Why the Expense Base Matters
Policies are not always comparable units of business. One policy may insure several coverages, provide a longer period of protection, or require more servicing than another. Allocating every expense equally per policy can therefore produce a different result from allocating by coverage unit, exposure, or another measure of activity. The selected base determines how an expense pool is distributed among groups; it should reflect the activity that is expected to drive that expense.
This tutorial focuses on choosing and applying a base when policies differ in size or servicing needs. The allocation is an expense attribution method: it distributes a known pool across groups. It does not, by itself, prove that each group caused exactly the allocated amount or that the allocation is a complete estimate of future costs.
For example, a coverage unit might mean one covered vehicle or one separately rated coverage, depending on the stated rating plan. A service unit might mean one endorsement transaction or one inspection. An exposure unit measures the amount of risk insured over the chosen time interval. These are not interchangeable measures: the allocation depends on what each unit means and how it is counted.
The Allocation Formula
Suppose a total expense pool \(E\) is allocated across groups \(j=1,\ldots,k\) in proportion to a selected base. Let \(B_j\) be the amount of that base for group \(j\), and let \(B=\sum_{j=1}^{k}B_j\) be the total base. Provided \(B>0\), the expense rate per base unit is \(E/B\), and the allocation to group \(j\) is its base amount multiplied by that rate.
The formula follows directly from proportional allocation. If each base unit receives the same expense rate \(r_B\), group \(j\) receives \(r_BB_j\). Summing the group allocations gives
Thus the allocation reconciles to the pool, as long as the groups are exhaustive, nonoverlapping, and measured using the same base definition. If the total base is zero, this proportional formula cannot allocate the pool. A different base or an explicit allocation rule is then needed.
When policies differ in size, the per-policy expense implied for a group is not necessarily the same across groups. If group \(j\) has \(N_j\) policies, its average allocated expense per policy is \(E_j/N_j\). For a per-unit allocation, that average depends on the number of units per policy. For a per-exposure allocation, it depends on the amount of exposure per policy. This is often the useful comparison when translating an aggregate expense allocation into a rating load.
Choosing Among Bases
A practical choice begins with the expense’s expected driver, not with whichever base is easiest to calculate. A fixed cost incurred once for each issued policy may be reasonably allocated per policy. A cost that increases with the number of covered items may be better allocated per coverage unit. A cost associated with the time or amount of risk insured may be more closely related to exposure. If servicing intensity differs, a count of relevant transactions may be more informative than either policy count or exposure.
| Possible base | What is counted | When it may be useful | Question to check |
|---|---|---|---|
| Per policy | Policies | Costs incurred once for each policy | Are policies similar in the activity that drives this cost? |
| Per unit | Defined coverage or service units | Costs that vary with covered items or measured activities | Is a unit defined consistently and does it reflect the cost driver? |
| Per exposure | Insured exposure units over a specified period | Costs that vary with the amount or duration of insured risk | Are exposure amounts measured on a comparable basis? |
A single portfolio can have different expense components with different drivers. In that case, allocate each component using its own appropriate base, then add the component allocations for each group. Applying one base to the entire pool simply because it is convenient can shift expenses toward or away from groups whose policy size or activity differs.
Worked Example: Comparing Per-Policy and Per-Coverage-Unit Loads
A hypothetical insurer has a $54,000 expense pool to allocate across two groups. Group A has 300 policies with one coverage unit per policy. Group B has 200 policies with three coverage units per policy. Calculate the allocations under a per-policy base and a per-coverage-unit base, then find each group’s expense per policy under each method.
Method. Apply the allocation formula separately to each proposed base. The groups have \(300+200=500\) policies. Their coverage-unit counts are \(300\times1=300\) for A and \(200\times3=600\) for B, for \(300+600=900\) units in total.
Per-policy allocation. The rate is \( \$54{,}000/500=\$108\) per policy. Group A receives \(300\times\$108=\$32{,}400\). Group B receives \(200\times\$108=\$21{,}600\). The implied expense per policy is \(\$32{,}400/300=\$108\) for A and \(\$21{,}600/200=\$108\) for B.
