Why Calculation Order Matters
Equity-linked calculations are sensitive to what happens, and when. A premium may be paid before or after a period’s investment return; only part of it may buy units; a dollar charge may cancel units at a particular unit price; and a guarantee may set a floor on the total benefit rather than add another benefit on top of the fund. Reversing one step can change the result even when every individual rate is used correctly.
The focus here is on tracing a hypothetical policy’s mechanics rather than carrying out a full valuation or profit test. In each example, the return is deterministic as stated. There is no stochastic return distribution or mortality model in the calculation: where a death benefit is considered, the calculation is conditional on death occurring at the specified time.
A reliable calculation follows the contract’s sequence: identify the premium payment time, apply the allocation rate, purchase units at the applicable price, apply the return only to amounts invested for that return period, and then apply any scheduled fund or benefit deductions. A surrender deduction is applied to the surrender value as specified by the contract; it is not automatically the same thing as a fund charge.
Deriving the Fund and Unit Calculations
Suppose a premium is invested at the start of a period, the unit price at purchase is \(U_t\), and the allocation rate is \(a\). The premium buys \(aP/U_t\) units. If the unit price then increases by a deterministic rate \(r\), its end-period value is \(U_{t+1}=U_t(1+r)\). With no other flows, the premium’s end-period fund value is therefore
This shows why the return applies to the allocated premium, not the gross premium. If the same premium is instead credited at the end of the period, it buys units at the end-period price and earns no return for that period. The timing assumption changes the calculation, even though the premium and allocation rate are unchanged.
Similarly, if a dollar charge \(C\) is taken from the fund when the unit price is \(U\), then the remaining units are \(N-C/U\). Their value is
The identity verifies that cancelling \(C/U\) units at price \(U\) reduces the fund by exactly \(C\). Cancelling the same number of units using a different price will not remove the intended dollar amount.
Decide whether a premium or benefit is paid at the start, during, or end of the return period.
Apply the allocation rate before calculating units or fund growth.
Apply returns to the amounts exposed to them, then deduct any charges using the stated price and timing.
Apply a benefit floor or surrender deduction according to its definition, and distinguish the policyholder’s benefit from the insurer’s top-up.
Premium Allocation and Unit Cancellation
Worked Example: Allocated Premium and an End-of-Year Fund Charge
A hypothetical one-year unit-linked policy receives a premium of $1,500 at the start of the year. The allocation rate is 92%, the initial unit price is $25, and the unit price earns a deterministic 12% return over the year. At year-end, a $36 administration charge is deducted by cancelling units at the then-current unit price. There are no other cash flows or deductions. Calculate the year-end units and fund value. No mortality assumption is needed because the calculation is for the policy account over the year, not a contingent benefit.
Method. Apply the allocation rate, purchase units at the initial price, update the unit price for the return, and cancel units for the dollar charge at the year-end price.
The allocated premium is \(0.92(1{,}500)=\$1{,}380\). The unallocated part is \(1{,}500-1{,}380=\$120\); it does not buy units. Units purchased at $25 each are
The year-end unit price is \(25(1.12)=\$28\). Before the charge, the fund is \(55.2(28)=\$1{,}545.60\). The charge cancels
Thus the remaining units are \(55.2-1.285714\ldots=53.914285\ldots\), and their value is \(53.914285\ldots(28)=\$1{,}509.60\). As a check, the pre-charge value less the charge is \(1{,}545.60-36=\$1{,}509.60\). Final money is reported to the nearest cent; the unit count is shown to six decimal places.
Common mistake avoided: Investing the entire $1,500 would buy 60 rather than 55.2 units and incorrectly include the unallocated $120 in the fund. Another error is to cancel \(36/25=1.44\) units. At the year-end price of $28, that would remove \(1.44(28)=\$40.32\), not $36.
Return Timing, Benefit Floors, and Top-Ups
A minimum death benefit usually defines the total amount payable, not an additional amount payable in addition to the fund. If the benefit is the greater of the fund \(F\) and a guaranteed amount \(G\), the total benefit is \(\max(F,G)\). The insurer’s top-up above the fund is \(\max(G-F,0)\), so that
The equality holds in both cases. If \(F\geq G\), the top-up is zero and the fund itself is the benefit. If \(F<G\), the top-up is \(G-F\), bringing the total to \(G\). Adding \(G\) to \(F\) would count the fund twice.
Worked Example: Anniversary Premium, Return, and a Death-Benefit Floor
A hypothetical policy has a fund of $5,000 at the start of a year. The fund earns a deterministic 6% return during the year. A $4,000 premium is due at the year-end anniversary, immediately before a death-benefit claim is determined. The allocation rate on that premium is 95%. The death benefit at that time is the greater of the fund and a guaranteed $10,000. There are no other charges. Calculate the fund, total benefit, and insurer top-up. The calculation is conditional on death at the specified time, so no mortality probability is required.
Method. Grow only the opening fund through the year, then credit the allocated part of the anniversary premium. Apply the benefit floor to the resulting fund.
The opening fund grows to \(5{,}000(1.06)=\$5{,}300\). The premium amount allocated to the fund is \(4{,}000(0.95)=\$3{,}800\); the remaining $200 is not credited to the fund. Because the premium is credited after the year’s return, the end-year fund is
The total death benefit is \(\max(9{,}100,10{,}000)=\$10{,}000\). The insurer top-up above the fund is \(10{,}000-9{,}100=\$900\). The check is that the fund plus top-up is \(9{,}100+900=\$10{,}000\), exactly the contractual benefit. Reported to the nearest cent, the fund is $9,100.00, the total benefit is $10,000.00, and the top-up is $900.00.
