Extra Principal Payments and Recast Loan Payments

Why an Extra Principal Payment Creates a Choice

An extra principal payment reduces the amount of debt on which future interest is charged. It does not, by itself, determine what happens to the scheduled installments. Under a fixed interest rate, a borrower and lender can instead preserve the original installment amount and pay off the smaller balance sooner, or preserve the remaining term and recalculate a smaller installment. These are different cash-flow arrangements, even though both begin with the same reduced balance.

The calculations below assume a level-payment loan: payments are made at the end of each period, the periodic effective interest rate remains fixed, and the extra payment is made immediately after a scheduled installment. The extra amount is applied entirely to principal, with no fee, penalty, or other charge. If a contract applies the extra payment differently, the balance update must reflect that contract.

Definition: Let $B$ be the outstanding balance immediately after a scheduled payment, and let $E$ be an extra payment applied at that same time directly to principal. The new balance is $B'=B-E$. If the periodic effective interest rate is $i$, the scheduled payment is $P$, and the remaining term is $n$ periods, a level-payment loan that amortizes exactly over that term satisfies $B'=P a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$.

Update the Balance Before Choosing a Recast

The first step is always to identify the balance at the extra-payment date. Because the extra payment in the stated model is made immediately after a scheduled installment and is applied directly to principal, subtract it from the post-installment balance. Do not include another period of interest between that installment and the extra payment.

After the balance is updated, the two choices use different quantities as fixed inputs. For a shortened term, keep $P$ fixed and solve for the number of periods needed to amortize $B'$. For a recast payment, keep the remaining number of periods $n$ fixed and solve for $P$.

Fix the payment timing.
Determine whether the balance is immediately before or after the regular installment and extra payment. The order changes the amount to which interest is applied.

Update principal.
When the extra payment is made immediately after an installment and applied entirely to principal, set $B'=B-E$.

Choose what stays fixed.
For a shorter payoff period, retain the scheduled payment. For a recast, retain the remaining term.

Check the resulting schedule.
For a shortened term that is not an integer number of periods, distinguish the number of full scheduled payments from the final smaller payment.

Keeping the Payment: Solve for the Payoff

Suppose the scheduled payment $P$ is retained after the balance falls to $B'$. The equivalent number of level payments is obtained by setting the balance equal to the present value of those payments:

B'=P a_{\overline{n}|} =P\frac{1-v^n}{i}.

Rearranging gives $v^n=1-iB'/P$, and therefore the equivalent term is:

n=\frac{-\ln(1-iB'/P)}{\ln(1+i)}.

This logarithmic result is a useful way to describe the equivalent term, but it need not be an integer. Actual scheduled loan payments are generally made at discrete dates. If the calculated term lies between $k$ and $k+1$, the borrower makes $k$ full payments and then a smaller final payment at the next scheduled date.

Condition: The formula for $n$ requires $i>0$ and $0<1-iB'/P<1$. In particular, $P>iB'$, so the payment exceeds the interest due for one period on the reduced balance. If $P\leq iB'$, the balance does not amortize to zero through these payments. When $i=0$, the equivalent term is $B'/P$ periods instead.

Worked Example: Keep the Payment and Make a Smaller Final Payment

Immediately after a scheduled installment, a loan has a balance of $9,500. The borrower makes an extra $2,000 principal payment at that time. The scheduled payment remains $1,000 at the end of each month, and the monthly effective interest rate is 1%. Find the equivalent term, the actual number of full payments, and the final payment.

Method. Update the balance first, then solve the level-payment present-value equation. This method avoids treating a fractional equivalent term as a fractional scheduled installment.

Step 1: Reduce the balance.

$$ B'=9{,}500-2{,}000=7{,}500. $$

Step 2: Solve for the equivalent term. Here $i=0.01$ per month and $P=1{,}000$. Thus:

n=\frac{-\ln(1-0.01(7{,}500/1{,}000))}{\ln(1.01)} =\frac{-\ln(0.925)}{\ln(1.01)} \approx 7.835.

