1-q$. For successive changes $r_1,r_2,\ldots,r_n$, the cumulative factor is $\prod_{j=1}^{n}(1+r_j)$, and the cumulative percentage change is that factor minus 1.

Percentages must be entered as decimals when forming factors: an increase of 6% gives $r=0.06$ and factor $1.06$; a decrease of 6% gives $r=-0.06$ and factor $0.94$. The factor multiplies the level in force immediately before the change.

Calculating the Cumulative Change

Let $R_0$ be the starting rate level, and let $r_k$ be the percentage change at step $k$, expressed as a signed decimal. After the first change, $R_1=R_0(1+r_1)$. After the second, $R_2=R_1(1+r_2)$. Continuing in the same way gives:

R_n=R_0\prod_{k=1}^{n}(1+r_k)

The cumulative percentage change is measured relative to the starting level, so divide the final level by the starting level and subtract 1:

\text{Cumulative percentage change} =\frac{R_n-R_0}{R_0} =\prod_{k=1}^{n}(1+r_k)-1

This result assumes every stated change is a percentage change in the rate level then in force, and that the same rate basis is being followed through the sequence. If a problem instead supplies dollar changes, fixed surcharges, or percentages defined relative to a different base, those instructions must be handled as stated; they cannot automatically be treated as ordinary sequential percentage changes.

Write each change as a signed decimal.
Use a positive value for an increase and a negative value for a decrease.

Form a factor for each change.
For a signed change $r_k$, the factor is $1+r_k$.

Multiply the factors.
The product gives the final level as a multiple of the starting level.

Convert the product to a percentage change if needed.
Subtract 1 from the product, then express the result as a percentage.

Why Adding the Percentages Does Not Work

Adding the stated percentages treats every change as though it were measured against the original rate. But the second change is measured against the rate after the first change, the third against the rate after the second, and so on. For two changes $r$ and $s$, the final factor is:

$$ (1+r)(1+s)=1+r+s+rs $$

The additive calculation $r+s$ omits the cross-product $rs$. With more changes, the product of factors includes further interaction terms as well. Adding the percentages can be a rough approximation when changes are small, but it is not the exact cumulative rate change unless the omitted terms happen to sum to zero.

A percentage increase and decrease with the same numerical size illustrate the issue. If a level increases by $p$ and then decreases by $p$, its factor is $(1+p)(1-p)=1-p^2$, which is less than 1 whenever $p\ne0$. The simple sum is zero, but the actual result is a decrease.

Worked Example: Two Increases and a Decrease

A hypothetical commercial package rate begins at $500 per exposure. It increases by 8%, then increases by 5%, and later decreases by 4%. Find the cumulative percentage change and the final rate. Give the final rate to the nearest cent. Identify the mistake made by adding the three percentages.

Method. Convert the changes to factors in the stated sequence: $1.08$, $1.05$, and $0.96$. Multiply them to get the cumulative factor:

1.08\times1.05=1.134

1.134\times0.96=1.08864

The final rate is 1.08864 times the starting rate:

500\times1.08864=\$544.32

The cumulative percentage change is $1.08864-1=0.08864$, or 8.864%. By contrast, simply adding $8\%+5\%-4\%$ gives 9%. That is the common mistake: the 5% increase and 4% decrease are not calculated from the original $500 rate. The compounded result is checked by observing that $544.32 is $44.32 above $500, and $44.32/500=0.08864$. The final rate and cumulative percentage agree.

Applying and Reversing a Sequence

Once the cumulative factor is known, applying it to a starting rate is direct multiplication. To move from a later rate level back to an earlier one, use the reciprocal of the factor for the forward sequence. Reversing the direction does not mean changing every increase into a decrease of the same stated size. The reverse percentage is measured from the later, higher or lower, level and can therefore have a different magnitude.

\text{Reverse factor}=\frac{1}{\text{Forward cumulative factor}}

The reciprocal works because multiplying the forward factor by its reciprocal returns the original level: $R_0F(1/F)=R_0$, provided the forward factor $F$ is nonzero. Ordinary rate-change factors are positive when each change leaves a positive rate level.

Worked Example: A Decrease Does Not Undo an Equal Increase

A hypothetical liability coverage rate of $840 per unit increases by 12% at renewal and then decreases by 10% at the next revision. Calculate the final rate and cumulative percentage change. Compare the result with the additive calculation.

