Why the Form of a Provision Matters
A rate indication must provide enough premium to fund expected losses and expenses while also allowing for selected profit and contingency provisions. The arithmetic depends on how each provision is specified. A fixed dollar amount per exposure is added to the required premium; a provision stated as a percentage of premium depends on the premium being calculated and must be included in the equation for that premium.
This distinction matters even when two provisions have the same purpose. For example, a selected contingency allowance might be a fixed amount per exposure, while a target profit provision might be a percentage of premium. Treating both as additions to a preliminary loss-and-expense subtotal generally gives the wrong rate. The method below is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The equation assumes that \(L\), \(F\), and \(Q\) are all measured per the same exposure base; that the percentage provisions are fractions of the indicated premium \(R\); and that \(1-v-p-c>0\). It also assumes the listed provisions are not already included in another input. If a provision is defined on a different base, such as as a percentage of losses rather than premium, it cannot be inserted as \(p\) or \(c\) without first translating its effect.
Why Percentage Provisions Belong in the Denominator
The required premium has to fund every component, including components whose amounts are calculated from that very premium. The balance equation makes the dependency explicit:
Collect the terms that are proportional to \(R\) on the left. This gives \((1-v-p-c)R=L+F+Q\). Dividing by \(1-v-p-c\) yields the indicated-rate formula. This derivation also explains why simply adding a percentage to the preliminary cost is not equivalent: that shortcut calculates the percentage on the preliminary cost, not on the final premium.
A useful process is to identify each provision’s unit and base before calculating. Then place fixed amounts in the dollar subtotal and premium-based percentages in the denominator.
Record whether it is a fixed dollar amount per exposure or a fraction of premium. Check the stated base for every percentage.
Add expected loss cost, fixed expenses, and all fixed dollar profit or contingency provisions.
Add variable expense, profit, and contingency fractions that all apply to the same premium base.
Divide the fixed-dollar subtotal by one minus the combined percentage, then verify that the resulting dollar components sum to the indicated premium.
Worked Indications
Worked Example: Fixed Contingency and Profit as a Premium Percentage
A hypothetical insurance program has expected loss and loss adjustment expense cost of $248 per exposure and fixed expenses of $32 per exposure. The selected contingency provision is a fixed $5 per exposure. Variable expenses are 12% of premium, and the target profit provision is 8% of premium. Find the indicated rate and check the component totals.
Method and classification. The loss cost, fixed expense, and contingency amount are dollar amounts per exposure. Their subtotal is \(248+32+5=\$285\). Variable expenses and profit are both percentages of the same premium base, so their combined fraction is \(0.12+0.08=0.20\).
Substitute into the premium balance equation:
Check. Variable expenses are \(0.12(356.25)=\$42.75\), and profit is \(0.08(356.25)=\$28.50\). The total is \(248+32+5+42.75+28.50=\$356.25\), which agrees with the indicated rate. This method avoids treating the fixed contingency as though it were a premium percentage.
Worked Example: Fixed Profit and Percentage Contingency
For a hypothetical specialty coverage, expected loss cost is $415 per exposure and fixed expenses are $27 per exposure. The selected profit provision is a fixed $18 per exposure. Variable expenses equal 9% of premium, while the contingency provision equals 3% of premium. Determine the indicated rate, to the nearest cent.
Method and classification. The fixed-dollar subtotal is \(415+27+18=\$460\). The percentage provisions total \(0.09+0.03=0.12\) of premium. Thus, 88% of premium remains to fund the fixed-dollar subtotal.
Rounding only the final rate to the nearest cent gives $522.73 per exposure.
Check. At the unrounded rate, variable expenses are \(0.09(522.7272\ldots)=\$47.0454\ldots\), and contingency is \(0.03(522.7272\ldots)=\$15.6818\ldots\). Adding these to the $460 fixed-dollar subtotal gives \(460+47.0454\ldots+15.6818\ldots=522.7272\ldots\). The check uses the unrounded rate so intermediate rounding does not create a discrepancy.
