Purpose and projection conventions
A flexible-premium policy allows premiums to vary in amount and timing, subject to the policy’s terms. To project its account, it is not enough to list the premiums and deductions: the order of transactions matters. A premium credited before a month’s interest earns interest for that month under the convention used here; a premium credited after interest would not.
This tutorial uses a deterministic monthly projection. The policy is assumed to remain in force throughout each example, with no death, withdrawal, or other benefit payment during the projection. No mortality model or probability distribution is applied. Unless an example says otherwise, each stated deduction is taken at the end of the month, after interest is credited.
A premium load is the portion of a gross premium not credited to the account. For example, with a 5% load, 95% of the gross premium enters the account. A recurring deduction may combine several contractual items, such as a cost-of-insurance charge, an expense charge, and a rider charge. In a projection, record the total deduction once if the problem supplies it as a combined amount; do not subtract the same component twice.
The credited rate must match the projection period. If a contract states a monthly rate \(j_t\), use that rate directly for the month. If it instead states an effective annual rate \(i\) and credits monthly at an equivalent effective rate, the corresponding monthly rate is \(j=(1+i)^{1/12}-1\). Do not divide an effective annual rate by 12 unless the problem explicitly specifies that convention.
Why the monthly recurrence works
The recurrence follows from the stated transaction sequence, not from a universal rule about all policies. First add the portion of the premium credited to the account. The resulting balance is \(B_t\). Interest for the month is \(j_tB_t\), so the balance immediately before the deduction is \(B_t+j_tB_t=B_t(1+j_t)\). Finally, subtract the end-of-month deduction \(C_t\). Thus:
This also explains the importance of recording the intermediate balance. If a premium is credited after interest, the interest base changes. If a deduction is taken before interest, it changes the amount that earns interest. A projection must follow the policy’s stated timing; the convention above is an assumption for the examples, not a claim that every contract uses it.
Start with the prior month’s closing value, \(AV_{t-1}\).
Multiply the gross premium by one minus its load proportion, then add that amount to the account.
Multiply the pre-interest balance by \(1+j_t\), using the rate for that month.
Subtract the month-end deduction once, record the closing value, and use it as the next month’s opening value.
For a projection with several months, this process is applied recursively: the closing value in one row becomes the opening value in the next. A table should show enough detail to make the premium load, interest, and deduction independently verifiable.
Worked projections
Worked Example: Irregular Premiums and a Changing Monthly Rate
A hypothetical policy has an opening account value of $1,000. In month 1, the gross premium is $500; in month 3, it is $300; there is no premium in month 2. Every premium has a 5% load. The monthly credited rates are 1.00% in months 1 and 3 and 0.50% in month 2. The month-end deductions are $40, $45, and $50, respectively. Assume no mortality event or other transaction during the three-month projection; the policy remains in force. The projection is deterministic, with no probability distribution. Find each closing account value. Use the premium-before-interest and deduction-after-interest convention.
Method. For each month, calculate the net premium, pre-interest balance, interest, and closing value in that order. The common mistake to avoid is carrying the gross premium into the interest calculation without first applying the load.
| Month | Opening value | Gross premium and net amount | Pre-interest balance | Interest | Deduction | Closing value |
|---|---|---|---|---|---|---|
| 1 | $1,000.00 | $500; net $475 | $1,475.00 | $14.75 | $40 | $1,449.75 |
| 2 | $1,449.75 | $0; net $0 | $1,449.75 | $7.24875 | $45 | $1,411.99875 |
| 3 | $1,411.99875 | $300; net $285 | $1,696.99875 | $16.9699875 | $50 | $1,663.9687375 |
In month 1, the load is \(500(0.05)=25\), leaving a net premium of \(500-25=475\). The pre-interest balance is \(1,000+475=1,475\), and interest is \(1,475(0.01)=14.75\). The closing value is \(1,475+14.75-40=1,449.75\).
In month 2, there is no premium, so the pre-interest balance is $1,449.75. Interest is \(1,449.75(0.005)=7.24875\). After the $45 deduction, the closing value is \(1,449.75+7.24875-45=1,411.99875\).
In month 3, the net premium is \(300(0.95)=285\). The pre-interest balance is \(1,411.99875+285=1,696.99875\). Interest is \(1,696.99875(0.01)=16.9699875\), giving \(1,696.99875+16.9699875-50=1,663.9687375\). The final account value, rounded to cents, is $1,663.97. As a check, each row’s closing value equals its pre-interest balance plus interest less the listed deduction.
Worked Example: Rate Change with a Premium in a Later Month
A hypothetical policy begins with $800. It receives a gross premium of $200 in month 1 and $100 in month 3, with no other premiums during a four-month projection. The premium load is 10% in each month. The monthly credited rate is 0.50% in months 1 and 2, then 1.00% in months 3 and 4. A $30 deduction is taken at the end of each month. Assume the policy remains in force, with no mortality event or other account transaction; no probability distribution is used. Apply premiums before interest and deductions after interest.
Method. Carry the unrounded closing balance into the next row and change the credited rate only when the stated month changes. The common mistake to avoid is applying the new rate to the entire projection instead of only to months 3 and 4.
Month 1: The net premium is \(200(0.90)=180\). The pre-interest balance is \(800+180=980\). Interest is \(980(0.005)=4.90\). The closing value is \(980+4.90-30=954.90\).
