Estimating Policyholder Premium Impacts

From an Overall Selection to an Individual Premium

A selected overall rate change is a summary for a defined premium base; it is not necessarily the change each policyholder will experience. To estimate an individual impact, the actuary needs the policy’s current premium and the way the selected change is implemented. A uniform multiplier gives one result, while differing coverage components or rating-factor revisions can produce another.

The calculations in this tutorial are illustrations, not predictions of actual renewal bills. Unless stated otherwise, assume annual premiums, unchanged exposure, unchanged coverage and risk characteristics, and no changes in taxes, fees, commissions, or policy-term timing. These assumptions isolate the premium effect of the specified rate change.

Definition: For a policy with current premium $P_{\text{old}}$ and illustrative premium $P_{\text{new}}$, the dollar impact is $P_{\text{new}}-P_{\text{old}}$. The percentage impact is $(P_{\text{new}}-P_{\text{old}})/P_{\text{old}}$, provided $P_{\text{old}}>0$. A positive impact is an increase; a negative impact is a decrease.

If the selected change $r$ is applied as the same multiplicative change to the policy’s entire current premium, the illustrative calculation is:

P_{\text{new}}=P_{\text{old}}(1+r), \qquad \text{dollar impact}=rP_{\text{old}}.

Here $r$ is expressed as a decimal: an increase of 7% means $r=0.07$. The formula applies only when the full premium being compared is subject to that uniform change. It does not, by itself, account for changes in coverage, exposure, fees, or rating factors.

Uniform Changes: A Useful Illustration, Not a Universal Result

A uniform multiplier is useful when the question explicitly asks for the effect of applying the selected overall change to current premiums. It also provides a clear baseline for communication. However, it should not be mistaken for a conclusion that every policy’s filed premium will change by that percentage. The overall selection may be implemented through changes that vary by coverage, territory, or other rating characteristics.

Worked Example: Applying a Uniform Increase to a Small Policy Set

A hypothetical insurer selects an overall increase of 7.5%. Three annual premiums, before the change, are $840, $1,260, and $2,400. Assuming the full premium for each policy receives the same multiplier and all other characteristics remain fixed, calculate each illustrative new premium and impact. Check the aggregate result.

Method and assumption. Use the uniform multiplier $1+r=1+0.075=1.075$. This example assumes there are no fixed charges excluded from the rate change.

For the first policy:

P_{\text{new}}=\$840(1.075)=\$903.00,\qquad \text{impact}=\$903.00-\$840.00=\$63.00.

For the second policy:

P_{\text{new}}=\$1{,}260(1.075)=\$1{,}354.50,\qquad \text{impact}=\$1{,}354.50-\$1{,}260.00=\$94.50.

For the third policy:

P_{\text{new}}=\$2{,}400(1.075)=\$2{,}580.00,\qquad \text{impact}=\$2{,}580.00-\$2{,}400.00=\$180.00.

The old premiums total $\$840+\$1{,}260+\$2{,}400=\$4{,}500$. The new premiums total $\$903+\$1{,}354.50+\$2{,}580=\$4{,}837.50$, an increase of $\$337.50$. The aggregate check is $\$4{,}500(0.075)=\$337.50$, which agrees with the sum of the policy impacts. Each policy’s percentage impact is 7.5%, but the dollar impacts differ because the starting premiums differ.

Common mistake avoided: Adding 7.5 percentage points to the premium, rather than multiplying by $1.075$, does not define a valid dollar premium calculation. The rate change is a proportion of the existing premium.

Coverage Components Can Have Different Changes

When a policy premium contains multiple coverage components, calculate each component’s new premium under its own change and then add the results. For example, a property coverage adjustment need not equal a liability coverage adjustment. The total policy percentage impact is therefore generally not the simple arithmetic average of the component changes.

Suppose component $j$ has current premium $P_j$ and receives change $r_j$, with all other components held fixed. The old total is the sum of the current component premiums, and the new total is the sum of the adjusted component premiums. The policy-level percentage change is:

\frac{\sum_j P_j(1+r_j)-\sum_j P_j}{\sum_j P_j} = \frac{\sum_j P_jr_j}{\sum_j P_j}.

This is a premium-weighted average of the component changes. The weights are each component’s share of the old total premium; they are not equal unless the component premiums are equal.

The identity follows by subtracting the old total from the new total: $\sum_j P_j(1+r_j)-\sum_jP_j=\sum_jP_jr_j$. Dividing by the old total $\sum_jP_j$ gives the stated percentage change. This derivation assumes the component premiums are additive and the specified changes apply to those components without interaction.

Worked Example: Combining Three Coverage Changes

A hypothetical package policy has current annual premiums of $900 for property coverage, $600 for liability coverage, and $300 for equipment coverage. The illustrative changes are +10% for property, +4% for liability, and −5% for equipment. Assume the coverages remain unchanged and there are no separate fixed charges. Find the new total premium, the dollar impact, and the policy-level percentage impact.

Method. Apply each change to its own current component premium, then add. Property becomes $\$900(1.10)=\$990.00$; liability becomes $\$600(1.04)=\$624.00$; equipment becomes $\$300(0.95)=\$285.00$.

The old total is $\$900+\$600+\$300=\$1{,}800.00$. The new total is $\$990+\$624+\$285=\$1{,}899.00$. Therefore, the dollar impact is $\$1{,}899-\$1{,}800=\$99.00$, and the percentage impact is:

\frac{\$99}{\$1{,}800}=0.055=5.5\%.

