Fixed Expense Effects on the Permissible Ratio

Why Fixed Expenses Change the Permissible Ratio

In a loss ratio indication, the permissible loss ratio is the share of premium available for losses after providing for expenses and profit or contingencies. Some expenses are modeled as a percentage of premium; others are incurred as a fixed dollar amount per exposure. A fixed dollar amount does not stay the same percentage when premium changes. Consequently, the permissible loss ratio can differ between risks or periods with different premiums, even when their variable expense and profit provisions are identical.

The central calculation is to divide the fixed expense per exposure by the premium per exposure at the applicable rate level. The result is a premium-relative provision, which must be included alongside variable expense and profit provisions when calculating the permissible loss ratio.

Definition: Let $P>0$ be premium per exposure at the applicable rate level, $F\geq 0$ be fixed expense per exposure, $e$ be the variable expense ratio, and $q$ be the profit and contingencies provision as a ratio of premium. Under the model that losses, variable expenses, and profit provisions are expressed relative to premium, the fixed expense provision as a ratio of premium is $F/P$. The permissible loss ratio is $$ PLR(P)=1-e-q-\frac{F}{P}. $$

All quantities must be on a consistent basis. For example, if $F$ is an annual fixed expense per policy-year, $P$ must be annual premium per policy-year, and the exposure unit must be the same in both amounts. The fixed amount is not the total fixed expense for an entire book unless the premium denominator is also the total premium for that same book.

Deriving the Formula

The formula follows from allocating premium among its intended uses. For one exposure, write $L$ for the amount allocated to losses. In this simplified model, variable expenses are $eP$, fixed expenses are $F$, and the profit and contingencies provision is $qP$. A premium that exactly supports those amounts satisfies

$$ P=L+eP+F+qP. $$

Divide both sides by the positive premium $P$, and then solve for the loss share:

1=\frac{L}{P}+e+\frac{F}{P}+q \quad\Longrightarrow\quad \frac{L}{P}=1-e-q-\frac{F}{P}.

The share $L/P$ is the permissible loss ratio, giving the stated formula. This derivation also explains why the fixed expense is divided by premium: it is a dollar amount in the allocation, while the permissible loss ratio is a proportion of premium.

The model assumes the fixed expense $F$, variable expense ratio $e$, and profit provision $q$ are specified for the same exposure and rate level being analyzed. It does not assume that fixed expenses actually vary with premium; rather, their ratio to premium varies when premium changes. If the expense amount itself changes, $F$ must be updated before applying the formula. Other provisions, such as taxes or assessments, should be included in the appropriate ratio if they are part of the stated pricing model.

Worked Example: Comparing Two Premium Levels

A hypothetical equipment policy has fixed expenses of $F=\$48$ per policy-year, a variable expense ratio $e=0.24$, and a profit and contingencies provision $q=0.06$. Calculate the permissible loss ratio at annual premiums of $\$600$ and $\$800$ per policy-year. Check each result by allocating the premium.

Method. Convert the fixed dollar expense into a premium ratio separately at each premium, then subtract all provisions from 1.

At a premium of $\$600$, the fixed expense ratio is

\frac{F}{P}=\frac{48}{600}=0.08.

Thus the permissible loss ratio is

$$ PLR(600)=1-0.24-0.06-0.08=0.62. $$

Check the allocation in dollars: variable expenses are $0.24(600)=\$144$; fixed expenses are $\$48$; the profit and contingencies provision is $0.06(600)=\$36$; and the permitted loss amount is $0.62(600)=\$372$. Their total is $\$144+\$48+\$36+\$372=\$600$.

At a premium of $\$800$, the fixed expense ratio is

\frac{F}{P}=\frac{48}{800}=0.06, \qquad PLR(800)=1-0.24-0.06-0.06=0.64.

The dollar check gives variable expenses of $0.24(800)=\$192$, fixed expenses of $\$48$, profit and contingencies of $0.06(800)=\$48$, and permitted losses of $0.64(800)=\$512$. The total is $\$192+\$48+\$48+\$512=\$800$.

