The Two Payment Questions
A fixed-rate loan links four quantities: the amount borrowed, the interest rate, the payment, and the number of payments. Fix any three and the fourth follows. In practice people arrive with one of two questions.
"What will my payment be?" — you know the amount, rate and term, and need the payment. This is the standard amortization formula.
Payment = P × i ÷ (1 − (1 + i)−n)
"How long until this is paid off?" — you know what you can afford and want the term. This requires rearranging for n, which needs a logarithm:
n = −log(1 − P×i ÷ PMT) ÷ log(1 + i)
In both, P is the loan amount, i the monthly rate (annual rate divided by 12) and n the number of monthly payments.
The Payment Must Exceed the Interest
Look at the second formula: if PMT is less than or equal to P × i, the logarithm's argument becomes zero or negative and there is no solution. This is not a mathematical curiosity — it describes a real and common situation. A $20,000 balance at 6.5% accrues about $108 of interest in the first month. Paying $100 means the balance grows, and the loan never ends.
This is the trap behind credit card minimum payments, which are often set at 1–2% of the balance plus interest. The balance falls, but so slowly that a $5,000 debt can take over twenty years to clear.
How the Payment Changes the Term
A $20,000 loan at 6.5%:
| Monthly payment | Payoff time | In years | Total interest |
|---|---|---|---|
| $300 | 82.9 months | 6.9 years | $4,881 |
| $400 | 58.5 months | 4.9 years | $3,388 |
| $500 | 45.2 months | 3.8 years | $2,602 |
| $600 | 36.9 months | 3.1 years | $2,117 |
| $800 | 26.9 months | 2.2 years | $1,549 |
| $1000 | 21.2 months | 1.8 years | $1,226 |
The relationship is strongly non-linear. Raising the payment from $300 to $400 saves nearly three years and over $1,300 in interest; raising it from $800 to $1,000 saves only five months. The early increases are worth far more, which is why even a modest addition to a struggling payment has an outsized effect.
Where the Payment Goes
Every payment splits into interest, calculated on the current balance, and principal, which is the remainder. Because the balance falls each month, the interest portion shrinks and the principal portion grows. On a long loan this makes the early years feel unproductive — on a 30-year mortgage at 6.5%, roughly 78% of the first payment is interest, and it takes about 18 years before principal exceeds interest in a single payment.
A practical consequence: extra payments early are worth far more than extra payments late, because each dollar of principal removed cancels every future interest charge that dollar would have generated.
The Last Payment Is Usually Smaller
Because the number of payments must be a whole number and the payment is rounded to cents, the final payment rarely lands exactly on zero. Lenders normally adjust the last payment up or down by a few dollars. The schedule above shows this explicitly rather than hiding the rounding.
What Changes the Payment
| Change | Effect on payment | Effect on total interest |
|---|---|---|
| Borrow less | Falls proportionally | Falls proportionally |
| Longer term | Falls | Rises substantially |
| Lower rate | Falls | Falls substantially |
| Pay extra each month | Unchanged | Falls, and the term shortens |
| Biweekly instead of monthly | Effectively 13 payments a year | Falls, typically shortening a 30-year term by 4–5 years |
Frequently Asked Questions
Why is my actual payment higher than this?
This calculates principal and interest only. A mortgage payment usually also includes property tax and insurance held in escrow, and may include PMI and HOA fees. A car payment may include financed fees.
Does paying extra reduce my required payment?
Normally no. Extra principal shortens the term rather than lowering the payment. Some lenders offer a "recast" that recalculates the payment on the reduced balance, usually for a fee.
Is it better to shorten the term or pay extra on a longer one?
Paying extra on a longer term gives almost the same interest saving with the flexibility to fall back to the lower required payment if circumstances change. A shorter term usually carries a lower rate, which is the trade-off.
What payment can I afford?
Lenders generally look for total debt payments under 36–43% of gross monthly income, and housing costs under 28%. Those are ceilings, not targets.