Changing Installment Amounts Mid-Loan

Why a Changed Installment Requires a New Calculation

When a borrower changes the scheduled installment on an existing loan, the outstanding balance does not reset. The amount owed at the change date is the starting point for the revised schedule. From that date forward, each period’s interest is calculated on the balance then outstanding, and the new installment is applied on the specified payment date.

The central question is not simply whether the new installment is larger or smaller than the old one. It is whether it is large enough to cover the interest accruing on the current balance. If it is, the principal falls; if it only covers the interest, the balance remains level; if it is smaller, the balance grows. This test also determines whether a final adjusted installment can eventually repay the loan.

Definition: Let $B$ be the outstanding loan balance immediately after an installment on the change date. Let $i$ be the effective interest rate per payment period, and let $C$ be the new installment, paid at the end of each period. The balance after $k$ new installments is the outstanding amount immediately after the $k$th installment. This setup assumes a constant rate, regular equal-length periods, and payments at period-end.

Balance After the Installment Amount Changes

For each payment period, interest is added to the opening balance and the installment is then subtracted. If $B_k$ is the balance immediately after the $k$th new installment, the recurrence is

B_0=B,\qquad B_k=(1+i)B_{k-1}-C.

Applying the recurrence repeatedly gives a convenient direct formula. The original balance accumulates for $k$ periods. Each new installment reduces the balance, but an earlier installment has a greater effect at the later valuation date because it avoids more future interest.

B_k=B(1+i)^k-C\left((1+i)^{k-1}+(1+i)^{k-2}+\cdots+1\right) =B(1+i)^k-C\,s_{\overline{k}|}.

Here $s_{\overline{k}|}$ is the accumulated value of an annuity-immediate of $k$ unit payments at rate $i$. This formula applies when the new payments are equal and occur at the end of each period. If the rate or payment dates change, the appropriate period-by-period accumulation must be used instead.

Find the change-date balance.
Use the old loan schedule through the last payment made before the new installment amount takes effect. Be precise about whether the balance is just before or just after that payment.

Match the rate and payment period.
Use the effective rate for one interval between scheduled payments. For example, a monthly payment schedule requires a monthly effective rate, not an annual rate applied once per month.

Test the installment.
Compare $C$ with $iB$, the interest due for the first revised period. If $C>iB$, the balance decreases initially and the loan can be amortized by a finite number of equal installments, with a final adjustment if needed.

Calculate the revised balance or payoff.
Use the balance formula or recurrence for the requested number of periods. If the balance after a regular installment is positive, accrue one more period of interest to find the amount due on the next scheduled date.

Why the Formula Holds

The balance formula follows by accounting for every cash flow at the same date. At the date immediately after the $k$th revised installment, the starting balance $B$ has accumulated to $B(1+i)^k$. The first new installment has accumulated for $k-1$ periods, the second for $k-2$, and the $k$th for zero periods. Subtracting their accumulated values gives

B_k=B(1+i)^k-C\sum_{j=0}^{k-1}(1+i)^j.

The geometric sum is $s_{\overline{k}|}=\big((1+i)^k-1\big)/i$ when $i\ne 0$, which yields the stated formula. Equivalently, the recurrence follows directly from one period’s account: opening balance plus interest, less the payment. The two methods must give the same result and provide a useful arithmetic check.

Does the New Installment Amortize the Loan?

The first period’s interest is $iB$. Therefore, the balance after the first installment is $B_1=B(1+i)-C$. If $C=iB$, then $B_1=B$, and the same calculation repeats every period: the balance stays level indefinitely. If $C<iB$, then $B_1>B$; the installment does not even cover all of the first period’s interest, so the balance grows.

If $C>iB$, the first payment reduces principal. Under a constant positive rate and regular equal installments, the balance continues to be reduced and the loan is eventually repaid. For a specified integer number $n$ of equal installments to amortize the balance exactly, the transaction-date equation is $B=C\,a_{\overline{n}|}$, or $C=B/a_{\overline{n}|}$. If this condition does not hold for an integer $n$, a common arrangement is to make the regular installment until the next scheduled date’s payoff is smaller than that installment, then pay the smaller adjusted amount on that date.