Per-coverage-unit allocation. The rate is \( \$54{,}000/900=\$60\) per coverage unit. Group A receives \(300\times\$60=\$18{,}000\). Group B receives \(600\times\$60=\$36{,}000\). The implied expense per policy is \(\$18{,}000/300=\$60\) for A and \(\$36{,}000/200=\$180\) for B. Group B’s higher per-policy load follows from its three times as many units per policy.
Check and mistake avoided. The allocations under each method total \(\$54{,}000\): \(\$32{,}400+\$21{,}600=\$54{,}000\), and \(\$18{,}000+\$36{,}000=\$54{,}000\). A common denominator error is to divide the unit-based pool by 500 policies instead of 900 coverage units. That would produce \(\$108\) per unit and allocate \(\$32{,}400\) to A and \(\$64{,}800\) to B, totaling \(\$97{,}200\), not the stated pool. Match the denominator to the selected base.
Allocating Separate Expense Components
When a pool contains costs with different drivers, splitting it into components can avoid forcing one base to represent all expenses. The allocation rate is calculated separately for each component using that component’s base. The resulting group allocations are then added. The complete method is:
State which expense amount is allocated by policy, unit, exposure, or another specified measure.
Add the groups’ counts using the same definition and measurement period for each group.
Divide that component’s expense amount by its own total base.
Multiply each group’s base by the component rate, add components by group, and verify that all allocations sum to the original pool.
Worked Example: Combining Policy, Coverage, and Service Bases
A hypothetical insurer separates a $70,000 expense pool into three components: $24,000 allocated per policy, $28,000 allocated per coverage unit, and $18,000 allocated per service transaction. Group M has 250 policies, 500 coverage units, and 600 transactions. Group N has 150 policies, 900 coverage units, and 300 transactions. Find each group’s component allocations, combined allocation, and combined expense per policy.
Method. Calculate a separate rate for each component. There are \(250+150=400\) policies, \(500+900=1{,}400\) coverage units, and \(600+300=900\) transactions.
Policy component. The rate is \(\$24{,}000/400=\$60\) per policy. M receives \(250\times\$60=\$15{,}000\); N receives \(150\times\$60=\$9{,}000\).
Coverage-unit component. The rate is \(\$28{,}000/1{,}400=\$20\) per unit. M receives \(500\times\$20=\$10{,}000\); N receives \(900\times\$20=\$18{,}000\).
Service-transaction component. The rate is \(\$18{,}000/900=\$20\) per transaction. M receives \(600\times\$20=\$12{,}000\); N receives \(300\times\$20=\$6{,}000\).
Combine and convert to per-policy amounts. M’s total allocation is \(\$15{,}000+\$10{,}000+\$12{,}000=\$37{,}000\), or \(\$37{,}000/250=\$148\) per policy. N’s total is \(\$9{,}000+\$18{,}000+\$6{,}000=\$33{,}000\), or \(\$33{,}000/150=\$220\) per policy.
Check and mistake avoided. The policy component reconciles to \(\$15{,}000+\$9{,}000=\$24{,}000\); the unit component to \(\$10{,}000+\$18{,}000=\$28{,}000\); and the transaction component to \(\$12{,}000+\$6{,}000=\$18{,}000\). Combined, \(\$37{,}000+\$33{,}000=\$70{,}000\). A common mistake is to calculate one rate using the total number of policies, units, and transactions together. Those unlike bases cannot be added meaningfully; each component needs its own denominator.
Exposure Bases and Policy Duration
A per-policy allocation treats every policy as one unit, even if the policies insure different amounts of exposure or are in force for different periods. An exposure base can reflect those differences when it is defined consistently. For instance, if an exposure unit is one insured item-year, a policy covering two items for half a year represents one item-year. The measure should be specified before calculating the allocation; the formula does not decide what counts as an exposure unit.
Worked Example: Comparing Per-Policy and Per-Exposure Allocation
A hypothetical portfolio has two groups and a $40,000 expense pool. Group R has 120 policies, each averaging 0.5 exposure unit. Group S has 80 policies, each averaging 1 exposure unit. Allocate $28,000 of the pool by exposure and $12,000 per policy. Find each group’s total allocation and average expense per policy.
