Introduction
An equity-linked policy fund is built from transactions that occur at particular times. Premiums may purchase units, investment performance changes their price, and charges may be met by cancelling units. The order of those events matters: a premium invested at the start of a period participates in that period’s return, while one invested at the end does not.
A reliable projection therefore records both the number of units and the unit price at every event. Multiplying them gives the fund value, but the fund value alone does not show how it will respond to the next return or charge.
Set the Timeline Before Calculating
The contract or projection instructions determine when each transaction occurs. One useful timeline might be: allocate a premium at the start of the period, cancel an immediate charge, apply the period’s investment return, and cancel an end-of-period charge. Another contract could apply the charge after the return. Neither ordering should be assumed without being stated.
For a single fund with period return \(r\), the unit price changes from \(P\) to \(P(1+r)\). If \(r\) is negative, the price falls. Unless the problem says otherwise, treat this as the change in the unit price before any separately specified policy charge. A premium allocation charge is also distinct: if the gross premium is \(G\) and the allocation charge rate is \(c\), the amount used to buy units is \(G(1-c)\).
Write down the opening unit count and price, and calculate the opening fund value if needed.
Add premium-funded units or cancel charge-funded units using the price that applies at that instant.
Change the unit price by the period return. Units already held participate in this price movement.
Apply any remaining premiums or charges at their stated prices, then verify that fund value equals units times price.
Why the Unit Projection Works
The key calculation follows directly from the meaning of a unit price. If a premium allocation of \(A\) dollars buys units at price \(P\), then the number of units purchased must be \(A/P\), because those units have value \((A/P)P=A\). Similarly, cancelling \(C/P\) units removes value \((C/P)P=C\) at that instant.
For example, suppose a period starts with \(N\) units at price \(P\), a net premium allocation \(A\) is made immediately, and the price then earns return \(r\). Before any later charge, the new fund value is
This identity shows why a start-of-period allocation earns the full period return: both the opening fund and the new allocation are present before the price changes. If the premium is instead allocated after the return, the fund value becomes \(NP(1+r)+A\), not \((NP+A)(1+r)\). The difference is \(Ar\), the return on the premium that would have been invested for the period.
These calculations assume a positive unit price, transactions at the stated price, and enough units to meet any charge. If a charge exceeds the available fund, a projection needs the contract’s rules for fund exhaustion; simply cancelling a negative number of units is not a valid continuation.
Worked Example: Two Annual Returns and Charges
A hypothetical policy starts with 20 units priced at $10. At time 0, a gross premium of $500 is subject to a 4% allocation charge, and the remainder buys units. At the end of year 1, the unit price has earned a 10% return; then a $22 charge is deducted by cancelling units. At the start of year 2, a $220 premium is fully allocated to units. The year 2 return is -10%, after which a $19.80 charge is deducted. Find the fund value after each year. Assume all stated transactions occur at the specified time and there are no other charges.
Method. Track units and unit price separately, applying each charge at the price available when the charge is taken.
The allocation charge on the first premium is \(0.04(500)=\$20\). Thus, \(\$500-\$20=\$480\) buys \(480/10=48\) units. Including the initial 20 units gives \(20+48=68\) units at time 0, worth \(68(10)=\$680\).
The year 1 return changes the price to \(10(1.10)=\$11\). Before the charge, the fund is \(68(11)=\$748\). The $22 charge cancels \(22/11=2\) units, leaving \(68-2=66\) units. The year 1 fund value is \(66(11)=\$726\).
At the start of year 2, the $220 premium buys \(220/11=20\) units. The holding becomes \(66+20=86\) units, worth \(86(11)=\$946\) before investment performance. The -10% return changes the price to \(11(0.90)=\$9.90\), so the fund before the charge is \(86(9.90)=\$851.40\). The $19.80 charge cancels \(19.80/9.90=2\) units, leaving 84 units.
The final fund value is \(84(9.90)=\mathbf{\$831.60}\). As a check, the charge reduced the post-return fund by exactly $19.80: \(\$851.40-\$19.80=\$831.60\). The common mistake avoided here is deducting a dollar charge directly from the unit count; a dollar amount must first be divided by the applicable unit price.
Transaction Timing in Practice
A projection should identify each price used for a premium or charge, not merely list the annual return. A start-of-period charge is valued at the start price if that is the specified timing; an end-of-period charge is valued after the period’s return. If a premium and charge occur at the same time, use the contract’s prescribed order. For instance, a charge taken before the premium may leave fewer units than the same charge taken after it if the fund is close to exhaustion.
