How Interest Works: Compounding, APR and Loans

Simple versus compound interest

Simple interest is paid only on the original amount: 10,000 at 5% a year earns 500 every year, 25,000 in total after 30 years. Compound interest is paid on the balance, including interest already earned, so the same 10,000 grows to 10,000 × 1.05³⁰ = 43,219 after 30 years. In the first year the two are identical; the gap comes entirely from time, which is why the years you leave money alone matter more than almost anything else.

A quick way to feel this is the rule of 72: divide 72 by the annual rate to get roughly how many years it takes to double. At 5% that is 72 ÷ 5 = 14.4 years; the exact figure is ln 2 ÷ ln 1.05 = 14.2. The rule works for debts too: a credit card balance at 24% doubles in about three years if nothing is paid.

Nominal rate, effective rate and APR

A nominal annual rate compounded more often than once a year earns slightly more than it says. At 5% compounded monthly, each month adds 5 ÷ 12 percent, and over a year that comes to (1 + 0.05 ÷ 12)¹² − 1 = 5.116%. That is the effective annual rate. The difference is real but small, so the rate itself and the time invested matter far more than the compounding frequency a bank advertises.

For loans, the annual percentage rate (APR) folds in fees such as arrangement charges, so it is usually higher than the interest rate used to work out the payment. When comparing loans, compare APRs; two loans with the same rate but different fees do not cost the same. Rules on what APR must include differ between countries, so read what a lender's figure covers.

How a loan payment is built

Most mortgages and car loans use equal monthly payments. The payment is P × r ÷ (1 − (1 + r)⁻ⁿ), where P is the amount borrowed, r the monthly rate and n the number of months. For 300,000 at 6% over 30 years, r is 0.5% and n is 360, giving 1,798.65 a month: 647,515 paid in total, 347,515 of it interest.

The payment stays fixed, but its make-up does not. In the first month, interest on 300,000 at 0.5% is 1,500, so only 298.65 reduces the balance. After ten years of payments the balance is still about 251,057: a third of the term has gone, but only a sixth of the loan. This is why an early extra payment, which goes straight to the balance, saves so much interest later.

Ways to pay less interest

A shorter term raises the payment but cuts interest sharply. The same 300,000 at 6% over 15 years costs 2,531.57 a month and 155,683 in interest, less than half of the 30-year figure. Paying 200 extra every month on the 30-year loan instead clears it in about 279 months, a little over 23 years, and saves roughly 91,000 in interest.

Some lenders also offer equal principal repayments: the same slice of principal every month plus interest on what is left. The first payment is higher (2,333.33 in the example), but payments fall every month and total interest drops to 270,750. Before paying early, check for a prepayment fee, and compare the loan's rate with what the money could safely earn elsewhere.

Inflation and the real return

A return has to be set against inflation to show what it buys. The real return is (1 + nominal) ÷ (1 + inflation) − 1, so 5% in a year with 3% inflation is a real gain of about 1.94%, not 2%. Over long periods the gap compounds just like interest, and an account paying less than inflation is losing purchasing power even while its balance grows.

Taxes work the same way: interest is often taxed each year, which lowers the rate that actually compounds. A calculator that ignores tax and inflation shows an upper bound, useful for comparing options, not a promise of what the money will be worth.

• Compare loans by APR, not by the headline rate
• Check whether a rate is nominal or effective, and how often it compounds
• Look at the amortisation table, not just the monthly payment
• Extra payments early in a loan save the most interest
• Ask about prepayment fees before paying early
• Subtract inflation and tax before judging a return