Reprojecting After Premium Changes

Why a Premium Change Requires a New Projection

A flexible-premium policyholder may change the amount or frequency of future premiums. The account value already accumulated does not reset when that happens. Instead, it becomes the starting value for a revised projection, using the future premium schedule and the applicable charges and credited rates. This makes the change date an important boundary: past cash flows determine the account value at that date, while the new schedule determines the projection after it.

A revised projection can show whether the account is expected to grow, decline, or become insufficient to meet charges. It is not a guarantee of future performance: the result depends on the assumed credited rate, charges, premium timing, and any other policy terms. The examples below use hypothetical policies with stated, fixed assumptions.

Definition: A reprojection after a premium change is a forward projection that begins with the policy account value at the change date and applies the revised future premium schedule, credited rates, and charges. It does not recalculate the account value before the change date.

The Monthly Projection Method

For a simple monthly model, let $A_{m-1}$ be the account value at the start of month $m$, $P_m$ the gross premium paid at the beginning of that month, $L$ the premium load rate, $j$ the effective monthly credited rate, and $C_m$ the charge deducted at the end of the month. The amount added to the account from the premium is $P_m(1-L)$. Under this ordering, the monthly recurrence is

A_m=\bigl(A_{m-1}+P_m(1-L)\bigr)(1+j)-C_m.

This recurrence assumes the premium is credited at the start of the month, the monthly interest is credited after the premium, and the charge is deducted at month-end. A different contractual transaction order requires a different recurrence. If the stated credited rate is annual rather than monthly, convert it to the monthly rate consistent with the contract; do not use an annual rate directly in a monthly calculation.

Key rule: At a change date, carry forward the account value immediately before the changed schedule takes effect. Then specify the premium in each future month, including months with no premium. Keep the load, credited rate, charge, and transaction timing consistent throughout the projection.

When the premium, load, credited rate, and monthly charge remain constant for $n$ months, write $q=P(1-L)$ and $g=1+j$. The recurrence becomes $A_m=gA_{m-1}+gq-C$. Repeated substitution gives a convenient segment formula.

Start with the recurrence.
$A_m=gA_{m-1}+gq-C$, with $A_0$ the account value at the start of the segment.

Substitute repeatedly.
After two months, $A_2=g^2A_0+(gq-C)(1+g)$. Each month adds another term from the same geometric sequence.

Sum the sequence.
For $j\ne 0$, $A_n=g^nA_0+(gq-C)(g^n-1)/(g-1)$. If $j=0$, each month changes the account by $q-C$, so $A_n=A_0+n(q-C)$.

The derivation follows directly by repeated substitution: each period multiplies the earlier account by $g$, while the net addition $gq-C$ is accumulated for the remaining periods. Thus the additions have multipliers $1,g,\ldots,g^{n-1}$, whose sum is $(g^n-1)/(g-1)$ when $g\ne1$. The formula is useful for a constant-schedule segment, but a projection with changing premiums or charges should be split into segments or calculated month by month.

Worked Example: Changing from Monthly to Quarterly Payments

A hypothetical policy has account value $A_0=\$0$. For the next three months, the credited rate is 1% per month, the premium load is 5%, and the month-end charge is $\$10$. Compare two schedules: (i) a gross premium of $\$100$ at the beginning of each month, or (ii) one gross premium of $\$300$ at the beginning of month 1 and no premium in months 2 and 3. Find each month-3 account value. Round final values to cents. Method: apply the monthly recurrence separately to each schedule. This avoids treating a quarterly payment as though it arrived in three equal monthly instalments.

The net premium for each monthly payment in schedule (i) is $100(1-0.05)=\$95$. Therefore:

\begin{aligned} A_1&=(0+95)(1.01)-10=85.95,\\ A_2&=(85.95+95)(1.01)-10=180.95(1.01)-10=172.7595,\\ A_3&=(172.7595+95)(1.01)-10=267.7595(1.01)-10=260.437095. \end{aligned}

The month-3 value under monthly payments is $\$260.44$, rounded to cents.