The weighted-change check gives the same result. The old-premium weights are $900/1{,}800=0.5$, $600/1{,}800=1/3$, and $300/1{,}800=1/6$. Thus:

0.5(10\%)+\frac{1}{3}(4\%)+\frac{1}{6}(-5\%) =5.0\%+1.3333\%-0.8333\%=5.5\%.

The check confirms that the net increase is $99.00, or 5.5% of the old total. Common mistake avoided: Taking the unweighted mean of $10\%$, $4\%$, and $-5\%$ would give 3%, which is wrong because the three coverage premiums are not equally sized.

Rating-Factor Changes Affect Different Policies Differently

A policy premium may be calculated from a base rate, an exposure amount, and rating factors for the policy’s characteristics. To compare old and new premiums, hold the policy’s exposure and characteristics fixed unless the problem says they change, and calculate using the old and new rate plan values. If a factor itself changes, the policy’s impact can differ from the change in the base rate.

Calculation rule: If a policy premium is the product of a base rate, exposure, and multiplicative rating factors, its new-to-old premium ratio is the product of the corresponding new-to-old ratios, provided the same premium formula applies in both periods and the exposure is unchanged.

For example, if only the base rate changes by $r$, the premium changes by $r$ under those assumptions. If the territory factor also changes, the ratio is instead the base-rate ratio multiplied by the new territory factor divided by the old territory factor. The policy’s overall percentage change is calculated from the resulting premium ratio, not by simply adding the percentage changes in the base rate and factor.

Worked Example: Base Rate and Territory Relativity Changes

A hypothetical insurer increases a base rate by 5%. At the same time, a territory relativity changes from 0.90 to 0.95. For a policy with 10 exposure units and no other changing factors, the old base rate is $100 per exposure unit. Calculate the old and new annual premiums and the policy-level impact. Then compare with a policy whose territory relativity changes from 1.10 to 1.15, with the same base rate and exposure.

Method. Use premium = base rate × exposure × territory relativity. For the first policy, the old premium is $\$100(10)(0.90)=\$900.00$. The new base rate is $\$100(1.05)=\$105$, so the new premium is $\$105(10)(0.95)=\$997.50$. Its dollar impact is $\$997.50-\$900.00=\$97.50$, and its percentage impact is $\$97.50/\$900=10.8333\%$.

The ratio check is:

\frac{P_{\text{new}}}{P_{\text{old}}} =1.05\left(\frac{0.95}{0.90}\right) =1.108333\ldots,

so the increase is 10.8333%, as calculated from the premiums.

For the second policy, the old premium is $\$100(10)(1.10)=\$1{,}100.00$. The new premium is $\$105(10)(1.15)=\$1{,}207.50$. The dollar impact is $\$107.50$, and the percentage impact is $\$107.50/\$1{,}100=9.7727\%$. Its ratio is $1.05(1.15/1.10)=1.097727\ldots$, confirming the result.

Both policies receive the same 5% base-rate increase, yet their illustrative total changes differ because their territory-factor ratios differ. The calculations do not establish an overall portfolio change; that would depend on the relevant premium weights and the full implementation. Common mistake avoided: Adding 5% to the territory relativity’s percentage-point change ignores that the relativity enters as a multiplicative factor.

What the Estimate Can and Cannot Say

An illustrative policy impact is most useful when its basis is explicit. State whether it assumes unchanged exposure and coverages, which premium components receive the change, and whether fixed charges are included. A policy with an unchanged fixed fee and a changed variable premium does not receive the same percentage change on its total bill as on its variable premium alone.

Likewise, an overall selected change alone does not identify each policy’s result. Without implementation detail and policy-level rating information, the actuary can calculate a uniform-multiplier illustration but cannot infer the distribution of actual changes across policyholders. Even when a set of policy estimates is available, the aggregate comparison should be checked on a consistent premium base: sum the old premiums, sum the new premiums, and divide the difference by the old total.

Use dollars to show the size of an individual policy’s change.
Use percentages to compare changes across policies with different starting premiums.
Weight component changes by old premium when calculating a policy-level percentage from additive components.
Separate assumptions from results. A held-constant exposure or unchanged fee is an assumption, not a consequence of the selected rate change.

These distinctions make policyholder illustrations more informative without implying that a portfolio-average selection applies identically to every risk. They are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, assume annual premiums and hold exposures and coverage constant unless the problem states otherwise.

A uniform 6% increase applies to two policies with current premiums of $1,250 and $980. Find each new premium and their combined dollar increase. (Answer: $1,325.00 and $1,038.80; combined increase $133.80.)
A policy has current coverage premiums of $700 and $500. The first increases by 8% and the second decreases by 2%. Find the new total and total percentage impact. (Answer: New total $1,246.00; increase $46.00, or 3.8333%.)
A policy’s current premium is $840. Its base rate increases by 4%, while its territory relativity changes from 1.05 to 1.10. No other factor changes. Find the new premium and percentage impact. (Answer: $915.20; increase $75.20, or 8.9524%.)
A policy has two variable coverage premiums of $400 and $600, changing by +12% and −3%, respectively, plus a $25 fixed charge that does not change. Find the old total, new total, and total percentage impact. (Answer: Old total $1,025.00; new total $1,055.00; increase $30.00, or 2.9268%.)
Two policies have old premiums of $500 and $1,500. Their respective changes are +10% and +2%. Find the combined percentage impact and explain why it is not the simple average of the two policy changes. (Answer: Combined increase $80.00 on $2,000.00, or 4.0000%; the percentage is premium-weighted, not an equal-weight average.)

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