Why Variable Expenses Need a Premium Base
Some expenses increase directly with the premium charged. A hypothetical insurer may pay a commission equal to a stated percentage of premium, remit a premium tax, or incur another expense that is assessed as a percentage of premium. These amounts are different from fixed expenses per exposure: when premium changes, the dollar amount of a variable expense changes too.
That distinction matters in a rate indication. If the amount needed for losses and fixed expenses is known, it is not enough to add a percentage of that amount when the percentage expense is defined as a share of the final premium. The variable expense must be calculated on the premium that includes the variable expense itself. This tutorial develops that relationship and shows how to apply it. The treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Examples of variable expenses include commissions, premium taxes, and other expenses explicitly specified as percentages of premium. A fixed expense per exposure, such as a stated dollar amount for issuing a policy, is not a variable expense ratio. A fee may be fixed or premium-based depending on how it is defined; its label alone does not determine the treatment.
For a rate indication, consider the non-variable amount \(B\) per exposure. It may include an expected loss cost and a fixed expense per exposure, provided both have been placed on the appropriate prospective basis. Let \(R\) be the required premium per exposure and \(e\) the applicable variable expense ratio. The budget identity is \(R=B+eR\): the premium funds the non-variable amount and variable expenses calculated on that premium.
The corresponding variable expense dollars are \(eR\), or equivalently \(Be/(1-e)\). The condition \(e<1\) is necessary for a finite positive solution when \(B>0\). This equation isolates variable expenses; it does not include a provision for profit or contingency.
Building the Variable Expense Ratio
When several expenses are stated as percentages of the same premium base, add their rates to obtain the total variable expense ratio. For instance, if commission is 9% of premium and premium tax is 2% of that same premium, the combined variable expense ratio is 11%. The dollar expenses are then calculated on the premium, not on the loss cost or the amount before variable expenses.
Check the stated basis before combining expense rates. If one rate applies to written premium and another to earned premium, they cannot automatically be added for an indication until they have been expressed on a compatible basis. Likewise, if an expense is calculated on a different base—for example, premium net of another expense—use the problem's specified relationship rather than treating it as a percentage of gross premium.
Worked Example: Commission, Tax, and a Premium-Based Service Expense
A hypothetical coverage has an expected loss cost of $420 per exposure and fixed expenses of $60 per exposure. Commission is 10% of premium, premium tax is 3% of premium, and a service expense is 2% of premium. All three percentages apply to the same gross premium base. Find the premium required to fund these amounts and the variable expense dollars.
Method. Add the rates that share a premium base, then solve the premium budget equation. This avoids applying the expense percentages only to the loss and fixed-expense amount.
Step 1: Find the non-variable amount. The loss cost and fixed expense total \(B=\$420+\$60=\$480\) per exposure.
Step 2: Find the variable expense ratio.
Step 3: Solve for premium.
Rounded to the nearest cent, the required premium is $564.71 per exposure.
Step 4: Calculate and check the expense dollars. Commission is \(0.10(\$564.705882\ldots)=\$56.470588\ldots\), tax is \(0.03(\$564.705882\ldots)=\$16.941176\ldots\), and service expense is \(0.02(\$564.705882\ldots)=\$11.294118\ldots\). Together they are \(0.15(\$564.705882\ldots)=\$84.705882\ldots\), or $84.71 to the nearest cent. The budget checks: \(\$480+\$84.705882\ldots=\$564.705882\ldots\). The method avoids treating the variable expenses as percentages of $480 instead of percentages of premium.
Why the Premium Equation Holds
The formula follows directly from the definition of a variable expense ratio. If the premium is \(R\), then expenses equal to a proportion \(e\) of that premium are \(eR\). The premium must cover both the non-variable amount \(B\) and those expenses. Thus the required premium satisfies \(R=B+eR\). Subtracting \(eR\) from each side gives \((1-e)R=B\), and dividing by \(1-e\) gives \(R=B/(1-e)\), provided \(e\ne1\).
This derivation explains why the variable expense contribution is not simply \(eB\). The ratio is defined on the premium \(R\), which is larger than \(B\) when \(B>0\) and \(e>0\). Substituting the solution for \(R\) gives the expense amount:
The total premium can therefore be separated into its non-variable amount \(B\) and variable expense contribution \(Be/(1-e)\). This is a budget identity, not an assumption that the dollar expenses stay constant when the premium changes.
Worked Example: Multiple Variable Expense Rates on One Base
A hypothetical policy has an expected loss cost of $268 per exposure and fixed expenses of $42 per exposure. The applicable commission rate is 9%, premium tax is 2.5%, and another premium-based expense is 6.5%. Each rate applies to gross premium. Calculate the required premium and each variable expense amount.
Method. Combine the rates, solve the budget equation, and then apply each expense rate to the solved premium. The common mistake to avoid is calculating the expenses on $310 alone.
Step 1: Combine the non-variable amounts and expense rates.
Step 2: Calculate the required premium.
Rounded to the nearest cent, the required premium is $378.05 per exposure.
