Recasting Payments After a Lump-Sum Prepayment

What a Recast Changes

A borrower may make a lump-sum payment toward principal and ask the lender to reduce the regular installments while keeping the original maturity date. This is a recasting of the installments: the loan balance falls, but the scheduled payment dates and the number of remaining installments do not change.

The key is to value the loan at the date of the prepayment. First determine the balance immediately before the lump-sum principal payment. Subtract the prepayment to get the reduced balance. Then find the level installment whose value at that same date equals the reduced balance. This uses the earlier material on loan balances and present value, but the focus here is the revised installment.

Definition: A recast installment is a new level periodic payment that amortizes the loan's reduced balance over the remaining original payment dates, at the specified loan rate. A recast does not, by itself, shorten the term or change the maturity date.

The Recast Equation

Assume the prepayment is made immediately after a scheduled installment. Let $B$ be the loan balance at that time, $C$ the additional principal payment, $n$ the number of scheduled installments remaining, and $R$ the new installment. Let $i$ be the effective interest rate per payment period. Assume the rate stays fixed, payments are made at the end of each period, and there are no fees, penalties, or other cash flows. The reduced balance is $B-C$.

Theorem: Under these assumptions, the recast installment $R$ satisfies $$ B-C=R a_{\overline{n}|} \qquad\text{and hence}\qquad R=\frac{B-C}{a_{\overline{n}|}}, $$ where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$. If $i=0$, the annuity factor is $n$, so $R=(B-C)/n$.

The factor $a_{\overline{n}|}$ is for an annuity-immediate: one payment at the end of each of the next $n$ periods. The timing matters. If the quoted balance is before a scheduled installment, first establish which payments remain and what balance is being valued; do not treat a balance before a payment as though it were immediately after that payment.

Why the Equation Holds

At the prepayment date, the remaining installments are valued by discounting each one to that date. For a level installment $R$, the first payment is one period away, the second is two periods away, and the last is $n$ periods away. Thus their present value is the sum below.

R v+R v^2+\cdots+R v^n =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i} =R a_{\overline{n}|}.

Here $1-v=i/(1+i)=iv$, which gives the middle simplification. The reduced balance must equal the present value of the payments that repay it, so $B-C=R a_{\overline{n}|}$. This also shows why a prepayment is not generally divided equally among the remaining installments: changing principal changes the amount on which future interest accrues, so the installment change depends on the annuity factor.

Fix the valuation date.
Use the date at which the balance and prepayment are both measured. Confirm whether a scheduled installment has just been made.

Count the remaining payments.
Use the number of scheduled installments left to the original maturity, not the number of periods since the loan began.

Find the reduced balance.
Subtract the principal prepayment from the balance at that date.

Use the correct period rate.
Calculate the annuity-immediate factor using the rate per installment period, then divide the reduced balance by that factor.

Worked Recast Calculations

Worked Example: Recasting Three Annual Installments

Immediately after a scheduled annual installment, a loan has a balance of $8,000. The borrower pays an additional $1,200 toward principal at that date. The effective annual interest rate is 10%, and three annual installments remain. Find the recast annual installment, keeping the original maturity date.

Method. The date is immediately after a scheduled installment, so the $8,000 balance is the balance to which the extra principal payment is applied. There are still three end-of-year installments to make. The effective rate per annual payment period is $i=0.10$.

The reduced balance is $$ B-C=8{,}000-1{,}200=6{,}800. $$ The discount factor and annuity factor are $$ v=\frac{1}{1.10}=0.9090909,\qquad a_{\overline{3}|}=\frac{1-(1.10)^{-3}}{0.10} =\frac{1-0.7513148}{0.10} =2.48685199. $$ Therefore, $$ R=\frac{6{,}800}{2.48685199}=2{,}734.38. $$ The recast installment is $2,734.38 per year, rounded to the nearest cent. As a check, $2{,}734.38(2.48685199)$ is approximately $6,800, with the small difference attributable to rounding the installment to cents. The method avoids changing the three-payment schedule or applying the prepayment as though it reduced each installment by an equal amount of principal.

Worked Example: Finding the Prepayment for a Target Installment

Immediately after a scheduled annual installment, a loan balance is $9,000. The effective annual interest rate is 10%, and three annual installments remain. What lump-sum principal prepayment at that date would allow the loan to be recast to annual installments of $3,000?

Method. First calculate the present value of the target installments at the change date. The required prepayment is the current balance less that present value.