For schedule (ii), the net premium paid at the beginning of month 1 is $300(1-0.05)=\$285$. There are no further premiums:

\begin{aligned} A_1&=(0+285)(1.01)-10=277.85,\\ A_2&=(277.85+0)(1.01)-10=270.6285,\\ A_3&=(270.6285+0)(1.01)-10=263.334785. \end{aligned}

The month-3 value under the single beginning-of-quarter payment is $\$263.33$. It is higher than the monthly-schedule value because more premium enters the account earlier and earns interest for longer, even though the total gross premiums paid over the three months are equal. The check is that both schedules use the same load, monthly credited rate, and three month-end charges.

Reprojecting When the Premium Amount Changes

A change in premium amount affects only the months in which the revised amount is paid. To reproject, first establish the account value at the change date; then apply the new net premium from its effective payment date. If a charge depends on age, net amount at risk, or another changing policy feature, its value may also need to be recalculated in each month. A projection that holds charges fixed is appropriate only when that is an explicit assumption.

Worked Example: Increasing the Monthly Premium Mid-Projection

At the start of month 1, a hypothetical policy has account value $\$400$. The policyholder pays a gross monthly premium of $\$80$ for months 1 and 2, then changes it to $\$140$ beginning in month 3. The load is 10%, the credited rate is 0.5% per month, and the fixed charge is $\$15$ at each month-end. Calculate the account value at the end of month 6, rounding the final value to cents. Method: apply the recurrence month by month, changing the premium only from month 3. This avoids either applying the new amount retroactively or continuing the old amount after the change.

The net premiums are $80(0.90)=\$72$ in months 1 and 2 and $140(0.90)=\$126$ in months 3 through 6. The successive calculations are:

\begin{aligned} A_1&=(400+72)(1.005)-15=459.36,\\ A_2&=(459.36+72)(1.005)-15=519.0168,\\ A_3&=(519.0168+126)(1.005)-15=633.242884,\\ A_4&=(633.242884+126)(1.005)-15=748.03909842,\\ A_5&=(748.03909842+126)(1.005)-15=863.4092939121,\\ A_6&=(863.4092939121+126)(1.005)-15=979.3563403816605. \end{aligned}

Thus the projected month-6 account value is $\$979.36$. As a check on the timing, the first two calculations use a net premium of $\$72$, and the remaining four use $\$126$; all six months include the stated interest credit and charge.

For comparison, if the $\$80$ gross premium had continued in all six months, the same recurrence would give $A_6=\$760.6417915228$. The difference is $\$979.3563403816605-\$760.6417915228=\$218.7145488588605$, or $\$218.71$ to cents. This difference reflects the larger net premiums in months 3 through 6 and the interest subsequently credited on those amounts.

Premium Frequency, Depletion, and Coverage

Changing payment frequency is not just a change in the total premium over a year. It changes when money enters the account. A quarterly premium paid at the start of a quarter can earn interest during that quarter; three separate monthly premiums enter at three different times. A payment calendar must therefore identify the actual payment months. If no premium is due in a month, set $P_m=0$ for that month rather than spreading a less-frequent payment across months.

To assess depletion under a constant schedule, project until the account can no longer meet the stated charges. With no credited interest and a constant net premium $q$ and charge $C$, the projected month-end value after $n$ months is $A_0+n(q-C)$, provided the recurrence is being used as an unconstrained arithmetic projection. When this value crosses zero, a practical policy projection must apply the contract’s insufficient-value and grace-period provisions. The negative arithmetic result is a signal that the assumed charges cannot all be met; it is not an account balance that can necessarily occur or proof of the exact date coverage terminates.

