The one formula behind the monthly payment
Lenders choose a payment that stays the same every month and brings the balance to exactly zero on the last one. For a loan of P at a monthly rate r over n months:
M = P × r ÷ (1 − (1 + r)−n)
For this example: P = $20,000, r = 0.08 ÷ 12 = 0.006667 and n = 48, which gives M = $488.26.
Why the first payments are mostly interest
Interest is recalculated every month on whatever you still owe. In month one you owe the full $20,000, so interest is $133.33 and only about $354.93 of your payment reduces the debt. By the last payment the balance is tiny, the interest is a couple of dollars, and nearly the whole payment is principal. Drag the slider above from first to last and watch the bar flip colour.
Dragging a slider and watching the balance react teaches more than any formula; ahaboo is built on that idea too, down to a leaf you can zoom into to follow photosynthesis.
What extra payments actually do
An extra payment skips the interest step entirely: it all goes to principal. That lowers next month's balance, so next month's interest is lower, so more of your regular payment goes to principal too. The effect compounds in your favour. Tap "+$100" above: this loan finishes months early and the saving shows under the bar.
Daily vs monthly interest
Mortgages generally use the monthly method shown here. Many auto and personal loans accrue simple interest daily: balance × annual rate ÷ 365 for each day between payments. The totals end up nearly identical if you pay on the due date, but paying a few days early trims a little interest, and paying late adds some. Credit cards also charge daily, on your average daily balance.
Watch out for precomputed interest
A minority of installment loans, often in subprime auto lending, calculate all the interest at the start and add it to what you owe. Paying early may then save less than you'd expect. The Consumer Financial Protection Bureau suggests asking a lender directly whether a loan uses simple or precomputed interest before you sign.
Try it with your own loan
Put your own numbers in the loan calculator or the amortization calculator, and use the loan payoff calculator to see what extra payments would save you.