Why the Premium Base Matters
When expenses are allocated by premium, each portion of the total expense is assigned in proportion to the premium attributed to the relevant policy, class, or coverage. The method therefore gives more expense dollars to a group with more premium. Its central calculation is straightforward: divide the expenses being allocated by the premium measure selected as the allocation base.
The calculation is simple; choosing the denominator requires care. Earned premium, written premium, and on-level premium can describe different amounts for the same book of business. The resulting expense ratio and class allocations may differ, even when the expense total does not. A valid calculation must identify what the premium total measures and use a premium base that is appropriate to the expense and the ratemaking purpose.
The calculation assumes the specified expenses are to be allocated in proportion to premium, the selected premiums are measured on a consistent basis, and the total premium is positive. It does not establish that premium caused each expense or that every cost is naturally proportional to premium. It is an allocation rule. If expenses are not assigned on this basis, or the premium amounts do not match the relevant population or period, the formula does not by itself make the allocation appropriate.
Calculating and Interpreting the Ratio
Suppose an insurer selects an expense total of $300,000 and a corresponding premium base of $2,500,000. The ratio is \(300{,}000/2{,}500{,}000=0.12\), or 12%. In this allocation, 12 cents of expense is assigned for each dollar of the selected premium base. If the expenses and premiums are stated on different accounting or ratemaking bases, that arithmetic still produces a number, but the number may not represent a meaningful provision.
The denominator must be specified rather than treated as a generic “premium.” For example, written premium generally relates to policies written during a period, while earned premium relates to the portion of premium recognized for coverage provided during a period. On-level premium adjusts historical premium to a selected rate level for a ratemaking comparison. Which measure to use depends on what the expense amount represents and what question the calculation is intended to answer. Do not substitute one merely because it is available.
The same ratio applies to every premium amount in a single proportional allocation. A class with 20% of the selected premium receives 20% of the expenses assigned under that allocation. If the selected base changes the relative premium shares among classes, their dollar allocations can change as well. If the base is multiplied by the same factor for every class, the ratio changes inversely but the class shares—and therefore the allocated expense dollars—stay the same.
This identity is an important check: when the class premiums add to the selected total, allocated expenses should add back to the expense total. If they do not, check for a missing class, a duplicated amount, inconsistent units, or rounding.
Worked Example: Calculating a Ratio From a Matched Premium Total
A hypothetical insurer assigns $168,000 of acquisition expenses and $42,000 of premium taxes to a book of small commercial policies. The expenses relate to coverage provided during a year. The selected earned premium for that same book and year is $1,750,000. Calculate the combined premium-based expense ratio and the expense provision on $80,000 of premium on the same basis.
Method. Add only the expenses included in the stated allocation, then divide by the matching earned premium. The method avoids using a written-premium denominator simply because it appears in the same financial report.
Step 1: Total the expenses. The specified acquisition expense and premium tax are both included: \(168{,}000+42{,}000=\$210{,}000\).
Step 2: Calculate the ratio. \(\$210{,}000/\$1{,}750{,}000=0.12\), so the premium-based expense ratio is 12.00%.
Step 3: Apply it to the stated premium. \(0.12\times\$80{,}000=\$9{,}600\). The provision allocated to $80,000 of premium is $9,600.
Check. The allocated amount is 12% of $80,000, as required. The expense and premium bases refer to the same book and coverage period; using an unrelated written-premium amount would answer a different calculation.
Why the Allocation Formula Works
The allocation formula follows directly from the proportional rule. Each segment receives the same fraction of expense per dollar of selected premium. If the common fraction is \(r_p\), segment \(j\), with premium \(P_j\), receives \(E_j=r_pP_j\). Across all segments, the allocations must total \(E\). Therefore \(r_p\sum_jP_j=E\). When the selected segment premiums sum to \(P\), solving gives \(r_p=E/P\), and substituting this ratio gives \(E_j=E(P_j/P)\).