The three-payment factor at 10% is $$ a_{\overline{3}|} =\frac{1-(1.10)^{-3}}{0.10} =2.48685199. $$ The target installment has present value $$ 3{,}000(2.48685199)=7{,}460.55597\approx 7{,}460.56. $$ Thus the prepayment is $$ C=9{,}000-7{,}460.55597 =1{,}539.44403 \approx 1{,}539.44. $$ The required lump-sum principal prepayment is $1,539.44, rounded to the nearest cent. Check: the balance after the payment is $9{,}000-1{,}539.44=7{,}460.56$, which is the rounded present value of three $3,000 installments. The prepayment is not found by comparing the target payment with the old payment and multiplying the difference by three; the correct value is the present value of the target payment stream.

Worked Example: Recasting Two Annual Installments

Immediately after a scheduled annual installment, a loan balance is $10,000. A $2,000 lump-sum payment is applied directly to principal. The effective annual interest rate is 10%, and two annual installments remain. Find the recast installment and verify it by valuing the two payments.

Method. The reduced balance is $10{,}000-2{,}000=8{,}000$. The annual payment rate is 10%, and the payment dates are one and two years after the prepayment date.

The annuity-immediate factor is $$ a_{\overline{2}|} =\frac{1-(1.10)^{-2}}{0.10} =\frac{1-0.82644628}{0.10} =1.73553719. $$ Therefore, $$ R=\frac{8{,}000}{1.73553719}=4{,}609.52381. $$ The recast installment is $4,609.52 per year, rounded to the nearest cent. For a direct check before rounding, the present value is $$ 4{,}609.52381\left(\frac{1}{1.10}+\frac{1}{1.10^2}\right) =4{,}609.52381(1.73553719) =8{,}000. $$ This is the reduced balance, as required. The original maturity date is retained because both remaining payments are still included.

Rate and Timing Checks

The interest rate in the formula must match the payment period. For example, if an annual nominal rate $i^{(m)}$ is convertible $m$ times per year and installments are made at those conversion dates, the rate per installment period is $i^{(m)}/m$. If instead an effective annual rate $i$ is given and there are $m$ equal periods per year, the equivalent rate per period is $(1+i)^{1/m}-1$, not $i/m$. Use that periodic rate in $a_{\overline{n}|}$.

The formula does not apply unchanged if the loan rate varies, the remaining payments are not level, payment dates change, or the prepayment is subject to a fee or penalty. In those cases, the value equation must reflect the actual rates, cash flows, and dates. For the fixed-rate, level-payment recast considered here, the assumptions are what make one annuity factor sufficient.

Common Exam Trap

A tempting shortcut is to divide the lump-sum prepayment equally among the remaining installments. In the first example, that shortcut says that $1,200 divided by three reduces each payment by $400. The original installment corresponding to an $8,000 balance is $8{,}000/2.48685199$, or about $3,216.92. Subtracting $400 would suggest a recast installment of about $2,816.92. This is wrong: the $400 is not the payment reduction that has the same present value as a $1,200 principal reduction.

The correct installment reduction is the prepayment divided by the annuity factor: $$ \frac{1{,}200}{2.48685199}=482.54. $$ So the recast installment is approximately $3{,}216.92-482.54=2{,}734.38$, consistent with the direct calculation. A useful check is to verify that the present value of the revised installments equals the balance after the prepayment, not the balance before it.

Key takeaway: For a fixed-rate loan with level installments, a lump-sum principal prepayment followed by a recast over the original remaining term gives $R=(B-C)/a_{\overline{n}|}$. Use the balance and payment count at the correct date, and use the rate per payment period.

Check Your Understanding

For each problem, assume the lump-sum payment is applied immediately after a scheduled installment, with no fees or other cash flows.

A loan balance is $6,000, a $1,000 principal prepayment is made, and two annual installments remain. The effective annual rate is 0%. Find the recast installment. (Answer: $2,500.00 per year)
A loan balance is $12,000. The borrower prepays $2,000, with two annual installments remaining at an effective annual rate of 10%. Find the recast installment. (Answer: $5,761.90 per year)
A loan balance is $9,000, the effective annual rate is 10%, and three annual installments remain. Find the prepayment needed to recast the loan to installments of $3,000 per year. (Answer: $1,539.44)
A loan balance is $15,000. The borrower prepays $5,000, with two annual installments remaining at an effective annual rate of 5%. Find the recast installment. (Answer: $5,378.05 per year)
A loan balance is $20,000. The borrower prepays $4,000, with five annual installments remaining at an effective annual rate of 0%. Find the recast installment and state whether the original maturity date changes. (Answer: $3,200.00 per year; the original maturity date does not change)

What You'll Learn