Fixed Expense Ratio Formulations

Why a Fixed Expense Can Have a Changing Ratio

A fixed expense is a dollar amount that does not vary directly with premium over the range and period being analyzed. But its ratio to premium can change whenever premium changes. For example, if an insurer incurs the same fixed expense for a group of policies but premium per policy rises, that expense is a smaller percentage of premium. The expense amount has not become variable; only the denominator of the ratio has changed.

The phrase “fixed expense ratio” therefore needs a defined premium base. The denominator might be a specified premium measure for the policies being rated. The expense amount and premium base must cover a consistent set of policies, exposures, and time periods. This tutorial focuses on that calculation and interpretation, aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For a stated group and period, a fixed expense ratio is the fixed expense amount divided by the selected premium base. If $F$ is the fixed expense and $P$ is that premium base, then $r_F=F/P$. Expressed as a percentage, the ratio is $100F/P\%$. The ratio is meaningful only when $F$ and $P$ refer to a consistent scope and the chosen premium base is stated.

The word “fixed” describes how the expense behaves in dollars, not how its ratio behaves. Fixed expenses are not automatically a constant percentage of premium. A percentage may be used as a rate formulation, but the percentage calculated from a fixed dollar amount depends on the premium base.

Per-Exposure and Aggregate Formulations

For a homogeneous group with $N$ exposures, let $f$ be the fixed expense per exposure and $p$ the selected premium base per exposure. Then total fixed expense is $F=Nf$, and total premium base is $P=Np$. If the same exposure count applies to both, the aggregate ratio can also be written per exposure:

r_F=\frac{F}{P}=\frac{Nf}{Np}=\frac{f}{p}.

This cancellation is useful, but only when the expense and premium figures use the same exposure basis. If there are different exposure counts, policy terms, or groups, calculate the ratio from totals or carefully reconcile the denominators before simplifying.

Key distinction: A fixed expense per exposure is a dollar amount, such as $60 per exposure. A fixed expense ratio is a percentage, such as 12% of a stated premium base. The percentage is derived by dividing the dollar amount by premium per exposure; it is not an intrinsic characteristic of the fixed expense.

The choice of premium base matters. If the fixed expense is $60 per exposure, and the selected base is $500 of premium per exposure, the ratio is $60/500=12\%$. If instead the analysis uses a different premium base of $600 per exposure, the ratio is $60/600=10\%$. The expense amount has not changed, but the denominator has.

Why the Ratio Moves When Premium Changes

Suppose the fixed expense amount $F$ remains unchanged, while the selected premium base changes from $P$ to $P'$. The old and new ratios are $r_F=F/P$ and $r_F'=F/P'$. If premium changes by a factor $k$, so that $P'=kP$, then:

r_F'=\frac{F}{kP}=\frac{1}{k}r_F.

Thus, if the premium base rises by 8%, its new level is $1.08P$, and the fixed expense ratio is divided by 1.08—not reduced by 8 percentage points. If the premium base falls, the ratio rises, all else equal. This relationship assumes the fixed expense amount remains constant for the comparison. If the expense amount also changes, use the full ratio $F'/P'$; the inverse relationship alone no longer determines the result.

Define the scope.
Identify which exposures and period are included in the fixed expense amount.

Name the premium base.
Specify the premium measure used as the denominator and ensure it covers the same scope.

Calculate the ratio.
Divide the fixed expense by the selected premium base, then convert to a percentage if required.

Interpret any change.
Check whether the expense amount, premium base, or both changed before explaining a different ratio.

Worked Applications

Worked Example: Converting Expense Per Exposure to a Ratio

A hypothetical insurer has 2,400 earned exposures during a rating period. Fixed expenses allocated to this group total $180,000. The selected premium base is $600 per exposure. Calculate the fixed expense per exposure and the fixed expense ratio. Then recalculate the ratio if the premium base per exposure increases by 8%, with exposure count and fixed expenses unchanged.

Method and scope. Use the same 2,400 exposures and period for both totals. The fixed expense is held constant for the comparison; the premium base is the stated amount per exposure.

Step 1: Calculate fixed expense per exposure.

f=\frac{\$180{,}000}{2{,}400}=\$75.

Step 2: Calculate the original aggregate premium base.

P=2{,}400\times \$600=\$1{,}440{,}000.

Step 3: Calculate the original ratio.

r_F=\frac{\$180{,}000}{\$1{,}440{,}000}=0.125=12.5\%.

Step 4: Update premium after an 8% increase. New premium per exposure is $\$600\times1.08=\$648$. The new aggregate premium base is $2{,}400\times\$648=\$1{,}555{,}200$. The fixed expense remains $180,000, so:

r_F'=\frac{\$180{,}000}{\$1{,}555{,}200}=0.1157407\ldots=11.57\%.

Check and common mistake. The inverse-factor check gives $12.5\%/1.08=11.57407\ldots\%$, consistent with the aggregate calculation. A common mistake is to subtract 8 percentage points from 12.5%, giving 4.5%. The premium rose by 8% proportionally, so the ratio is divided by 1.08; it does not fall by 8 percentage points.

Worked Example: How the Premium Base Changes the Ratio

For a hypothetical portfolio, fixed expenses are $96,000 for a period. The selected gross premium base is $800,000. An analyst also wants to report the ratio against a different defined premium base of $720,000. Calculate both ratios. Then find the ratio against the $800,000 base after that base increases by 5%, assuming fixed expenses do not change.

Method and scope. Divide the same fixed expense amount by each stated premium base. The two initial results are both mathematically valid ratios, but they describe different denominators and should be labeled accordingly.

Step 1: Ratio using the gross premium base.

