Introduction
An indicated rate-level factor and a percentage change describe the same adjustment in different forms. The factor is the multiplier applied to the current rate level; the percentage change reports how far that multiplier is above or below 1.000. A factor of 1.08, for example, represents an 8% increase—not a 108% increase.
That distinction matters when communicating a rate indication, checking a calculation, or combining successive adjustments. The conversion is simple, but confusing the multiplier with the change can produce a large error in a written conclusion. This tutorial treats the indicated factor as an input and focuses on interpreting and reporting it. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
From a Factor to a Percentage Change
The conversion uses a deterministic multiplicative model: the indicated overall rate level is obtained by multiplying the current rate level by a single factor \(F\). Thus, the factor describes the new level relative to the old level. The percentage change instead measures the difference between those levels relative to the old level. The same comparison can be made with premium amounts if they reflect the same exposures and the same current-rate basis.
This model assumes that \(F\) applies to the current level being compared and that the numerator and denominator use a consistent basis. It gives an overall change; it does not, by itself, specify how rates for individual classes or other rating cells should change. If the factor is already stated relative to current rates, do not apply an earlier adjustment to it again unless the question explicitly asks for a comparison to that earlier rate level.
Why subtract 1?
Let the current rate level be \(R\). By definition of the factor, the indicated level is \(FR\). The absolute change is therefore \(FR-R=(F-1)R\). To express that difference as a proportion of the current level, divide by \(R\), which is positive:
Multiplying this proportion by 100 converts it to a percentage. The subtraction of 1 removes the unchanged current level, represented by a factor of 1.000. This is why the percentage change is not \(100F\%\).
Confirm that the factor compares the indicated level with the current level.
Calculate \(F-1\) to obtain the change as a proportion of current rates.
Multiply the proportion by 100 and report “increase” or “decrease” according to its sign.
A factor above 1.000 must produce an increase; a factor below 1.000 must produce a decrease.
Worked Example: Converting a Factor Above 1.000
A hypothetical regional insurer obtains an indicated rate-level factor of 1.0875 relative to its current rates. State the percentage change and explain what the factor means. Treat the factor as exact; no interest, mortality, or probability model is involved.
Method. Convert the factor to a percentage change by subtracting 1.000 and multiplying by 100.
Step 1: Find the change as a proportion. \(1.0875-1.0000=0.0875\).
Step 2: Express the proportion as a percentage. \(0.0875\times100=8.75\%\).
The indicated change is an 8.75% increase. The factor 1.0875 means the indicated level is 1.0875 times the current level, or 108.75% of that level. It does not mean an increase of 108.75%.
Check. Starting with a hypothetical current rate of $100.00, the indicated rate is \(100.00\times1.0875=\$108.75\). The difference is $8.75, and \(\$8.75/\$100.00=8.75\%\). The result agrees with the factor conversion.
Common mistake avoided. Reporting “a 108.75% increase” confuses the indicated level as a percentage of current rates with the percentage change from current rates.
Interpreting Factors Below 1.000
The same conversion applies when \(F\) is less than 1.000. Subtracting 1 then gives a negative number. That negative sign indicates that the indicated level is below the current level. In ordinary communication, report the magnitude as a decrease rather than leaving the reader to interpret a negative percentage.
Worked Example: Converting a Factor Below 1.000
A hypothetical specialty insurer compares $2,400,000 of current-level premium with $2,246,400 of indicated premium on the same stated basis. Determine the indicated factor and report the percentage change. Treat the premium amounts as exact for this exercise; no interest, mortality, or probability model is involved.
Method. First calculate the indicated-to-current ratio, then convert that factor to a percentage change.
Step 1: Calculate the factor. \[ F=\frac{\$2{,}246{,}400}{\$2{,}400{,}000}=0.936. \] The indicated level is 0.936 times the current level.
Step 2: Subtract 1. \(0.936-1.000=-0.064\).
Step 3: Convert to a percentage. \(-0.064\times100=-6.4\%\). The indicated change is a 6.4% decrease.