Important condition: A final adjusted payment on a scheduled date is appropriate only when the revised installment is large enough to amortize the balance, such as when $C>iB$. A payment equal to the interest leaves the balance unchanged; a smaller payment causes it to grow. Neither case produces a later scheduled date at which the loan is paid off by the stated installment and a final adjustment.

Worked Example: Finding the Balance Before and After a Payment Change

A hypothetical loan has a principal of $10,000 and an effective annual interest rate of 5%. The borrower pays $2,500 at the end of each of the first two years. Immediately after the second payment, the scheduled annual installment changes to $1,800. Find the balance immediately after the change and the final adjusted payment, assuming the new installment is paid at each year-end.

Method. First value the old schedule through the second payment, then apply the new schedule from that date. This avoids treating the new installment as though it had been in force from the start.

The balance after the first year’s interest and payment is

$$ 10{,}000(1.05)-2{,}500=8{,}000. $$

The balance immediately after the second $2,500 payment is

$$ 8{,}000(1.05)-2{,}500=5{,}900. $$

The first year of interest under the new schedule is $0.05(5{,}900)=\$295$. Since $1,800 exceeds $295, the new installment amortizes the balance. After three new installments, the balance is

\begin{aligned} B_3 &=5{,}900(1.05)^3-1{,}800\big((1.05)^2+1.05+1\big)\\ &=5{,}900(1.157625)-1{,}800(3.1525)\\ &=6{,}829.9875-5{,}674.5\\ &=\$1{,}155.4875. \end{aligned}

This is the amount immediately after the third new installment. One year later, immediately before the next payment, the payoff is

1{,}155.4875(1.05)=\$1{,}213.261875.

Thus the borrower makes three $1,800 installments and then a final payment of $1,213.26 at the fourth scheduled date after the change. The final payment is smaller than $1,800 because the balance just before it is smaller. As a check, after the third installment the positive balance has accrued one year of interest to the stated final amount. Money is rounded only at the final step, to the nearest cent.

Common mistake avoided: Counting the balance before the second old payment as the change-date balance would leave one old installment unaccounted for. The new schedule begins with the balance after the second old payment, which is $5,900.

Worked Example: A Changed Installment That Produces a Final Payment

A hypothetical borrower has a balance of $1,000 immediately after a scheduled payment. The effective interest rate is 10% per year, and the borrower will now pay $300 at each year-end. Find the number of regular $300 installments and the final adjusted payment.

Method. Use the recurrence and track the balance after each installment. The first-period interest is $0.10(1{,}000)=\$100$, so the $300 installment exceeds the interest and reduces principal.

After the first installment, the balance is $1{,}000(1.10)-300=\$800$. After the second, it is $800(1.10)-300=\$580$. After the third, it is $580(1.10)-300=\$338$. The borrower has made three regular installments.

At the next year-end, the balance due before payment is

338(1.10)=\$371.80.

The payoff at that date is less than the scheduled $300? No: $371.80 is greater than $300, so a fourth regular installment would not pay off the loan. After making a fourth $300 installment, the balance would be $371.80-300=\$71.80$; it would then accrue to $71.80(1.10)=\$78.98$ at the fifth scheduled date. Therefore, the correct schedule is four regular installments of $300 followed by a final payment of $78.98 at the fifth year-end.

The check is that the fourth regular installment leaves $71.80 immediately after payment, and one further year of interest makes that amount $78.98. This example illustrates why the balance after three installments is not itself the final payoff: the final amount is due one payment period later.

Common mistake avoided: Calling $371.80 the final payment would ignore the fourth scheduled $300 installment. The adjusted payment occurs only after the regular installments have been made.

