Pay Off a Loan

Where each payment goes

A fixed-rate loan is amortized: paid off by equal monthly payments. The yearly rate is divided by 12 to get the monthly rate i. Each month the lender first charges interest on what you still owe,

interest = balance × i,

and whatever is left of your payment goes to principal, shrinking the balance. Early on the balance is large, so most of the payment is interest. As the balance shrinks, the interest shrinks and more of the same payment goes to principal. Watch the red part of the bars fade year by year.

The payment formula

For an amount A, monthly rate i and n monthly payments, the level payment that brings the balance to exactly zero is

M = A · i / [1 − (1 + i)⁻ⁿ]

It comes from setting the present value of all n payments equal to the amount borrowed: the bank is handing you today’s value of your future payments. At 0% interest it is simply A/n. A spreadsheet’s =PMT does the same calculation.

Why extra payments are so strong

Every extra dollar goes straight to principal, and principal you no longer owe never charges interest again, month after month, for the rest of the loan. That is why a modest extra payment can cut years off a long loan. Before paying extra, check that the loan has no prepayment penalty and tell the lender to apply the extra to principal.

How the table is figured

As on a lender’s statement, the payment is rounded to the cent, each month’s interest is rounded to the cent, and the last payment is whatever is left, so it can differ from the others by a few cents (or by a lot, when extra payments finish the loan early). The page treats the rate as a simple yearly rate with no fees. A loan’s official APR also counts certain fees, so it can be a little higher than the note rate.

Check yourself

1. Find the monthly payment on a $20,000 loan at 6% for 4 years, and the total interest.
   Answer

   i = 0.005 and n = 48, so M = 20,000(0.005) / (1 − 1.005⁻⁴⁸) ≈ $469.70. Forty-eight payments of $469.70 come to $22,545.60, so the interest is about $2,546. (On the lender’s schedule the last payment differs by a few cents; the table gives $2,545.66.) =PMT(6%/12, 48, −20000).

2. A $200,000 mortgage has a 6.6% rate. How much interest is charged in the first month?
   Answer

   200,000 × 0.066/12 = $1,100. If the payment is $1,277.32 (30 years), only $177.32 of the first payment goes to principal.

3. Load the home example (30 years). Then try the 15-year version. Why is the 15-year payment not twice as large, even though it is paid off in half the time?
   Answer

   The payments are $1,896.20 and $2,613.32: only about 1.4 times as much. The 30-year payment is mostly interest at first, so halving the term moves far more of each payment to principal. Total interest falls from $382,636.71 to $170,398.28, less than half.

4. On the 30-year home example, how much interest does $300 extra a month save?
   Answer

   The loan is paid off in 250 months instead of 360, 9 years and 2 months sooner, and the interest falls to $247,519.43: a saving of $135,117.28. The extra payments themselves add up to $74,700 (249 months of $300), so each extra dollar saved about $1.80 of interest.

A bit of history: In 1831 neighbors in Frankford, Pennsylvania, formed the Oxford Provident Building Association, the first building and loan in America. Members paid in a little each month, and the pooled money was lent to members to build or buy homes; the first loan went to a lamplighter, Comly Rich.

“The borrower is servant to the lender.” — Proverbs 22:7