Step 3: Calculate each expense. Commission is \(0.09(\$378.048780\ldots)=\$34.024390\ldots\). Tax is \(0.025(\$378.048780\ldots)=\$9.451220\ldots\). The other expense is \(0.065(\$378.048780\ldots)=\$24.573171\ldots\). Their sum is \(\$68.048780\ldots\), which also equals \(0.18R\).
Check. The non-variable amount plus variable expenses is \(\$310+\$68.048780\ldots=\$378.048780\ldots\), matching the calculated premium. Rounded to cents, the three expenses are $34.02, $9.45, and $24.57, totaling $68.04; the one-cent difference from subtracting rounded totals is due to rounding. The unrounded amounts reconcile exactly.
Combining Ratios Across Classes
When expense ratios differ by class, the overall ratio depends on the premium distribution across classes. It is a premium-weighted average, not generally a simple average and not necessarily an exposure-weighted average. Let \(w_j\) be the share of the applicable premium base in class \(j\), and let \(e_j\) be that class's variable expense ratio. The weights sum to 1, and the combined ratio is \(\sum_j w_j e_j\).
This result follows because class \(j\) contributes \(e_j\) times its premium. If total premium is \(R\), class \(j\)'s premium is \(w_jR\), so its expense is \(e_jw_jR\). Summing class expenses gives \(R\sum_j w_je_j\), establishing the weighted ratio.
Worked Example: Premium-Weighted Expense Ratios
A hypothetical portfolio has two classes. Class A is expected to produce 70% of the applicable premium and has variable expenses of 12% of its premium. Class B is expected to produce 30% and has variable expenses of 18%. The portfolio's non-variable funding requirement is $172,000. Find the combined variable expense ratio, required premium, and total variable expenses. Assume the stated premium shares apply to the prospective premium being indicated.
Method. Calculate the ratio using premium shares, then apply the combined ratio to the portfolio's premium. The method avoids taking an unweighted average of the class rates.
Step 1: Calculate the combined ratio.
The combined variable expense ratio is 13.8%.
Step 2: Solve for total premium.
Rounded to the nearest cent, the required premium is $199,535.96.
Step 3: Verify the expense contribution. The class premiums are \(0.70R=\$139{,}675.1740\ldots\) for A and \(0.30R=\$59{,}860.7889\ldots\) for B. Class A's expenses are \(0.12(0.70R)=\$16{,}761.0209\ldots\); class B's are \(0.18(0.30R)=\$10{,}774.9420\ldots\). Total variable expenses are \(\$27{,}535.9629\ldots\), also equal to \(0.138R\). Adding the non-variable requirement gives \(\$172{,}000+\$27{,}535.9629\ldots=\$199{,}535.9629\ldots\), so the result checks.
Common Mistakes and Practical Checks
A frequent error is to compute variable expenses as a percentage of the non-variable amount. In the second worked example, adding 18% of $310 gives \(\$310+\$55.80=\$365.80\). But 18% of that proposed premium is \(\$65.844\), not $55.80. The proposed amount therefore does not fund the stated expenses: \(\$310+\$65.844=\$375.844\), which exceeds $365.80. The correction is to solve \(R=310+0.18R\), giving \(R=\$378.048780\ldots\).
Before calculating a variable expense provision, check the following:
- Identify the base. Confirm whether each percentage is applied to written premium, earned premium, gross premium, or another specified amount.
- Separate fixed and variable items. Include fixed expenses as dollar amounts in \(B\); do not add them to the variable expense ratio.
- Use appropriate weights. For class-specific ratios, weight by the relevant premium shares unless the problem specifies a different calculation.
- Check the result. Confirm that non-variable amounts plus calculated variable expenses equal the required premium before rounding.
A variable expense ratio of zero leaves the premium equal to the non-variable amount. For a positive ratio below 1 and a positive non-variable amount, the required premium exceeds that amount. If the ratio is 1 or greater, the equation has no finite positive solution for a positive \(B\); the assumed expense structure cannot be funded by a finite premium under this model.
Check Your Understanding
For each problem, identify the applicable variable expense ratio and use the stated premium base.
- A hypothetical policy has a loss cost of $210 per exposure and fixed expenses of $30. A commission of 8% applies to gross premium. Find the required premium and variable expense amount. (Answer: premium $260.87; variable expense $20.87.)
- A coverage has a non-variable requirement of $510 per exposure. Commission is 7%, tax is 3%, and another expense is 5%, all of gross premium. Find the required premium. (Answer: $600.00.)
- Two hypothetical classes have prospective premium shares of 25% and 75%, with variable expense ratios of 8% and 16%, respectively. The non-variable requirement is $430. Find the combined ratio and required premium. (Answer: combined ratio 14%; premium $500.00.)
- A hypothetical book has loss cost $380 per exposure, fixed expenses $40 per exposure, and a variable expense ratio of 11%. Calculate the required premium and variable expense dollars, rounded to cents. (Answer: premium $471.91; variable expenses $51.91.)
- A policy has a loss cost of $205 and a fixed expense of $18 per exposure. Commission is 9% and premium tax is 2%, both applied to gross premium. Calculate the required premium and total variable expenses, rounded to cents. (Answer: premium $250.56; variable expenses $27.56.)