Why Extra Principal Payments Matter
A scheduled loan payment covers the interest accrued for the period and reduces principal by the amount left over. A borrower may also pay extra principal. That extra amount reduces the balance on which future interest is calculated, but its effect on the scheduled payment depends on the loan terms: the scheduled payment may stay fixed, or the lender may recalculate it over the remaining term.
To analyze the change, be precise about when the extra payment is applied. In this tutorial, payments occur at the end of each interest period. Interest accrues first; the scheduled payment is then made; and any extra principal payment is applied immediately afterward on the same date. There are no fees, prepayment penalties, or changes to the interest rate. Calculations retain full precision until final dollar amounts are rounded to cents.
When there is no extra payment, set \(E_k=0\). A positive \(E_k\) reduces principal immediately; it is not a credit toward the next scheduled installment unless the loan agreement explicitly treats it that way. The recurrence assumes the extra amount is applied to principal on the stated date and that interest is not charged on it after that date.
How the Extra Payment Changes the Balance
Suppose a borrower makes an extra principal payment after a scheduled payment. The scheduled payment has already been applied, so first calculate the balance that would remain under the ordinary schedule. Then subtract the extra principal. This order prevents an extra payment from being mistaken for a substitute for the scheduled installment.
Multiply the opening balance by \(j\). The scheduled payment is not subtracted before the period’s interest is calculated under the stated end-of-period convention.
Subtract the scheduled payment from the balance plus accrued interest.
Subtract the extra payment from the balance remaining after the scheduled installment.
Calculate the next period’s interest using the balance after both payments.
This process explains why an extra principal payment reduces future interest: future interest is calculated on a smaller balance. It does not, by itself, determine whether the next scheduled payment changes. That depends on whether the loan continues with the original payment or is recast.
Worked Example: Keep the Scheduled Payment and Pay Off Earlier
A loan has a balance of $1,000 immediately before a period begins. The effective rate is 10% per period, and the scheduled payment is $400 at each period end. After the first scheduled payment, the borrower pays an additional $100 of principal. The scheduled payment remains $400. Find the balances, total interest, final payment, and interest saved compared with making no extra payment.
Method. Apply the recurrence period by period. Since the scheduled payment stays fixed, continue it until the remaining balance, including that period’s interest, is less than $400.
Period 1. Interest is \(1{,}000(0.10)=\$100\). The scheduled payment leaves a balance of \(1{,}000+100-400=\$700\). The extra payment then reduces it to \(700-100=\$600\).
Period 2. Interest is \(600(0.10)=\$60\). After the $400 scheduled payment, the balance is \(600+60-400=\$260\).
Period 3. Interest is \(260(0.10)=\$26\). The amount due is \(260+26=\$286\), so the borrower pays $286 to close the loan rather than making another $400 payment.
Total interest with the extra payment is \(100+60+26=\$186\). To compare, without the extra payment the balances after the first and second $400 payments would be \(1{,}000+100-400=\$700\), then \(700+70-400=\$370\). The third period’s interest would be \(370(0.10)=\$37\), and the final payment would be \(370+37=\$407\). Total interest without the extra payment is \(100+70+37=\$207\), so interest saved is \(207-186=\$21\).
Check. With the extra payment, total cash paid is \(400+100+400+286=\$1{,}186\). This equals the original $1,000 principal plus $186 of interest. The common mistake is to pay the full $400 in period 3: the amount due is only $286, so charging another $114 would overpay the balance.
Recasting the Scheduled Payment
A lender may instead keep the original payoff date and recalculate the scheduled payment after an extra principal payment. If the balance immediately after the extra payment is \(B\), the effective rate per remaining payment period is \(j\), and \(n\) payments remain, equal end-of-period payments \(R\) satisfy a present-value equation. Define \(a_{\overline{n}|}\) at rate \(j\) as the present value of \(n\) payments of one at the ends of successive periods.
The reason is that the balance on the recast date must equal the present value of the remaining payments: each future installment is discounted by the number of periods until it is paid. Thus \(B=R\sum_{k=1}^{n}(1+j)^{-k}=R\,a_{\overline{n}|}\). Dividing both sides by \(a_{\overline{n}|}\) gives the recast formula. If the lender does not recast, the formula does not describe the borrower’s new scheduled payment; the original installment continues instead.
Worked Example: Recalculate Two Remaining Installments
Immediately after a scheduled payment, a loan balance is $1,100. Two end-of-period payments remain, and the effective rate is 10% per period. The borrower pays $100 of extra principal immediately. The lender recasts the loan so the remaining two installments are equal. Find the old scheduled payment, the recast installment, and total interest from the date just before the extra payment through repayment.
Method. First find the balance after extra principal: \(1{,}100-100=\$1{,}000\). Use the present-value formula separately for the original and reduced balances. The extra payment is immediate, so it is not discounted.
The two-period annuity factor is \(a_{\overline{2}|}=1/1.10+1/1.10^2=10/11+100/121=210/121\). The old scheduled installment, based on $1,100, is \(1{,}100/(210/121)=\$633.333\ldots\), or $633.33 to cents.
After the extra payment, the recast installment is \(1{,}000/(210/121)=\$576.190476\ldots\), or $576.19 to cents. The installment is smaller because the lender is spreading the lower balance over the same two remaining periods.
To check the recast, after the first new installment the balance is \(1{,}000(1.10)-576.190476\ldots=\$523.809524\ldots\). Next-period interest is \(523.809524\ldots(0.10)=\$52.380952\ldots\). Adding that interest to the remaining balance gives \(523.809524\ldots+52.380952\ldots=\$576.190476\ldots\), exactly the final installment before rounding.