Month 2: There is no premium, so the pre-interest balance is $954.90. Interest is \(954.90(0.005)=4.7745\). The closing value is \(954.90+4.7745-30=929.6745\).
Month 3: The net premium is \(100(0.90)=90\), making the pre-interest balance \(929.6745+90=1,019.6745\). At 1.00%, interest is \(1,019.6745(0.01)=10.196745\). The closing value is \(1,019.6745+10.196745-30=999.871245\).
Month 4: There is no premium. Interest at 1.00% is \(999.871245(0.01)=9.99871245\). The closing value is \(999.871245+9.99871245-30=979.86995745\). The final account value, rounded to cents, is $979.87. The rate change is reflected at month 3, while the previous closing balance is preserved as the opening balance for that month.
Worked Example: Changing Both the Premium and the Deduction
A hypothetical policy starts with $2,500. Gross premiums are $400 in month 1, $600 in month 3, and $200 in month 5; there are no other premiums in the five-month projection. Each premium has a 5% load. The monthly credited rate is 0.75% for months 1 through 3 and 0.50% for months 4 and 5. The deduction is $55 in months 1 and 2, then $65 in months 3 through 5. Assume the policy remains in force, with no mortality event or other transaction; the projection is deterministic and uses no probability distribution. Premiums precede interest, and deductions are taken after interest.
Method. Project each month separately, updating the premium and deduction when their amounts change. The common mistake to avoid is treating a premium holiday as a month with no account activity: interest and deductions still apply.
Month 1: Net premium \(=400(0.95)=380\); pre-interest balance \(=2,500+380=2,880\); interest \(=2,880(0.0075)=21.60\); closing value \(=2,880+21.60-55=2,846.60\).
Month 2: There is no premium. Interest \(=2,846.60(0.0075)=21.3495\); closing value \(=2,846.60+21.3495-55=2,812.9495\).
Month 3: Net premium \(=600(0.95)=570\); pre-interest balance \(=2,812.9495+570=3,382.9495\); interest \(=3,382.9495(0.0075)=25.37212125\); closing value \(=3,382.9495+25.37212125-65=3,343.32162125\).
Month 4: There is no premium. Interest \(=3,343.32162125(0.005)=16.71660810625\); closing value \(=3,343.32162125+16.71660810625-65=3,295.03822935625\).
Month 5: Net premium \(=200(0.95)=190\); pre-interest balance \(=3,295.03822935625+190=3,485.03822935625\); interest \(=3,485.03822935625(0.005)=17.42519114678125\); closing value \(=3,485.03822935625+17.42519114678125-65=3,437.46342050303125\). The final account value, rounded to cents, is $3,437.46. In month 2, despite no premium, the balance increased by interest and decreased by the deduction, confirming that a premium holiday does not suspend the other monthly transactions.
Projection checks and limits
A well-documented projection makes it possible to locate a timing or arithmetic error without recomputing every month. For each row, verify the following:
- The net premium equals the gross premium multiplied by one minus the load proportion.
- The pre-interest balance equals the prior closing balance plus the net premium.
- The interest amount uses the credited rate for that particular month and the correct interest base.
- The closing value equals the pre-interest balance plus interest less the deduction.
- The next row’s opening value equals the previous row’s closing value.
Do not silently round intermediate balances to cents if the problem asks for a precise projection. Retaining full precision through the calculation avoids small accumulated differences; round the requested reported result at the end. Also distinguish an account projection from a projection of the death benefit: the account value is the balance produced by these transactions, while the policy’s benefit may depend on separate contract terms.
The recurrence is only valid while the stated transaction sequence can be carried out. If the account cannot support a deduction, simply allowing the recurrence to produce a negative balance is generally not a valid description of policy behavior. Whether charges are limited, a grace period applies, or coverage terminates depends on the contract rules and must be handled using those rules. Do not apply the ordinary recurrence beyond that point without an explicit assumption authorizing it.
Check Your Understanding
Use the premium-before-interest, deduction-after-interest convention in each question. Unless stated otherwise, assume a deterministic projection, no mortality event or other transaction, and that the policy remains in force over the stated months. Round final dollar answers to cents.
- A policy opens with $500. At the start of the month it receives a $100 gross premium with a 5% load. The monthly credited rate is 1.00%, and a $20 deduction is taken at month-end. Find the closing account value. (Answer: $580.95)
- A policy opens with $1,000 and pays no premiums for two months. The monthly credited rate is 0.50% in both months, and the month-end deduction is $25 each month. Find the account value after month 2. (Answer: $959.90)
- A policy opens with $700. It receives a $200 gross premium in month 1 and a $100 gross premium in month 2; both have a 10% load. The monthly credited rates are 1.00% and 0.50%, respectively, and each month has an $18 deduction. Find the month 2 closing value. (Answer: $947.60)
- A policy opens with $1,200. It receives a $300 gross premium in month 2 only, subject to a 4% load. The monthly credited rate is 0.50% in month 1 and 1.00% in months 2 and 3. The deductions are $20 in month 1 and $25 in months 2 and 3. Find the closing value after month 3. (Answer: $1,453.38)
- A policy opens with $2,000. It receives a $500 gross premium in month 1 and a $250 gross premium in month 4, each with an 8% load. The monthly credited rate is 0.75% in months 1 and 2 and 0.40% in months 3 and 4. The deduction is $40 in months 1 through 3 and $60 in month 4. Find the month 4 closing value. (Answer: $2,566.87)