Deriving the Indicated Rate Change

Introduction

An indicated premium requirement is not itself a percentage rate change. To express the result as a rate change, compare the premium required at the indicated rate level with the premium produced by current rates for the same exposure base. The comparison is a ratio: it measures how much the current premium must be multiplied by to reach the required premium.

The key is to keep the two premium amounts comparable. They must correspond to the same policies or exposure base, coverage, and premium definition. Once that condition is met, the percentage change follows directly from the multiplicative relationship between the current and indicated rate levels. This algebra is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $P_{\text{cur}}$ be the premium produced by current rates for a specified exposure base, and let $P_{\text{ind}}$ be the required premium for that same exposure base at the indicated rate level. The indicated rate factor is $P_{\text{ind}}/P_{\text{cur}}$. The indicated percentage rate change is the indicated rate factor minus one, expressed as a percentage.

Deriving the Rate Change

Assume that a single uniform rate change $c$ is applied to the current rates and that the relevant exposures, coverage, and rating mix remain fixed. Under this assumption, premium for the specified base changes in direct proportion to the rate level. Therefore, the premium at the new rate level is the current premium multiplied by $1+c$.

P_{\text{ind}}=P_{\text{cur}}(1+c)

Solve for $c$ by dividing both sides by $P_{\text{cur}}$, which must be positive, and then subtracting one:

\frac{P_{\text{ind}}}{P_{\text{cur}}}=1+c \qquad\Longrightarrow\qquad c=\frac{P_{\text{ind}}}{P_{\text{cur}}}-1

Multiplying $c$ by 100 expresses it as a percentage. For example, if $c=0.075$, the indicated rate change is a 7.5% increase. If $c=-0.04$, it is a 4.0% decrease. The rate factor and the percentage change are related but are not the same quantity: a factor of 1.075 means a 7.5% increase, not a 107.5% increase.

The derivation is a consequence of how a uniform rate change acts on a fixed base. If every applicable rate is multiplied by $1+c$, each policy's rate-based premium is multiplied by that same factor, and their total is multiplied by it as well. Setting that total equal to the required premium gives the equation above. The algebra does not require a probability distribution, mortality assumption, or interest assumption; it is a deterministic comparison of premium amounts.

Establish the comparison base.
Confirm that current and indicated premiums refer to the same exposures, coverage, and premium components.

Calculate the indicated rate factor.
Divide required premium by premium at current rates.

Convert the factor to a percentage change.
Subtract one from the factor, then multiply by 100 to report a percentage.

Check the result.
Multiply the current premium by one plus the unrounded rate change. The result should equal the required premium, apart from any stated rounding.

What Must Be Comparable

The formula is simple, but the inputs require care. $P_{\text{cur}}$ is not just any premium total on hand; it is the premium at the current rate level for the exposure base being evaluated. $P_{\text{ind}}$ is the premium required for that same base under the selected indication. If one amount is based on one set of risks and the other on a different set, their ratio combines the effect of rate level with differences in exposure or mix.

Premium components must also be treated consistently. For example, if a fixed policy fee does not change with the rates, it should not be treated as though it scales proportionally with a uniform rate change. The analyst must define the premium basis being compared and handle any non-rate-based components appropriately. A single ratio is appropriate when the indicated change is intended to apply uniformly to the included rate-based premium.

The calculation is not a direct description of the premium collected on a future set of policies. It is the indicated rate-level change for the stated comparison base. If exposures, coverage choices, rating mix, or applicable premium components differ, the ratio alone does not isolate a rate-level effect.

Worked Example: Converting a Total Premium Requirement

A hypothetical insurer evaluates a fixed book of small equipment policies. Current rates produce $1,240,000 of premium for that book, while the selected indication requires $1,339,200. Calculate the indicated rate factor and percentage rate change. The amounts are already on the same exposure and coverage basis. This is a deterministic calculation; no interest rate, mortality assumption, or probability distribution is used.