Method. The increase factor is $1.12$, and the decrease factor is $0.90$. Their product is:

1.12\times0.90=1.008

The final rate is:

840\times1.008=\$846.72

The cumulative percentage change is $1.008-1=0.008$, or an increase of 0.8%. Simply adding the stated percentages gives $12\%-10\%=2\%$, which is incorrect because the 10% decrease applies to the increased level. Check the calculation: $846.72 is $6.72 above $840, and $6.72/840=0.008$. Thus the final rate corresponds exactly to the 0.8% compounded increase.

Worked Example: Calculating the Reverse Change

A hypothetical equipment rate is increased by 10% and then decreased by 10%. Starting from $1,000, find the resulting rate. Then calculate the single percentage increase required to move from that resulting rate back to $1,000. Give the reverse percentage to four decimal places.

Method. The forward factors are $1.10$ and $0.90$, so the forward cumulative factor is:

1.10\times0.90=0.99

The resulting rate is $1000\times0.99=\$990$, a net decrease of 1%. To return from $990 to $1,000, calculate the needed increase relative to $990, not relative to $1,000:

\text{Reverse factor}=\frac{1000}{990}=\frac{1}{0.99}=1.010101\ldots

The reverse percentage is $1.010101\ldots-1=0.010101\ldots$, or 1.0101% to four decimal places. A common mistake is to claim that a 1% increase reverses the 1% decrease. In fact, $990\times1.01=\$999.90$, not $1,000. Using the unrounded reverse factor gives $990\times(1/0.99)=\$1,000$, confirming the result.

Exam Traps and Practical Checks

When a problem describes several effective rate changes, keep the signed percentages and their factors visible until the multiplication is complete. This makes it easier to catch sign errors and to distinguish a factor from a percentage. A cumulative factor of $1.07$ means the final level is 107% of the starting level, or an increase of 7%; it does not mean a 107% increase.

Adding instead of compounding. Calculate the product of the factors first. Add the percentages only if the question explicitly asks for their arithmetic sum, not for the net change in rate level.
Using the wrong sign. A decrease of 7% has factor $0.93$, not $1.07$.
Reversing with the same percentage. Use the reciprocal of the forward factor. A forward increase of 10% is reversed by a decrease of $1-1/1.10$, approximately 9.0909%, not 10%.
Reporting the factor as the change. Subtract 1 from the cumulative factor to find the cumulative percentage change.
Rounding too early. Preserve enough precision in the factor product, then round only the final requested rate or percentage.

For scalar rate levels, multiplication of the factors gives the same final product in any order. Nevertheless, retain the stated chronology when setting up a problem: it identifies the base to which each percentage applies and helps reveal whether a stated change is actually a percentage of the current level or something else. If all changes are ordinary percentage changes on the successive rate levels, their product is the exact cumulative result.

Key takeaway: Convert each successive percentage change into a factor, multiply the factors to obtain the final level relative to the starting level, and subtract 1 to express the cumulative percentage change. For the reverse movement, take the reciprocal of the cumulative factor.

Check Your Understanding

For each problem, calculate using the stated successive changes rather than adding the percentages.

A hypothetical inland marine rate of $600 increases by 7% and then decreases by 3%. Find the cumulative percentage change and final rate. Give the rate to the nearest cent. (Answer: cumulative increase 3.79%; final rate $622.74)
A hypothetical policy rate decreases by 8% and then increases by 8%. Find the net percentage change. (Answer: decrease of 0.64%)
A hypothetical fleet rate of $250,000 increases by 4%, increases by 4% again, and then decreases by 2%. Find the cumulative percentage change and final rate. (Answer: cumulative increase 5.9968%; final rate $264,992)
A hypothetical rate level is increased by 12% and then decreased by 5%. What percentage change, applied to the resulting level, would return it to its starting level? Give the answer to three decimal places. (Answer: decrease of 6.015%)
A hypothetical rate of $2,400 decreases by 15%, increases by 10%, and then increases by 8%. Calculate the cumulative factor, cumulative percentage change, and final rate. Give the final rate to the nearest cent. (Answer: cumulative factor 1.0098; cumulative increase 0.98%; final rate $2,423.52)

What You'll Learn