Allocating Shared Expenses Across Heterogeneous Policies

Why Allocation Basis Matters

A portfolio can contain policy groups with different numbers of policies, premium volumes, and sums assured. Some expenses, such as a hypothetical central administration cost, cannot be attributed directly to one group. They must be allocated across groups before group asset shares can be calculated. The total expense is fixed, but the allocation basis determines how much each group bears.

An allocation is an accounting rule, not a change in the cash expense incurred by the insurer. If one group is assigned a larger share, another group must receive a correspondingly smaller share. The combined asset share is therefore unchanged by the choice of basis, while each group’s asset share can change materially.

Definition: For policy groups $g=1,\ldots,m$, let $C$ be a shared expense for a stated period and let $D_g$ be the group’s value of a chosen allocation driver. If $\sum_{j=1}^{m}D_j>0$, the allocation weight and allocated expense for group $g$ are $$ w_g=\frac{D_g}{\sum_{j=1}^{m}D_j}, \qquad C_g=Cw_g. $$ The weights sum to 1, so the allocated expenses sum to $C$. Common drivers include policy count, total premium, and total sum assured.

The driver must be defined consistently. A premium basis normally uses each group’s total premium volume for the relevant period, not the premium per policy by itself. A sum-assured basis uses the aggregate sum assured for the group. A count basis uses the number of policies included in the allocation. The chosen measurement date and period should be stated, particularly if policies enter or leave during the year.

If a group has a zero value for a driver, its allocation under that driver is zero. If every group has a zero value, the formula is undefined; an alternative allocation rule must be specified. The driver is a modelling choice and should reflect the stated basis, not be changed partway through a calculation.

Incorporating the Allocation into an Asset Share

Use the following annual timing convention in the calculations below. The opening aggregate asset share and premiums are available at the start of the year and earn the effective annual investment return $i$. Claims, direct expenses, and the allocated shared expense are paid at year-end. Premiums and group cash flows are stated as deterministic amounts for the example year; no mortality distribution or other probability model is assumed.

Asset-share update: If group $g$ begins with aggregate asset share $A_g$, receives premium income $P_g$, pays claims $K_g$ and direct expenses $E_g$, and is allocated shared expense $C_g$, then its year-end aggregate asset share is $$ A_g^{\text{end}}=(A_g+P_g)(1+i)-K_g-E_g-C_g. $$ Its year-end per-policy asset share is $A_g^{\text{end}}/N_g$, where $N_g$ is the number of policies used for that per-policy measure. The formula assumes the stated timing; a different cash-flow timing requires different accumulation factors.

Direct expenses are already attributed to a group and should not be included again in the shared-expense pool. Conversely, if an expense is included in $C$, it must be allocated once, and only once. All groups must use the same expense period and allocation basis for the stated comparison.

Worked Example: Count Basis Versus Premium Basis

A hypothetical insurer has two groups. Group A has 100 policies and Group B has 50. Each group receives total annual premiums of $120,000. At the start of the year, their aggregate asset shares are $500,000 and $300,000, respectively. The effective annual investment return is 4%. Year-end claims are $20,000 for A and $15,000 for B; direct expenses are $5,000 and $3,000. A further $30,000 of shared administration expense is paid at year-end. Compare allocating it by policy count and by total premium. Treat the stated cash flows as fixed amounts; no mortality distribution is assumed.

Method. First accumulate each group’s opening asset share and premium, then deduct claims, direct expenses, and the allocation. The common error to avoid is allocating the shared expense and then also deducting the full $30,000 separately from each group.

Count basis. Total policy count is $100+50=150$. The weights are $100/150=2/3$ for A and $50/150=1/3$ for B. The allocations are $30{,}000(2/3)=\$20{,}000$ and $30{,}000(1/3)=\$10{,}000$.

For A, the amount accumulated before year-end outgo is $(500{,}000+120{,}000)(1.04)=620{,}000(1.04)=\$644{,}800$. Its closing aggregate asset share is $644{,}800-20{,}000-5{,}000-20{,}000=\$599{,}800$, or $\$599{,}800/100=\$5{,}998.00$ per policy.

For B, the accumulated amount is $(300{,}000+120{,}000)(1.04)=420{,}000(1.04)=\$436{,}800$. Its closing aggregate asset share is $436{,}800-15{,}000-3{,}000-10{,}000=\$408{,}800$, or $\$408{,}800/50=\$8{,}176.00$ per policy.

Premium basis. Total premiums are $120{,}000+120{,}000=\$240{,}000$, so each group has weight $120{,}000/240{,}000=1/2$. Each is allocated $\$15{,}000$. A’s closing aggregate asset share becomes $644{,}800-20{,}000-5{,}000-15{,}000=\$604{,}800$, or $\$6{,}048.00$ per policy. B’s becomes $436{,}800-15{,}000-3{,}000-15{,}000=\$403{,}800$, or $\$8{,}076.00$ per policy.

