Calculating Variable Expense Ratios

Why Variable Expense Ratios Matter

Some expenses increase or decrease with the amount of premium written or earned. Commissions, premium taxes, and certain billing or assessment charges are common examples. To include these costs in a rate analysis, an actuary often expresses them as a proportion of premium: a variable expense ratio. The key calculation is straightforward, but only after the premium denominator and the expense base are made consistent.

A commission rate may be stated as a percentage of gross premium, while a tax may be calculated on premium after commission. Those stated percentages are not automatically additive. Likewise, when combining experience from books of different sizes, component ratios must be combined using their premium volumes rather than by taking a simple average. These distinctions make the result interpretable and prevent a provision from being understated or overstated.

Definition: For an expense component $j$, its variable expense ratio relative to premium base $P$ is $r_j=E_j/P$, where $E_j$ is the expense amount attributable to that component and $P$ is the specified premium amount. The ratio is meaningful only when the expense amount and premium denominator refer to a consistent population, time period, and premium basis.

Calculating Component Ratios

Suppose a hypothetical insurer has $P$ dollars of premium on the selected basis. If commissions are $C$, premium taxes are $T$, and other premium-linked expenses are $O$, their ratios relative to that same premium base are

r_C=\frac{C}{P},\qquad r_T=\frac{T}{P},\qquad r_O=\frac{O}{P}.

For example, if $C$ is already calculated as a fixed percentage $c$ of $P$, then $C=cP$, so $r_C=cP/P=c$. This identity is why a commission rate stated on the same premium base can be used directly as a commission ratio. The same reasoning applies to a tax or other expense quoted as a percentage of that same base.

The phrase “same premium base” is essential. The denominator could be written premium, earned premium, or another defined premium measure. The choice should match the expense data and the purpose of the analysis. A ratio using earned premium should not be combined casually with another ratio using written premium. If figures are drawn from different periods or populations, first place them on a consistent basis or calculate ratios separately and aggregate them appropriately.

An expense described as “premium-linked” is not necessarily calculated on gross premium. The governing expense rule specifies its base. If a tax is calculated on premium after commission, for instance, its quoted percentage is applied to that net amount. To express that tax as a proportion of gross premium, calculate the tax dollars first and divide by gross premium.

Worked Example: Adding Components on a Common Premium Base

A hypothetical program has $800,000 of written premium. Its commission is 12% of written premium, its premium tax is 2.5% of written premium, and a premium-linked assessment is 0.5% of written premium. Calculate the dollars for each expense, each expense ratio, and the total variable expense ratio. Assume all three components apply to the same written premium without overlap.

Method. Calculate each expense using its stated base, divide each amount by the same $800,000 premium denominator, and then add the component ratios.

Commission. The commission is $0.12\times \$800{,}000=\$96{,}000$. Its ratio is $\$96{,}000/\$800{,}000=0.12=12\%$.

Premium tax. The tax is $0.025\times \$800{,}000=\$20{,}000$. Its ratio is $\$20{,}000/\$800{,}000=0.025=2.5\%$.

Assessment. The assessment is $0.005\times \$800{,}000=\$4{,}000$. Its ratio is $\$4{,}000/\$800{,}000=0.005=0.5\%$.

The total expense is $\$96{,}000+\$20{,}000+\$4{,}000=\$120{,}000$. Therefore, the total variable expense ratio is $\$120{,}000/\$800{,}000=0.15=15\%$. As a check, the component ratios also sum to $12\%+2.5\%+0.5\%=15\%$. This addition is valid because every component uses the same premium denominator.

Common mistake avoided: adding quoted percentages is not justified merely because they are all called expense rates. Here, addition works because each percentage is explicitly applied to the same written premium.

Converting Different Premium Bases

If a component uses a base other than the selected denominator, express its dollar amount relative to the selected denominator before combining ratios. Consider a hypothetical tax calculated on gross premium less commission. Let $P$ be gross premium, $c$ the commission rate on gross premium, and $t$ the tax rate on premium after commission. Commission dollars are $cP$, so the tax base is $P-cP=(1-c)P$. Tax dollars are $t(1-c)P$, and the effective tax ratio relative to gross premium is therefore $t(1-c)$, not $t$.

