Allocating Investment Income Across Policy Funds

Why Investment Income Needs an Allocation Rule

An asset-share projection tracks the amounts attributable to a policy or group of policies over time. When several policy funds or cohorts share an investment pool, the total investment income is known at pool level, but the portion assigned to each asset share depends on an allocation rule. That rule should reflect the stated investment arrangement and cash-flow timing; it should not be chosen after seeing which allocation produces a preferred result.

Two approaches are useful to distinguish. Under fund-specific attribution, each fund is credited with the income earned on its own invested amount. Under pooled allocation, the total income earned by a shared pool is distributed among funds or cohorts using stated weights. A common earned return is a special case: when the same rate applies to every fund and the same investment periods are recognized, applying that rate to each fund's correctly timed amounts gives the same allocation as the pooled calculation.

Definition: Let $A_j$ be the opening asset share for fund or cohort $j$, $I$ the investment income earned by a pooled account over the period, and $w_j$ its nonnegative allocation weight. If $\sum_j w_j>0$, the pooled-income allocation is $$ I_j=I\frac{w_j}{\sum_k w_k}. $$ The weights must be specified by the method—for example, opening balances, fund units, or exposure adjusted for cash-flow timing. The allocated amounts reconcile to the pool because $\sum_j I_j=I$.

The allocation is only as meaningful as its basis. Opening-balance weights are straightforward when balances remain invested for the whole period. If contributions or withdrawals occur during the period, a method based only on opening balances may fail to reflect how long each amount was invested. A unitized fund may instead assign income through changes in unit values; another stated method may use balances adjusted for the dates of cash flows.

Fund-Specific Returns and a Common Earned Return

Suppose each fund has its own investment return. If fund $j$ has opening balance $A_j$ and earns effective annual rate $i_j$ for a full year, its investment income is $A_j i_j$, assuming no within-year cash flows and no other changes to the fund. These amounts need not be proportional to opening balances if the fund returns differ.

A pooled common-rate calculation instead first totals income and balances, then applies one rate to each balance. If total income is $I$ and total opening balance is $A=\sum_j A_j$, the common earned rate is $r=I/A$. Crediting $rA_j$ to each fund preserves total income, but it generally does not preserve the fund-specific income attribution.

Worked Example: Fund Returns Versus a Pooled Common Rate

A hypothetical insurer tracks two policy funds for one year. Fund North opens with $120,000 and earns an effective annual return of 4%. Fund South opens with $80,000 and earns an effective annual return of 10%. There are no contributions, withdrawals, or expenses. Calculate fund-specific income, then compare it with allocating the combined income using one common rate based on opening balances.

Method. First apply each fund's stated return to its own opening balance. Then calculate the pooled common rate and allocate the same total income proportionally to opening balances.

North's fund-specific income is $$ 120{,}000(0.04)=\$4{,}800. $$ South's fund-specific income is $$ 80{,}000(0.10)=\$8{,}000. $$ Thus, pooled income is $I=\$4{,}800+\$8{,}000=\$12{,}800$, and total opening assets are $\$120{,}000+\$80{,}000=\$200{,}000$.

The common earned rate is $$ r=\frac{12{,}800}{200{,}000}=0.064=6.4\%. $$ Using opening balances as allocation weights, North receives $12{,}800(120{,}000/200{,}000)=\$7{,}680$, and South receives $12{,}800(80{,}000/200{,}000)=\$5{,}120$.

The fund-specific ending amounts are $\$120{,}000+\$4{,}800=\$124{,}800$ and $\$80{,}000+\$8{,}000=\$88{,}000$. The pooled common-rate ending amounts are $\$127{,}680$ and $\$85{,}120$. Both methods give total ending assets of $\$212{,}800$, but assign different amounts to each fund.

Check and trap. The pooled allocations sum to $\$7{,}680+\$5{,}120=\$12{,}800$, as required. The common mistake is to assume a common rate reproduces fund-specific attribution merely because total income is preserved. It does not: the common-rate allocation shifts income from the higher-return fund to the lower-return fund.

Why Proportional Allocation Reconciles

A useful allocation rule must preserve the pool total. For nonnegative weights, define each fund's share of total weight as $w_j/\sum_k w_k$. These shares sum to one, so multiplying each by $I$ distributes all and only the income available in the pool.

\sum_j I_j =\sum_j I\frac{w_j}{\sum_k w_k} =I\frac{\sum_j w_j}{\sum_k w_k} =I.

This also explains the connection with a common earned return. Suppose the weights are opening balances, so $w_j=A_j$, and the common rate is defined as $r=I/\sum_k A_k$. Then

I_j=I\frac{A_j}{\sum_k A_k} =\frac{I}{\sum_k A_k}A_j =rA_j.

Thus, proportional allocation by opening balances is exactly equivalent to applying the common earned rate to each opening balance. This equivalence requires the same balances and period to be used in both calculations. It does not establish that opening balances are the right weights when money enters or leaves at different times.

Cash-Flow Timing in a Cohort Allocation

For a common effective annual rate $i$, an amount invested at time $t$, measured in years from the start of the year, accumulates for $1-t$ years. Its end-of-year investment income is the amount multiplied by $(1+i)^{1-t}-1$. This allows cohort income to reflect the actual timing of opening assets and later cash flows.

