Why the Gross Rate Must Be Solved
A rate must fund more than expected claims. It also has to fund expenses that vary with premium, such as commissions or premium taxes. Because those expenses are calculated as a percentage of the premium itself, the premium is not simply expected losses plus fixed expenses. It must be large enough to fund those costs as well. The gross rate formula expresses that balance directly.
The formula is useful when the rating unit is an exposure and the information is already expressed on a consistent per-exposure basis. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. Its key algebraic feature is the denominator: premium-based expenses reduce the share of each premium dollar available to pay expected losses and fixed expenses.
Deriving the Gross Rate Formula
Assume that expected losses and fixed expenses are stated in dollars per exposure, and that variable expenses equal a constant proportion \(v\) of gross premium. At a gross rate \(R\), the premium-funded variable expenses are therefore \(vR\) per exposure. A rate that exactly funds the specified amounts must satisfy a balance equation: premium equals expected losses plus fixed expenses plus variable expenses.
Collect the terms involving \(R\) on the left. If \(1-v\) is positive, dividing by it gives the indicated gross rate.
The denominator has a direct interpretation. For each dollar of premium, the fraction \(v\) pays variable expenses, leaving \(1-v\) dollars to fund expected losses and fixed expenses. The rate must provide \(L+F\) dollars from that remaining share, so \(R(1-v)=L+F\).
If there are several variable expenses, each stated as a proportion of the same premium base, add their ratios before using the formula. For example, a commission ratio of 12% and a premium tax ratio of 3% give \(v=0.12+0.03=0.15\). This addition is appropriate only when both expenses are percentages of the same premium amount and apply to the same rate basis.
Applying the Formula Consistently
Before calculating, check that each amount has the same unit. If \(L\) is expected losses per vehicle-year, \(F\) must also be dollars per vehicle-year, and the resulting \(R\) is dollars per vehicle-year. A total fixed expense for an entire book cannot be added directly to a per-exposure loss cost; it must first be allocated to the relevant exposures.
The variable ratio must also match the premium being solved for. If the commission is 10% of gross premium, use 0.10 as part of \(v\). Do not treat that expense as a fixed dollar amount, and do not calculate it from \(L+F\) before determining the gross rate. The premium base is \(R\), not merely the amount left after subtracting variable expenses.
Express expected losses and fixed expenses in dollars per the same exposure unit and time period.
Add the applicable expense proportions when they are all based on the same gross premium.
Use \(R=(L+F)/(1-v)\), provided the assumptions hold and \(v<1\).
Calculate \(vR\) and verify that \(L+F+vR=R\), allowing for any stated rounding.
Worked Example: A Basic Gross Rate Indication
A hypothetical single-vehicle policy has an expected loss cost of $420 per vehicle-year and an allocated fixed expense of $36 per vehicle-year. Variable premium expenses equal 25% of gross premium. Find the indicated gross rate and verify it. Use exact arithmetic; report the rate to the nearest cent.
Method. Both the loss cost and fixed expense are per vehicle-year, and the variable expense is a percentage of the gross rate. Thus \(L=420\), \(F=36\), and \(v=0.25\).
The indicated rate is $608.00 per vehicle-year. To check it, the variable expense is \(0.25(608)=152\) dollars. The total funded amount is \(420+36+152=608\) dollars, which equals the rate.
Common mistake avoided: Adding a 25% variable expense to $456 by multiplying $456 by 1.25 gives $570, but that uses the expense ratio on the wrong base. At $570, the variable expense would be \(0.25(570)=142.50\), leaving only $427.50 for losses and fixed expense—not the required $456.
Multiple Variable Expenses and a Full Reconciliation
When premium-based expenses are listed separately, first identify their bases. If each is a proportion of gross premium, their ratios form the total variable premium expense ratio. The numerator remains the sum of the per-exposure amounts that are not proportional to premium, such as expected losses and fixed expense provisions. A loss-related amount expressed as dollars per exposure belongs in that numerator; it is not a premium ratio merely because it is an expense.
Worked Example: Combining Commission and Premium Tax
A hypothetical equipment policy has expected losses of $420 per insured unit-year, loss-related expenses of $30 per unit-year, and fixed expenses of $60 per unit-year. Commission is 12% of gross premium, and a premium tax is 3% of gross premium. Find and verify the indicated gross rate.
Step 1: Combine the per-exposure costs. The loss-related expense is a dollar amount per exposure, so it is included with expected losses and fixed expenses:
Step 2: Combine the variable ratios. Commission and tax use the same gross premium base, so
Step 3: Solve.