Check using the premium difference. The difference is \(\$2{,}246{,}400-\$2{,}400{,}000=-\$153{,}600\). Relative to current-level premium, \(-\$153{,}600/\$2{,}400{,}000=-0.064=-6.4\%\). This confirms both the magnitude and the direction.
Common mistake avoided. Saying “a 0.936% decrease” treats the factor as a percentage change. The factor is 93.6% of the current level; the decrease is the remaining 6.4%.
Factors and Successive Changes
When a question asks for the total effect of successive rate changes, factors multiply because each change applies to the level produced by the preceding change. Percentages stated for separate steps are not generally added to obtain the total percentage change. After multiplying the factors, subtract 1 from the product to express the result as a single change relative to the starting level.
Keep the reference point clear. A factor can describe a change relative to current rates, while a product of factors can describe the combined effect relative to an earlier level. The percentage is meaningful only when its comparison level is stated.
Worked Example: Combining Two Successive Rate Changes
A hypothetical insurer first implements a 4% increase from a stated starting rate level. A later analysis indicates a further 7.5% increase relative to the rates after that first change. Find the combined change from the original starting level. Assume each percentage applies to the level immediately before it; no interest, mortality, or probability model is involved.
Method. Convert each increase to a factor, multiply the factors, then convert the combined factor to a percentage change from the original level.
Step 1: Convert the first increase to a factor. \(1+0.04=1.04\).
Step 2: Convert the second increase to a factor. \(1+0.075=1.075\).
Step 3: Multiply the factors. \(1.04\times1.075=1.118\).
Step 4: Convert the combined factor. \(1.118-1=0.118\), so the total change from the original level is an 11.8% increase.
Check with a starting rate. If the original rate is $100.00, the first increase produces \(\$100.00\times1.04=\$104.00\). The second produces \(\$104.00\times1.075=\$111.80\). The total increase is \(\$11.80/\$100.00=11.8\%\), consistent with the combined factor.
Common mistake avoided. Adding 4% and 7.5% gives 11.5%, but the second 7.5% applies to $104.00, not to the original $100.00. The factor calculation correctly accounts for that change in base.
Common Reporting Traps
The arithmetic is short, so most errors arise from interpreting the factor or losing track of the reference level. Before reporting a result, state whether it is a factor or a percentage change and name the level against which the percentage is measured.
| Given | Correct interpretation | Typical error |
|---|---|---|
| \(F=1.12\) | 12% increase from current rates | Calling it a 112% increase |
| \(F=0.91\) | 9% decrease from current rates | Calling it a 91% decrease |
| Successive factors \(1.04\) and \(1.075\) | Combined factor \(1.118\), or an 11.8% total increase | Adding the percentages and reporting 11.5% |
- Do not omit the reference level. “A 6% decrease” should indicate that the decrease is relative to the specified current or starting level.
- Do not change the sign without changing the wording. A result of \(-6.4\%\) can be reported as a 6.4% decrease, not a 6.4% increase.
- Do not round prematurely. Keep the supplied factor or unrounded ratio through the subtraction and percentage conversion, then round the reported result as requested.
- Check direction before finalizing. A factor above 1.000 cannot imply a decrease under this model, and a factor below 1.000 cannot imply an increase.
Check Your Understanding
For each problem, state the factor or percentage change requested and check that the direction agrees with the comparison described.
- A hypothetical product has an indicated factor of 1.064 relative to current rates. What is the percentage change? (Answer: 6.4% increase)
- A hypothetical portfolio has current-level premium of $850,000 and indicated premium of $816,000 on the same basis. Find the factor and percentage change. (Answer: Factor 0.96; 4% decrease)
- A hypothetical indicated factor is 0.9875. Report the percentage change to two decimal places. (Answer: 1.25% decrease)
- A hypothetical rate level increases by 3% and then by 6% relative to the level after the first change. Find the combined factor and total percentage change from the original level. (Answer: Factor 1.0918; 9.18% increase)
- A current rate is $240.00, and an indicated factor of 1.125 applies to that rate. Find the indicated rate and the dollar increase. (Answer: Indicated rate $270.00; increase $30.00, or 12.5%)