Compound interest, worked out step by step

Simple interest is paid only on the money you put in. Compound interest is paid on the money you put in and on the interest it has already earned. The difference looks small at first and becomes large over time.

This guide works through the arithmetic one step at a time, so you can check the numbers yourself with nothing more than a phone calculator. The examples are illustrations of how the maths works, not predictions of what any real account or investment will pay.

A worked example

Suppose you save 1,000 at 5 percent a year, compounded annually.

• After year 1: 1,000 × 1.05 = 1,050
• After year 2: 1,050 × 1.05 = 1,102.50
• After year 3: 1,102.50 × 1.05 = 1,157.63

After 10 years the balance is about 1,628.89. With simple interest, you would have 1,500. The extra 128.89 is interest earned on interest.

The general formula is: final amount = starting amount × (1 + rate) ^ number of years.

Doing it yourself

To use the formula on a calculator:

1. Turn the rate into a decimal. 5 percent becomes 0.05.
2. Add one. 1 + 0.05 = 1.05. This is the growth factor for one year.
3. Raise it to the number of years. For 10 years, 1.05 ^ 10 is about 1.6289.
4. Multiply by your starting amount. 1,000 × 1.6289 gives about 1,628.89.

Now stretch the time. Left alone at the same rate, that 1,000 becomes about 2,653 after 20 years and about 4,322 after 30 years. Notice that the balance grows by roughly 1,000 in the second decade and by almost 1,700 in the third, even though the rate never changed. That accelerating curve is the signature of compounding.

How often interest is added

Many accounts add interest monthly or daily rather than once a year. The more often interest is added, the sooner it starts earning interest of its own. The formula becomes: final amount = starting amount × (1 + rate ÷ n) ^ (n × years), where n is the number of times interest is added each year.

For 1,000 at 5 percent for 10 years, compounding monthly gives about 1,647, compared with about 1,629 for annual compounding. The difference is real but modest. This is why savings products often quote an effective annual rate, which already takes compounding into account and makes it easier to compare like with like. The exact name for that figure varies between countries.

Adding money regularly

Most people save a little at a time rather than in one lump. Suppose you add 100 at the end of every month for 10 years, earning 5 percent a year compounded monthly. You would pay in 12,000 in total, and the balance would reach about 15,500. The roughly 3,500 difference is interest, and a growing share of it is interest on interest.

The rule of 72

A quick way to estimate how long money takes to double: divide 72 by the annual interest rate. At 6 percent, money doubles in about 12 years; at 8 percent, about 9 years. It is an approximation, but a useful one.

The same trick works for inflation. If prices rise at 3 percent a year, they double in roughly 24 years, which means money kept as cash loses about half of its buying power over that period.

It works against you too

Debt compounds in the same way. A credit card balance at a high annual rate, left unpaid, can grow surprisingly fast, because interest is charged on previous interest. As an illustration, a 1,000 balance at an annual rate of 20 percent, compounded monthly with nothing paid off, would grow to about 1,219 in a year. Real cards have their own rules, charges and minimum payments, so the exact figures will differ. Paying more than the minimum each month is one of the most effective ways to stop that growth.

Common misconceptions

• “A higher rate always means a better deal.” Not if fees, charges or conditions eat into it. Compare the effective return after costs.
• “The early years don’t matter much.” They matter most, because money added early has the longest time to compound.
• “Growth is guaranteed.” Compounding at a fixed rate applies to things like savings accounts. Investments such as shares do not grow at a steady rate and can fall in value.

Three things that matter most

1. Time. Starting earlier gives compounding more years to work.
2. Rate. Small differences in the rate make large differences over decades.
3. Fees. Annual fees on savings or investments compound too, in the wrong direction.

To see how much fees matter, compare 10,000 growing at 5 percent a year for 30 years, which reaches about 43,200, with the same sum growing at 4 percent because a 1 percent annual fee is deducted. That reaches only about 32,400. A fee that sounds small takes roughly a quarter of the final balance.

What it means for you

The maths is certain; the inputs are not. Future interest rates, investment returns and inflation cannot be known in advance, so any long-range projection is only as good as its assumptions. Free calculators from official bodies, such as the one linked below, let you try your own numbers. This guide is general information, not personal financial advice.