What an Amortization Schedule Shows
A loan payment does not usually reduce the amount owed by the full payment amount. Part of each payment covers interest accrued since the prior payment; the rest repays principal. An amortization schedule lays out this split installment by installment, making the changing loan balance visible.
To build a schedule, use the interest rate for the payment interval, not automatically the annual rate quoted in the loan terms. The schedule also depends on payment timing. The standard setup here assumes payments are made at the end of each period, interest is charged once per period at a fixed rate, and there are no fees, skipped payments, or extra principal payments. Amounts are calculated without intermediate rounding unless stated otherwise.
Let \(B_{k-1}\) be the balance at the start of period \(k\), \(i\) the effective interest rate for one payment period, and \(R_k\) the payment at the end of that period. The row calculations are:
These calculations give a row-by-row method, not a substitute for identifying the correct rate and payment timing. If a quoted rate is nominal, or if compounding and payment frequencies differ, convert it to an effective rate per payment period before calculating interest.
Why the Row Formulas Work
Interest is charged on the amount outstanding during the period. Under the stated fixed-rate, end-of-period model, the balance \(B_{k-1}\) therefore accumulates to \(B_{k-1}(1+i)\) just before the payment. The payment then removes \(R_k\). Equivalently, the accumulated balance consists of the original principal plus interest, so the amount of principal still owed after payment is:
Rearranging shows why the schedule separates the payment into interest and principal:
Thus, \(iB_{k-1}\) pays the period’s interest, and the remainder \(R_k-iB_{k-1}\) reduces principal. This assumes the payment is at the end of the period and that no other charges or transactions change the balance. A payment at the beginning of a period, a variable rate, or a fee charged to the loan requires a different treatment.
Constructing Each Row
Use the same order in every period. First find the beginning balance. Next calculate interest using the periodic rate. Subtract interest from the payment to find principal repaid, then subtract that principal from the beginning balance. The ending balance becomes the next period’s beginning balance.
Match the rate period to the interval between payments. For example, a nominal annual rate \(i^{(12)}\) convertible monthly gives a monthly rate of \(i^{(12)}/12\).
Multiply the beginning balance by the periodic rate. Do not use the original loan amount after the first period unless the balance has not changed.
Subtract interest from the payment. The difference is principal repaid. If the payment does not cover the interest, the unpaid interest may increase the amount owed; that case is not ordinary positive amortization.
Subtract principal repaid from the beginning balance. Use this ending balance as the next row’s beginning balance.
Worked Example: A Two-Installment Loan at an Annual Rate
A borrower receives $2,100 and agrees to make two equal end-of-year payments of $1,210. The effective annual interest rate is 10%. Construct the schedule and find total interest. Use the annual rate directly because each payment period is one year.
Method and assumptions. Interest accrues annually at 10%, and each payment is made at year-end. There are no fees or additional transactions. For each row, calculate interest on the beginning balance, then allocate the remaining payment to principal.
Year 1. Beginning balance is $2,100.00. Interest is \(0.10(2{,}100.00)=\$210.00\). Principal repaid is \(\$1{,}210.00-\$210.00=\$1{,}000.00\). Ending balance is \(\$2{,}100.00-\$1{,}000.00=\$1{,}100.00\).
Year 2. Beginning balance is $1,100.00. Interest is \(0.10(1{,}100.00)=\$110.00\). Principal repaid is \(\$1{,}210.00-\$110.00=\$1{,}100.00\). Ending balance is \(\$1{,}100.00-\$1{,}100.00=\$0.00\).
| Year | Beginning balance | Interest | Payment | Principal repaid | Ending balance |
|---|---|---|---|---|---|
| 1 | $2,100.00 | $210.00 | $1,210.00 | $1,000.00 | $1,100.00 |
| 2 | $1,100.00 | $110.00 | $1,210.00 | $1,100.00 | $0.00 |
Check. The two interest charges total \(\$210.00+\$110.00=\$320.00\). Total payments are \(\$2{,}420.00\), which equal the original $2,100.00 principal plus $320.00 interest. The ending balance is zero, as required for full repayment.
Common mistake avoided: Subtracting the entire first payment from the original balance would give $890.00. That incorrectly treats the $210.00 interest charge as principal repayment.
When the Quoted Rate Is Not the Periodic Rate
Suppose a loan quotes a nominal annual rate \(i^{(m)}\) convertible \(m\) times per year, with payments made at the same frequency. The rate for each payment period is \(i^{(m)}/m\). Use that rate in every schedule row. If compounding and payment frequencies differ, convert the quote to an effective rate for the payment interval rather than dividing by the payment frequency without justification.
Worked Example: Monthly Payments Under a Nominal Annual Quote
A $8,000 loan has a nominal annual rate of 12% convertible monthly. The borrower makes end-of-month payments of $700. Find the ending balance after the first three payments. Assume the quoted rate remains fixed and that there are no fees or extra payments.
Method and assumptions. Payments and compounding are both monthly, so the periodic rate is \(0.12/12=0.01\), or 1% per month. Calculate each month’s interest from that month’s beginning balance, and carry the unrounded balance into the next month.
