How an amortization schedule works

A schedule has one row for each installment, and six columns. Take a loan of $25,000 at 6.00% a year, repaid in 36 monthly payments. The first row reads: installment 1, due one month after funding, payment $760.55, interest $125.00, principal $635.55, balance $24,364.45.

• Number. The installment’s position in the sequence, 1 to 36.
• Due date. The day the payment is due. Monthly loans fall on the same day each month.
• Payment. The total the borrower pays that period.
• Interest. The part of the payment that is the cost of borrowing for that period.
• Principal. The part that reduces the amount owed.
• Balance. The principal still owed after that payment.

Interest plus principal always equals the payment. Each row’s balance is the previous balance less that row’s principal. The last row’s balance is zero. Read down the interest column and it falls every month. Read down the principal column and it rises. That is what amortization means: the debt is paid down gradually, and the mix inside each payment shifts as it goes.

The rate on a note is an annual rate. For a monthly loan, each month’s interest is one twelfth of the annual rate applied to the balance outstanding at the start of that month. At 6.00% a year, the monthly rate is 0.50%. On a balance of $25,000, the first month’s interest is $25,000 × 0.005, or $125.00.

After the first payment the balance is $24,364.45, so the second month’s interest is $24,364.45 × 0.005, which rounds to $121.82. The interest is recomputed every period on whatever is still owed. It is never charged on the original amount after the balance has fallen. Each figure is rounded to the cent when it is posted, which is why the final payment needs a small adjustment, described below.

An amortizing loan has the same payment every period. That payment is the one amount which, paid 36 times with interest charged monthly on the falling balance, brings the balance to exactly zero on the last row. There is a closed formula for it. In words: multiply the principal by the periodic rate, then divide by one minus the quantity one plus the periodic rate raised to the negative power of the number of payments.

For $25,000 at 0.50% a month over 36 payments, that gives $760.55. You do not need to compute it by hand. The loan calculator returns the payment and the full schedule for any amount, rate and term.

Three things move the payment. A larger principal raises it in proportion. A higher rate raises it, more than proportionally over long terms. A longer term lowers it, but increases the total interest paid, because the balance stays higher for longer.

In the first month of the example, $125.00 of the $760.55 payment is interest and $635.55 is principal. By the last month, interest is $3.78 and principal is $756.67. The payment did not change. What changed is the balance it is charged on.

Interest is the rent on the money still borrowed. Early on, nearly all of the original $25,000 is still borrowed, so the rent is high and little is left over for principal. As principal is repaid, the rent falls and the leftover grows. On a three-year loan at 6.00%, the crossover where principal first exceeds interest happens in month 1. On a thirty-year mortgage at the same rate, it does not happen until well past the midpoint. The longer the term, the longer the interest-heavy stretch.

Total interest on the example is about $2,379.80, which is the sum of the interest column. Total payments come to about $27,380, which is the principal plus that interest.

The schedule assumes every payment arrives on its due date. When one does not, three things happen, and none of them is automatic in the table itself.

• Interest keeps running on the full balance. The borrower has not reduced the principal, so the next period’s interest is charged on a balance that should have been lower.
• A late fee may apply once the grace period in the note has passed. It is either a flat amount or a percentage of the overdue installment, and it is paid before anything else.
• The schedule does not change by itself. The missed row stays due. The borrower is now behind by one payment, plus the fee, and stays behind until it is caught up.

A loan record that posts interest as it falls due, records each payment as it arrives, and shows the arrears separately from the schedule is the only way to know the exact position. The guide to what to do when a borrower pays late covers the practical steps.

A payment larger than the scheduled amount, after fees and the interest due are covered, goes to principal. That lowers the balance below what the schedule expected, so every later interest figure is smaller than planned. Two conventions exist for what happens next, and the note should say which applies.

• Keep the payment, shorten the term. The borrower continues to pay $760.55. Because interest is lower, more of each payment is principal, and the loan ends early. Total interest falls the most under this convention.
• Keep the term, lower the payment. The remaining installments are recomputed on the new balance so the loan still ends on the original maturity date. Each payment is smaller. Total interest falls, but by less.

An extra $5,000 paid with the first installment of the example reduces the balance to $19,364.45. Under the first convention, the loan ends about seven months early and the borrower saves several hundred dollars of interest. Under the second, the payment drops to roughly $604 and the saving is smaller. Check the note’s prepayment clause before paying ahead, because some notes charge a fee for it.

Because each period’s interest is rounded to the cent, the level payment does not land on exactly zero after 36 rows. The difference is a few cents in either direction. The last row absorbs it: its payment is set to whatever clears the balance exactly, so on the example the final payment is $760.45 instead of $760.55.

That is why a schedule is trued up at the end rather than carrying a fractional balance. A loan is paid when the balance is zero, not when the last scheduled payment has been made. The two are the same only if nothing was ever late or early.

The method is the same for quarterly, semiannual or annual payments. The period rate is the annual rate divided by the number of periods in a year: 1.50% a quarter, 3.00% a half-year, 6.00% a year at a 6.00% annual rate. The payment is found with the same formula using that period rate and the number of periods.

A three-year loan paid quarterly has 12 rows. A loan paid once at maturity has one row and no amortization at all. Interest on that kind of loan accrues by the day, at the annual rate divided by 365, on the full principal until it is repaid. Less frequent payments mean the balance stays higher for longer between payments, so total interest is slightly higher than for the same loan paid monthly.