Why the Cancellation Price Matters
An equity-linked policy may deduct a regular monetary charge from its unit account by cancelling units. The charge is stated in dollars, but the number of units removed depends on the unit price used for the cancellation. The remaining units then continue to participate in subsequent unit-price movements.
The key discipline is to keep the timing and price labels explicit. A fund value measured at one price cannot automatically be reduced by a charge calculated using a different price. This distinction is essential when a charge is applied between valuation dates or when a question reports both a prior fund value and a cancellation price.
The amount \(C/p_c\) is a number of units, not a dollar amount. The formula assumes the charge is deducted in full, the policy permits fractional units, and no other transaction occurs at the same instant. If the contract specifies a different rounding convention, use that convention. If there are too few units to cover the charge, the formula does not describe the actual deduction in full; the contract must specify how the shortfall is handled.
Deriving the Fund-Value Effect
The fund immediately before cancellation is \(Np_c\), because it consists of \(N\) units each worth \(p_c\). The cancellation removes \(C/p_c\) units. Multiplying the remaining units by the same price gives the immediate post-charge fund:
Thus, at the cancellation price, the fund falls by exactly \(C\) dollars. This conclusion depends on valuing both the pre-charge and post-charge balances at \(p_c\). If the account was previously valued at a different price \(p\), then its previous value \(Np\) is not necessarily its value immediately before cancellation. Likewise, after the charge, a later price movement changes the fund value.
For example, suppose the units are subsequently valued at \(p_v\), with no further transactions or unit changes. The post-charge value at that later price is \((N-C/p_c)p_v\). Relative to holding all \(N\) units at that same later price, the value difference is:
The charge still cancelled \(C/p_c\) units, but the later value of those cancelled units is \(Cp_v/p_c\), which need not equal \(C\). The monetary deduction was \(C\) at the cancellation price; the units no longer held would have risen or fallen with the price after cancellation.
Record the unit balance immediately before the charge and the price specifically used to cancel units.
Divide the charge by the cancellation price: units cancelled equal \(C/p_c\).
Subtract the units cancelled from the pre-charge balance.
Multiply remaining units by the unit price at that date, not automatically by the cancellation price.
Worked Calculations
Worked Example: A Charge Between Two Valuation Dates
A hypothetical policy holds 90 units when the unit price is $10.00. The price rises to $12.50 before a $125 charge is deducted by cancelling units. It later rises to $15.00. Calculate the units cancelled, the immediate post-charge fund at the cancellation price, and the later fund value. Assume no other transactions and no unit rounding.
Method. First move the account to the cancellation date’s price, then cancel units using that price, and finally value the remaining units at the later price.
At the initial price, the fund is \(90\times \$10.00=\$900.00\). Before the charge at the cancellation date, its value is \(90\times \$12.50=\$1,125.00\).
The charge cancels \(\$125/\$12.50=10\) units. The post-charge balance is \(90-10=80\) units. Immediately after the charge, valued at $12.50 per unit, the fund is \(80\times \$12.50=\$1,000.00\), which is also \(\$1,125-\$125=\$1,000\).
At the later price of $15.00, the fund is \(80\times \$15.00=\$1,200.00\). As a check, without the charge the 90 units would then be worth \(90\times \$15.00=\$1,350.00\). The difference is $150.00, equal to the 10 cancelled units valued at $15.00 each. The $150.00 later-value difference is not the $125.00 charge; it reflects the subsequent price increase.
Common mistake avoided: subtracting $125 from the initial $900 fund. That combines a prior valuation with a charge taken later at $12.50 per unit. The account first changes in value to $1,125 at the cancellation date.
Worked Example: Charge Taken Before or After a Price Increase
A hypothetical account has 100 units at $10.00 per unit. The unit price increases by 10% to $11.00 over the period. A $100 charge is taken either at the start of the period or at its end. Compare the resulting end-of-period funds. Assume no other transactions and allow fractional units.
Charge at the start. The cancellation price is $10.00, so \(\$100/\$10.00=10\) units are cancelled. The balance is \(100-10=90\) units. At the end price of $11.00, the fund is \(90\times \$11.00=\$990.00\).
Charge at the end. Before the charge, the 100 units are worth \(100\times \$11.00=\$1,100.00\). The end-date charge cancels \(\$100/\$11.00=100/11\) units, or approximately 9.0909 units. The remaining balance is \(100-100/11=1,000/11\) units. Its end-date value is \((1,000/11)\times \$11.00=\$1,000.00\).
The end fund is $990.00 when the charge is taken at the start and $1,000.00 when it is taken at the end. The $10.00 difference is consistent with the $100 charge taken at the start being followed by a 10% increase: the account has $900 after the initial deduction, and that amount grows to $990.
Check. With an end-date deduction, the $1,100 pre-charge fund falls by exactly $100 to $1,000. With a start-date deduction, the $900 post-charge fund grows by 10% to $990.