This result follows directly by dividing tax dollars by gross premium:

\frac{t(P-cP)}{P} =\frac{t(1-c)P}{P} =t(1-c).

The calculation assumes the tax is actually assessed on premium net of the stated commission, and that the commission rate applies to the gross premium $P$. If the expense rule defines a different base, substitute that base rather than using this expression automatically.

Worked Example: A Tax Based on Premium After Commission

A hypothetical product has $500,000 of gross written premium. Commissions are 13% of gross premium. A premium tax is 2% of premium after commission, and a separate assessment is 0.6% of gross premium. Find each component’s ratio relative to gross premium and the total variable expense ratio.

Method. Use gross written premium as the common denominator. Calculate the tax on its specified net-of-commission base, then divide the tax dollars by gross premium.

Commission. Commission dollars are $0.13\times \$500{,}000=\$65{,}000$. Relative to gross premium, the commission ratio is $\$65{,}000/\$500{,}000=13\%$.

Tax base and tax. Premium after commission is $\$500{,}000-\$65{,}000=\$435{,}000$. The tax is $0.02\times \$435{,}000=\$8{,}700$. Relative to gross premium, the tax ratio is $\$8{,}700/\$500{,}000=0.0174=1.74\%$.

Assessment. The assessment is $0.006\times \$500{,}000=\$3{,}000$, giving a gross-premium ratio of $\$3{,}000/\$500{,}000=0.6\%$.

Total variable expenses are $\$65{,}000+\$8{,}700+\$3{,}000=\$76{,}700$. Dividing by gross premium gives $\$76{,}700/\$500{,}000=0.1534=15.34\%$. The component ratios check the same result: $13\%+1.74\%+0.6\%=15.34\%$.

Common mistake avoided: adding the quoted commission, tax, and assessment rates gives $13\%+2\%+0.6\%=15.6\%$, which is too high. The quoted 2% tax rate applies to $435,000, not to the $500,000 gross-premium denominator.

Combining Ratios Across Books

When experience is available for multiple groups, the overall ratio is total expense divided by total premium. This gives a premium-weighted average of the group ratios. To see why, let group $k$ have premium $P_k$ and expense ratio $r_k$. Its expense is $E_k=r_kP_k$. Across all groups, the aggregate ratio is

r_{\mathrm{all}} =\frac{\sum_k E_k}{\sum_k P_k} =\frac{\sum_k r_kP_k}{\sum_k P_k}.

The expression assigns each group a weight equal to its share of total premium, $P_k/\sum_kP_k$. Since these weights sum to one, the aggregate ratio is a weighted average. A group with more premium has more influence on the overall result. A simple average gives equal influence to groups of unequal premium size and generally does not reproduce the aggregate expense-to-premium ratio.

Specify the denominator.
Choose the premium measure, population, and period. Confirm that the expense amounts correspond to that basis.

Calculate component dollars.
Apply each expense rule to its stated base. Do not assume every quoted percentage uses gross premium.

Convert and aggregate.
Divide each component by the common denominator. For multiple groups, divide total expense by total premium, or use the equivalent premium-weighted average.

Reconcile.
Check that the sum of component expense dollars divided by total premium agrees with the total ratio calculated from component ratios.

Worked Example: Aggregating Ratios for Unequal Books

A hypothetical insurer reports the following written-premium experience for two independent segments. Segment A has $1,200,000 of premium, $144,000 of commission, $24,000 of premium tax, and $12,000 of other premium-linked expense. Segment B has $800,000 of premium, $112,000 of commission, $16,000 of tax, and $8,000 of other expense. All listed expenses use written premium as their denominator. Calculate the component ratios and the overall variable expense ratio.

Method. Sum the dollars for each expense component and divide by total premium. Also calculate each segment’s total ratio as a check.

Total premium. $\$1{,}200{,}000+\$800{,}000=\$2{,}000{,}000$.

Commission ratio. Total commission is $\$144{,}000+\$112{,}000=\$256{,}000$. The aggregate commission ratio is $\$256{,}000/\$2{,}000{,}000=12.8\%$.