This derivation also explains two useful checks. First, a class’s allocation share must equal its share of total selected premium. Second, summing the allocations must recover the original expense total. The derivation does not show that the allocation is the best possible method for every expense; it shows why the formula is correct when the specified rule is proportional allocation by the selected premium base.
Worked Example: Comparing Earned and On-Level Premium Bases
A hypothetical insurer is assigning $450,000 of expenses to three classes. The expenses are to be allocated by premium. The class earned premiums for the accounting period are $1,200,000, $800,000, and $1,600,000. For a separate ratemaking comparison, the corresponding on-level premiums are $1,500,000, $1,000,000, and $1,500,000. Calculate the ratio and class allocations using each base, then explain what the difference means.
Method. Calculate each base’s total, divide the same expense amount by that total, and allocate in proportion to each class’s premium share. The two bases are alternatives for comparison, not amounts to combine.
Step 1: Earned-premium calculation. Total earned premium is \(\$1{,}200{,}000+\$800{,}000+\$1{,}600{,}000=\$3{,}600{,}000\). The ratio is \(\$450{,}000/\$3{,}600{,}000=0.125\), or 12.50%.
Class A receives \(0.125\times\$1{,}200{,}000=\$150{,}000\). Class B receives \(0.125\times\$800{,}000=\$100{,}000\). Class C receives \(0.125\times\$1{,}600{,}000=\$200{,}000\). The allocations total \(\$150{,}000+\$100{,}000+\$200{,}000=\$450{,}000\).
Step 2: On-level-premium calculation. Total on-level premium is \(\$1{,}500{,}000+\$1{,}000{,}000+\$1{,}500{,}000=\$4{,}000{,}000\). The ratio is \(\$450{,}000/\$4{,}000{,}000=0.1125\), or 11.25%.
Class A receives \(0.1125\times\$1{,}500{,}000=\$168{,}750\). Class B receives \(0.1125\times\$1{,}000{,}000=\$112{,}500\). Class C receives \(0.1125\times\$1{,}500{,}000=\$168{,}750\). The allocations total \(\$168{,}750+\$112{,}500+\$168{,}750=\$450{,}000\).
Interpretation and check. Class C has the largest earned-premium share, so it receives the largest earned-basis allocation. On the on-level basis, A and C have equal premium and receive equal allocations. Both methods allocate the full $450,000, but they distribute it differently because the class shares differ. The lower on-level ratio is a consequence of its larger total denominator; it is not, by itself, evidence that the expenses changed.
Allocation by Premium Does Not Mean Equal Dollars per Policy
A constant expense ratio means equal expense per dollar of the selected premium, not equal expense per policy. Policies with different premium amounts receive different allocated expense dollars under the same ratio. This distinction matters when a question asks for a class total, a per-policy amount, or a percentage of premium: these are related but different quantities.
Worked Example: From Class Premium Shares to Per-Policy Allocations
A hypothetical specialty insurer allocates $240,000 of expenses across two classes using annual written premium. Class R has 600 policies and $1,200,000 of written premium. Class S has 200 policies and $800,000 of written premium. Calculate the expense ratio, the expense assigned to each class, and the average allocated expense per policy. Then check the totals.
Method. First calculate the ratio from total premium. Apply that ratio to each class premium, and only then divide each class allocation by its policy count. The common mistake is to treat a premium-based allocation as if it assigns equal expense to each policy.
Step 1: Total the premium. \(\$1{,}200{,}000+\$800{,}000=\$2{,}000{,}000\). The expense ratio is \(\$240{,}000/\$2{,}000{,}000=0.12\), or 12.00%.
Step 2: Allocate by class premium. Class R receives \(0.12\times\$1{,}200{,}000=\$144{,}000\). Class S receives \(0.12\times\$800{,}000=\$96{,}000\).