Method. Divide the indicated premium requirement by premium at current rates to obtain the factor, then subtract one.

First calculate the factor:

\frac{P_{\text{ind}}}{P_{\text{cur}}} = \frac{\$1{,}339{,}200}{\$1{,}240{,}000} = 1.08

Then convert the factor to a rate change:

c=1.08-1=0.08=8.0\%

Check. Applying the factor to current premium gives $\$1{,}240{,}000(1.08)=\$1{,}339{,}200$, equal to the indicated requirement. The indicated change is an 8.0% increase. The check confirms the unrounded arithmetic, so no intermediate rounding was needed.

Common mistake avoided. The premium difference is $99,200, but that dollar amount is not a percentage. It must be divided by the current premium: $\$99{,}200/\$1{,}240{,}000=0.08$. Dividing the difference by the indicated premium would use the wrong denominator.

Aggregation Before Comparing

When the book is divided into segments, the uniform indicated change for the combined book is found by totaling the premium amounts at each rate level and then taking their ratio. This is equivalent to weighting each segment's rate change by its share of current premium, not taking an unweighted average of segment changes.

c_{\text{overall}} = \frac{\sum_j P_{\text{ind},j}}{\sum_j P_{\text{cur},j}}-1 = \sum_j \left( \frac{P_{\text{cur},j}}{\sum_k P_{\text{cur},k}} \right)c_j

Here $c_j=P_{\text{ind},j}/P_{\text{cur},j}-1$ is the segment rate change. The second equality follows by writing $P_{\text{ind},j}=P_{\text{cur},j}(1+c_j)$, summing over segments, and dividing by total current premium. Thus, segments with larger current premiums receive greater weight in the combined result.

Worked Example: Combining Two Segments

A hypothetical insurer has two classes of marine equipment. For the same exposure base in each class, current rates produce $600,000 in Class A and $400,000 in Class B. The selected requirements are $642,000 and $424,000, respectively. Find the uniform indicated change for the combined book. The premium amounts are comparable within each class and are all rate-based. No interest rate, mortality assumption, or probability distribution is involved.

Method. Add the class premiums at each level, then compare the totals. This avoids giving equal weight to classes with different current premium volumes.

Total current premium is $\$600{,}000+\$400{,}000=\$1{,}000{,}000$. Total indicated premium is $\$642{,}000+\$424{,}000=\$1{,}066{,}000$. Therefore,

c_{\text{overall}} = \frac{\$1{,}066{,}000}{\$1{,}000{,}000}-1 = 0.066 = 6.6\%

For a check by class, Class A requires $642/600-1=7.0\%$, and Class B requires $424/400-1=6.0\%$. The combined result is $(0.60)(7.0\%)+(0.40)(6.0\%)=4.2\%+2.4\%=6.6\%$. It is not the simple average of the two changes, which would be 6.5%, because the classes have different current premium weights. Applying 1.066 to the combined current premium gives $1,066,000 as required.

Common mistake avoided. Do not average class percentage changes equally unless the class current premiums are equal. Aggregate premium amounts first, or use the current-premium weights shown above.

From a Premium Requirement to a Rate Change

In some indications, the required premium is supplied directly. In others, it is calculated from projected losses and expenses or from a selected indication factor. Whatever method produced $P_{\text{ind}}$, the final conversion uses the same ratio, provided the required and current premiums use the same base. This example shows the algebra after the required premium is assembled; it does not replace the analysis that establishes the underlying loss and expense provisions.

Worked Example: Calculating Required Premium Before the Rate Change

For a hypothetical service-contract portfolio, projected losses and loss adjustment expenses are $720,000, fixed expenses are $90,000, variable expenses are 12% of premium, and the selected profit and contingency provision is 8% of premium. Assume these percentages are additive proportions of premium, and that the fixed expenses and projected losses are stated for the same exposure base as the $900,000 premium at current rates. Calculate the required premium and indicated percentage change. No interest rate, mortality assumption, or probability distribution is used.