Worked Example: Tracking Several New Installments

A hypothetical loan has a principal of $8,000 and an effective annual rate of 5%. The borrower pays $2,000 at each of the first two year-ends. Immediately after the second payment, the annual installment changes to $1,000. Find the balance immediately after three installments at the new amount.

Method. Establish the balance on the change date, then use that balance as $B_0$ for the revised schedule. The balance after the first old payment is $8{,}000(1.05)-2{,}000=\$6{,}400$. After the second old payment, it is

6{,}400(1.05)-2{,}000=6{,}720-2{,}000=\$4{,}720.

The first period’s interest after the change is $0.05(4{,}720)=\$236$. The new $1,000 installment exceeds that amount. Applying the balance formula for three revised installments gives

\begin{aligned} B_3 &=4{,}720(1.05)^3-1{,}000\big((1.05)^2+1.05+1\big)\\ &=4{,}720(1.157625)-1{,}000(3.1525)\\ &=5{,}463.99-3{,}152.50\\ &=\$2{,}311.49. \end{aligned}

The balance is therefore $2,311.49 immediately after the third $1,000 revised installment. A recurrence check gives $4,720 to $3,956 after the first new installment, to $3,153.80 after the second, and to $2,311.49 after the third. The direct formula and period-by-period calculation agree. Money is rounded only at the final step, to the nearest cent.

Common mistake avoided: The two $2,000 payments determine the change-date balance, but they are not part of the three-payment revised schedule. Mixing the old and new installment amounts in one annuity term would value the wrong cash flows.

Common Exam Traps

When a payment amount changes, most errors come from mismatched dates or from assuming that any positive installment must eventually repay the debt. Keep the balance date, interest period, and payment date explicit. The following incorrect calculation shows a frequent balance error.

For the $10,000 loan in the first example, an incorrect calculation of the balance after two $2,500 payments might be

10{,}000(1.05)^2-2{,}500(1+1.05)=\$3{,}208.

This is wrong: $10{,}000(1.05)^2=11{,}025$, while $2{,}500(1+1.05)=5{,}125$, and their difference is $5,900, not $3,208. The corrected balance is also confirmed period by period: the first balance is $8,000 and the second is $5,900.

Using an annual rate for a monthly period. The rate in the recurrence must match the interval between payments.
Forgetting the final accrual period. A balance immediately after the last regular installment must accrue interest before the next scheduled payoff date.
Applying the new installment too early. Use the old schedule up to the stated change date, then switch to the new amount.
Assuming every positive payment amortizes. Compare the installment with the interest due, $iB$, before concluding that a finite payoff date exists.
Rounding balances repeatedly. Retain unrounded values in intermediate calculations and round only the final monetary result.

Key takeaway: A revised installment schedule starts with the outstanding balance on the change date. Use the effective rate per payment period, test $C>iB$ before claiming the loan will amortize, and distinguish the balance just after an installment from the payoff due at the next scheduled date.

Check Your Understanding

For each problem, treat the stated balance as immediately after the last payment before the changed installment takes effect. Use the stated effective rate per payment period.

A balance of $3,000 is subject to an effective annual rate of 4%. The new annual installment is $500. Find the balance immediately after two new installments. (Answer: $2,224.80)
A balance of $5,000 is subject to an effective annual rate of 6%, and the new annual installment is $400. Find the balance immediately after five new installments. (Answer: $4,436.29)
A balance of $2,000 is subject to an effective annual rate of 5%, and the new annual installment is $500. After three regular installments, what final adjusted payment is due at the next year-end? (Answer: $775.95)
A balance of $8,000 is subject to an effective annual rate of 5%. What happens to the balance if the new annual installment is $400? What if it is $390? (Answer: At $400, the balance stays at $8,000; at $390, it grows to $8,010 after the first installment.)
A balance of $1,000 is to be repaid in exactly three equal year-end installments at an effective annual rate of 5%. Find the installment, rounded to the nearest cent. (Answer: $367.21)

What You'll Learn