Step 3: Calculate average expense per policy. For Class R, \(\$144{,}000/600=\$240\) per policy. For Class S, \(\$96{,}000/200=\$480\) per policy. These averages differ because the method allocates by premium, and the class premium per policy differs.
Check. The class allocations sum to \(\$144{,}000+\$96{,}000=\$240{,}000\). Each allocation is 12% of its class premium. Assigning the same dollar amount to every policy would instead be a count-based allocation, not the specified premium-based method.
Choosing and Checking the Premium Measure
Before calculating a premium-based ratio, make the basis explicit. In a ratemaking application, an on-level premium base can support comparison at a selected rate level; an earned-premium base can align with expenses measured for coverage provided; a written-premium base can be relevant when the expense and analysis are tied to policies written. These are general distinctions, not automatic selection rules. The problem’s stated purpose and the expense measurement determine which base is appropriate.
Check that the expense amount and premium denominator refer to a compatible:
- Population: include the same classes, coverages, or policy groups in the expenses and premium base.
- Period: do not pair one period’s expenses with another period’s premium without a stated reason.
- Premium definition: identify whether the base is written, earned, on-level, or otherwise adjusted, and apply that definition consistently.
- Scope: include only the expenses assigned by the problem and use the premium amount that corresponds to that assignment.
- Unit: convert amounts reported in thousands or other units before dividing or adding them.
If the selected premium total is zero, \(E/P\) is undefined. If the premium total is very small, the resulting ratio can be unusually large and sensitive to small changes in the inputs. Neither situation is repaired by dividing by a different amount without justification; the analyst must reconsider the data or the allocation basis.
Common Exam Trap: Using the Wrong Denominator
Suppose a question specifies $180,000 of expenses and $1,500,000 of on-level premium. The correct ratio on that stated base is \(\$180{,}000/\$1{,}500{,}000=12.00\%\). A common incorrect calculation adds the expenses to the denominator first: \(\$180{,}000/(\$1{,}500{,}000+\$180{,}000)=10.71\%\), rounded to two decimals. That is not the premium-based expense ratio defined here. It uses a different denominator and therefore answers a different question.
Another trap is using written premium when the problem explicitly selects earned or on-level premium. The arithmetic may look reasonable, but it changes the basis. Read the denominator condition carefully, state the selected premium measure, and verify that the final allocated expenses add back to the specified expense total.
Check Your Understanding
For each problem, identify the premium basis before calculating. Amounts and scenarios are hypothetical.
- A hypothetical insurer allocates $96,000 of expenses using $800,000 of earned premium. Calculate the ratio and the provision on $25,000 of earned premium. (Answer: 12.00%; $3,000)
- A hypothetical book has $210,000 of allocated expenses. Class T has $700,000 of the selected premium, and Class U has $1,050,000. Calculate the ratio and each class’s expense allocation. (Answer: 12.00%; Class T $84,000; Class U $126,000)
- A hypothetical insurer has $360,000 of expenses and selects $2,400,000 of on-level premium. It also reports $2,100,000 of written premium, but the problem requires the on-level basis. Calculate the required ratio and the expense provision on $50,000 of on-level premium. (Answer: 15.00%; $7,500)
- A hypothetical insurer allocates $300,000 across two classes. Class V has $1,000,000 of earned premium and 500 policies; Class W has $1,500,000 and 300 policies. Calculate the ratio, each class allocation, and each class’s average allocation per policy. (Answer: 12.00%; Class V $120,000 and $240 per policy; Class W $180,000 and $600 per policy)
- A hypothetical insurer has $240,000 of expenses. Its earned-premium totals are $1,200,000 for Class M and $800,000 for Class N; its on-level totals are $1,000,000 for Class M and $1,000,000 for Class N. Calculate the ratio and class allocations under each basis, then state why the allocations differ. (Answer: Earned basis, 12.00%, M $144,000 and N $96,000; on-level basis, 12.00%, M $120,000 and N $120,000; the total premium is the same but the class premium shares differ.)