Separating Fund Deductions from Insurer Expenses

Why the distinction matters

In an equity-linked contract, a charge may be taken directly from the policyholder’s fund. The insurer may also pay expenses, such as administration costs or commission. These are different transactions: a fund deduction reduces the policyholder’s account, while an insurer expense is a cash outflow of the insurer. Confusing them can distort both the projected benefit and the profit test.

The distinction is especially important when a benefit depends on the fund value after charges, or when a guarantee requires the insurer to pay more than the fund can provide. The fund projection determines the account available for benefits. The profit calculation separately records the insurer’s charge income, expenses, and any benefits paid from insurer resources.

Definition: A fund deduction is an amount or number of units removed from a policyholder’s fund under the contract. An insurer expense is a cost paid by the insurer, such as an administration cost or commission. A fund deduction may generate income for the insurer, but it is not itself an insurer expense.

The examples below use hypothetical contracts. Unless stated otherwise, fund charges are remitted to the insurer, expenses are paid by the insurer, and there are no taxes, other cash flows, or additional charges. The timing of each item is stated explicitly; changing that timing can change the result.

Separate the fund account from the profit account

Let $F_t$ be the fund value at the start of a period, $j_t$ the effective investment return during that period, and $C_t$ a monetary charge deducted at the end of the period. If there are no premiums, benefits, or other fund movements, the fund just before the deduction is $F_t(1+j_t)$, and the fund after the deduction is

F_{t+1}=F_t(1+j_t)-C_t.

If the charge is a proportion $c_t$ of the fund just before deduction, then $C_t=c_tF_t(1+j_t)$, so $F_{t+1}=F_t(1+j_t)(1-c_t)$. A fixed amount and a proportional charge can both apply, provided the contract specifies how each is calculated.

The fund recurrence does not include an insurer expense unless the contract explicitly makes that expense a deduction from the policyholder’s account. In the profit account, a charge remitted to the insurer is income, while an expense paid by the insurer is an outflow. In a simple period with no other insurer cash flows, profit is charge income less insurer expenses and any benefits payable from insurer resources.

Worked Example: A Proportional Fund Charge and a Fixed Expense

A hypothetical policy has a fund of $800 at the start of a year. The fund earns an effective annual return of 5%. At year-end, the insurer deducts 1.5% of the fund immediately before charges, plus a fixed $6 charge. The insurer also incurs a separate $18 servicing expense during the year. Assume both charges are remitted to the insurer at year-end, the expense is paid at year-end, and there are no claims or other cash flows. Find the closing fund and the insurer’s profit from these items.

Method. First project the fund to the point just before the deductions. Then calculate each deduction from its stated base. Finally, enter charge income and the expense separately in the profit account.

The fund before charges is

$$ 800(1.05)=840. $$

The percentage charge is 1.5% of $840:

$$ 0.015(840)=12.60. $$

Total fund deductions are $12.60 + $6 = $18.60. Therefore the closing fund is

$$ 840-12.60-6=821.40. $$

The insurer receives $18.60 of charge income and pays an $18.00 expense. Its profit from these items is

$$ 18.60-18.00=0.60. $$

Check. The account falls by $18.60 from its $840 pre-charge value, giving $821.40. The expense is not deducted again from the fund: it reduces insurer profit, not the policyholder’s account under the stated contract. The common mistake is to deduct the expense from the fund as well, which would incorrectly give a closing fund of $803.40.

Why the two calculations are separate

The separation follows from the two different ledgers being measured. To justify the fund recurrence, start with the account value $F_t$. Investment at return $j_t$ changes its value to $F_t+F_tj_t=F_t(1+j_t)$. Removing a monetary charge $C_t$ then leaves $F_t(1+j_t)-C_t$. An insurer expense does not appear in this identity unless the contract directs that the expense be charged to the account.

For the insurer, each remitted fund deduction is a receipt, and each expense payment is an outflow. Thus, when these are the only insurer cash flows in a period, the net cash profit is $C_t-E_t$, where $E_t$ is the insurer expense. If a guarantee or another benefit also requires insurer resources, its cost must be included separately. This accounting distinction prevents one item from being counted twice or placed in the wrong account.

Key result: Project the policy fund using contractual fund movements. Project insurer profit using insurer receipts and payments. A charge can affect both calculations—by reducing the fund and creating insurer income—but an insurer expense affects profit only unless it is explicitly funded by a contractual deduction.

Period profits and discounted value

In a profit test, period profits are assigned to their payment times. If $P_t$ denotes profit paid at the end of year $t$, and the effective annual discount rate is $i$, the present value of profits over $n$ years is

\operatorname{NPV}=\sum_{t=1}^{n}\frac{P_t}{(1+i)^t}.

This formula discounts the end-of-year profit in year $t$ by $t$ years. A fund charge can contribute to $P_t$ as insurer income, while expenses and insurer-funded benefits reduce it. The fund value itself is not automatically a profit cash flow: its treatment depends on the contract and on which benefits or releases are included in the profit test.

Worked Example: Fund Projection and Two-Year Profit NPV

A hypothetical policy starts with a $1,000 fund. It earns 5% at the end of each year. A fund charge of $28 is deducted at the end of year 1, and $24.34 is deducted at the end of year 2. The insurer incurs expenses of $8 at the end of year 1 and $10 at the end of year 2. There are no lapses, claims, or other cash flows. The effective annual discount rate is 4%. Calculate each year’s closing fund and profit, then calculate the NPV of profit.

Method. Project the fund with the 5% investment return, then subtract the charge due at that year-end. Calculate profit as charge income less expense. Discount each resulting end-of-year profit for its own time period.