The following table summarizes the basic operations. Each row assumes the transaction occurs at the displayed price.
| Event | Unit-count change | Price used |
|---|---|---|
| Allocate net premium \(A\) | Add \(A/P\) units | Price when premium is invested |
| Deduct charge \(C\) from units | Subtract \(C/P\) units | Price when charge is taken |
| Apply period return \(r\) | No immediate change in units | New price is \(P(1+r)\) |
Worked Example: A Charge Before Investment Performance
A hypothetical contract begins with 80 units at $10. A $500 premium is paid at the start of year 1, with 10% deducted as an allocation charge. A $50 administration charge is then taken from units, also at the start-of-year price. The unit price earns 10% during year 1. At the start of year 2, a $220 premium is fully allocated at the then-current price. The price then earns -10% during year 2, after which a $19.80 charge is taken. Calculate the fund value after each year.
Method. Apply the start-of-year premium and charge before that year’s return, then apply the year-end charge after the return.
The year 1 premium allocation is \(500(1-0.10)=\$450\), which buys \(450/10=45\) units. There are \(80+45=125\) units before the administration charge. At $10 per unit, the $50 charge cancels \(50/10=5\) units, leaving 120. The 10% return raises the price to \(10(1.10)=\$11\). The year 1 fund value is \(120(11)=\mathbf{\$1,320}\).
At the start of year 2, the $220 premium buys \(220/11=20\) units, bringing the holding to \(120+20=140\). The -10% return lowers the price to \(11(0.90)=\$9.90\). Before the year-end charge, the fund is \(140(9.90)=\$1,386\). The charge cancels \(19.80/9.90=2\) units, leaving 138 units. The year 2 fund value is \(138(9.90)=\mathbf{\$1,366.20}\).
The check is that the end-of-year charge reduces the pre-charge value by exactly $19.80: \(\$1,386-\$19.80=\$1,366.20\). The timing distinction is essential: the $50 charge is converted to units at $10, not at the post-return price of $11.
Worked Example: Premium Invested After the Period Return
A hypothetical policy has 100 units at $10 at the start of year 1. The unit price earns 20% during the year. Immediately after that return, a $600 premium is fully invested, and a $24 charge is then deducted from units at the same price. During year 2, the unit price falls by 25%, after which a charge of $18 is deducted. Find the fund value at the end of each year. Assume no other cash flows.
Method. Apply the year 1 return only to units held during that period; then invest the premium and apply the charge at the resulting price.
The year 1 price becomes \(10(1.20)=\$12\). The original 100 units are worth \(100(12)=\$1,200\). The premium buys \(600/12=50\) units, and the $24 charge cancels \(24/12=2\) units. The final year 1 holding is \(100+50-2=148\) units, so the fund value is \(148(12)=\mathbf{\$1,776}\).
The year 2 price becomes \(12(1-0.25)=\$9\). Before the charge, the 148 units are worth \(148(9)=\$1,332\). The $18 charge cancels \(18/9=2\) units, leaving 146 units. The year 2 fund value is \(146(9)=\mathbf{\$1,314}\). The check is \(1,332-18=1,314\).
A common timing error is to apply the year 1 return to the $600 premium as though it had been invested at the start. That would incorrectly calculate \((\$1,000+\$600)(1.20)-\$24=\$1,896\). The correct year 1 value is \(\$1,200+\$600-\$24=\$1,776\), because the premium enters after the return.
Common Projection Errors
Most incorrect fund projections are not caused by difficult arithmetic; they result from using the right arithmetic at the wrong time. Before finalizing a projection, check the transaction sequence and the price attached to each event.
- Using one balance instead of tracking units and price. Record both, since a charge changes units while investment performance changes price.
- Applying returns to premiums invested later. Only units present during the investment period participate in that period’s return.
- Using the wrong price for a charge. Convert the charge to units using the price when the charge is deducted.
- Ignoring premium allocation deductions. Only the amount allocated to units buys units; the gross premium may be larger.
- Applying a percentage return to a dollar charge as if it were a unit count. A charge stated in dollars must be divided by the unit price to find the units cancelled.
Check Your Understanding
For each problem, state the transaction order and track units and prices before finding the final fund value.
- A policy has 50 units at $8. A $200 premium is fully allocated at the start of the period, the unit price then earns 5%, and a $12 charge is taken afterward. Find the ending fund value. (Answer: $618.00)
- A policy has 120 units at $6. At the start of the period, a $300 premium is subject to a 2% allocation charge, then a $30 charge is deducted at the same price. The unit price subsequently falls by 5%. Find the ending fund value. (Answer: $934.80)
- A policy has 90 units at $4. The unit price rises by 25%, then a $45 charge is deducted. Find the ending unit count and fund value. (Answer: 81 units and $405.00)
- A policy has 200 units at $3. The price rises by 10%, after which a $330 premium is invested and a $33 charge is deducted, both at that new price. Find the ending fund value. (Answer: $957.00)
- A policy starts with 40 units at $10. At time 0, a $200 premium has a 10% allocation charge. The price then rises by 10%, and a $22 charge is deducted at the end of year 1. At the start of year 2, a fully allocated $110 premium is added at the current price. The price then falls by 10%, followed by a $19.80 charge. Find the fund value after year 2. (Answer: $633.60)