Rate Changes and Variable Expense Provisions

Introduction

A percentage-based expense provision is a fraction of a specified premium base. If the premium rate changes for an unchanged portfolio, the premium base changes, so the dollar amount of that expense changes too. The selected percentage does not change merely because the rate changed: it remains the same fraction of the new base, provided the expense rule and the base itself are unchanged.

This distinction matters when comparing proposed rates with current rates. A common error is to hold the expense dollars fixed after increasing the premium, or to change the selected expense percentage just to make its dollar amount match the new premium. The correct calculation keeps the percentage provision fixed and recalculates the dollars from the appropriate premium base.

Definition: Let $P$ be the premium base to which a particular variable expense applies, $V$ the dollar amount of that expense, and $e$ its selected percentage provision. The relationship is $V=eP$. A rate change can change $P$ and $V$, while $e$ remains unchanged if the expense percentage and the definition of its base remain unchanged.

How a Rate Change Affects the Expense

Suppose a portfolio has $N$ unchanged exposures and the rate per exposure changes from $R_0$ to $R_1$. If the premium base consists only of that rated premium, then the before-and-after premiums are $P_0=NR_0$ and $P_1=NR_1$. Define the rate change factor as $k=R_1/R_0$. It follows that $P_1=kP_0$.

If the selected percentage provision is $e$ both before and after the change, the expense amounts are $V_0=eP_0$ and $V_1=eP_1$. Substituting $P_1=kP_0$ gives the key relationship:

$$ V_1=eP_1=e(kP_0)=k(eP_0)=kV_0 $$

Thus, when the premium base rises by 6%, the expense dollars also rise by 6%, assuming the same exposures, percentage provision, and premium base. The expense ratio remains $V_1/P_1=e$. The dollar amount and the percentage provision answer different questions: one measures the expense in money, while the other measures the expense relative to its base.

Assumptions for the proportional result: The comparison uses the same exposure portfolio; the rate change scales the relevant premium base proportionately; the expense percentage is unchanged; and no other charge or adjustment is included in that base. If the portfolio mix, coverage terms, expense rule, or base changes, the new expense must be calculated from the actual new base rather than by automatically applying the old total-premium change.

Why the Percentage Stays the Same

The result follows directly from the definition of the provision. If the expense is a fixed fraction $e$ of premium, then dividing the expense by its matching premium base returns that fraction. After a rate change, both numerator and denominator scale by $k$, so their ratio is unchanged:

\frac{V_1}{P_1} = \frac{kV_0}{kP_0} = \frac{V_0}{P_0} = e

The cancellation requires $k$ to be the same multiplier for the expense and premium bases. It does not say that every expense amount in a rate calculation changes by the rate-change factor. A fixed dollar expense per policy, for example, is not defined as a constant fraction of premium and should not be scaled this way. Nor does the result authorize using the same percentage if the underlying commission schedule or tax rule changes.

Identify the base.
Determine which premium amount the expense percentage applies to. Do not assume that every amount billed to a policyholder is part of that base.

Calculate the new base.
Apply the rate change to the relevant premium, keeping the exposure count and other specified factors consistent with the comparison.

Apply the selected percentage.
Multiply the new base by the unchanged provision to find the new expense dollars.

Check the result.
Divide the new expense dollars by the new base. The ratio should equal the selected percentage, subject only to rounding.

Worked Before-and-After Calculations

Worked Example: Commission Dollars After a Rate Increase

A hypothetical portfolio has 2,400 identical exposures. The current rate is $500 per exposure, and the selected commission provision is 12% of premium. A proposed rate is 7.5% higher. Assume exposure count, commission percentage, and premium basis are unchanged. Calculate the before-and-after premium and commission, and verify the percentage provision.

Method and trap avoided. Apply the rate change to the premium first, then apply the commission percentage. The common mistake is to leave the commission dollars at their old value even though the premium base has increased.

Current premium. There are 2,400 exposures at $500 each:

P_0=2{,}400\times \$500=\$1{,}200{,}000

Current commission. Apply the 12% provision to the current premium:

V_0=0.12\times \$1{,}200{,}000=\$144{,}000

Proposed rate and premium. A 7.5% increase gives a rate-change factor of $1.075$. The new rate is:

R_1=\$500\times 1.075=\$537.50

The proposed premium is therefore:

P_1=2{,}400\times \$537.50=\$1{,}290{,}000

Proposed commission. Keep the selected percentage at 12% and apply it to the new base:

V_1=0.12\times \$1{,}290{,}000=\$154{,}800

Check. The commission dollars increased by $\$154{,}800-\$144{,}000=\$10{,}800$, which is $ \$10{,}800/\$144{,}000=0.075$, or 7.5%. Also, $\$154{,}800/\$1{,}290{,}000=0.12$. Both checks agree with the stated assumptions.

Worked Example: A Rate Decrease and a Separate Fixed Expense

A hypothetical book has a premium base of $840,000 and a percentage-based expense of $67,200. The rate is reduced by 5%, with exposures unchanged. A separate fixed expense budget is $42,000 and does not vary with premium. Calculate the selected percentage provision and the new variable expense. Then compare the fixed expense as a percentage of premium before and after the change.

Method and trap avoided. Calculate the percentage provision from the variable expense and its base, then apply it to the new premium. Do not reduce the fixed dollar expense by 5%: its stipulated amount is unchanged.