At the end of year 1, the fund before the charge is

$$ 1{,}000(1.05)=1{,}050. $$

After the $28 deduction, the year-1 closing fund is $1{,}050-28=\$1{,}022$. The insurer’s year-1 profit is $28-8=\$20$.

During year 2, the fund before the second charge is

$$ 1{,}022(1.05)=1{,}073.10. $$

After the $24.34 deduction, the year-2 closing fund is $1{,}073.10-24.34=\$1{,}048.76$. The insurer’s year-2 profit is $24.34-10=\$14.34$.

The NPV at 4% is

\begin{aligned} \operatorname{NPV} &=\frac{20}{1.04}+\frac{14.34}{(1.04)^2}\\ &=19.230769\ldots+\frac{14.34}{1.0816}\\ &=19.230769\ldots+13.258136\ldots\\ &=32.488905\ldots \end{aligned}

Rounded to the nearest cent, the NPV is $32.49. The intermediate fund values reconcile to the stated charges, and the year-2 profit is $14.34 before discounting. The common mistake is to discount the year-2 profit by one year instead of two, or to treat the fund charges as expenses rather than income.

Fund deductions and a guaranteed benefit

The order of events matters when a benefit is compared with the fund. If a contractual charge is deducted before the benefit is measured, the relevant fund value is the post-charge amount. A minimum benefit may then create a payment from insurer resources, equal to the excess of the guaranteed amount over the available fund when that excess is positive.

Worked Example: A Post-Charge Fund and a Death Guarantee

A hypothetical one-year policy has a $500 fund at the start of the year. The fund earns an effective annual return of 4%. At year-end, an $8 charge is deducted before any death claim is settled. If death occurs during the year, the beneficiary receives the greater of $518 and the post-charge fund. The probability of death during the year is 0.03; otherwise, the policy remains in force and no benefit is paid at year-end. The insurer incurs a $5 servicing expense for each policy in force at the start of the year, paid at year-end regardless of whether death occurs. Assume the charge is collected for every policy and calculate expected insurer profit per policy.

Method. Determine the post-charge fund first. Compare that value with the guaranteed death benefit to find the insurer-funded top-up. Then take expected guarantee cost using the death probability and subtract both that expected cost and the expense from charge income.

The fund before the charge is

$$ 500(1.04)=520. $$

After the $8 charge, the fund is $520-8=\$512$. If death occurs, the total benefit is $518, so the top-up paid from insurer resources is

\max(518-512,0)=6.

The expected guarantee cost is $0.03(6)=\$0.18$ per policy. Charge income is $8 and the expense is $5 per policy, so expected profit is

$$ 8-5-0.18=2.82. $$

The expected insurer profit is $2.82 per policy. In the death state, the fund supplies $512 of the $518 benefit and the insurer supplies the $6 top-up. In the survival state, the fund remains $512. The common mistake is to compare the guarantee with the $520 pre-charge fund; the contract specifies that the charge is taken first, so the top-up is $6, not zero.

Common errors to avoid

Putting an insurer expense into the fund recurrence. Do this only if the policy terms explicitly deduct that cost from the account. Otherwise, it is a profit-test outflow, not a fund movement.
Treating a fund deduction as an insurer expense. When the deduction is remitted to the insurer, it is a receipt in the profit account, even though it reduces the policyholder’s fund.
Using the wrong charge base. A percentage charge on the pre-charge fund is calculated before subtracting that charge or any other end-of-period deduction, unless the contract specifies another base.
Comparing a guarantee with the wrong fund value. Apply contractual deductions in their specified order before identifying any insurer-funded top-up.
Discounting a profit for the wrong number of years. An end-of-year-$t$ profit is divided by $(1+i)^t$, not by $(1+i)$ for every period.

These distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: the calculation must respect the contract’s cash-flow timing and keep policy account movements distinct from insurer receipts and payments.

Check Your Understanding

For each problem, keep the fund projection and insurer profit calculation separate. Unless stated otherwise, assume there are no other cash flows.

A fund of $600 earns 3% for one year. At year-end, a 2% charge is taken from the fund just before a fixed $4 deduction; the percentage is applied to the fund immediately before either deduction. The insurer incurs a separate $9 expense. Find the closing fund and profit from these items. (Answer: Closing fund $601.64; profit $7.36.)
At the ends of years 1 and 2, a hypothetical policy generates fund-charge income of $15 and $17, respectively, and insurer expenses of $7 and $11. Profits are paid at each year-end, and the effective annual discount rate is 5%. Find the two-year NPV, rounded to the nearest cent. (Answer: $13.06.)
A fund of $900 earns 4% for one year. A charge of 1% of the pre-charge fund is deducted at year-end. If death occurs, the benefit is $940; the death probability is 0.02. The insurer pays a $4 servicing expense per policy at year-end and collects the charge regardless of death. Find expected insurer profit per policy, including any guarantee top-up. (Answer: $5.09.)
A fund of $1,200 earns a return of $-10\%$ during the year. A $12 charge is deducted at year-end. The insurer separately pays $20 of expenses, and there are no benefits or other receipts. Find the closing fund and insurer profit. (Answer: Closing fund $1,068; profit $-\$8$.)
A policy has 100 units at the start of a year, with a unit price of $10. The unit price rises by 6% during the year. At year-end, the insurer deducts 5 units at the then-current unit price, and incurs a separate $40 expense. There are no other cash flows. Find the remaining fund value and the insurer’s profit from the charge and expense. (Answer: Remaining fund value $1,007; profit $13.)

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