Check and common mistake. Both allocations sum to premium. The fixed expense remains $\$48$, but its ratio falls from $0.08$ to $0.06$, so the permissible loss ratio rises from $0.62$ to $0.64$. A common mistake is to carry the $0.08$ fixed expense ratio from the first calculation into the second. That would incorrectly give $1-0.24-0.06-0.08=0.62$ at $\$800$, even though $\$48/\$800=0.06$.

How Premium Changes Affect the Ratio

Write $c=1-e-q$ for the share of premium remaining after variable expenses and the profit and contingencies provision, before allowing for fixed expenses. Then

PLR(P)=c-\frac{F}{P}.

When $F>0$, increasing premium while holding $F$, $e$, and $q$ fixed decreases the fixed expense ratio and increases the permissible loss ratio. For $P_2>P_1>0$,

\frac{F}{P_2}<\frac{F}{P_1} \quad\Longrightarrow\quad PLR(P_2)>PLR(P_1).

The strict inequalities require $F>0$. If $F=0$, the fixed expense ratio is zero at every positive premium, and the permissible loss ratio is constant as premium changes. For $F>0$, $PLR(P)$ approaches $c$ as premium grows, but it remains below $c$ for every finite positive premium. This limiting value is useful when checking whether a target ratio is attainable.

Worked Example: Finding the Premium for a Target Ratio

A hypothetical policy has fixed expense $F=\$75$ per exposure, variable expense ratio $e=0.20$, and profit and contingencies provision $q=0.05$. Find the positive premium per exposure that gives a permissible loss ratio of $T=0.65$.

Method. Set the permissible loss ratio equal to the target and solve for premium. First calculate the share before fixed expenses:

$$ c=1-e-q=1-0.20-0.05=0.75. $$

The target equation is $T=c-F/P$. Rearranging gives $F/P=c-T$, so

P=\frac{F}{c-T} =\frac{75}{0.75-0.65} =\frac{75}{0.10} =\$750.

Check by substituting the premium: $F/P=75/750=0.10$, and $PLR(750)=1-0.20-0.05-0.10=0.65$, exactly the target. In dollars, variable expenses are $\$150$, fixed expenses are $\$75$, profit and contingencies are $\$37.50$, and permitted losses are $0.65(750)=\$487.50$. These amounts sum to $\$750$.

Check and common mistake. The fixed expense percentage is $10\%$ at the solved premium. Treating $\$75$ as though it were already a $10\%$ provision, without first solving for the premium, would not establish that the target is met. The result depends on comparing the fixed dollar amount with the unknown premium.

Result: If $F>0$, a positive finite premium achieves a target permissible loss ratio $T$ exactly when $T<c=1-e-q$. In that case, $$ P=\frac{F}{1-e-q-T}. $$ If $F=0$, then $PLR(P)=c$ for every $P>0$: the target is attained at every positive premium when $T=c$, and is not attained at any positive premium when $T\ne c$.

The boundary condition matters. With positive fixed expenses, a target equal to $c$ would require $F/P=0$, which cannot hold for finite positive $P$. A target above $c$ would require a negative fixed expense ratio. Neither is possible under this model. When $F=0$, however, the ratio equals $c$ for every premium, so the positive-fixed-expense condition cannot be applied.

Worked Example: Isolating the Fixed Expense Effect

Two hypothetical rating groups have the same fixed expense of $\$36$ per exposure, variable expense ratio $e=0.25$, and profit and contingencies provision $q=0.05$. Group A has premium of $\$400$ per exposure; Group B has premium of $\$600$. Calculate each permissible loss ratio and explain the difference. For a further comparison, suppose each group has an adjusted loss ratio of $0.62$; calculate the simple loss ratio indication factor as adjusted loss ratio divided by permissible loss ratio.

Method. Calculate the premium-relative fixed expense separately for each group. The common share before fixed expenses is

$$ c=1-0.25-0.05=0.70. $$

For Group A, $F/P=36/400=0.09$, so $PLR_A=0.70-0.09=0.61$. The permitted loss dollars are $0.61(400)=\$244$; variable expenses are $\$100$, fixed expenses are $\$36$, and profit and contingencies are $\$20$. The total, $\$244+\$100+\$36+\$20$, is $\$400$.