Check. Under either basis, the combined closing asset share is $\$599{,}800+\$408{,}800=\$1{,}008{,}600$, or $\$604{,}800+\$403{,}800=\$1{,}008{,}600$. The shared expense totals $30,000 under both allocations. The per-policy results differ because the groups have different policy counts, even though their premium totals are equal.

Allocating by Sum Assured

A sum-assured basis may be appropriate when a stated allocation rule relates the shared cost to the amount insured. It is not interchangeable with a count or premium basis: groups with the same policy count can have very different aggregate sums assured, and high premiums do not necessarily imply the largest sum assured.

Worked Example: Allocating by Aggregate Sum Assured

A hypothetical portfolio has 40 policies in Group A, each with a sum assured of $100,000, and 60 policies in Group B, each with a sum assured of $50,000. Opening aggregate asset shares are $250,000 and $180,000. Premiums received at the start of the year total $80,000 for A and $78,000 for B. The effective annual return is 5%. Year-end claims are $28,000 and $24,000, and direct expenses are $4,000 and $6,000. Allocate a $18,000 shared expense by aggregate sum assured. The cash flows are fixed inputs; no mortality distribution is assumed.

Method. Calculate each group’s total sum assured, use those amounts to obtain weights, and apply the annual asset-share update. The trap is using the number of policies as the weight even though the stated basis is sum assured.

A’s aggregate sum assured is $40(100{,}000)=\$4{,}000{,}000$. B’s is $60(50{,}000)=\$3{,}000{,}000$. Total sum assured is $7,000,000, giving weights $4/7$ and $3/7$. The exact expense shares are $18{,}000(4/7)=\$10{,}285.714\ldots$ and $18{,}000(3/7)=\$7{,}714.285\ldots$. To the nearest cent, allocate $10,285.71 to A and $7,714.29 to B; these rounded amounts total $18,000.00.

For A, the accumulated opening asset share and premium are $(250{,}000+80{,}000)(1.05)=330{,}000(1.05)=\$346{,}500$. Its closing aggregate asset share is $346{,}500-28{,}000-4{,}000-10{,}285.71=\$304{,}214.29$. Per policy, this is $304{,}214.29/40=\$7{,}605.36$, rounded to the nearest cent.

For B, the corresponding accumulation is $(180{,}000+78{,}000)(1.05)=258{,}000(1.05)=\$270{,}900$. Its closing aggregate asset share is $270{,}900-24{,}000-6{,}000-7{,}714.29=\$233{,}185.71$. Per policy, this is $233{,}185.71/60=\$3{,}886.43$, rounded to the nearest cent.

Check. The combined closing asset share is $537,400. Independently, $(250{,}000+180{,}000+80{,}000+78{,}000)(1.05)-28{,}000-24{,}000-4{,}000-6{,}000-18{,}000=\$537{,}400$. This confirms that the shared expense has been deducted once and in full.

Why the Combined Total Is Unchanged

The conservation property follows directly from the allocation formula. Since $C_g=Cw_g$ and $w_g=D_g/\sum_j D_j$, summing allocations across groups gives:

\sum_{g=1}^{m} C_g =\sum_{g=1}^{m} C\frac{D_g}{\sum_{j=1}^{m}D_j} =C\frac{\sum_{g=1}^{m}D_g}{\sum_{j=1}^{m}D_j} =C.

To see the consequence for asset shares, define $B_g=(A_g+P_g)(1+i)-K_g-E_g$, the group’s year-end asset share before the shared expense. Then $A_g^{\text{end}}=B_g-C_g$. Summing across groups yields $\sum_g A_g^{\text{end}}=\sum_g B_g-C$. The combined total depends on the total expense, not on how that expense is distributed across groups.

However, comparing two allocation bases $r$ and $s$ gives $A_{g,r}^{\text{end}}-A_{g,s}^{\text{end}}=-C(w_{g,r}-w_{g,s})$. The group assigned a larger weight under basis $r$ has a lower asset share under that basis by the difference in its allocated expense. For a per-policy comparison, divide that group-level difference by its policy count.

Worked Example: Comparing Three Bases Across Groups

A hypothetical portfolio has Groups A, B, and C with 20, 30, and 10 policies. Their total annual premiums are $20,000, $60,000, and $30,000; sums assured per policy are $100,000, $50,000, and $150,000. Opening aggregate asset shares are $80,000, $100,000, and $50,000. Premiums are received at the start of the year, the effective annual return is 5%, and year-end claims and direct expenses are respectively $3,000 and $2,000 for A, $12,000 and $6,000 for B, and $10,000 and $4,000 for C. A $12,000 shared expense is paid at year-end. Compare policy-count, total-premium, and aggregate-sum-assured bases. Treat the claims and expenses as fixed amounts; no mortality distribution is assumed.

Method. First calculate the asset share before the shared expense for each group. Then repeat the allocation using each driver. This separates the group’s own experience from the effect of the allocation rule.