Tax ratio. Total tax is $\$24{,}000+\$16{,}000=\$40{,}000$. The aggregate tax ratio is $\$40{,}000/\$2{,}000{,}000=2.0\%$.

Other-expense ratio. Total other expense is $\$12{,}000+\$8{,}000=\$20{,}000$. Its aggregate ratio is $\$20{,}000/\$2{,}000{,}000=1.0\%$.

Total expense is $\$256{,}000+\$40{,}000+\$20{,}000=\$316{,}000$. Thus the overall variable expense ratio is $\$316{,}000/\$2{,}000{,}000=15.8\%$. The component ratios also sum to $12.8\%+2.0\%+1.0\%=15.8\%$.

Aggregation check. Segment A’s total expense is $\$180{,}000$, so its ratio is $\$180{,}000/\$1{,}200{,}000=15\%$. Segment B’s total is $\$136{,}000$, so its ratio is $\$136{,}000/\$800{,}000=17\%$. Weighting by premium gives $(0.60)(15\%)+(0.40)(17\%)=9.0\%+6.8\%=15.8\%$, matching the aggregate calculation.

Common mistake avoided: the simple average $(15\%+17\%)/2=16\%$ is not the overall ratio. It gives the two segments equal weights even though their premium volumes differ.

Common Traps and Practical Checks

Variable expense ratios are inputs to rate analysis, so a small denominator error can affect every premium amount to which the provision is applied. Keep the calculation tied to the actual expense rule and the chosen premium measure. Do not add a dollar expense twice if it already includes an assessment or commission, and do not treat a fixed dollar charge as premium-linked merely because it appears in the same expense ledger.

Mixing premium measures. A ratio based on earned premium and one based on written premium may describe different bases. Identify the intended denominator before combining them.
Adding rates with different bases. First calculate component dollars, then convert each to the common denominator.
Using an unweighted average. Aggregate expenses and premiums, or weight group ratios by each group’s share of total premium.
Misreading the tax base. A percentage applied to premium net of commission is not the same effective percentage of gross premium.
Reporting only a total. Showing component dollars and ratios makes the calculation easier to audit and helps reveal a missing or duplicated expense.

A concise justification for summing ratios is that, when every component is measured against the same $P$, total expense is $\sum_j E_j$, and

\frac{\sum_j E_j}{P} =\sum_j\frac{E_j}{P} =\sum_j r_j.

If the components use different bases, the middle equality does not establish that the quoted rates can simply be added. Convert each expense amount to the chosen denominator first. These calculations are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the common premium denominator and keep expense bases consistent.

A hypothetical book has $840,000 of written premium. Commission is 11% of written premium, tax is 2.5% of written premium, and a service assessment is 0.5% of written premium. Find each expense amount and the total variable expense ratio. (Answer: commission $92,400; tax $21,000; assessment $4,200; total ratio 14.0%)
Two hypothetical segments have the following experience: Segment A has $300,000 premium, $36,000 commission, $5,000 tax, and $4,000 other expense; Segment B has $700,000 premium, $98,000 commission, $14,000 tax, and $7,000 other expense. All expenses use the stated premium base. Find the aggregate commission, tax, other-expense, and total ratios. (Answer: 13.4%, 1.9%, 1.1%, and 16.4%, respectively)
A hypothetical portfolio has $400,000 of gross premium. Commission is 9% of gross premium, tax is 3% of premium after commission, and another expense is 0.4% of gross premium. Find the tax dollars and total ratio relative to gross premium. (Answer: tax base $364,000; tax $10,920; total ratio 12.13%)
One hypothetical book has $400,000 of premium and a 14% total variable expense ratio; a second has $600,000 of premium and an 18% ratio. Calculate the aggregate ratio, and compare it with the simple average. (Answer: aggregate ratio 16.4%; simple average 16.0%, which is not the aggregate ratio)
A hypothetical program has $1,500,000 of gross written premium, $180,000 of commission, tax of 2.2% of gross premium, and another premium-linked expense of 0.75% of gross premium. Calculate each component ratio and total expense dollars. (Answer: commission 12.0%; tax 2.2%; other expense 0.75%; total ratio 14.95%; total expense $224,250)

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