Why Use an Expense Constant Per Policy?
Some expenses are incurred for each policy, rather than for each unit of exposure or as a percentage of premium. For example, a hypothetical insurer may incur a similar administrative cost to issue and service each policy, whether the policy covers one exposure unit or several. A flat expense constant represents this cost as a fixed dollar amount per policy.
The distinction matters when policies have different exposure amounts. The dollar constant is the same for each policy, but its contribution to the premium per exposure unit is larger for a smaller policy. Treating the constant as though it were an exposure-based load changes the allocation and can distort rates by policy size.
An exposure unit might be a vehicle-year, a specified amount of payroll, or another measure used to rate coverage. Let \(e\) be the number of exposure units on a policy, \(L\) the expected loss cost per exposure unit, and \(P\) the gross premium for the policy. The expected loss provision is then \(eL\). The expense constant \(C\), by contrast, is added once per policy.
Putting the Constant into a Gross Premium
Suppose variable expenses are a proportion \(v\) of gross premium and the target profit and contingency provision is a proportion \(q\) of gross premium. For this calculation, assume that \(eL\) is the expected loss provision, \(C\) is the expense constant for one policy, and all amounts are measured over the same policy term. Assume also that the variable expense and profit proportions apply to the entire gross premium.
Rearranging gives the gross premium and the corresponding premium per exposure unit:
Here \(R\) is the gross rate per exposure unit for that policy, given the stated inputs. The condition \(v+q<1\) is necessary for a positive finite gross-up denominator. This formulation does not apply unchanged if, for example, an expense is a fixed dollar amount per exposure instead of per policy, or if a percentage provision is based on something other than gross premium. Those provisions must be represented on their actual bases.
Why the Gross-Up Formula Works
The formula follows directly from balancing the premium against the provisions funded by that premium. Expected losses and the per-policy constant together require \(eL+C\) dollars. Variable expenses and profit require \(vP+qP\) dollars, so the total required premium satisfies \(P=eL+C+vP+qP\). Collecting the terms involving \(P\) yields \((1-v-q)P=eL+C\). Dividing by \(1-v-q\) gives the stated premium.
Dividing that premium by \(e\) gives \(R=(L+C/e)/(1-v-q)\). This shows the key allocation effect: the constant contributes \(C/e\) dollars per exposure unit. As \(e\) increases, \(C/e\) decreases, even though the total dollar constant remains \(C\).
Divide the applicable fixed expense budget by the number of policies receiving that allocation, if the constant is not already given.
Multiply the loss cost per exposure unit by the policy's exposure amount: \(eL\).
Divide \(eL+C\) by \(1-v-q\), provided the percentage provisions are based on gross premium and \(v+q<1\).
Divide the policy premium by \(e\). The per-exposure rate includes \(C/e\), not \(C\).
Worked Applications
Worked Example: Calculating the Constant and Gross Premium
A hypothetical program has an annual fixed expense budget of $360,000 allocated evenly across 12,000 policies. The expected loss cost is $210 per exposure unit. A policy has 2.5 exposure units. Variable expenses are 18% of gross premium, and the profit and contingency provision is 5% of gross premium. Calculate the policy's expense constant, gross premium, and gross rate per exposure unit.
Model and assumptions. Use the gross-premium formulation above. All amounts are for one annual policy term; there is no interest or mortality adjustment. The loss cost is an assumed expected value, not a probability distribution specified for this calculation. The percentage provisions are based on gross premium.
Step 1: Find the constant.
Step 2: Calculate the loss provision.
Step 3: Calculate the gross-up denominator and premium.
Step 4: Calculate the rate.
Rounded to the nearest cent, the gross premium is $720.78 and the gross rate is $288.31 per exposure unit. As a check, using the unrounded premium, variable expense is \(0.18(\$720.7792\ldots)=\$129.7403\ldots\), and the profit and contingency provision is \(0.05(\$720.7792\ldots)=\$36.0390\ldots\). The provisions total \( \$525+\$30+\$129.7403\ldots+\$36.0390\ldots=\$720.7792\ldots\), matching the premium. The method avoids treating the $30 constant as a percentage or as a charge per exposure unit.
Worked Example: Comparing Policies with Different Exposure Amounts
A hypothetical insurer uses a $42 expense constant per policy, a loss cost of $120 per exposure unit, variable expenses of 20% of gross premium, and a profit provision of 5% of gross premium. Find the gross premium and rate per exposure unit for a policy with one exposure unit and a policy with three exposure units.
Model and assumptions. Use the same gross-premium model for both policies. All amounts are for the same policy term. No interest or mortality adjustment is assumed, and the loss cost is an assumed expected value rather than a specified probability distribution.
For both policies, the denominator is \(1-0.20-0.05=0.75\).
One-unit policy. Its loss provision is \(1(\$120)=\$120\), so
Three-unit policy. Its loss provision is \(3(\$120)=\$360\), so
Rounded to the nearest cent, the premiums are $216.00 and $536.00; the rates are $216.00 and $178.67 per exposure unit, respectively. The three-unit policy has a higher total premium, but a lower rate per unit because its single $42 constant is spread across three units. As a check, its provisions are $360 in losses, $42 in fixed expense, $107.20 in variable expense, and $26.80 in profit, totaling $536. The common error this method avoids is multiplying $42 by three: the constant is per policy, not per exposure unit.
Worked Example: Checking a Portfolio Allocation
A hypothetical book has a fixed expense budget of $480,000 and 16,000 policies. Assume that the book contains 8,000 policies with 0.8 exposure units each and 8,000 policies with 4 exposure units each. The loss cost is $85 per exposure unit, variable expenses are 22% of gross premium, and profit is 3% of gross premium. Calculate the premium for each policy type and check that the portfolio totals fund the stated provisions.
