Loan Payoff Between Scheduled Payment Dates

Why the Payoff Date Matters

A loan balance stated on a scheduled payment date is not automatically the amount needed to settle the loan on a different date. Interest may accrue between those dates. If the payoff date coincides with an installment date, the amount also depends on whether the installment has been paid or is still due. A correct calculation therefore begins with the balance’s precise timing and the lender’s stated accrual convention.

The method here applies to a loan with scheduled installments and a known balance immediately after a scheduled payment. It calculates the amount needed to settle the loan at a later date, before any fees or penalties unless those are separately specified. The key is to distinguish a balance that has already had a payment subtracted from one measured just before that payment.

Definition: Let $B_k^+$ be the outstanding balance immediately after the $k$-th scheduled installment. A payoff between payment dates is found by accumulating $B_k^+$ from that date to the payoff date under the stated interest-accrual convention, provided no intervening payment has been made. If an installment is due at the payoff date, distinguish the balance immediately before that installment, $B_{k+1}^-$, from the balance immediately after it, $B_{k+1}^+$.

Accruing a Balance Over Part of a Payment Period

Let $j$ denote the effective interest rate for one scheduled payment period, and let $u$ be the fraction of that period from the balance date to the payoff date. The value of $u$ is determined by the contract’s day-count convention. For example, if a monthly period is treated as 30 days and 12 days have elapsed, then $u=12/30$. The payoff formula depends on how interest accrues over that fraction.

Under a simple prorated convention, interest is proportional to the elapsed fraction. Under a fractional-compound convention, the accumulation factor for a fraction of a period is defined as a fractional power. These are different conventions and can produce slightly different payoffs. Use the convention stated in the problem or loan terms; do not choose one merely because it is familiar.

Payoff between scheduled dates: If no installment is due or paid between the balance date and payoff date, then $$ \text{Payoff}=B_k^+(1+ju) $$ under simple prorated accrual, and $$ \text{Payoff}=B_k^+(1+j)^u $$ under fractional-compound accrual. Here $u$ is the elapsed fraction of a payment period. These formulas assume no additional fees, penalties, or payments; add such amounts only if the terms expressly require them.

If the contract instead specifies a constant effective annual rate $i$ and fractional-year compounding, the corresponding accumulation over $t$ years is $(1+i)^t$. A quoted annual rate does not by itself establish the partial-period method: confirm whether the contract calls for fractional compounding, daily simple accrual, or another specified convention.

Why the Accumulation Formula Holds

The formula follows from the definition of an accumulation factor. Over one full payment period, a balance $B$ grows to $B(1+j)$. Under fractional compounding, a fraction $u$ of a period has accumulation factor $(1+j)^u$, so the balance becomes $B(1+j)^u$. Under simple prorating, the full-period interest $Bj$ is multiplied by the elapsed fraction $u$, giving interest $Bju$ and an accumulated balance $B+Bju=B(1+ju)$.

The same timing logic explains how to handle an installment due on the payoff date. If the scheduled installment is $R_{k+1}$, the balance immediately before it is the balance accumulated since the previous payment. After the installment is paid, that amount is reduced by $R_{k+1}$:

B_{k+1}^- = B_k^+(1+j), \qquad B_{k+1}^+ = B_{k+1}^- - R_{k+1}.

Rearranging the second equation gives $B_{k+1}^-=B_{k+1}^++R_{k+1}$. Thus, if a ledger gives the balance after the due installment but that installment has not actually been paid, add the installment to recover the amount due immediately before it. Do not add the installment to a balance that is already measured before payment.

Locate the starting balance.
Confirm the date of the balance and whether it is immediately before or after the scheduled installment on that date.

Determine elapsed time.
Convert the number of elapsed days or other time units to a fraction of a payment period using the stated day-count convention.

Apply the specified accrual method.
Use simple prorating or fractional compounding as stated. If an installment has been paid during the interval, update the balance at that date before accruing further interest.

Check for an installment due.
If none is due at the payoff date, do not add a future installment. If one is due, identify whether the amount already calculated is before or after that installment.

Reconcile the result.
Check that the payoff equals the balance under the stated timing convention, with any due installment included exactly once.

Worked Payoff Calculations

Worked Example: Simple Accrual Between Monthly Dates

A hypothetical loan has a balance of $20,000 immediately after a scheduled payment. The effective rate per monthly period is $j=0.006$. The contract uses simple prorated accrual on a 30-day monthly basis. A borrower requests a payoff 17 days later, before the next installment is due. Find the payoff amount.

Method. Since no installment is due or paid during the 17 days, accrue the $20,000 balance for $u=17/30$ of a period using simple prorating.

First calculate the elapsed fraction: $u=17/30=0.566666\ldots$. The interest is $20{,}000(0.006)(17/30)=68$. Therefore, the payoff is $20{,}000+68=\$20{,}068.00$.

Check. The full-period interest would be $20{,}000(0.006)=\$120$. Seventeen-thirtieths of $120 is $68, consistent with the calculated accrued interest. The final answer is rounded to the nearest cent.

Common mistake avoided. Adding the next scheduled installment would be incorrect: it is not yet due, and the requested payoff is the accrued balance on the stated date.