Method. The premium must fund projected losses, fixed expenses, variable expenses, and the profit and contingency provision. If $P_{\text{ind}}$ is required premium, the amount remaining after variable expenses and the stated provision is $P_{\text{ind}}(1-0.12-0.08)$. Set that amount equal to projected losses plus fixed expenses:

P_{\text{ind}}(1-0.12-0.08) = \$720{,}000+\$90{,}000

The available proportion is $1-0.12-0.08=0.80$, and the amount to be funded is $\$810{,}000$. Therefore,

P_{\text{ind}} = \frac{\$810{,}000}{0.80} = \$1{,}012{,}500

Now compare this with the $900,000 premium at current rates:

c = \frac{\$1{,}012{,}500}{\$900{,}000}-1 = 1.125-1 = 0.125 = 12.5\%

Check. The required premium, multiplied by the 80% remaining proportion, funds $\$1{,}012{,}500(0.80)=\$810{,}000$, which equals losses plus fixed expenses. Applying the 12.5% rate change to the current premium gives $\$900{,}000(1.125)=\$1{,}012{,}500$. The calculations agree.

Common mistake avoided. The variable expense and profit provisions are percentages of premium, so they reduce the proportion of premium available to fund losses and fixed expenses. Subtracting 20% from $810,000 would not solve for required premium; the correct operation is division by 0.80.

Signs, Denominators, and Limits

A positive indicated change occurs when $P_{\text{ind}}>P_{\text{cur}}$; a negative change occurs when $P_{\text{ind}}<P_{\text{cur}}$. Equal amounts imply a zero change. The denominator is premium at current rates because the question is how much the current rate level must change relative to its present level.

For example, if required premium is $1,176,000 and current premium is $1,200,000, then $(1{,}176{,}000/1{,}200{,}000)-1=0.98-1=-0.02$, a 2.0% decrease. The result is not a 2.0% increase merely because the absolute difference is 2% of the required premium; the direction is determined by which amount is larger.

A useful final check is to reconstruct the indicated premium:

P_{\text{cur}}(1+c)=P_{\text{ind}}

If this equality fails by more than rounding, recheck the factor, the subtraction of one, the percentage conversion, and the input basis. A calculated ratio should not be interpreted as a uniform rate change when the underlying comparison changes exposures, mix, coverage, or non-rate-based premium items.

Key takeaway: For a comparable exposure base and a uniform rate-level change, divide required premium by premium at current rates to obtain the indicated factor, then subtract one to obtain the indicated percentage change. Aggregate premium amounts before comparing segments, and verify the result by multiplying current premium by one plus the change.

Check Your Understanding

For each problem, use comparable premium amounts unless the problem states otherwise. Answers are given for checking.

A hypothetical portfolio has $840,000 of premium at current rates and requires $907,200 for the same exposure base. Find the indicated factor and percentage change. (Answer: factor 1.08; 8.0% increase.)
Current premium is $1,050,000 and required premium is $1,018,500 for the same policies and coverage. Find the percentage change and verify its direction. (Answer: 0.97 factor; 3.0% decrease.)
Two segments have current premiums of $250,000 and $750,000, with indicated requirements of $270,000 and $780,000. Calculate the combined change and explain why an unweighted average is not the combined result. (Answer: 5.0% overall increase; the segment changes are 8.0% and 4.0%, and the overall result weights them by current premium.)
A hypothetical book has projected losses of $560,000, fixed expenses of $40,000, variable expenses of 10% of premium, and a profit provision of 10% of premium. Current premium for the same base is $750,000. Calculate required premium and the indicated percentage change. (Answer: required premium $750,000; 0.0% change.)
For a fixed exposure base, the indicated factor is 1.045 and current premium is $1,600,000. Calculate the required premium and the dollar difference from current premium, then state the rate change. (Answer: required premium $1,672,000; difference $72,000; 4.5% increase.)

What You'll Learn