What an Amortization Table Records
A scheduled loan payment is level when the same payment amount is due at each specified payment date. An amortization table shows how each payment is applied over time: first to the interest accrued during the period, then to the loan principal. The table makes the changing balance visible and provides a direct way to check that the loan is repaid as intended.
The standard level-payment schedule assumes a fixed periodic effective interest rate, payments at the end of each period, and no additional fees, prepayments, missed payments, or changes in the rate. The opening balance for the first period is the original principal. Each later opening balance is the preceding period’s closing balance. If payments are due at the beginning of periods instead, the timing differs and the ordinary-immediate schedule below does not apply unchanged.
Let \(L\) be the original loan principal, \(i\) the effective interest rate per payment period, \(n\) the number of end-of-period payments, and \(R\) the level payment. If the loan is fully repaid by those payments, its value at the start is the present value of the payment stream:
This payment formula determines the amount to enter in each row; it does not by itself produce the period-by-period allocations. For a nominal annual rate convertible \(m\) times per year, use the periodic rate \(i^{(m)}/m\) when payments occur at those same \(m\) dates per year. A rate quoted for another compounding or payment frequency must first be converted to the effective rate for one payment period.
Building the Rows
For each period, use the balance just before that period’s interest accrues. The calculation is performed in this order:
Begin with the original loan principal in period 1. In each later period, carry forward the prior period’s closing balance.
Multiply the opening balance by the effective interest rate per payment period.
Subtract the interest charge from the scheduled payment to find principal repaid.
Add the period’s interest to the opening balance and subtract the payment. Carry this result into the next row.
The bookkeeping identity behind these steps is
It holds because interest increases the amount owed during the period, while the payment reduces it. Since principal repaid is the payment less interest, the same closing balance can be found by subtracting principal repaid from the opening balance. This identity is a useful row-by-row check, not a substitute for using the correct opening balance and interest rate.
Worked Example: A Two-Payment Loan With Exact Dollar Allocations
A hypothetical borrower receives a loan of $2,100 at an effective annual rate of 10%. The loan is repaid by two equal payments, each made at the end of a year. Construct the schedule. Assume there are no fees or extra payments.
Method. Find the level payment as the present value of two end-of-year payments, then apply the row calculations. This example avoids rounding during the schedule because its amounts are exact whole dollars.
The annuity factor is
Therefore the payment is \(R=2{,}100/(210/121)=1{,}210\). In period 1, interest is \(2{,}100(0.10)=210\), so principal repaid is \(1{,}210-210=1{,}000\). The closing balance is \(2{,}100+210-1{,}210=1{,}100\).
In period 2, interest is \(1{,}100(0.10)=110\). Principal repaid is \(1{,}210-110=1{,}100\), leaving \(1{,}100+110-1{,}210=0\).
| Period | Opening balance | Payment | Interest | Principal repaid | Closing balance |
|---|---|---|---|---|---|
| 1 | $2,100.00 | $1,210.00 | $210.00 | $1,000.00 | $1,100.00 |
| 2 | $1,100.00 | $1,210.00 | $110.00 | $1,100.00 | $0.00 |
Check. The first closing balance becomes the second opening balance, and the last closing balance is zero. Also, \(1{,}210(210/121)=2{,}100\), confirming that the two payments have the required present value.
Precision and Rounding in a Schedule
A practical table usually displays money to the nearest cent, but calculations should use unrounded values internally. If each intermediate balance is rounded before it is used again, rounding error can accumulate. A final displayed row can also appear not to add exactly because its separately rounded entries need not preserve an identity to the cent.
The next example keeps additional digits during the calculations, then rounds the displayed schedule to cents. It is important to distinguish rounding a full-precision result from subtracting two values that have already been rounded.
Worked Example: Three Payments and Full-Precision Calculations
A hypothetical loan of $1,000 bears interest at an effective annual rate of 10% and is repaid with three equal payments at the ends of years 1, 2, and 3. Find the payment and construct the schedule, reporting the table to cents.
Method. First calculate the exact scheduled payment from the present value, then roll the balance forward at full precision. Do not round the balance or interest before calculating the next row.
The annuity factor and payment are
Period 1 begins with $1,000. The interest is \(1{,}000(0.10)=\$100\), principal repaid is \(R-100\approx \$302.1148036254\), and the closing balance is \(1{,}100-R\approx \$697.8851963746\).
Period 2 interest is \(0.10(697.8851963746)\approx \$69.7885196375\). Principal repaid is \(R-69.7885196375\approx \$332.3262839879\). The closing balance is approximately \(697.8851963746+69.7885196375-402.1148036254=\$365.5589123867\).
