Why Loss Costs and Expenses Must Be Combined
A projected loss cost is not, by itself, the premium needed to insure an exposure. The premium must also provide for expenses, and some expenses depend on the premium ultimately charged. For example, a commission stated as a percentage of premium cannot be calculated until the premium is known. The required premium is therefore found by solving a balance equation, rather than by simply adding every expense percentage to a loss cost.
This tutorial focuses on combining projected loss costs with selected expense provisions. It does not add profit or contingency provisions. Any cost amounts and rates below are hypothetical, and the calculations treat the inputs as selected prospective values.
The Premium Balance Equation
Let \(L\) be the projected loss cost per exposure, \(F\) the fixed expense per exposure, \(R\) the required premium per exposure, and \(v\) the total variable expense ratio expressed as a decimal proportion of premium. With no other provisions in the calculation, the premium must cover the loss cost, fixed expense, and variable expense:
The variable expense is \(vR\), not \(vL\) or \(v(L+F)\), because its stated base is premium. Rearranging the balance equation gives:
This formula applies when the variable expense is a proportion of the same premium \(R\) being solved for, and \(v<1\). It does not directly apply if an expense percentage uses a different base, such as losses or payroll; that expense must first be expressed on its stated base and incorporated consistently. If there are multiple premium-based expense provisions, add their rates to obtain \(v\), provided they all apply to the same premium base.
Converting Policy Expenses to an Exposure Basis
A fixed expense given per policy cannot be added directly to a loss cost per exposure. If the average policy has \(n\) exposures and the fixed expense is \(C\) dollars per policy, the allocated fixed expense per exposure is \(C/n\). This conversion assumes the stated average number of exposures is an appropriate allocation basis for the policies being rated. If policy sizes vary materially, a single average may not describe every policy, but it can still be used when the problem specifies an average-based calculation.
The units are part of the calculation: dollars per policy divided by exposures per policy gives dollars per exposure. Once the per-exposure premium \(R\) is known, multiplying by \(n\) gives the premium for a policy with \(n\) exposures.
Convert policy-level fixed expenses to dollars per exposure when the loss cost is per exposure.
Add the selected expense percentages that use premium as their base to obtain \(v\).
Use \(R=(L+F)/(1-v)\) to account for variable expenses being a fraction of the premium itself.
Calculate the variable expense as \(vR\) and confirm that loss cost plus fixed and variable expenses equals \(R\).
Why the Denominator Is \(1-v\)
The denominator follows directly from the definition of a premium-based expense. Suppose a premium \(R\) must cover \(L+F\) dollars of loss and fixed expense, plus \(vR\) dollars of variable expense. Starting with the balance equation:
Subtracting \(vR\) from both sides leaves the portion of premium available after variable expenses. Factoring out \(R\), then dividing by \(1-v\), gives:
Thus the denominator represents the share of premium left after the variable expense provision. Simply multiplying \(L+F\) by \(1+v\) does not solve the balance equation: it calculates the variable expense as a percentage of the loss-and-fixed-expense subtotal, not as a percentage of premium.
Worked Example: Fixed Policy Expense and Variable Expense
A hypothetical coverage has a projected loss cost of $720 per exposure. The selected fixed expense is $60 per policy, the average policy has 2 exposures, and variable expenses are 15% of premium. Calculate the required premium per exposure and verify the result. Treat the inputs as selected prospective values for a one-year period; no interest rate, mortality assumption, or probability distribution is required for this deterministic calculation.
Method: convert the policy expense, then solve the premium balance. First convert $60 per policy to a per-exposure amount:
The numerator of the premium formula is the projected loss cost plus fixed expense:
Here \(v=0.15\), so the share of premium remaining after variable expenses is \(1-0.15=0.85\). The indicated premium is:
Rounded to the nearest cent, the required premium is $882.35 per exposure. Check the balance using the unrounded result: variable expense is \(0.15\times \$882.352941\ldots=\$132.352941\ldots\). Then:
The balance reconciles. Common mistake avoided: the fixed expense must be converted from dollars per policy to dollars per exposure before addition, and the 15% must be applied to the premium being solved for.
Combining Multiple Expense Provisions
Several expense items may each be stated as a percentage of premium. For example, a hypothetical expense selection could include a commission percentage and a premium-tax percentage. If both percentages apply to the same premium base, their rates are added before solving for \(R\). Fixed expense dollars remain separate in the numerator; they are not included in the variable expense ratio.
Worked Example: Combining Variable Expense Rates
A hypothetical product has a projected loss cost of $460 per exposure. Fixed administrative expense is $84 per policy, and the average policy has 3 exposures. The selected commission provision is 9% of premium and the premium-tax provision is 3.5% of premium. Find the premium per exposure and check the allocation. Treat the inputs as prospective one-year values; no interest rate, mortality assumption, or probability distribution is needed.