For a cohort with opening assets $A_j$ and cash flows $C_{jm}$ invested at times $t_{jm}$, where positive $C_{jm}$ denotes a contribution, the income attributed under the common-rate assumption is

I_j=A_j i+\sum_m C_{jm}\left((1+i)^{1-t_{jm}}-1\right).

This expression assumes the opening assets are invested for the full year, each stated contribution is invested from its stated time until year-end, and the same effective annual rate applies throughout. A withdrawal can be represented as a negative cash flow at its withdrawal time. If fund returns vary by fund or over time, this common-rate expression is not the correct attribution unless that common-rate assumption is actually part of the method.

Worked Example: Allocating Cohort Income With a Midyear Contribution

Two hypothetical policy cohorts invest in a shared account for one year at a common effective annual return of 21%. Cohort A opens with $100,000 and contributes another $20,000 exactly halfway through the year. Cohort B opens with $200,000 and has no further cash flows. Assume the annual effective accumulation factor applies consistently over fractional years. Calculate each cohort's income and ending asset share.

Method. Apply the annual rate to opening assets for the full year. For the halfway contribution, use the half-year accumulation factor. Since $(1.21)^{1/2}=1.10$, money invested halfway through the year earns 10% by year-end.

Cohort A earns $\$100{,}000(0.21)=\$21{,}000$ on its opening assets. Its halfway contribution earns $\$20{,}000(1.10-1)=\$2{,}000$. Therefore its total income is $\$21{,}000+\$2{,}000=\$23{,}000$.

Cohort B earns $\$200{,}000(0.21)=\$42{,}000$. The shared account's total income is $\$23{,}000+\$42{,}000=\$65{,}000$. Cohort A's ending assets are $\$100{,}000+\$20{,}000+\$23{,}000=\$143{,}000$, and Cohort B's are $\$200{,}000+\$42{,}000=\$242{,}000$. Total ending assets are $\$385{,}000$, equal to opening assets of $\$300{,}000$, plus the $\$20{,}000$ contribution, plus $\$65{,}000$ of income.

Check and trap. The income allocations reconcile to the pooled amount. Crediting the halfway contribution with a full year's return would incorrectly assign $\$4{,}200$, rather than $\$2{,}000$, to that contribution. Applying the common rate only to opening balances would also miss its investment income.

Choosing and Documenting the Allocation Basis

The calculation should make clear whether income is attributed to separate funds or allocated from one pool. When a pool-level amount is distributed, state both the weights and the timing basis used to construct them. In particular, distinguish these choices:

Opening-balance weights: suitable when the prescribed method allocates in proportion to opening assets and the balances represent the relevant invested base.
Fund-specific income: suitable when each fund's own return and invested amount determine its income credit.
Cash-flow-adjusted cohort weights: suitable when a stated method recognizes when each cohort's assets entered or left the pool.
Fund units or unit values: suitable when the investment arrangement attributes changes in value through the units held by each fund.

If the method uses weights $w_j$, the allocation should be performed using those weights even if another method seems more intuitive. If the method instead states one common earned rate, apply that rate to each eligible amount for its actual investment period. The result from one approach should not be substituted for another without explaining the change in attribution.

A practical reconciliation is to calculate the total income credited to policy funds and compare it with the income available in the pool. A difference indicates an arithmetic error, an omitted fund, or an inconsistent treatment of timing. Reconciliation is necessary but not sufficient: as the first worked example shows, two methods can produce the same aggregate total while assigning different amounts to individual funds.

Key takeaway: A pooled allocation must reconcile to total pool income, but the allocation to each fund depends on the stated basis. A common earned rate matches proportional allocation only when the same balances, cash-flow dates, and investment period are recognized.

These techniques support asset-share projections in which investment experience is assigned across policy funds or cohorts. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the allocation basis and check that the fund credits reconcile to total investment income.

A pooled account earns $18,000. Two cohorts have opening balances of $150,000 and $300,000, and the stated method allocates income in proportion to opening balances. Find each allocation and the common earned rate. (Answer: $6,000 and $12,000; common rate 4.0%.)
Two separate funds earn their own returns for a full year without cash flows. Fund C has $90,000 at 3%; Fund D has $60,000 at 8%. Find fund-specific income, total income, and the common rate that would preserve total income on the combined opening balance. (Answer: Fund C $2,700, Fund D $4,800, total $7,500, common rate 5.0%.)
A pooled account earns a common effective annual return of 10%. Cohort E opens with $80,000 and contributes $20,000 halfway through the year. Cohort F opens with $120,000 and has no other cash flows. Find each cohort's income, using $(1.10)^{1/2}$ for the half-year accumulation factor. (Answer: E earns $8,000 + $20,000(\sqrt{1.10}-1) = $8,976.18; F earns $12,000; total income is $20,976.18.)
A pool earns $24,000 in one year. Opening balances are $240,000 and $160,000; the stated allocation basis is opening balances. Find the allocations, the implied common rate, and the ending amounts if no other cash flows occur. (Answer: allocations $14,400 and $9,600; common rate 6.0%; ending amounts $254,400 and $169,600.)
A fund has $50,000 invested for a full year and receives a $10,000 contribution exactly halfway through the year. The effective annual return is 21%, with $(1.21)^{1/2}=1.10$. Find total income and explain the error in applying 21% to both amounts. (Answer: income is $10,500 + $1,000 = $11,500; applying 21% to the halfway contribution overstates its income by $1,100.)

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