Worked Example: Several Provisions on a Single Premium Base
A hypothetical commercial class has expected loss cost of $630 per exposure, fixed expenses of $41 per exposure, and a fixed contingency allowance of $9 per exposure. Variable expenses are 7% of premium, the selected profit provision is 6% of premium, and a second contingency provision is 2% of premium. Find the indicated rate to the nearest cent and verify it.
Method and classification. The dollar subtotal is \(630+41+9=\$680\). All three percentage provisions use premium as their base, so their combined fraction is \(0.07+0.06+0.02=0.15\). The remaining fraction is \(1-0.15=0.85\).
Check. Variable expenses are \(0.07(800)=\$56\), profit is \(0.06(800)=\$48\), and the percentage contingency is \(0.02(800)=\$16\). Together with the dollar subtotal, the total is \(680+56+48+16=\$800\). The $9 fixed contingency is already included in the $680 subtotal, so it must not be added a second time.
Common Mistakes and Interpretation
A frequent error is to calculate a premium-based provision from the loss-and-expense subtotal and then add it. In the first example, that shortcut would take 8% profit as \(0.08(285)=\$22.80\) and produce a rate of \(285+22.80=\$307.80\), before even considering the variable expenses. That cannot be correct for the stated model: the profit provision is 8% of the final premium, not 8% of the preliminary subtotal. The correct rate is $356.25 because variable expenses and profit together consume 20% of premium.
Other checks help catch setup errors:
- Do not put a fixed dollar load in the denominator. A $5-per-exposure contingency adds $5 to the dollar subtotal; it is not a five-percentage-point premium load.
- Do not put a premium percentage in the dollar subtotal. A 6% profit provision is \(0.06R\), not a fixed amount until \(R\) has been determined.
- Use the specified percentage base. A percentage of losses is not automatically a percentage of premium. Translate it using the stated definition before combining it with premium-based fractions.
- Check for duplicate provisions. If an expense or contingency amount is already included in a selected input, adding it again overstates the indication.
- Check the denominator. The denominator must be positive for a finite positive indication under this model. If the combined premium percentages are 100% or more, the formula does not yield a finite positive rate for a positive fixed-dollar requirement.
The indication is a cost balance under the selected assumptions, not a statement that the provisions will be realized exactly. It depends on the chosen loss cost, expense assumptions, and profit and contingency loads. The practical value of separating the bases is that an actuary can change one assumption and see precisely where it enters the calculation.
Check Your Understanding
For each hypothetical indication, classify the stated provisions, calculate the rate per exposure, and check that your result is consistent with the percentage bases.
- A program has expected losses of $300 per exposure, fixed expenses of $25, and fixed contingency of $4. Variable expenses are 10% of premium and profit is 5% of premium. What is the indicated rate? (Answer: $387.06 per exposure.)
- A coverage has expected loss cost of $520, fixed expenses of $36, and fixed profit of $14 per exposure. Variable expenses are 8% of premium and contingency is 4% of premium. Find the indicated rate to the nearest cent. (Answer: $647.73 per exposure.)
- A hypothetical class has expected loss cost of $180 and fixed expenses of $20 per exposure. Variable expenses are 6% of premium, profit is 7% of premium, and contingency is a fixed $3 per exposure. Find the indicated rate to the nearest cent. (Answer: $233.33 per exposure.)
- Expected losses are $740 per exposure. Fixed expenses are $48 and fixed contingency is $12. Variable expenses are 11% of premium and profit is 5% of premium. Calculate the rate and state the combined percentage provision. (Answer: $952.38 per exposure; combined percentage provision is 16%.)
- A hypothetical portfolio has expected losses of $950 per exposure and fixed expenses of $55. Variable expenses are 9% of premium, profit is 6% of premium, and contingency consists of both a fixed $10 per exposure and a 2.5% premium provision. Find the indicated rate to the nearest cent. (Answer: $1,230.30 per exposure.)