Model and assumptions. Apply the constant equally per policy and use the gross-premium formulation for each type. Amounts refer to one consistent policy term; no interest or mortality adjustment applies. The loss cost is an assumed expected value, with no separate probability distribution specified.
Step 1: Calculate the constant and denominator.
Step 2: Calculate the premium for a 0.8-unit policy. Its expected loss provision is \(0.8(\$85)=\$68\), so
Step 3: Calculate the premium for a 4-unit policy. Its expected loss provision is \(4(\$85)=\$340\), so
The corresponding rates are \(\$130.6667\ldots/0.8=\$163.3333\ldots\) and \(\$493.3333\ldots/4=\$123.3333\ldots\) per exposure unit. The larger policy has the lower rate per unit because the $30 constant is spread over more exposure.
Step 4: Reconcile the portfolio using unrounded premiums. Total premium is
Total exposure is \(8{,}000(0.8)+8{,}000(4)=38{,}400\) units, giving expected losses of \(38{,}400(\$85)=\$3{,}264{,}000\). The fixed expense provision is \(16{,}000(\$30)=\$480{,}000\). Variable expenses are \(0.22(\$4{,}992{,}000)=\$1{,}098{,}240\), and profit is \(0.03(\$4{,}992{,}000)=\$149{,}760\). These provisions sum to \(\$3{,}264{,}000+\$480{,}000+\$1{,}098{,}240+\$149{,}760=\$4{,}992{,}000\), matching total premium. The reconciliation confirms that the constant is allocated once to each policy.
Per-Policy Constant Versus Fixed Expense per Exposure
A fixed expense load per exposure unit has a different structure. If \(f\) is the fixed expense allocated per exposure unit, the policy's fixed expense provision is \(fe\), not \(C\). Under the same percentage-of-premium assumptions, the gross premium would be \(P=e(L+f)/(1-v-q)\), and the rate per exposure unit would be \((L+f)/(1-v-q)\). That rate does not vary with \(e\) solely because of the fixed expense allocation.
| Expense basis | Fixed expense provision for a policy | Fixed expense per exposure unit |
|---|---|---|
| Per policy | \(C\) | \(C/e\) |
| Per exposure unit | \(fe\) | \(f\) |
These are different allocation choices, not two interchangeable ways of writing the same charge. For example, if a $30 amount is a per-policy constant, a four-unit policy receives $30 of fixed expense, not $120. If instead the intended provision is $30 per exposure unit, that policy receives $120. The basis must match how the expense is being allocated.
Common Mistakes and Checks
- Multiplying the constant by exposure. In \(eL+C\), the loss provision scales with \(e\), but the per-policy constant does not. Multiplying \(C\) by \(e\) changes the assumed expense basis.
- Dividing by the wrong denominator. If \(v\) and \(q\) are both proportions of gross premium, the denominator is \(1-v-q\). This expression should not be used without adjustment when a provision is defined on another base.
- Calling the total premium a rate. The policy premium is \(P\); the rate per exposure unit is \(P/e\). Keep the units visible, especially when comparing policies of different sizes.
- Using an inconsistent policy count. The budget and number of policies used to calculate \(C=F/N\) must refer to the same allocation population and period.
- Rounding too early. Retain unrounded premiums for a portfolio reconciliation, then round the reported dollar premium or rate at the final step.
Check Your Understanding
For each problem, keep the expense basis and premium basis explicit. Unless stated otherwise, assume all amounts apply to the same policy term, percentage provisions are based on gross premium, and there is no interest or mortality adjustment.
- A hypothetical fixed expense budget of $525,000 is allocated evenly over 15,000 policies. Loss cost is $140 per exposure unit, and a policy has 2 exposure units. Variable expenses are 16% and profit is 4% of gross premium. Find the constant, gross premium, and rate per exposure unit, rounded to cents. (Answer: \(C=\$35\); premium \(=\$393.75\); rate \(=\$196.88\) per exposure unit.)
- A policy has a $24 constant and loss cost of $95 per exposure unit. Variable expenses are 25% and profit is 5% of gross premium. Calculate the gross premium and rate per exposure unit for policies with 0.5 and 2 exposure units. Round to cents. (Answer: For 0.5 unit, premium \(=\$102.14\) and rate \(=\$204.29\); for 2 units, premium \(=\$305.71\) and rate \(=\$152.86\) per exposure unit.)
- A hypothetical expense budget of $216,000 is spread over 9,000 policies. The loss cost is $72 per exposure unit, the policy has 1.5 exposure units, variable expenses are 19% of gross premium, and profit is 6% of gross premium. Find the constant, gross premium, and rate per exposure unit. (Answer: \(C=\$24\); premium \(=\$176.00\); rate \(=\$117.33\) per exposure unit.)
- A book has 2,000 one-unit policies and 1,000 three-unit policies. Its total fixed expense budget is $150,000, allocated per policy. With loss cost $100 per unit, variable expenses of 20% of premium, and profit of 5% of premium, calculate each type's gross premium. Then compare those premiums with the premiums that would result from mistakenly using the portfolio's fixed expense per exposure as the per-policy constant. (Answer: Correct constant \(=\$50\) per policy and fixed expense per exposure \(=\$30\). Correct premiums are \(\$200.00\) for a one-unit policy and \(\$466.67\) for a three-unit policy. The mistaken premiums are \(\$173.33\) and \(\$440.00\), respectively.)
- A two-unit policy has a gross premium of $300. Loss cost is $100 per exposure unit, variable expenses are 15% of premium, and profit is 5% of premium. Infer the per-policy constant. Then find the premium for a four-unit policy with the same loss cost and provisions. (Answer: \(C=\$40\); the four-unit policy premium is \(\$550.00\).)