How Interest Works: Simple, Compound, and Loan Payments

Simple interest grows in a straight line

Simple interest is calculated only on the original principal: interest = principal × annual rate × years. $5,000 at 4.5% for three years earns $675, the same $225 each year.

It is used for some short-term loans, certain bonds, and as a quick estimate. Because interest is never added to the balance, it understates growth whenever interest would really be reinvested.

Compound interest earns interest on interest

With compound interest, each period's interest is added to the balance, and the next period's interest is calculated on the larger amount. The balance after t years is P × (1 + r/n)^(n × t), where r is the annual rate and n is how many times a year interest is added.

At 5% a year, $10,000 grows to about $16,289 after ten years with annual compounding — roughly $1,289 more than simple interest would give. The gap widens every year, which is why time is the most powerful input in any savings calculation.

Compounding frequency matters, but less than you'd think

Compounding more often helps, with diminishing returns. At 5% for ten years, moving from annual to monthly compounding raises $16,289 to about $16,470; daily or continuous compounding adds only another $17 or so.

This is why banks quote an annual percentage yield (APY): it folds the compounding frequency into a single effective annual rate, making accounts with different frequencies easy to compare.

How a loan payment is built

A standard installment loan is designed so that every monthly payment is the same and the balance reaches exactly zero at the end. The payment is P × i ÷ (1 − (1 + i)^−n), where i is the monthly rate (the annual rate ÷ 12) and n is the number of payments.

Each payment first covers the interest on the current balance; the rest reduces the principal. Early on, interest takes a large share. As the balance falls, the interest portion shrinks and more of each payment goes to principal. That is why extra payments early in a loan save the most interest.

What calculators usually leave out

• Fees: origination fees, account fees, and closing costs change the true cost of borrowing.
• Taxes and insurance, which are often bundled into mortgage payments.
• Variable rates: most calculators assume the rate stays fixed for the whole term.
• Rounding: lenders round each payment to the cent, so the final payment is often slightly different.