From Premium Paid to Units Purchased
In a hypothetical equity-linked policy, a premium paid by the policyholder need not be invested in full. The contract may deduct a fixed amount, apply a percentage allocation charge, or use different allocation rates for different portions of a premium. The remaining amount is used to purchase units at the applicable unit price. Keeping these steps separate is essential: the premium paid is a cash amount received, while the amount invested is the portion actually used to buy units.
This tutorial focuses on the allocation calculation at the time a premium is applied. It does not assume that the unit price stays unchanged after that time. The key inputs are the premium, the contractual charges and allocation rates, the amount directed to each fund, and each fund's unit price at allocation.
State the Contract's Calculation Order
There is no single universal charge formula for every contract. The contract specifies which charges apply, the base to which each percentage is applied, and the order of operations. Those details determine \(I\). For example, a percentage charge might be calculated on the gross premium, or on the premium remaining after a fixed charge. Those are different instructions and generally produce different invested amounts.
For a simple arrangement with a percentage allocation charge \(c\) applied to the gross premium, the amount invested is \(P(1-c)\). Equivalently, the allocation rate is \(a=1-c\), so the invested amount is \(Pa\). Here \(c\) and \(a\) are proportions, not percentages: a 4% charge means \(c=0.04\), and a 96% allocation rate means \(a=0.96\). Once the invested amount is determined, divide it by the unit price.
If a fixed charge \(F\) is first deducted and a percentage allocation rate \(a\) then applies to the remainder, the invested amount is \((P-F)a\), provided the contract states that order. If different portions of a premium have different allocation rates, calculate the invested amount for each portion separately and add the results. If the premium is split among funds, calculate each fund's share and divide it by that fund's own unit price.
Record the amount paid and identify whether the contract applies each charge to the gross premium or to an amount after another deduction.
Apply fixed charges, percentage charges, allocation rates, and any tiering in the specified order.
If there are multiple funds, apply the stated split to the total amount invested and check that the proportions add to 100%.
Divide each fund's invested amount by its unit price. Multiply the resulting units by that price to verify the amount allocated.
Why the Unit Formula Holds
At the allocation time, the price \(U\) is the cost of one unit. If \(N\) units are purchased, their total value at that price is \(NU\). When the whole invested amount \(I\) is used for the purchase, value conservation gives \(NU=I\). Since a unit price is positive, division by \(U\) gives \(N=I/U\). This reasoning also shows why using the gross premium in place of \(I\) overstates the units whenever a charge is deducted before investment.
The check \(NU=I\) is useful even when the unit count is rounded for reporting. In that case, the rounded units multiplied by the unit price may differ slightly from the invested amount; any such difference should be consistent with the stated rounding precision.
Worked Allocations
Worked Example: A Percentage Charge on the Gross Premium
A hypothetical policyholder pays a single premium of $1,250. The contract deducts a 4% premium allocation charge from the gross premium, and the remainder purchases units in one fund at a unit price of $2.50. Calculate the charge, the invested amount, and the units purchased. Give the unit count to three decimal places.
Method. Apply the percentage charge to the gross premium first, then divide the amount left for investment by the unit price. The charge is
The amount invested is therefore
At $2.50 per unit, the number purchased is
Check. \(480.000(2.50)=\$1{,}200.00\), which matches the invested amount. The policyholder paid $1,250, but only $1,200 bought units. The common mistake avoided here is dividing the gross premium by $2.50; that would give 500 units and incorrectly invest the $50 charge.
Worked Example: A Fixed Charge Before an Allocation Rate
A hypothetical contract receives a premium of $2,000. It first deducts a fixed charge of $20. The contract then applies a 95% allocation rate to the amount remaining. Units are purchased at $3.30 each. Find the invested amount, the total charges, and the unit count. Give units to three decimal places.