Exposure component. R has \(120\times0.5=60\) exposure units, and S has \(80\times1=80\), for \(60+80=140\) units. The rate is \(\$28{,}000/140=\$200\) per exposure unit. R receives \(60\times\$200=\$12{,}000\); S receives \(80\times\$200=\$16{,}000\).
Per-policy component. There are \(120+80=200\) policies. The rate is \(\$12{,}000/200=\$60\) per policy. R receives \(120\times\$60=\$7{,}200\); S receives \(80\times\$60=\$4{,}800\).
Combined amounts. R receives \(\$12{,}000+\$7{,}200=\$19{,}200\), or \(\$19{,}200/120=\$160\) per policy. S receives \(\$16{,}000+\$4{,}800=\$20{,}800\), or \(\$20{,}800/80=\$260\) per policy.
Check and mistake avoided. The exposure allocations sum to \(\$28{,}000\), the policy allocations sum to \(\$12{,}000\), and the combined allocations sum to \(\$40{,}000\). Allocating all $40,000 per policy would assign \(\$24{,}000\) to R and \(\$16{,}000\) to S. That is a different allocation method, not a harmless shortcut: it ignores the groups’ different exposure per policy.
Interpretation and Common Traps
An allocation base is a modeling choice that should be supported by the expense’s expected driver and by reliable, consistently defined data. Per-policy allocation is simple, but it gives the same amount to every policy. Per-unit allocation differentiates policies according to the counted units, while per-exposure allocation differentiates them according to the specified exposure measure. Neither greater detail nor a more complicated base guarantees a better allocation if the base is poorly measured or unrelated to the cost.
Keep the distinctions between the following quantities clear:
- Expense pool: the total amount to allocate.
- Base total: the sum of the selected measure across all groups.
- Rate per base unit: the expense pool divided by that base total.
- Group allocation: the rate per base unit multiplied by the group’s base.
- Per-policy amount: the group allocation divided by its policy count.
A frequent exam trap is to report an allocation rate as if it were an expense per policy. For example, a rate of $20 per coverage unit is not automatically $20 per policy. If a group averages three units per policy, its allocated amount from that component averages \(3\times\$20=\$60\) per policy. Another trap is to use one group’s base as the denominator instead of the total base for all groups. Always calculate the total first and then check that the allocations reconcile.
Selecting and applying expense bases for heterogeneous policies is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. In a rate indication, the resulting expense allocation must ultimately be used consistently with the other rate components, but the allocation calculations here concern the expense base itself.
Check Your Understanding
For each problem, identify the appropriate base stated in the problem, calculate the requested allocations, and check the totals.
- A $19,200 pool is allocated by coverage unit. Group X has 160 policies with two units each; Group Y has 80 policies with four units each. Find the rate per unit, each group’s allocation, and each group’s average allocation per policy. (Answer: $30 per unit; X receives $9,600 and $60 per policy; Y receives $9,600 and $120 per policy.)
- A $22,500 pool is allocated per policy. Group A has 250 policies and Group B has 125. Find the rate per policy and each group’s allocation. (Answer: $60 per policy; A receives $15,000 and B receives $7,500.)
- A $25,000 pool has a $9,000 component allocated per policy and a $16,000 component allocated per service transaction. Group A has 100 policies and 200 transactions; Group B has 200 policies and 400 transactions. Find the combined allocation to each group and the average per policy. (Answer: A receives $8,333.33, or $83.33 per policy; B receives $16,666.67, or $83.33 per policy, rounded to cents.)
- A $36,000 pool is allocated by exposure unit. Group R has 90 policies averaging 0.8 exposure unit per policy; Group S has 120 policies averaging 1 exposure unit per policy. Find the rate per exposure unit, each group’s allocation, and its average allocation per policy. (Answer: $187.50 per exposure unit; R receives $13,500, or $150 per policy; S receives $22,500, or $187.50 per policy.)
- A $20,000 servicing pool is allocated by service visits. Group P has 100 policies averaging 1.5 visits per policy; Group Q has 200 policies averaging 0.5 visit per policy. Find the rate per visit, each group’s allocation, and the average allocation per policy. (Answer: $80 per visit; P receives $12,000, or $120 per policy; Q receives $8,000, or $40 per policy.)