Worked Example: Two Charges Across Changing Prices
A hypothetical policy starts with 150 units at $8.00 each. The price first rises by 25% to $10.00, when a $120 charge is deducted. The price then falls by 10% to $9.00, when a $90 charge is deducted. Find the unit balance after each charge and the final fund. Assume no other transactions and no unit rounding.
The initial fund is \(150\times \$8.00=\$1,200.00\). The first price increase gives a cancellation-date price of \(\$8.00\times 1.25=\$10.00\). Immediately before the first charge, the fund is \(150\times \$10.00=\$1,500.00\).
The first charge cancels \(\$120/\$10.00=12\) units, leaving \(150-12=138\) units. The immediate post-charge fund is \(138\times \$10.00=\$1,380.00\).
After the 10% price decline, the price is \(\$10.00\times 0.90=\$9.00\). Before the second charge, the fund is \(138\times \$9.00=\$1,242.00\). The second charge cancels \(\$90/\$9.00=10\) units, leaving \(138-10=128\) units. The final fund is \(128\times \$9.00=\$1,152.00\).
Check. At the final price, the 12 units cancelled at $10.00 would have been worth \(12\times \$9.00=\$108.00\), and the 10 units cancelled at $9.00 would be worth $90.00. The total difference from the no-charge fund of \(150\times \$9.00=\$1,350.00\) is \(\$108+\$90=\$198\). The calculated final fund is $1,152, and \(\$1,350-\$1,152=\$198\).
Timing, Sufficiency, and Exam Traps
When several events occur in sequence, apply them in their stated order. A price movement changes the value of the units currently held; a monetary charge then removes units at its stated cancellation price. Reversing those steps can produce a different unit balance and a different subsequent fund value. For each transaction, record the number of units before it, the price used for it, and the number of units after it.
A full charge can be funded only if \(C/p_c\leq N\), equivalently if \(C\leq Np_c\). If this condition fails, the account does not have enough value at the cancellation price to fund the charge solely by cancelling its existing units. Whether the charge is partly collected, carried forward, paid from another source, or triggers a policy consequence is contract-specific. Do not silently allow the unit balance to become negative.
| Quantity | Meaning | When it is used |
|---|---|---|
| \(N\) | Units immediately before the charge | Unit balance at the transaction instant |
| \(p_c\) | Cancellation price per unit | Converting the monetary charge to units |
| \(C/p_c\) | Units cancelled | Subtracting from the pre-charge unit balance |
| \(Np_c-C\) | Immediate post-charge fund value | Only when valued at the cancellation price |
| \((N-C/p_c)p_v\) | Post-charge value at a later price \(p_v\) | Valuing the remaining units at another date |
A frequent exam error is to write “fund after charge = previous fund minus charge” without checking that the previous fund and charge are measured at the same transaction price. For instance, if \(N=1\), a prior price is $10.00, the cancellation price is $20.00, and the charge is $5.00, the prior fund is $10.00 but the cancellation-date fund before the charge is $20.00. The charge cancels \(5/20=0.25\) units, leaving 0.75 units worth \(0.75\times \$20.00=\$15.00\) at the cancellation price. Subtracting $5 from the prior $10 value would incorrectly give $5. The correct calculation uses the cancellation-date value: \(\$20-\$5=\$15\).
This unit-level approach is useful for checking hypothetical equity-linked policy calculations, where charges reduce the number of units participating in future unit-price movements. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The essential check is always to verify that each charge is converted at the specified price and each fund value is reported at the price for the stated valuation time.
Check Your Understanding
For each problem, assume charges are deducted in full, no other transactions occur unless stated, and fractional units are permitted.
- A policy holds 72 units at a cancellation price of $15.00. A $90 charge is deducted. How many units remain, and what is the immediate fund value at the cancellation price? (Answer: 66 units; $990.00.)
- A policy has 80 units at $12.00 per unit. The price rises to $15.00 before a $75 charge is deducted. What is the immediate fund after the charge, and what is it worth later if the price rises to $18.00? (Answer: 75 units; $1,125.00 immediately after the charge; $1,350.00 later.)
- An account holds 50 units at $20.00. A $100 charge is taken either before or after the price increases to $22.00. Find the end fund under each timing. (Answer: $990.00 if taken before the increase; $1,000.00 if taken after it.)
- A policy starts with 100 units at $10.00. The price falls by 20% to $8.00, a $64 charge is taken, and then the price rises by 25%. Find the final unit balance and fund. (Answer: 92 units; $920.00.)
- An account has 5 units at the $18.00 cancellation price, and the requested charge is $100. Can all of the charge be funded solely by cancelling these units? State the maximum fund available for the charge and the shortfall. (Answer: No; $90.00 is available and the shortfall is $10.00, subject to the contract’s treatment of the shortfall.)