Worked Example: Fractional Compounding Within a Monthly Period

A hypothetical loan has a balance of $48,000 immediately after a payment. The effective rate per monthly period is $j=0.005$. The loan terms specify fractional compounding within each 30-day period. The payoff date is 12 days after the balance date and before the next installment. Calculate the payoff to the nearest cent.

Method. The elapsed fraction is $u=12/30=0.4$. Use fractional compounding, not simple prorating: $48{,}000(1.005)^{0.4}$.

The accumulation factor is $(1.005)^{0.4}\approx1.001997008$. The accrued interest is therefore $48{,}000(1.001997008-1)\approx \$95.8564$. Adding that interest to the starting balance gives $48{,}000+\$95.8564=\$48{,}095.8564$, or $48,095.86 to the nearest cent.

Check. The factor is greater than 1, so the payoff exceeds the starting balance. The accrued interest is also less than the full-period interest of $48{,}000(0.005)=\$240$, as expected for less than half of a period.

Common mistake avoided. Using $1+0.005(0.4)$ would apply simple prorating, a different convention. The problem specifies fractional compounding, so the fractional power is required.

Worked Example: A Scheduled Installment Is Due at Payoff

Immediately after a hypothetical loan’s scheduled payment, its balance is $12,500. The effective rate per monthly period is $j=0.008$. At the next scheduled payment date, the regular installment of $900 is due but has not been paid. Find the payoff required immediately before that installment. Also find the balance that would remain if the borrower paid the installment first.

Method. The date is one full period after the stated balance date. Accrue the balance for the full period to find the amount immediately before the due installment: $12{,}500(1.008)=\$12{,}600$.

If the installment is paid first, subtract it once: $12{,}600-900=\$11{,}700$. So the two consistent settlement paths are: pay $12,600 to settle immediately before the installment, or pay the $900 installment and then $11,700 to settle the remaining balance.

Check. The total paid under the second path is $900+11{,}700=\$12{,}600$, the same as settling immediately before the installment. The final answers are exact to the cent.

Common mistake avoided. The $12,600 is already the balance before the installment. Adding $900 to it would count the installment twice. Conversely, if starting with the after-installment balance of $11,700 while the installment remains unpaid, add $900 to recover the $12,600 due before payment.

Common Timing and Calculation Traps

The most important exam check is to attach a timing label to every balance: before payment, after payment, or between payments. A number without that label can be used in the wrong recurrence even when the arithmetic is correct. In particular, the amount owed immediately before a scheduled installment is not the same as the remaining balance after the installment.

Using the wrong partial-period convention. Simple prorating gives $B(1+ju)$; fractional compounding gives $B(1+j)^u$. Follow the specified convention.
Mixing time units. If $j$ is a monthly rate, express elapsed time as a fraction of a month under the stated day-count basis. Do not insert a number of days directly as $u$.
Adding a future payment. A scheduled installment that is not yet due is not part of the payoff solely because it will be due later.
Double counting a due installment. Add it only when converting an after-installment balance to an unpaid, before-installment amount. Do not add it to a balance already calculated before payment.
Assuming the balance date is the payoff date. Interest accrued after the stated balance date must be included if the payoff occurs later and the contract requires accrual.

For a quote on a date when a payment is due, identify the exact settlement sequence. If the borrower pays the installment and then settles, the two cash amounts are the installment plus the after-installment balance. If the borrower settles immediately before making that installment, the payoff is the before-installment balance. These totals agree under the standard balance recurrence when no additional charges or timing adjustments apply.

Key takeaway: Start from a balance with a clearly identified date and payment status, accrue it using the specified convention, and include a due installment exactly once. A payoff between payment dates is not automatically the next scheduled payment plus the balance; the payment’s due status and the balance’s timing determine the calculation.

Check Your Understanding

For each problem, use the stated accrual convention and treat fees as zero unless the problem specifies otherwise.

A loan balance is $16,400 immediately after a payment. The effective rate per monthly period is $j=0.0075$, with simple prorated accrual on a 30-day basis. The payoff is requested 8 days later, before the next installment is due. Find the payoff. (Answer: $16,432.80)
A loan balance is $27,500 immediately after a payment. The effective rate per monthly period is $j=0.006$. The contract uses simple prorating over 30 days. A payoff is requested 22 days after the balance date; the next installment is due 8 days later. What amount is due on the payoff date? (Answer: $27,621.00)
A balance of $31,000 is stated immediately after a scheduled payment. The effective rate per monthly period is $j=0.004$, and the contract specifies fractional compounding within a 30-day period. Find the payoff 15 days later, before the next installment, to the nearest cent. (Answer: $31,061.94)
Immediately after a payment, a loan balance is $8,500. The effective rate per payment period is $j=0.009$. At the next payment date, the scheduled installment of $700 is due and unpaid. Find the payoff immediately before the installment and the remaining balance if the installment is paid first. (Answer: Payoff $8,576.50; after-installment balance $7,876.50)
A balance of $42,000 is stated immediately after a monthly payment. The effective rate per period is $j=0.005$, with simple prorated accrual on a 30-day basis. The payoff is requested 18 days later; the next $1,300 installment is due 12 days after the payoff date. Find the amount due at payoff and explain whether to include the installment. (Answer: $42,126.00; do not include the installment because it is not yet due)

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