The indicated rate is $600.00 per unit-year. Commission is \(0.12(600)=72\) dollars, and tax is \(0.03(600)=18\) dollars. The total is \(420+30+60+72+18=600\) dollars, confirming the result.
Common mistake avoided: Applying the 15% ratio to $510 produces $76.50 of variable expenses and a supposed rate of $586.50. But 15% of that proposed rate is only $87.975, not $76.50; the calculation does not satisfy the premium balance. The variable expenses must be calculated from the gross rate being solved for.
Different Loss Costs Across Rating Groups
The formula can also be applied separately to groups when each group has its own expected loss cost or fixed expense per exposure. If the variable expense ratio is common across the groups, it appears in each group’s denominator. A resulting group rate should still be checked using that group’s own per-exposure costs.
Worked Example: Indicating Rates for Two Groups
A hypothetical insurer is pricing two categories of forestry equipment. Category A has an expected loss cost of $280 per unit-year, while Category B has an expected loss cost of $460 per unit-year. Each category has a fixed expense provision of $40 per unit-year. Both have variable expenses equal to 20% of gross premium. Find each indicated rate, then check the average rate for a portfolio with two Category A exposures and three Category B exposures.
Category A. Its per-exposure costs total \(280+40=320\) dollars. Therefore:
At $400, variable expenses are \(0.20(400)=80\) dollars. The total \(280+40+80=400\) dollars matches the rate.
Category B. Its per-exposure costs total \(460+40=500\) dollars. Therefore:
At $625, variable expenses are \(0.20(625)=125\) dollars. The total \(460+40+125=625\) dollars matches the rate.
Portfolio check. The portfolio has five exposures, so its average indicated rate per exposure is
The average per-exposure losses are \(\{2(280)+3(460)\}/5=(560+1{,}380)/5=388\), and the average fixed expense is $40. Thus average non-variable costs are \(388+40=428\). Variable expenses at the average rate are \(0.20(535)=107\), and \(428+107=535\). The portfolio check agrees.
Common mistake avoided: Using the simple, unweighted mean of $400 and $625 would give $512.50. That is not the average rate for this portfolio because the two categories have different numbers of exposures.
Conditions, Limits, and Exam Traps
The formula is an algebraic balance under stated assumptions, not a universal shortcut. It applies when the specified losses and fixed expenses can be represented as amounts per exposure and the relevant variable expenses are proportional to the same gross premium \(R\). The ratio \(v\) must be less than 1. If \(v=1\), no portion of premium remains to fund positive losses and fixed expenses; if \(v>1\), the denominator is negative and the resulting rate is not a meaningful solution to the intended pricing problem.
The formula does not, by itself, decide how expenses should be classified or allocated. Those inputs must already be appropriate for the rating unit and population under consideration. Nor does it automatically handle a fixed dollar charge per policy that is added separately to a rate, a variable expense tied to a different base, or a cost that changes nonlinearly with premium. In those situations, write the actual balance equation using the relevant bases rather than forcing the amounts into \(F\) or \(v\).
A frequent exam trap is to confuse a premium expense ratio with a markup on non-variable costs. If \(v\) is 20% of premium, the premium needed to fund a $100 cost is \(100/0.80=125\), not \(100(1.20)=120\). At a $120 premium, 20% is $24, leaving $96 for the $100 cost. The check \(R(1-v)=L+F\) is an efficient way to detect this error.
Check Your Understanding
For each problem, identify the per-exposure amount and premium-based ratio before calculating. Assume the stated expenses apply to the same gross premium unless the problem says otherwise.
- A hypothetical policy has expected losses of $350 per exposure and fixed expenses of $50 per exposure. Variable expenses are 20% of gross premium. What is the indicated rate? (Answer: $500 per exposure)
- A hypothetical coverage has expected losses of $275 per policy-year, loss-related expenses of $25 per policy-year, and fixed expenses of $40 per policy-year. Commission is 10% and premium tax is 5%, both of gross premium. What is the indicated rate? (Answer: $400 per policy-year)
- A hypothetical rating group has expected losses of $560 per unit-year and fixed expenses of $64 per unit-year. Its total variable premium expense ratio is 18%. Find the indicated rate to the nearest cent. (Answer: $760.98 per unit-year)
- A hypothetical rate is $720 per exposure, and variable premium expenses are 22% of that rate. If fixed expenses are $70 per exposure, what expected loss cost is consistent with the rate balance? (Answer: $491.60 per exposure)
- A hypothetical coverage has expected losses of $360 per exposure, variable expenses of 12% of gross premium, and a target rate of $500 per exposure. What fixed expense per exposure is implied by the balance? (Answer: $80 per exposure)