The loan therefore requires seven full payments of $1,000, followed by a smaller payment at the eighth monthly due date. To calculate that final payment, first find the balance after seven full installments. With $s_{\overline{7}|}=((1.01)^7-1)/0.01$:

(1.01)^7=1.072135352,\qquad s_{\overline{7}|}=7.213535211, $$ $$ B_7=7{,}500(1.072135352)-1{,}000(7.213535211) =827.479930.

The remaining balance accrues one month of interest before the eighth due date. The final payment is:

$$ 1.01B_7=1.01(827.479930)=835.754729. $$

Rounded to the nearest cent, the borrower makes seven payments of $1,000 and a final payment of $835.75. As a check, the total of these future payments is $7(1{,}000)+835.754729=7{,}835.754729$. Subtracting the post-extra-payment balance of $7,500 gives $335.754729 of future interest, consistent with the balance being fully repaid.

Keeping the Term: Recast the Payment

If the remaining term is fixed at $n$ periods instead, the new level payment $P'$ must have present value equal to the reduced balance. Therefore:

B'=P'a_{\overline{n}|} \qquad\Longrightarrow\qquad P'=\frac{B'}{a_{\overline{n}|}} =\frac{B'i}{1-v^n}.

This is a payment recalculation, not a change to the interest rate. The term and periodic rate stay the same; only the installment changes to amortize the lower balance over the remaining periods.

Worked Example: Recast the Payment Over the Original Remaining Term

Immediately after a regular monthly installment, a loan balance is $10,000. An extra $2,000 is applied directly to principal. The monthly effective interest rate is 1%, and 12 monthly payment periods remain. Calculate the level payment that amortizes the reduced balance in exactly 12 periods.

Method. Hold the term fixed and divide the updated balance by the annuity-immediate present-value factor. This avoids leaving the original payment unchanged when the question requires the original term to be preserved.

The new balance is:

$$ B'=10{,}000-2{,}000=8{,}000. $$

For $i=0.01$, $v=1/1.01$. The 12-period annuity factor is:

v^{12}=(1.01)^{-12}\approx 0.887449225, \qquad a_{\overline{12}|}=\frac{1-v^{12}}{0.01} \approx 11.2550775.

The recast payment is:

P'=\frac{8{,}000}{11.2550775}\approx 710.790.

Rounded to the nearest cent, the revised monthly payment is $710.79. As a check, the present value of 12 payments of approximately $710.790 at 1% per month is $710.790(11.2550775)\approx \$8{,}000$, the updated principal balance.

Why the Annuity Equation Works

The prospective balance equation follows directly from the loan’s period-by-period balance recurrence. Let $B_k$ be the balance immediately after the $k$th future scheduled payment, starting from $B_0=B'$. Each period, interest is added and the payment is subtracted:

B_k=(1+i)B_{k-1}-P.

For one period, $B_1=B'(1+i)-P$. Repeating the recurrence gives:

B_k=B'(1+i)^k-P\left(1+(1+i)+\cdots+(1+i)^{k-1}\right).

At the end of period $n$, an exactly amortized loan has $B_n=0$. Dividing the equation $0=B'(1+i)^n-P\sum_{j=0}^{n-1}(1+i)^j$ by $(1+i)^n$ discounts every future payment back to the date of the updated balance:

B'=P\sum_{j=1}^{n}(1+i)^{-j} =P(v+v^2+\cdots+v^n) =Pa_{\overline{n}|}.

Thus the annuity equation is the present-value form of the same balance recurrence. It also explains why, with the payment unchanged, reducing the balance generally shortens the time to payoff; with the term unchanged, the same reduction instead lowers the payment.

Worked Example: Compare the Two Choices After the Same Extra Payment

After a scheduled monthly installment, a loan balance is $9,500. The borrower pays an extra $2,000 directly to principal. The monthly effective rate is 1%, the scheduled payment is $1,000, and the borrower is considering a 12-month recast. Compare the remaining payments and future interest under (a) keeping the $1,000 payment and (b) recasting over 12 months.