Common mistake avoided: Applying the return to the anniversary premium would give \((5{,}000+3{,}800)(1.06)=\$9{,}328\), which incorrectly gives the end-year premium a full year of investment return. A second error is to add the fund to the guaranteed benefit and report \(9{,}100+10{,}000=\$19{,}100\). The guarantee is a floor on the total benefit; it is not an extra payment of $10,000.
Surrender Deductions and Sequential Returns
For a surrender deduction stated as a percentage of the fund at surrender, first calculate the fund after all returns and relevant fund transactions, then apply the percentage once. If the deduction rate is \(s\), the surrender payment is \(F(1-s)\). This differs from cancelling units for a fixed dollar fund charge, and it should not be deducted a second time if the stated surrender value is already net of the deduction.
Worked Example: Two Returns and a Percentage Surrender Deduction
A hypothetical policy receives a $3,000 premium at the start of year 1. The allocation rate is 90%, and the initial unit price is $30. The deterministic unit returns are +10% in year 1 and −5% in year 2. The policy is surrendered at the end of year 2, when a 4% deduction is applied to the fund value. There are no other premiums or charges. Calculate the gross fund and net surrender payment. The calculation is deterministic; no mortality assumption or probability distribution is used.
Method. Convert the allocated premium to units, update the unit price sequentially for each year’s return, and apply the surrender deduction to the resulting fund.
The amount allocated is \(3{,}000(0.90)=\$2{,}700\). At $30 per unit, the policy purchases \(2{,}700/30=90\) units. The end of year 1 unit price is \(30(1.10)=\$33\). The end of year 2 price is \(33(0.95)=\$31.35\). Therefore the gross fund at surrender is
The 4% surrender deduction is \(0.04(2{,}821.50)=\$112.86\). The net surrender payment is \(2{,}821.50-112.86=\$2{,}708.64\), equivalently \(2{,}821.50(0.96)=\$2{,}708.64\). The two methods agree, and the final values are exact to the cent.
Common mistake avoided: The returns compound sequentially: the price multiplier is \(1.10(0.95)=1.045\), not \(1+0.10-0.05=1.05\). Also, once the $112.86 deduction has been subtracted to obtain the net surrender payment, subtracting it again would understate the payment.
Exam Checks That Prevent Double Counting
Before accepting an equity-linked benefit calculation, check what each amount represents. A fund value is an asset supporting a benefit; a top-up is the additional amount needed to meet a floor. A surrender deduction reduces the surrender payment; it is not automatically another fund charge. An allocation rate determines how much of a premium enters the fund, while a charge may later reduce that fund. These items have different contractual roles, so a calculation should not combine them merely because they all appear as dollar amounts.
| Item | Correct interpretation | Common error |
|---|---|---|
| Premium allocation | Apply the allocation rate before unit purchase or fund credit. | Invest the gross premium. |
| Dollar fund charge | Cancel units using the price when the charge is deducted. | Use the purchase price or subtract the charge twice. |
| End-period premium | Credit it after the return if the contract specifies end-period timing. | Give it a return for a period before it was invested. |
| Benefit floor | Set the total benefit to the greater of fund and guarantee. | Add the fund and the guarantee. |
| Surrender deduction | Apply it to the specified surrender base once. | Deduct it both from units and again from the payment. |
A useful final check is to write a short cash-flow identity. For a floor benefit, confirm that fund plus insurer top-up equals the total benefit. For a dollar fund charge, confirm that the fund falls by the stated dollar amount. For a surrender deduction, confirm that net payment plus deduction equals the gross fund if that is the contract’s stated basis. These checks often expose errors that are not apparent from the arithmetic alone.
Check Your Understanding
For each hypothetical contract, follow the stated timing and give the requested final value. Round money to the nearest cent and unit counts to six decimal places where needed.
- A $2,400 premium is paid at the start of a year. The allocation rate is 94%, the initial unit price is $16, and the deterministic annual return is 5%. A $20 charge is deducted at year-end by cancelling units at the year-end price. Find the year-end fund value and units remaining. (Answer: fund $2,348.80; 139.809524 units)
- A fund of $7,200 earns a deterministic 4% during the year. A $1,500 premium is credited at year-end with a 96% allocation rate, after the return. The death-benefit floor is $9,500. Find the fund, total benefit, and insurer top-up at that time. (Answer: fund $8,928.00; benefit $9,500.00; top-up $572.00)
- A policy has 200 units at a unit price of $12. The unit returns are −8% and then +10% in two successive years. At the end of year 2, a 3% surrender deduction applies to the fund. Find the net surrender payment. (Answer: $2,355.94)
- An opening fund of $10,000 earns a deterministic 5% annual return. A $2,000 premium has a 90% allocation rate. Compare the year-end fund if the premium is invested at the start of the year with the fund if it is credited at year-end. Assume no other cash flows or charges. (Answer: start-of-year fund $12,390.00; year-end-credit fund $12,300.00; difference $90.00)
- At claim time, a hypothetical policy has a fund of $13,250 and a guaranteed total death benefit of $15,000. State the total benefit and the insurer top-up, and identify the incorrect result from adding the fund to the guarantee. (Answer: total benefit $15,000; top-up $1,750; incorrect double-counted result $28,250)