Worked Example: Estimating Depletion After a Premium Reduction

At a change date, a hypothetical policy has account value $\$250$. The policyholder then pays $\$30$ gross at the beginning of every month. The load is 5%, the credited rate is 0% per month, and the month-end charge is $\$38$. Find the number of complete months for which the projected month-end account remains positive, and identify the first month-end at which the unconstrained projection is nonpositive. Method: calculate the constant monthly net change and check the values around zero. This avoids declaring an exact-zero month when the projection actually crosses zero between month-ends.

The net premium is $30(1-0.05)=\$28.50$, so the account changes by $28.50-38=-\$9.50$ per month. The unconstrained projected value after $n$ months is

$$ A_n=250-9.50n. $$

At month 25, $A_{25}=250-9.50(25)=250-237.50=\$12.50$. At month 26, $A_{26}=250-9.50(26)=250-247=\$3.00$. At month 27, $A_{27}=250-9.50(27)=250-256.50=-\$6.50$. The account remains positive for 26 complete months, and the unconstrained projection first becomes nonpositive at the end of month 27. It does not equal zero at any month-end.

In month 27, the account before the charge is $\$3+\$28.50=\$31.50$, which is $\$6.50$ short of the $\$38$ charge. In practice, the contract’s rules for an insufficient account value and any grace period determine what happens next. This projection identifies the shortfall; it does not by itself determine the coverage termination date.

Common Projection Errors

A reliable reprojection makes the change date, payment dates, and transaction order explicit before calculating. Common errors include:

Restarting from zero. The new projection starts from the account value at the change date, not from the policy’s original issue value.
Applying a changed premium to the wrong months. A new amount affects payments on or after its effective date, not premiums already paid.
Ignoring payment timing. A single quarterly payment is not equivalent to three monthly payments when interest is credited between payment dates.
Using the gross premium as the account deposit. If a load applies, add $P_m(1-L)$, not $P_m$, to the account.
Treating a negative projected value as an actual account balance. It signals an unmet charge under the assumed projection; contract provisions determine the policy consequences.

For example, in the first worked example, an incorrect month-2 calculation under monthly premiums might write $(85.95+95)(1.01)=172.6595$. The multiplication is wrong: $85.95+95=180.95$, and $180.95(1.01)=182.7595$. After the $\$10$ charge, the correct account value is $\$172.7595$, not $\$162.6595$. Carrying full precision between months also prevents rounding differences from accumulating.

Key takeaway: Reproject from the account value at the change date, then apply the revised net premiums in their actual payment months, with the stated crediting and charge order. A projected depletion point is a funding warning; contractual lapse and coverage timing require the applicable policy provisions.

Check Your Understanding

For each hypothetical policy, use the stated monthly transaction order and round final account values to cents where needed.

A policy starts a month with $\$600$. A $\$50$ gross premium is paid at the beginning of the month, the load is 4%, the monthly credited rate is 0.5%, and the month-end charge is $\$8$. What is the ending account value? (Answer: $\$643.24$)
A policy starts with $\$100$, has no credited interest, a 10% load, and a $\$12$ month-end charge. The gross premium is $\$50$ in month 1 and $\$80$ in months 2 and 3. What is the month-3 account value? (Answer: $\$253.00$)
A policy starts with zero account value. It receives a gross premium of $\$200$ at the beginning of month 1 and none in month 2. The load is 6%, the credited rate is 1% per month, and the month-end charge is $\$5$. What is the month-2 account value? (Answer: $\$181.73$)
A policy starts with $\$180$, then receives a $\$20$ gross monthly premium. The load is 5%, the credited rate is 0%, and the monthly charge is $\$25$. How many complete months have a positive projected month-end value, and when does the projection first equal zero? (Answer: 29 complete months; it equals zero at the end of month 30.)
A policy starts with $\$500$. It receives $\$100$ gross monthly for months 1–3, then the gross premium falls to $\$40$ for months 4–6. The load is 5%, the credited rate is 0.5% per month, and the month-end charge is $\$10$. What is the month-6 account value? (Answer: $\$861.78$)

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