How Compound Interest Works — A Complete Guide

A saver puts £1,000 into an account paying 5 percent interest a year. At the end of the first year the account holds £1,050: the original £1,000 plus £50 of interest. In the second year the 5 percent is calculated not on £1,000 but on £1,050, so the interest is £52.50. In the third year it is calculated on £1,102.50, and it is £55.13. Each year's interest is added to the balance, and the next year's interest is calculated on the larger balance. That is compound interest: interest earned on interest, as well as on the original amount.

This guide explains what compound interest means, how it differs from simple interest, how the formula works term by term, and what compounding frequency does, with three worked examples including the Rule of 72, and the mistakes to avoid.

What compound interest means

Compound interest is interest calculated on the balance as it stands at the start of each period, after earlier interest has been added. In the first period the balance is just the starting amount, so compound and simple interest give the same answer. From the second period onwards they part company, because the compound balance now includes the first period's interest. Growth is steady as a percentage, so the pounds added get bigger as the balance grows. Over short periods the difference is small; over long periods it is large.

How compound interest differs from simple interest

Simple interest is calculated only on the original amount, called the principal, for the whole period. At 5 percent a year on £1,000 it is £50 every year, and £150 over three years. The simple interest guide on this site works through that method in full. The two methods share the same starting point and rate; only the balance they work from is different.

Compound interest recalculates each period on the new balance: £50 in year one, £52.50 in year two, and £55.13 in year three. After three years the compound balance is £1,157.63, while the simple-interest balance would be £1,150. The gap is only £7.63 after three years, but it grows every year, and after ten years at a higher rate it becomes hundreds of pounds, as the second worked example shows.

The formula, term by term

The compound interest formula is A = P(1 + r)^t. A is the final amount: the balance at the end, including the original money and all interest added. P is the principal: the amount at the start. The letter r is the interest rate for one compounding period, written as a decimal: 5 percent becomes 0.05, and 10 percent becomes 0.10. The letter t is the number of compounding periods, usually the number of years when interest is compounded yearly. The brackets mean 1 + r is worked out first, and the small t means the result is multiplied by itself t times.

Take £2,000 at 4 percent a year, compounded yearly, for 2 years. Here P is 2,000, r is 0.04, and t is 2, so A = 2,000 × (1.04)^2. Working it out: 1.04 squared is 1.0816, and 2,000 times 1.0816 is £2,163.20. The interest earned is £163.20. Simple interest would be £160, giving £2,160. The compound total is £3.20 higher, and that £3.20 is exactly the second year's interest on the first year's £80: 4 percent of £80 is £3.20. Every compound calculation can be understood as a chain of single-year calculations, and the formula is a short way of writing the whole chain at once.

What compounding frequency does

Compounding frequency is how often interest is added to the balance: yearly, every six months, monthly, or daily. The more often interest is added, the sooner it starts earning interest of its own, so a higher frequency gives a slightly larger final amount at the same headline rate. The headline rate is called the nominal rate. The rate actually earned over a full year, once compounding is taken into account, is called the effective rate, and it is always a little higher than the nominal rate when compounding happens more than once a year.

Consider £1,000 at a nominal rate of 12 percent a year. Compounded yearly, the balance after one year is £1,120. Compounded monthly, the monthly rate is 1 percent, and the balance is £1,000 × (1.01)^12, which is £1,126.83. The interest is £126.83, which is £6.83 more, and the effective rate is 12.68 percent. The rate itself and the length of time matter far more than frequency: moving from yearly to monthly compounding changes the answer far less than a higher rate or more years.

Worked example 1: £1,000 at 5 percent for 3 years

The starting amount is £1,000, the rate is 5 percent a year, compounded yearly for 3 years. Working year by year, with figures rounded to the nearest penny: in year 1, the interest is 5 percent of £1,000, which is £50, and the balance becomes £1,050. In year 2, the interest is 5 percent of £1,050, which is £52.50, and the balance becomes £1,102.50. In year 3, the interest is 5 percent of £1,102.50, which is £55.13, and the balance becomes £1,157.63.

The formula gives the same answer in one step: A = 1,000 × (1.05)^3. The value of 1.05 cubed is 1.157625, and 1,000 times that is £1,157.63 to the nearest penny. The total interest is £157.63, against £150 of simple interest, so compounding has added £7.63: £2.50 from the second year's interest on the first year's £50, and £5.13 from the third year's interest on the £102.50 already added.