Method: combine rates with the same base, then solve. Convert the policy expense:
Both variable provisions apply to premium, so their combined rate is:
Add loss cost and fixed expense, then divide by the share of premium remaining:
Rounded to the nearest cent, the premium is $557.71 per exposure. Using the unrounded premium, commission is \(0.09\times \$557.714285\ldots=\$50.194285\ldots\), and premium tax is \(0.035\times \$557.714285\ldots=\$19.52\). The combined variable expense is $69.714285… per exposure. The check is:
This confirms the solution. Common mistake avoided: adding 9% and 3.5% is appropriate here because both are percentages of the same premium. If their bases differed, their percentages could not be combined this way without first reconciling the bases.
Per-Exposure Premium Versus Per-Policy Premium
The premium formula produces a premium on the same unit as the loss cost and fixed expense supplied to it. If the loss cost is per exposure, the result is per exposure. To obtain a policy premium for a policy with a specified number of exposures, multiply the per-exposure result by that number. Keep the fixed expense conversion consistent: the expense per policy is divided by the average exposures per policy when solving per exposure, then the resulting rate is multiplied by the policy’s exposures when converting back to a policy amount.
Worked Example: Converting the Indication Back to a Policy Premium
For a hypothetical vehicle coverage, the projected loss cost is $540 per vehicle-year. Fixed expense is $36 per policy, the average policy has 1.5 vehicle-years, and variable expenses are 8% of premium. Calculate the required premium per vehicle-year and the premium for a policy with 1.5 vehicle-years. Treat the inputs as selected prospective amounts for one year; no interest rate, mortality assumption, or probability distribution is used.
Method: solve per exposure and convert to the policy basis. The fixed expense per vehicle-year is:
The amount to be covered before variable expenses is \(\$540+\$24=\$564\) per vehicle-year. The premium-based expense rate is \(v=0.08\), leaving \(1-v=0.92\) of premium. Therefore:
Rounded to the nearest cent, this is $613.04 per vehicle-year. For a policy with 1.5 vehicle-years, use the unrounded rate:
The policy premium is $919.57, rounded to the nearest cent. Check at the policy level: projected losses are \(1.5\times \$540=\$810\); fixed expense is $36; and variable expense is \(0.08\times \$919.565217\ldots=\$73.565217\ldots\). Thus:
The policy-level check agrees. Common mistake avoided: multiplying the loss cost by the number of exposures while leaving the fixed expense at a per-exposure amount would mix units and understate the policy’s fixed expense.
Common Errors and Practical Checks
The algebra is short, but errors often come from using the wrong base or mixing policy and exposure units. In the first worked example, one incorrect approach would multiply the $750 loss-and-fixed-expense subtotal by \(1.15\), obtaining $862.50. That calculation treats the variable expense as 15% of $750. But the provision is 15% of the premium, so it must satisfy \(R=750+0.15R\). Solving gives \(R=750/0.85=\$882.35\), not $862.50.
A second common error is to divide a fixed policy expense by the number of policies instead of the average exposures per policy. The conversion must yield the same unit as the loss cost. A third is to round too early: retaining full precision until the final premium and then checking with unrounded values avoids small apparent differences in the balance.
| Input or calculation | Correct treatment | Common error |
|---|---|---|
| Fixed expense per policy; loss cost per exposure | Divide by average exposures per policy | Add the policy amount directly to the per-exposure loss cost |
| Variable expense stated as a percentage of premium | Include its rate in \(v\), then divide by \(1-v\) | Apply the rate only to losses or to losses plus fixed expense |
| Multiple variable provisions | Add rates only when they share the same premium base | Combine percentages that use different bases |
| Result needed per policy | Multiply the per-exposure premium by the policy’s exposures | Report the per-exposure rate as the policy premium |
The calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. In practice, the result depends on the selected loss cost, expense amounts, allocation basis, and variable expense rates. The equation organizes those assumptions; it does not determine whether the assumptions themselves are appropriate.
Check Your Understanding
For each problem, treat the stated amounts as hypothetical prospective selections. Unless a problem says otherwise, assume a one-year period, no discounting, and no mortality or probability distribution assumption. Round final money amounts to the nearest cent.
- A projected loss cost is $680 per exposure. Fixed expense is $75 per policy, with an average of 2.5 exposures per policy. Variable expenses are 14% of premium. Find the required premium per exposure. (Answer: $825.58 per exposure)
- A hypothetical policy has a projected loss cost of $395 per exposure and fixed expense of $18 per exposure. Commission is 7% of premium and premium tax is 3% of premium. Find the premium per exposure and verify the variable expense rate used. (Answer: $458.89 per exposure; combined variable rate 10%)
- A projected loss cost is $620 per exposure. Fixed expense is $48 per policy and the average policy has 2 exposures. Variable expenses are 16% of premium. Find the premium per exposure. (Answer: $766.67 per exposure)
- A coverage has a projected loss cost of $310 per exposure, fixed expense of $42 per policy, an average of 3 exposures per policy, and variable expenses of 5.5% of premium. Find both the premium per exposure and the premium for an average policy. (Answer: $342.86 per exposure; $1,028.57 per average policy)
- A hypothetical product has a loss cost of $650 per exposure, fixed expense of $45 per exposure, and variable expenses of 12% of premium. Find the required premium per exposure and check that 12% of that premium, added to loss cost and fixed expense, balances to the premium. (Answer: $789.77 per exposure; variable expense $94.77 per exposure, with the unrounded components balancing)