Payments Made on the Balance Date

Why the Payment Date Needs Two Balance Values

A loan balance is a value at a particular time, but a payment made at that time creates a timing choice. Immediately before the payment is applied, the balance includes the amount due on that date. Immediately after the payment is applied, that amount has been paid and is no longer part of the balance. The two values differ by the payment amount.

This distinction matters whenever a question says only that the balance is “on” a payment date. The date alone does not say whether the payment has already been made. State the timing convention before calculating. Under the prospective method, the pre-payment balance includes the payment occurring on the valuation date among the remaining payments; the post-payment balance does not. Under the retrospective method, the pre-payment balance accumulates payments made strictly before the date; the post-payment balance also subtracts the payment on the date.

Definition: Let $t$ be a payment date, let $C_t$ be the payment due at $t$, and let $B_t^-$ and $B_t^+$ denote the balances immediately before and immediately after that payment, respectively. Assuming the payment is applied in full at $t$, $B_t^- - B_t^+ = C_t$. The superscript minus means “just before,” and the superscript plus means “just after.”

Assume throughout that payments occur at the stated times and that interest accrues between payment dates according to the stated rate. There is no additional interest interval between the instant immediately before a payment and the instant immediately after it. Thus, the balance changes at that instant by the payment amount, not by an extra period of interest.

Prospective Balances at the Payment Date

The prospective method values the payments still required from the valuation date. For payments at dates $s\ge t$, discount each amount to $t$ to find the pre-payment balance. For the post-payment balance, include only payments at dates $s>t$. This convention assumes the stated future payments will be made as scheduled and uses the applicable interest rate to value them.

B_t^-=\sum_{s\ge t} C_s v^{s-t}, \qquad B_t^+=\sum_{s>t} C_s v^{s-t}, \qquad v=\frac{1}{1+i}.

These expressions use an effective annual rate $i$ and annual payment times, so a payment one year after $t$ is discounted by $v$. If the time intervals are fractional, the model must specify how interest applies over those intervals; use the appropriate fractional-period accumulation and discount factors rather than silently treating a fraction of a year as a full year.

Justification: At $t$, the payment $C_t$ is already due and has discount exponent zero, so its value at $t$ is $C_t v^0=C_t$. Every later payment is discounted back to $t$. The pre-payment sum therefore equals the post-payment sum plus $C_t$, giving $B_t^- = B_t^+ + C_t$. This is why the two balances differ by exactly the payment on the date.

Worked Example: Two Payments and the Final Payment Date

A loan is repaid by payments of $450 at time 1 and $700 at time 2. The effective annual interest rate is 8%. Find the original loan amount and the balance at time 2 immediately before and immediately after the payment. Use the prospective method for the two balances.

Method. Discount each scheduled payment to time 0 for the original loan. At time 2, the final payment is due immediately, so it belongs in the pre-payment balance and not in the post-payment balance. This avoids treating a payment due now as if it were a future payment requiring discounting.

The discount factor is $v=1/1.08$. The present value of the first payment is $450/1.08=416.6666667$, and the present value of the second payment is $700/(1.08)^2=600.1371742$. Therefore,

L=\frac{450}{1.08}+\frac{700}{(1.08)^2} =416.6666667+600.1371742 =1{,}016.8038409.

Rounded to the nearest cent, the original loan amount is $1,016.80. At time 2, the only payment remaining immediately before the payment is the $700 due at time 2 itself:

$$ B_2^-=700. $$

Immediately after that payment, no further payments remain:

$$ B_2^+=0. $$

The check is $B_2^- - B_2^+=700-0=700$, exactly the payment at time 2. All monetary answers are rounded to the nearest cent.

Retrospective Balances at the Payment Date

The retrospective method accumulates the original loan to the valuation date and subtracts the accumulated value of payments already made. The key is to define “already made” relative to the instant being valued. Immediately before the date’s payment, only payments at earlier dates have been made. Immediately after, the payment at the valuation date is also subtracted.

B_t^-=L(1+i)^t-\sum_{s<t} C_s(1+i)^{t-s}, \qquad B_t^+=L(1+i)^t-\sum_{s\le t} C_s(1+i)^{t-s}.

Here $L$ is the original loan at time 0, payments occur at annual times, and the effective annual rate $i$ is constant. Each earlier payment is accumulated from its payment date to $t$. The payment at $t$ has accumulation factor $(1+i)^0=1$, so including it in the post-payment calculation reduces the balance by precisely $C_t$. The formulas do not apply unchanged if rates vary over time; in that case, accumulate each cash flow using the rates for the intervals it actually spans.

Worked Example: Using Past Payments to Find Both Balances

A loan of $1,000 is issued at time 0. The effective annual interest rate is 10%. Payments of $300 at time 1, $400 at time 2, and $528 at time 3 are scheduled to repay the loan. Find the balances immediately before and after the time-2 payment, using the retrospective method, and check them prospectively.

Method. For the pre-payment balance, accumulate the loan for two years and subtract only the time-1 payment accumulated for one year. For the post-payment balance, subtract the time-2 payment as well. This avoids counting the time-2 payment as past before it has been applied.