Period 3 interest is \(0.10(365.5589123867)\approx \$36.5558912387\). The principal repaid is \(R-36.5558912387\approx \$365.5589123867\), and the closing balance is zero before rounding. In particular, the full-precision period-3 principal amount rounds to $365.56.
| Period | Opening balance | Payment | Interest | Principal repaid | Closing balance |
|---|---|---|---|---|---|
| 1 | $1,000.00 | $402.11 | $100.00 | $302.11 | $697.89 |
| 2 | $697.89 | $402.11 | $69.79 | $332.33 | $365.56 |
| 3 | $365.56 | $402.11 | $36.56 | $365.56 | $0.00 |
Check. Using unrounded values, the closing balance after the third payment is exactly zero because \(R\) was calculated to amortize the original principal over three periods. The displayed period-3 payment, interest, and principal do not add to the cent: \(36.56+365.56=402.12\), while the payment rounds to $402.11. This one-cent display difference is a rounding effect. It is not a reason to change the correctly rounded full-precision principal amount.
Payment Frequency and the Periodic Rate
The interest rate used in each row must match the length of that row. For quarterly payments under a nominal annual rate convertible quarterly, each row represents one quarter and the rate per row is the nominal rate divided by four. Applying the annual nominal rate directly to a quarterly opening balance would overstate the interest.
Worked Example: A Two-Payment Loan at a Quarterly Rate
A hypothetical loan of $1,000 is repaid by two equal quarterly payments. The nominal annual rate is 8% convertible quarterly. Payments occur at the end of each quarter, with no fees or extra payments. Construct the schedule to cents.
Method. Convert the nominal rate to the rate for one payment period, determine the two-payment amount, and apply that quarterly rate to each opening balance.
The quarterly rate is \(i^{(4)}/4=0.08/4=0.02\). Thus
In quarter 1, interest is \(1{,}000(0.02)=\$20\). Principal repaid is approximately \(515.0495049505-20=\$495.0495049505\), leaving a closing balance of approximately \(1{,}000+20-515.0495049505=\$504.9504950495\).
In quarter 2, interest is \(0.02(504.9504950495)\approx \$10.0990099010\). Principal repaid is approximately \(515.0495049505-10.0990099010=\$504.9504950495\), which clears the balance.
| Quarter | Opening balance | Payment | Interest | Principal repaid | Closing balance |
|---|---|---|---|---|---|
| 1 | $1,000.00 | $515.05 | $20.00 | $495.05 | $504.95 |
| 2 | $504.95 | $515.05 | $10.10 | $504.95 | $0.00 |
Check. The calculations use the 2% quarterly effective rate, not 8% per quarter. At full precision, \(R a_{\overline{2}|}=1{,}000\), and the final unrounded balance is zero.
Checks and Common Exam Traps
A complete table should be internally consistent, use the stated payment dates and rate period, and show repayment at the end. These checks help catch errors before a schedule is used in a larger financial calculation.
- Use the preceding closing balance. Every row after the first starts with the prior row’s closing balance, not the original principal.
- Match rate and period. For monthly or quarterly rows, convert the quoted rate to the effective rate for one month or one quarter as appropriate.
- Apply interest before allocating the payment. Principal repaid is payment minus interest, not the payment itself.
- Keep precision internally. Carry unrounded values through the calculations, then round displayed dollar amounts to cents. Do not calculate a later row from rounded display values if unrounded values are available.
- Do not force a rounded row to balance by altering one entry. A cent-level discrepancy between displayed values can result from independent rounding. If a contract specifically requires payments rounded to cents, that is a different model: the actual rounded payments may leave a small residual that must be handled according to the stated terms.
A schedule is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it combines the present value of a level payment stream with period-by-period interest accumulation and balance accounting. In a longer problem, the schedule may also serve as a source of information about a balance or payment component; the first requirement is still to build each row from the correct opening balance and periodic rate.
Check Your Understanding
For each problem, assume end-of-period payments, no fees, no extra payments, and a fixed rate over the stated term. Report money to cents.
- A $1,100 loan has an effective annual rate of 10% and is repaid by two equal annual payments. Find the payment and the period-1 interest and closing balance. (Answer: Payment $576.19; period-1 interest $110.00; closing balance $633.81.)
- A $2,000 loan has an effective annual rate of 5% and two equal annual payments. Find the payment and the period-2 interest charge. (Answer: Payment $1,075.61; period-2 interest $51.22.)
- A $1,000 loan has a nominal annual rate of 8% convertible quarterly and two equal quarterly payments. Find the quarterly rate, payment, and period-1 closing balance. (Answer: Quarterly rate 2%; payment $515.05; closing balance $504.95.)
- A hypothetical $3,000 loan is repaid by three equal annual payments at an effective annual rate of 10%. Calculate the scheduled payment and the period-1 principal repaid. (Answer: Payment approximately $1,206.34; period-1 principal repaid approximately $906.34.)
- A loan has an opening balance of $4,800 for a period, an effective rate of 1.5% for that period, and a scheduled payment of $1,000 at the period end. Find the interest, principal repaid, and closing balance. (Answer: Interest $72.00; principal repaid $928.00; closing balance $3,872.00.)