Method. Use the same updated balance for both alternatives, then change only the quantity specified by each choice. The common mistake this avoids is comparing one option using the pre-extra-payment balance and the other using the post-extra-payment balance.

The balance for both options is $9{,}500-2{,}000=\$7{,}500$. Under option (a), the calculation in the shortened-term method gives seven payments of $1,000 and a final payment of $835.754729. Total future payments are:

7(1{,}000)+835.754729=7{,}835.754729, \qquad \text{future interest}=7{,}835.754729-7{,}500=335.754729.

Rounded to cents, this is $7,835.75 in future payments, including $335.75 of interest.

Under option (b), the 12-month annuity factor at 1% is approximately 11.2550775, so the recast payment is:

P'=\frac{7{,}500}{11.2550775}\approx 666.3657.

The unrounded total of 12 payments is approximately $12(666.3657)=\$7{,}996.39$, so future interest is approximately $7{,}996.39-7{,}500=\$496.39$. The recast offers a lower monthly payment, while preserving the 12-month term; keeping the larger payment pays off the balance earlier and produces less future interest. This comparison assumes no fees or other contract charges.

Common Timing and Calculation Traps

Applying the extra payment at the wrong point in the period. If the payment is immediately after a regular installment, reduce the post-installment balance directly. If it is made at another time, first determine how interest accrues to that date and how the contract applies the payment.
Reusing the old balance. Both alternatives begin with $B'=B-E$, not $B$. Using the old balance overstates the payment for a recast and the time needed to repay the loan with a fixed installment.
Changing both payment and term without being asked. A reduced balance permits different repayment arrangements, but each calculation must state what is held fixed.
Treating a noninteger term as a scheduled installment count. The logarithmic result is an equivalent term. For a discrete payment schedule, calculate the balance after the last full payment and accrue one more period of interest to find the final payment.
Mixing annual and payment-period rates. For monthly payments, use a monthly effective rate. If the given rate is an effective annual rate $i_{\text{annual}}$, the equivalent monthly effective rate is $(1+i_{\text{annual}})^{1/12}-1$, not automatically $i_{\text{annual}}/12$.

For example, using the first worked example’s balance of $7,500 but incorrectly treating the equivalent term $7.835$ as 7.835 full payments would suggest a payment schedule that does not occur on monthly due dates. The correction is to compute the balance after seven full $1,000 payments, then multiply that balance by $1.01$ to determine the amount due at the eighth date: $827.479930(1.01)=\$835.754729$, or $835.75 rounded to cents.

The distinction matters in practical loan analysis as well as in exam calculations. A lower recast payment can improve near-term cash flow, while maintaining the payment can reduce the time in debt. The appropriate comparison depends on the required payment schedule and any applicable contract terms, not just on the size of the extra principal payment.

Check Your Understanding

Assume each extra payment is made immediately after a scheduled installment, is applied entirely to principal, and incurs no fee. Use the stated periodic effective interest rate and round money to the nearest cent.

A balance of $6,000 is reduced by an extra $2,000. The monthly effective rate is 1%, and 8 months remain. Find the recast monthly payment. (Answer: $522.76)
After an extra principal payment, the balance is $4,000. The monthly effective rate is 1%, and the scheduled payment remains $550. Find the number of full $550 payments and the final payment. (Answer: 7 full payments, followed by a final payment of $324.31 at the eighth due date)
A balance of $12,000 is reduced by an extra $3,000. The monthly effective rate is 0.5%, and 24 months remain. Find the recast payment. (Answer: $398.88 per month)
Immediately before a monthly installment, a loan balance is $10,000. The monthly effective rate is 1%. A $1,000 scheduled payment is made, followed immediately by an extra $2,000 principal payment. If 10 months then remain, what is the recast payment? (Answer: $749.63 per month)

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