A’s amount before the shared expense is $(80{,}000+20{,}000)(1.05)-3{,}000-2{,}000=105{,}000-5{,}000=\$100{,}000$. B’s is $(100{,}000+60{,}000)(1.05)-12{,}000-6{,}000=168{,}000-18{,}000=\$150{,}000$. C’s is $(50{,}000+30{,}000)(1.05)-10{,}000-4{,}000=84{,}000-14{,}000=\$70{,}000$. The pre-allocation total is $320,000.

Policy-count basis. Total count is $20+30+10=60$, so the allocations are $12{,}000(20/60)=\$4{,}000$, $12{,}000(30/60)=\$6{,}000$, and $12{,}000(10/60)=\$2{,}000$. Closing aggregate shares are $96,000, $144,000, and $68,000. Per-policy shares are $4,800.00, $4,800.00, and $6,800.00.

Total-premium basis. Total premiums are $20{,}000+60{,}000+30{,}000=\$110{,}000$. The allocations, rounded to cents, are $12{,}000(20/110)=\$2{,}181.82$, $12{,}000(60/110)=\$6{,}545.45$, and $12{,}000(30/110)=\$3{,}272.73$. Closing aggregate shares are $97,818.18, $143,454.55, and $66,727.27. Per-policy shares, rounded to cents, are $4,890.91, $4,781.82, and $6,672.73.

Aggregate-sum-assured basis. Group totals are $20(100{,}000)=\$2{,}000{,}000$, $30(50{,}000)=\$1{,}500{,}000$, and $10(150{,}000)=\$1{,}500{,}000$. The total is $5,000,000, giving allocations of $4,800, $3,600, and $3,600. Closing aggregate shares are $95,200, $146,400, and $66,400. Per-policy shares are $4,760.00, $4,880.00, and $6,640.00.

Check. Each basis allocates $12,000. Each set of closing aggregate shares sums to $308,000, which is $\$320{,}000-\$12{,}000$. The group results vary, but the combined result does not.

Common Mistakes and Practical Checks

An allocation should be explicit enough that another analyst can reproduce it. Before applying it, record the expense amount and period, the groups included, the driver definition, and the data used to measure each group’s driver. Then verify both the group calculations and the total.

Using the wrong driver. If the basis is total premium, use group premium totals. Multiplying or dividing by policy count again changes the stated basis.
Allocating an expense more than once. Deduct the allocated group amounts, not the full shared expense from every group as well.
Omitting a group or part of the pool. The denominator must include all groups that share the stated expense under the allocation rule.
Confusing aggregate and per-policy amounts. Allocate the aggregate expense to groups first; divide by policy count only when calculating a per-policy result.
Ignoring timing. Apply investment accumulation only to cash flows that are invested during the period under the stated convention. The examples place the shared expense at year-end.

The allocation basis should be selected and documented for the purpose of the calculation. A count basis spreads costs equally per policy; a premium basis assigns more cost to groups with larger total premiums; a sum-assured basis assigns more cost to groups with greater aggregate sums assured. None is automatically correct in every setting. Whatever the basis, the allocation should reconcile to the original shared expense and should not alter the portfolio’s combined asset share.

Key takeaway: Calculate allocation weights from the stated group-level driver, allocate the shared expense once, and carry each group’s allocated amount through the asset-share cash flows. The allocations change group and per-policy results, but their sum must equal the common expense.

Check Your Understanding

For each problem, apply the stated allocation basis and the annual timing convention given in the question.

A shared expense of $15,000 is allocated by policy count between Group A with 24 policies and Group B with 36 policies. What expense is allocated to each group? (Answer: A: $6,000; B: $9,000.)
A $24,000 shared expense is allocated by total annual premiums. Group A has $90,000 of premiums and Group B has $150,000. Find each group’s allocation. (Answer: A: $9,000; B: $15,000.)
Group A has 20 policies with sum assured $150,000 each; Group B has 40 policies with sum assured $50,000 each. Allocate $25,000 by aggregate sum assured. (Answer: A: $15,000; B: $10,000.)
Group A starts with $120,000, receives $30,000 premium, and has $8,000 claims and $2,000 direct expenses. Group B starts with $80,000, receives $40,000 premium, and has $10,000 claims and $3,000 direct expenses. The effective annual return is 4%, both premiums are available at the start of the year, and year-end claims and expenses are fixed. Allocate a $12,000 shared expense by counts of 20 policies in A and 40 in B. Find both closing aggregate asset shares and their total. (Answer: A: $142,000; B: $103,800; total: $245,800.)
Before a shared expense, two groups have asset shares of $50,000 and $80,000. They have 20 and 80 policies, respectively, and each has total premiums of $100,000. Compare the group allocations and closing asset shares for a $10,000 expense under count and premium bases. (Answer: Count basis allocations: $2,000 and $8,000; closing shares: $48,000 and $72,000. Premium basis allocations: $5,000 each; closing shares: $45,000 and $75,000.)

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