For Group B, $F/P=36/600=0.06$, so $PLR_B=0.70-0.06=0.64$. The permitted loss dollars are $0.64(600)=\$384$; variable expenses are $\$150$, fixed expenses are $\$36$, and profit and contingencies are $\$30$. The total, $\$384+\$150+\$36+\$30$, is $\$600$.

Using the stated adjusted loss ratio, the simple indication factors are $0.62/0.61=1.0164$ for Group A and $0.62/0.64=0.9688$ for Group B, rounded to four decimal places. These calculations isolate the effect of different premium levels on the fixed expense ratio; they do not assert that adjusted loss experience is always identical across groups.

Check and common mistake. Group B's fixed expense remains $\$36$, not $9\%$ of its premium. Its fixed expense ratio is $6\%$, so its permissible loss ratio is three percentage points higher than Group A's. Reusing Group A's $9\%$ fixed expense ratio for Group B would incorrectly produce a permissible loss ratio of $0.61$.

Common Interpretation Traps

Using the wrong denominator. Divide fixed expense per exposure by premium per the same exposure—not by total premium from a differently sized group or period.
Holding a ratio fixed when the dollars are fixed. If $F$ is unchanged and $P$ changes, recalculate $F/P$. A fixed expense amount and a fixed expense ratio are different assumptions.
Mixing rate levels. Use premium at the applicable rate level. If the premium basis changes, recompute the fixed expense ratio using that premium basis.
Claiming strict monotonicity when $F=0$. For positive fixed expense, higher premium means a strictly higher permissible loss ratio. With zero fixed expense, the permissible loss ratio is constant.
Ignoring attainability. With $F>0$, a finite positive premium cannot produce $PLR=c$ or a value above $c$. Check the target before using the solved-premium formula.

The fixed expense provision is especially important when premiums vary substantially across risks or rating groups. The same dollar expense can consume a large portion of a small premium and a smaller portion of a large premium. An indication based on a permissible loss ratio should therefore reflect the expense amount and the premium level on a consistent basis. This topic is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Key takeaway: A fixed expense per exposure becomes a premium-relative provision only after division by premium per exposure. Under the stated model, $PLR(P)=1-e-q-F/P$; recalculate $F/P$ whenever the applicable premium changes.

Check Your Understanding

For each problem, use the fixed expense per exposure and premium on the same basis. Answers are provided after each question.

A hypothetical policy has $F=\$30$ per exposure, $e=0.15$, $q=0.05$, and premium $P=\$500$. Find the fixed expense ratio and permissible loss ratio. (Answer: fixed expense ratio $0.0600$; permissible loss ratio $0.7400$.)
A small commercial account has $F=\$64$, $e=0.22$, $q=0.03$, and premium $P=\$800$, all per exposure. Calculate the permissible loss ratio and verify the dollar allocation. (Answer: $PLR=0.6700$; permitted losses $\$536$, variable expenses $\$176$, fixed expenses $\$64$, profit and contingencies $\$24$, totaling $\$800$.)
A hypothetical coverage has $F=\$90$ per exposure, $e=0.25$, and $q=0.05$. What premium per exposure gives a permissible loss ratio of $0.60$? (Answer: $\$900$.)
A policy has $F=\$40$ per exposure, $e=0.18$, and $q=0.07$. Can any positive finite premium produce a permissible loss ratio of $0.75$? Explain using the limiting share. (Answer: No. $c=1-0.18-0.07=0.75$, and with $F>0$, $PLR(P)=0.75-40/P<0.75$ for every finite positive premium.)
A program has zero fixed expense, $F=0$, variable expense ratio $e=0.24$, and profit and contingencies provision $q=0.06$. For what positive premiums is a target permissible loss ratio of $0.70$ achieved? Would a target of $0.69$ be achieved? (Answer: $PLR=0.70$ at every positive premium; the target $0.69$ is not achieved at any positive premium.)

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