Worked example 2: £1,000 at 10 percent for 10 years

The starting amount is £1,000, the rate is 10 percent a year, compounded yearly, for 10 years. The formula gives A = 1,000 × (1.10)^10. The value of 1.10 to the power of 10 is 2.593742, so the final balance is £2,593.74 to the nearest penny, and the total interest is £1,593.74.

Milestones show the acceleration. After 1 year the balance is £1,100. After 5 years it is £1,610.51. After 10 years it is £2,593.74. The second five years add £983.23, while the first five add £610.51 — the same rate, applied to a much larger balance. Simple interest would give £100 a year, £1,000 in total, and a final balance of £2,000. Compounding adds £593.74 on top of that. Time is the most powerful ingredient in compounding: the extra interest needs years in which to earn interest of its own.

Worked example 3: the Rule of 72

The Rule of 72 is a shortcut for estimating how long money takes to double at a given compound rate. Divide 72 by the percentage rate, and the answer is roughly the number of years needed. At 6 percent a year, 72 divided by 6 is 12, so money takes about 12 years to double. Checking with the formula: £1,000 × (1.06)^12 is £2,012.20, close to double and slightly over, because the rule is an approximation. At 8 percent the rule gives 9 years, and the exact doubling time is 9.01 years. At 12 percent the rule gives 6 years, while the exact time is 6.12 years.

The rule works best for rates between about 6 and 10 percent, where it is usually within a few months of the exact answer. Outside that range it drifts, so for very low or very high rates the formula should be used instead. Its real value is as a sense-check: if a claim says money doubles in 5 years at 6 percent, the rule says 12 years, and the claim can be dismissed at once. The rule also works in reverse: dividing 72 by the number of years gives the approximate rate needed to double in that time.

Common mistakes to avoid

Confusing simple and compound interest is the commonest error, and it usually comes from using the simple-interest formula — principal times rate times time — for a compound problem. That formula is correct only when interest is never added to the balance. The difference is small over one or two years and large over ten.

Ignoring compounding frequency is the second error. A rate quoted as 12 percent a year does not tell the whole story until the frequency is known: compounded monthly, it earns an effective 12.68 percent. Comparing two accounts by headline rates alone, without checking frequency, compares unlike with unlike.

Mixing up nominal and effective rates is the third error. The nominal rate is the headline figure; the effective rate is what is actually earned over a year after compounding. The two are identical only when interest is compounded once a year.

Forgetting that interest joins the balance each period is the fourth error. It shows up as calculating every year's interest from the original principal, which is simply simple interest in disguise. The test is easy: if the interest figure is the same every year, the calculation is not compounding. In a genuine compound calculation it rises every period — £50, then £52.50, then £55.13.

Putting the percentage straight into the formula instead of the decimal is the fifth error, and the most dramatic. The formula needs r as a decimal, so 5 percent is 0.05. Putting 5 in its place gives 1,000 × (6)^3, which is £216,000. At 5 percent, money should roughly double in about 14 years by the Rule of 72, not multiply by 216 in three.

Frequently asked questions

Does compounding work the same way for money owed?

Yes, the mathematics is identical: interest charged on a balance that already includes earlier interest grows in the same accelerating way. That is why a debt left unpaid can grow faster than the headline rate suggests. The formula does not care whether the balance is savings or debt.

What happens if the interest rate changes part-way through?

The formula is applied in stages, one stage per rate. Work out the balance at the end of the first rate's period, then use that balance as the starting amount for the next rate. For example, £1,000 at 5 percent for 2 years becomes £1,102.50, and if the rate then rises to 6 percent, the third year is calculated on £1,102.50 at 6 percent.

Is monthly compounding always better for a saver than yearly compounding?

At the same nominal rate, yes, but only slightly: the 12 percent example earned £6.83 more in a year compounded monthly than yearly. In practice the headline rate matters far more than the frequency, and an account paying 12.5 percent compounded yearly beats one paying 12 percent compounded monthly.

Can the Rule of 72 be used for any interest rate?

It can be used as a rough estimate for any rate, but it is most accurate between about 6 and 10 percent. At very low rates it overstates the doubling time, and at very high rates it understates it, so for rates far outside that range the formula should be used instead.

This guide is general maths education, not financial advice. If you want to check a result, try the matching tools on this site: the Percentage Calculator handles the rate conversions the formula needs, and the Scientific Calculator handles the powers in the formula. The simple interest guide is the companion to this one, and reading the two together makes the difference between the methods clear.