Method. Follow the stated order: deduct the fixed charge, then apply the allocation rate to the remainder. The amount after the fixed charge is
The amount invested is
The percentage allocation deduction is \(1{,}980-1{,}881=\$99\), so total charges are
The number of units purchased is
Check. \(570.000(3.30)=\$1{,}881.00\), and \(2{,}000-119=\$1{,}881\). A common mistake is to take 95% of the gross $2,000 and then subtract $20. That would produce $1,880, not $1,881, because the 95% rate in this contract applies after the fixed deduction.
Worked Example: Tiered Rates and a Two-Fund Split
A hypothetical policy receives a premium of $2,000. The first $500 is allocated at 92%, and the remaining $1,500 is allocated at 97%. After applying those tiered rates, the invested amount is split 40% to Fund A and 60% to Fund B. The unit prices are $2.50 for Fund A and $1.50 for Fund B. Calculate the amount invested overall, the invested amount in each fund, and units purchased in each fund. Give unit counts to three decimal places.
Method. Calculate the invested amount for each premium tier before applying the fund split. The first tier contributes
and the second tier contributes
Thus, the total amount invested is
The fund amounts are \(0.40(1{,}915)=\$766\) for Fund A and \(0.60(1{,}915)=\$1{,}149\) for Fund B. They sum to $1,915, as required. The units purchased are
Check. Fund A's units have value \(306.400(2.50)=\$766.00\), and Fund B's have value \(766.000(1.50)=\$1{,}149.00\). Together they equal $1,915. The common mistake avoided is applying one rate to the whole premium: the two tiers have different allocation rates, so their invested amounts must be calculated separately before combining them.
Common Mistakes and Checks
The calculation is short, but a small interpretation error can change both the invested amount and the unit count. In an exam-style calculation, write down the sequence and the base for every percentage before substituting numbers.
- Using the premium paid as the invested amount. If the contract deducts a charge, first find what remains for investment.
- Confusing an allocation rate with a charge rate. A 94% allocation rate means 94% is invested; a 6% charge rate means 6% is deducted. They are complements only when they refer to the same base and there are no other deductions.
- Changing the percentage base. A rate applied to the gross premium is not generally the same as a rate applied after a fixed deduction.
- Using the wrong fund's unit price. For a split investment, calculate units separately using the unit price of each fund.
- Applying a tiered rate to the wrong portion. Identify each tier's amount before applying its rate, then add the invested tier amounts.
A final consistency check is to compare the amounts at each stage: charges plus total invested should reconcile to the premium when the contract's specified deductions are the only deductions, and the fund amounts should add to the total invested. For each fund, units multiplied by that fund's price should recover its allocation. These checks do not replace careful reading of the contract terms, but they often expose an arithmetic or interpretation error.
Check Your Understanding
For each hypothetical policy, follow the stated charge order and use the given unit price. Answers are shown so you can check both the invested amount and the units.
- A premium of $750 is subject to a 6% charge on the gross premium. Units cost $1.50. How much is invested and how many units are purchased? (Answer: $705 invested; 470.000 units.)
- A $1,600 premium has a fixed $16 charge deducted first. A 97% allocation rate then applies to the remainder. Units cost $3.20. Find the invested amount and units. (Answer: $1,536.48 invested; 480.150 units.)
- A $2,400 premium has a 90% allocation rate. The invested amount is split 25% to Fund X at $2.40 per unit and 75% to Fund Y at $1.80 per unit. Find the units in each fund. (Answer: $2,160 invested; 225.000 Fund X units and 900.000 Fund Y units.)
- A $2,000 premium is allocated at 92% for its first $500 and 97% for the remaining $1,500. All invested money buys units priced at $2.50. Find the invested amount and units. (Answer: $1,915 invested; 766.000 units.)
- A $1,000 premium first has a fixed $10 charge deducted, then a 5% charge applied to the remainder. Units cost $1.90. Find total charges and units. (Answer: $59.50 in total charges; $940.50 invested; 495.000 units.)