From the date before the extra payment, total cash paid is \(100+2(576.190476\ldots)=\$1{,}252.380952\ldots\). Subtracting the original $1,100 balance gives total interest of $152.380952..., or $152.38 to cents. A common error is to compare only the new installments with the old installments and omit the $100 extra payment when reconciling total cash paid.
Justifying the Interest Effect
The recurrence also gives a direct way to see the interest effect over the next period. If the same scheduled payment is made in two otherwise identical schedules, but one schedule has an extra principal payment \(E\) at the end of the current period, its next opening balance is lower by \(E\). The next period’s interest difference is therefore \(jE\). For example, reducing a balance by $100 when \(j=5\%\) reduces the following period’s interest by \(0.05(100)=\$5\).
Further interest savings depend on the future balances and payments. If the regular payment stays fixed, the lower balance changes how much of later payments goes to principal, and may produce an earlier final payment. If payments are recast, the installment changes instead. Therefore, multiplying the extra payment by the rate and the number of remaining periods is generally not a valid total-interest-savings calculation. The complete schedule, or an appropriate present-value comparison under the loan terms, is needed.
Worked Example: Update a Schedule After a Later Extra Payment
A loan has a $2,000 opening balance, an effective rate of 5% per period, and scheduled payments of $500 at each period end. After the second scheduled payment, an additional $100 of principal is paid. The scheduled payment stays $500 until the final payoff. Calculate the schedule, total interest, and interest saved relative to making no extra payment.
Method. Track interest on each opening balance, then subtract the scheduled payment and, in period 2 only, the extra payment.
Period 1: interest is \(2{,}000(0.05)=\$100\); the balance after the scheduled payment is \(2{,}000+100-500=\$1{,}600\).
Period 2: interest is \(1{,}600(0.05)=\$80\); after the scheduled payment, the balance is \(1{,}600+80-500=\$1{,}180\); after the extra $100 payment, it is $1,080.
Period 3: interest is \(1{,}080(0.05)=\$54\); after the $500 payment, the balance is $634. Period 4: interest is \(634(0.05)=\$31.70\); after the $500 payment, the balance is $165.70. Period 5: interest is \(165.70(0.05)=\$8.285\). The final amount due is \(165.70+8.285=\$173.985\), or $173.99 to cents.
Total interest before rounding is \(100+80+54+31.70+8.285=\$273.985\), or $273.99 to cents. Without the extra payment, the balances after periods 1 through 4 would be $1,600, $1,180, $739, and $275.95. The corresponding interest amounts are $100, $80, $59, $36.95, and \(275.95(0.05)=\$13.7975\). Total interest without the extra payment is $289.7475. Interest saved is \(289.7475-273.985=\$15.7625\), or $15.76 to cents.
Check. With the extra payment, total cash paid is \(4(500)+100+173.985=\$2{,}273.985\), equal to $2,000 principal plus $273.985 interest. The error to avoid is applying the extra $100 before period 2 interest: it is made after the second scheduled payment, so it first affects period 3 interest.
Common Traps and Schedule Checks
Small timing or accounting errors can change both the stated balance and the calculated interest. Before finalizing a schedule, confirm that the extra payment is treated according to its actual date and contractual application.
- Subtracting the extra payment too early. Under the convention used here, period interest is calculated on the opening balance. An extra payment after the period-end scheduled installment does not reduce that period’s interest.
- Replacing the installment with the extra amount. If the borrower owes a scheduled payment plus extra principal, both amounts are paid. Do not subtract only the extra amount.
- Assuming the installment automatically changes. An extra payment reduces the balance. It changes the scheduled payment only if the loan is recast or otherwise amended.
- Continuing to pay the full installment after payoff. The final amount due is the remaining balance plus interest for that period. It may be less than the usual installment.
- Using an annual rate as a per-period rate. Use the effective rate for the payment period. When payment periods are monthly, the appropriate monthly rate must be established from the stated rate convention before building the schedule.
A useful reconciliation is that total cash paid equals original principal plus total interest, provided there are no fees or other cash flows. For each period, also check that the closing balance is opening balance plus interest, less scheduled payment, less any extra principal payment. These checks catch sign errors and misplaced payments.
Check Your Understanding
Use the stated timing convention: interest accrues first, then the scheduled payment is made, then any extra principal payment on that date. Round dollar answers to cents where needed.
- A balance of $2,400 accrues interest at 1% per period. The scheduled payment is $500, followed immediately by $200 of extra principal. What is the balance after both payments? (Answer: $1,724.00)
- A $5,000 balance accrues interest at 2% per period. The scheduled payment is $1,200, followed by $300 of extra principal. Find the balance after both payments. (Answer: $3,600.00)
- A loan balance is $1,000 and the rate is 5% per period. A $400 scheduled payment is made at the end of period 1, followed by $100 of extra principal. The scheduled payment remains $400. Find the final payoff amount at the end of period 3, assuming no further extra payments. (Answer: $186.38)
- A $1,100 balance has two periods remaining at 10% per period. An immediate $100 extra principal payment is made and the loan is recast to two equal end-of-period installments. Find the installment to cents. (Answer: $576.19)
- A balance is reduced by an extra $200 payment at the end of a period. The rate is 5% per period, and the scheduled payment remains unchanged. By how much does this extra payment reduce interest in the next period? (Answer: $10.00)