Month 1. Interest is \(0.01(\$8{,}000.00)=\$80.00\). Principal repaid is \(\$700.00-\$80.00=\$620.00\). Ending balance is \(\$8{,}000.00-\$620.00=\$7{,}380.00\).
Month 2. Interest is \(0.01(\$7{,}380.00)=\$73.80\). Principal repaid is \(\$700.00-\$73.80=\$626.20\). Ending balance is \(\$7{,}380.00-\$626.20=\$6{,}753.80\).
Month 3. Interest is \(0.01(\$6{,}753.80)=\$67.538\). Principal repaid is \(\$700.00-\$67.538=\$632.462\). Ending balance is \(\$6{,}753.80-\$632.462=\$6{,}121.338\), or $6,121.34 to the nearest cent.
Check. Total interest for the three months is \(\$80.00+\$73.80+\$67.538=\$221.338\). Total principal repaid is \(\$620.00+\$626.20+\$632.462=\$1{,}878.662\). The balance check is \(\$8{,}000.00-\$1{,}878.662=\$6{,}121.338\), which agrees with the row calculation. Round only the reported final balance to cents.
Common mistake avoided: Using 12% as the monthly rate would charge \(\$960.00\) interest in the first month. The 12% figure is a nominal annual quote; the monthly rate here is 1%.
Reading and Checking a Schedule
A correctly constructed schedule has a simple reconciliation in each row: the payment equals interest plus principal repaid, and the beginning balance minus principal repaid equals the ending balance. These checks can catch arithmetic errors even when the rate and payment have been entered correctly.
Worked Example: A Complete Three-Period Schedule
A loan of $3,310 is repaid by three equal end-of-year payments of $1,331. The effective annual interest rate is 10%. Construct the schedule and find the total interest paid.
Method and assumptions. Use an annual periodic rate of 10%, with interest calculated on each year’s beginning balance. No intermediate rounding is required because each row produces exact amounts to cents.
Year 1. Interest is \(0.10(\$3{,}310.00)=\$331.00\). Principal repaid is \(\$1{,}331.00-\$331.00=\$1{,}000.00\). Ending balance is \(\$3{,}310.00-\$1{,}000.00=\$2{,}310.00\).
Year 2. Interest is \(0.10(\$2{,}310.00)=\$231.00\). Principal repaid is \(\$1{,}331.00-\$231.00=\$1{,}100.00\). Ending balance is \(\$2{,}310.00-\$1{,}100.00=\$1{,}210.00\).
Year 3. Interest is \(0.10(\$1{,}210.00)=\$121.00\). Principal repaid is \(\$1{,}331.00-\$121.00=\$1{,}210.00\). Ending balance is \(\$1{,}210.00-\$1{,}210.00=\$0.00\).
| Year | Beginning balance | Interest | Payment | Principal repaid | Ending balance |
|---|---|---|---|---|---|
| 1 | $3,310.00 | $331.00 | $1,331.00 | $1,000.00 | $2,310.00 |
| 2 | $2,310.00 | $231.00 | $1,331.00 | $1,100.00 | $1,210.00 |
| 3 | $1,210.00 | $121.00 | $1,331.00 | $1,210.00 | $0.00 |
Check. Total interest is \(\$331.00+\$231.00+\$121.00=\$683.00\). Total payments of \(\$3{,}993.00\) equal the original principal of $3,310.00 plus $683.00 interest. Each row also reconciles separately: for example, in Year 2 the $1,331.00 payment is $231.00 interest plus $1,100.00 principal.
Common mistake avoided: Calculating every year’s interest on $3,310.00 would overstate interest. Once principal is repaid, later interest is calculated on the reduced balance.
Common Schedule Errors
- Mixing rate units. A monthly payment requires a monthly effective rate. A nominal annual quote is not automatically the rate for one month.
- Using the payment as principal repayment. Interest is part of the payment. Subtract interest first to find the principal portion.
- Carrying forward the wrong amount. The next beginning balance is the prior ending balance, not the original principal minus total payments.
- Rounding too early. Rounding interest or balances in every row can introduce small differences. Keep adequate precision during calculation, then report money to the cent unless the problem specifies another convention.
- Ignoring the payment date. These formulas assume end-of-period payments. A beginning-of-period payment changes the time over which interest accrues.
Check Your Understanding
For each problem, use the stated payment timing and rate convention. Report money to the nearest cent.
- A $900 loan has a 6% effective annual rate and an end-of-year payment of $500. Find the interest, principal repaid, and ending balance for the first year. (Answer: interest $54.00; principal repaid $446.00; ending balance $454.00)
- A loan begins with a balance of $1,200. The effective monthly rate is 2%, and the end-of-month payment is $410. Find the ending balance after two payments. (Answer: $420.28)
- A $2,500 balance accrues interest at 4% per payment period. The payment is $900 at each period-end. Find the ending balance after three payments. (Answer: $2.72)
- A borrower receives $2,100 and makes two end-of-year payments of $1,210 at an effective annual rate of 10%. Find the total interest and verify full repayment. (Answer: total interest $320.00; ending balance $0.00)
- A loan has a nominal annual rate of 12% convertible monthly, a beginning balance of $8,000, and monthly end-of-period payments of $700. Find the ending balance after three payments. (Answer: $6,121.34)