Interest and the power of compounding

Lina and her grandfather each put €1,000 into savings accounts that pay 5% a year. Every year Lina takes her interest out and spends it. Her grandfather leaves his in. After 10 years Lina has received €500. Her grandfather’s account holds €1,629. Where did his extra €129 come from, and why does the gap keep growing?

Interest is the price of using money

Interest is the money paid for using someone else’s money. When you save, the bank pays you interest. When you borrow, you pay it. The interest rate is usually given per year.

With simple interest, interest is always calculated on the original amount only:

interest = amount × rate × years

Lina’s interest is 1,000 × 0.05 × 10 = €500: €50 every year, never more.

Compound interest: interest on interest

Her grandfather earns compound interest. Each year the interest is added to his balance, so next year’s 5% is taken from a bigger amount. Adding interest to the balance so that it earns interest too is called compounding.

Each year his balance is multiplied by 1.05 (that is, 100% + 5%):

The small number ¹⁰ means “multiply by 1.05 ten times”. So the shortcut is: final amount = amount × (1 + rate)^years. Any calculator or phone can do it.

Compounding works the same way on debts. If you owe €1,000 on a card at 1.5% a month and pay nothing, after 2 years you owe about 1,000 × 1.015²⁴ = €1,430.

How often interest is added matters

Interest can be added once a year, every month or even every day. The more often it is added, the sooner it starts earning interest of its own.

Take a “nominal” (stated) rate of 12% a year, added monthly. That means 1% each month. Over a year: 1.01¹² = 1.1268, so the true growth is 12.68%, not 12%. This true yearly rate is called the effective annual rate. On savings accounts it often appears as the APY or AER.

This is why a headline rate can mislead. A bank paying 3.95% added daily gives an effective rate of about 4.03%. That beats another bank paying 4.00% added once a year. To compare two offers, compare their effective yearly rates.

The Rule of 72: how fast does money double?

The Rule of 72 is a quick estimate:

years to double ≈ 72 ÷ yearly rate in %

The rule is very close between about 6% and 10%. At low rates, 70 works a little better: at 1%, money doubles in about 70 years, not 72. At very high rates the rule drifts further off. It also works for prices: at 4% inflation, prices double in about 18 years. It assumes the rate stays the same and all interest stays in.