Selected variable expense percentage.

e=\frac{\$67{,}200}{\$840{,}000}=0.08=8\%

New premium. A 5% decrease gives a factor of $0.95$:

P_1=\$840{,}000\times 0.95=\$798{,}000

New variable expense. The provision remains 8%:

V_1=0.08\times \$798{,}000=\$63{,}840

The decrease in variable expense dollars is $\$67{,}200-\$63{,}840=\$3{,}360$, equal to 5% of the original variable expense.

Fixed expense comparison. The fixed expense remains $42,000. Its ratio to premium before the rate change was:

\frac{\$42{,}000}{\$840{,}000}=0.05=5\%

After the decrease, its ratio is:

\frac{\$42{,}000}{\$798{,}000}\approx 0.052632=5.2632\%

Check. The variable expense ratio remains $ \$63{,}840/\$798{,}000=8\%$, while the fixed expense dollars remain $42,000 and its premium ratio increases. This confirms why the percentage-based result must not be applied to fixed dollars.

Worked Example: Applying the Percentage to the Correct Premium Base

A hypothetical insurer bills 800 policies. Each policy currently has $300 of covered premium and a separate $25 service fee. The stated commission is 15% of covered premium only; the service fee is excluded. Covered premium rates increase by 10%, while the service fee and policy count remain unchanged. Calculate the before-and-after commission and the after-change commission as a percentage of the total billed amount.

Method and trap avoided. Use covered premium as the commission base, not the total invoice. The common error is applying the commission to the fee simply because it appears on the same bill.

Current covered premium and fee.

P_0=800\times \$300=\$240{,}000

F_0=800\times \$25=\$20{,}000

The current total billed amount is $\$240{,}000+\$20{,}000=\$260{,}000$. The commission base is only the $240,000 covered premium, so current commission is:

V_0=0.15\times \$240{,}000=\$36{,}000

After-change premium. The rate factor is $1.10$, so covered premium becomes:

P_1=\$240{,}000\times 1.10=\$264{,}000

The fee remains $20,000, so the new total billed amount is $\$264{,}000+\$20{,}000=\$284{,}000$. Apply the unchanged 15% commission only to covered premium:

V_1=0.15\times \$264{,}000=\$39{,}600

Percentage of the total billed amount. Although the commission provision is 15% of covered premium, its ratio to total billed amount is:

\frac{\$39{,}600}{\$284{,}000}\approx 0.139437=13.9437\%

Check. Commission increased by $\$3,600$, which is 10% of the original $36,000, and $ \$39,600/\$264,000=15\%$. The 13.9437% ratio uses a different denominator and is not the selected commission provision.

Common Mistakes and Practical Checks

Before-and-after calculations are easiest to audit when the premium base is written explicitly. State what the percentage applies to, calculate that amount under each rate level, and then apply the provision. This also makes clear whether an observed difference comes from the rate, the base, the percentage, or the portfolio.

Holding variable expense dollars constant. If the base rises and the expense percentage is fixed, expense dollars rise in the same proportion. Recalculate them from the new base.
Changing the provision to preserve old dollars. Keeping $V$ fixed while increasing $P$ necessarily changes $V/P$. That is not consistent with an unchanged percentage-based expense provision.
Using the total bill without checking the contract or rule. A fee, surcharge, or other amount may not be part of the commission or tax base. Use the specified base, not an assumed one.
Applying the variable-expense rule to fixed dollars. A fixed amount per policy does not automatically scale with premium. Recalculate its ratio separately if a ratio is needed.
Assuming every portfolio change is a rate change. If exposure counts or the mix of policies change, total premium may not move by the rate factor alone. Calculate the new base from the new portfolio details.

The simple proportional result is most useful as a controlled comparison: hold exposures, terms, the expense rule, and the base definition constant, and change only the rate. In practice, verify that the selected percentage still represents the relevant commission, tax, or other premium-based expense. A change in the underlying expense arrangement may justify a new selection; a premium rate change by itself does not.

Key takeaway: For a percentage-based expense, calculate new expense dollars as the unchanged selected percentage multiplied by the new applicable premium base. When that base changes proportionately, the expense dollars change by the same factor, while the selected percentage provision stays constant. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the applicable premium base before calculating the expense.

A hypothetical portfolio has a $460,000 premium base and a 9.5% variable expense provision. Rates increase by 6%, with exposures and the expense provision unchanged. What are the new premium base and variable expense dollars? (Answer: $487,600 premium; $46,322 variable expense.)
A book has 1,500 exposures, a rate of $240 per exposure, and an 11.5% variable expense provision. The rate rises to $252. Calculate the old and new variable expense dollars and verify the percentage. (Answer: $41,400 old; $43,470 new; 11.5% in each case.)
A $720,000 premium base supports $54,000 of a premium-based expense. Rates then decrease by 12.5%, with all other assumptions unchanged. Find the provision, new base, and new expense dollars. (Answer: 7.5%; $630,000; $47,250.)
A hypothetical portfolio’s premium base is $2,200,000 and its selected variable expense provision is 13%. Premium increases by 20% because of a rate change, with exposures unchanged. Find the new expense dollars and the increase in expense dollars. (Answer: $343,200 new expense; $57,200 increase.)
A hypothetical book has $500,000 of covered premium and $20,000 in separate fees. A 10% expense applies only to covered premium. Covered premium rates increase by 8%; fees and exposure counts remain fixed. Find the new expense dollars and that amount as a percentage of the new total billed amount. (Answer: $54,000; $54,000/$560,000 = 9.6429%.)

What You'll Learn