\frac{\$96{,}000}{\$800{,}000}=0.12=12.00\%.

Step 2: Ratio using the alternative premium base.

\frac{\$96{,}000}{\$720{,}000}=0.133333\ldots=13.33\%.

Step 3: Increase the gross premium base by 5%. The new base is $\$800{,}000\times1.05=\$840{,}000$. Therefore:

\frac{\$96{,}000}{\$840{,}000}=0.1142857\ldots=11.43\%.

Check and common mistake. The new gross-base ratio is $12.00\%/1.05=11.42857\ldots\%$, as calculated directly. A common mistake is to compare 12.00% and 13.33% without noticing that the premium bases differ. A ratio should always be reported with its denominator identified; otherwise, the percentages may be mistaken for inconsistent calculations of the same quantity.

Worked Example: Combining Class Ratios Correctly

A hypothetical portfolio contains two classes for one period. Class A has 1,000 exposures, fixed expense of $50 per exposure, and a selected premium base of $500 per exposure. Class B has 500 exposures, fixed expense of $80 per exposure, and a selected premium base of $1,200 per exposure. Calculate each class ratio and the overall ratio. Then recalculate the overall ratio if Class A's premium base per exposure rises by 10%, with all exposure counts and fixed expenses unchanged.

Method and scope. Calculate class totals and divide total fixed expenses by total premium. Class ratios should not be averaged equally unless the class premium bases are equal.

Step 1: Calculate Class A totals and ratio.

F_A=1{,}000\times\$50=\$50{,}000,\qquad P_A=1{,}000\times\$500=\$500{,}000, $$ $$ r_A=\frac{\$50{,}000}{\$500{,}000}=10.00\%.

Step 2: Calculate Class B totals and ratio.

F_B=500\times\$80=\$40{,}000,\qquad P_B=500\times\$1{,}200=\$600{,}000, $$ $$ r_B=\frac{\$40{,}000}{\$600{,}000}=6.6667\%.

Step 3: Calculate the overall ratio from aggregate amounts. Total fixed expense is $\$50{,}000+\$40{,}000=\$90{,}000$; total premium is $\$500{,}000+\$600{,}000=\$1{,}100{,}000$. Thus:

r_{\text{overall}}=\frac{\$90{,}000}{\$1{,}100{,}000} =0.0818181\ldots=8.18\%.

Step 4: Increase Class A premium by 10%. Its new premium per exposure is $\$500\times1.10=\$550$, so its premium total becomes $1{,}000\times\$550=\$550{,}000$. Class B premium remains $600,000. Total premium is now $1,150,000, while total fixed expense remains $90,000:

r_{\text{overall}}'=\frac{\$90{,}000}{\$1{,}150{,}000} =0.0782608\ldots=7.83\%.

Check and common mistake. The new overall ratio lies between the class ratios after the change: Class A is $50{,}000/550{,}000=9.09\%$, while Class B remains $6.67\%$. The overall 7.83% is consistent with their combined totals. A simple average of the original class ratios, $(10.00\%+6.67\%)/2=8.335\%$, is not the portfolio ratio because the classes have different premium volumes.

Interpretation and Common Traps

A fixed expense ratio is useful for expressing a dollar expense in relation to a selected premium base. It can help compare expense burdens across rating scenarios, provided the same definition of the base is used. It does not establish that the underlying expense will rise or fall in proportion to premium.

For example, if a fixed expense is $75 per exposure, reducing premium per exposure from $600 to $540 increases the expense ratio from $75/600=12.5\%$ to $75/540=13.8889\%$, while the dollar expense stays at $75. Reporting the higher ratio as evidence that the expense itself increased would confuse a change in the denominator with a change in the expense.

Do not treat a derived ratio as a constant dollar expense. A 12% ratio on $500 of premium is $60; on $600 it is $72. Those amounts differ even though the percentage is unchanged.
Do not treat a fixed dollar amount as a constant percentage. A $60 expense is 12% of $500 but 10% of $600.
Do not mix scopes. An expense total for one set of policies should not be divided by premium for a different set or period without an explicit allocation and reconciliation.
Do not average class percentages indiscriminately. The portfolio ratio is total fixed expense divided by total premium. An arithmetic average of class ratios generally gives each class equal weight, even when their premium volumes differ.

Key takeaway: State the fixed expense amount, the selected premium base, and the shared scope before interpreting a fixed expense ratio. When fixed expenses are unchanged, the ratio moves inversely with the premium base; that change alone does not mean the dollar expense changed.

Check Your Understanding

For each problem, state the ratio against the specified premium base and keep the scope consistent.

A hypothetical book has fixed expenses of $45,000 and a selected premium base of $375,000 for the same period. What is the fixed expense ratio? (Answer: 12.00%.)
A hypothetical portfolio has 900 exposures, fixed expense of $72 per exposure, and premium base of $540 per exposure. Calculate total fixed expense, total premium base, and the ratio. (Answer: $64,800 fixed expense; $486,000 premium base; 13.33%.)
Fixed expenses are $132,000, and the selected premium base is $1,200,000. If the base falls by 10% and expenses stay unchanged, what is the new ratio? (Answer: 12.22%.)
Class X has 800 exposures, fixed expense of $60 per exposure, and premium base of $750 per exposure. Class Y has 400 exposures, fixed expense of $90 per exposure, and premium base of $900 per exposure. Calculate the overall ratio from aggregate totals. (Answer: 8.75%.)
A fixed expense amount is $50,000. Calculate its ratio using a gross premium base of $500,000 and a separately defined alternative base of $450,000. Why must the base be named when reporting either result? (Answer: 10.00% of gross premium and 11.11% of the alternative base; the ratios use different denominators and describe different measures.)

What You'll Learn