Immediately before the time-2 payment,

B_2^-=1{,}000(1.10)^2-300(1.10) =1{,}210-330 =880.

Immediately after the $400 payment,

$$ B_2^+=880-400=480. $$

Check prospectively. At time 2, the remaining payments immediately before the payment are $400 now and $528 one year later. Their value at time 2 is $400+528/1.10=400+480=880$. Immediately after the time-2 payment, only the time-3 payment remains, with value $528/1.10=480$. Both prospective values agree with the retrospective calculations. The check $880-480=400$ also matches the payment on the valuation date. Monetary answers are exact dollars in this example.

Level-Payment Loans: Keeping the Timing Explicit

For a level-payment loan with annual payment $R$, a constant effective annual rate $i$, and $n$ payments at times $1,2,\ldots,n$, the prospective method is especially direct. At a payment date $k$, the pre-payment balance includes the payment at $k$, while the post-payment balance includes only the $n-k$ later payments. The symbols $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$ are standard annuity factors; here the cash flows can also be written out to make the payment-date boundary explicit.

Worked Example: A Level Loan at an Interior Payment Date

A loan is repaid by four annual payments of $300, with the first payment one year after issue. The effective annual interest rate is 5%. Find the balances at time 2 immediately before and after the second payment. Also verify the pre-payment value retrospectively.

Method. First value the payments remaining from time 2. Include the second payment at time 2 in the pre-payment value, and exclude it from the post-payment value. For the retrospective check, first find the original loan as the present value of all four payments.

With $v=1/1.05$, we have $v=0.9523809524$ and $v^2=0.9070294785$. The pre-payment balance is

$$ B_2^-=300+300v+300v^2 =300(1+0.9523809524+0.9070294785) =857.8231293. $$

The post-payment balance excludes the payment at time 2:

$$ B_2^+=300v+300v^2 =300(0.9523809524+0.9070294785) =557.8231293. $$

To verify retrospectively, the original loan is

L=300(v+v^2+v^3+v^4) =300(3.5459505037) =1{,}063.785151.

At time 2, the accumulated original loan is $L(1.05)^2=1{,}172.823129$. The time-1 payment accumulates to $300(1.05)=315$. Thus the retrospective pre-payment balance is $1{,}172.823129-315=857.823129$, matching the prospective value. Subtracting the time-2 payment gives $557.823129$, matching the post-payment value. Rounded to the nearest cent, the balances are $857.82 before and $557.82 after. Their difference is $300.00, as required.

A Reliable Payment-Date Procedure

When a question asks for a balance “on” a payment date, do not choose a convention silently. A short timeline and a label such as “immediately before” or “immediately after” make the calculation unambiguous.

Mark the valuation instant.
Write down the date and whether the balance is required just before or just after the payment is applied.

Choose a method.
Prospectively, value remaining payments at the date. Retrospectively, accumulate the original loan and subtract payments already made.

Classify the payment on the date.
Include it in remaining payments for the pre-payment prospective balance; include it among past payments for the post-payment retrospective balance.

Check the difference.
Confirm that $B_t^- - B_t^+=C_t$. If both methods are available, compare them using the same timing convention.

The most common error is not an algebra error but a timing inconsistency: including the date’s payment in the prospective calculation while also treating it as already paid in the retrospective calculation. Those choices describe different instants. Another frequent error is to discount the payment due at the valuation date by one period. Its time from the valuation instant is zero, so its value there is the payment itself.

Balance requested	Prospective treatment of payment at $t$	Retrospective treatment of payment at $t$
Immediately before	Include it among remaining payments	Do not subtract it as a past payment
Immediately after	Exclude it from remaining payments	Subtract it as a payment already made

Key takeaway: A payment date has a pre-payment balance and a post-payment balance. Under consistent assumptions, the pre-payment balance exceeds the post-payment balance by exactly the payment made on that date.

Check Your Understanding

For each problem, distinguish the instant being valued before calculating.

A loan has payments of $250 at time 2 and $600 at time 3. The effective annual interest rate is 6%. Find the time-2 balances immediately before and after the payment. (Answer: Before, $816.04; after, $566.04.)
A $1,200 loan is issued at time 0 at an effective annual rate of 5%. A payment of $400 is made at time 1, and a payment of $500 is due at time 2. Using the retrospective method, find the balances at time 2 immediately before and after the payment. (Answer: Before, $903.00; after, $403.00.)
A loan is to be repaid by $700 at time 3 and $400 at time 4. The effective annual rate is 8%. Find the time-3 balances immediately before and after the payment. (Answer: Before, $1,070.37; after, $370.37.)
A loan has four annual payments of $200 at times 1 through 4, at an effective annual rate of 4%. Find both balances at time 2. (Answer: Before, $577.22; after, $377.22.)
A $5,000 loan accumulates at an effective annual rate of 6%. Payments of $1,000 at time 1 and $1,200 at time 2 are made. Find the time-2 balances retrospectively, immediately before and after the second payment. (Answer: Before, $4,558.00; after, $3,358.00.)

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