Compound Interest

With compound interest, each period earns interest on everything so far, including past interest. Growth snowballs.

Secondary
Advanced
Updated 14 November 2026

What you will learn

Calculate compound interest year by year.
Use the multiplier method.
Compare compound and simple interest.

Before you start: you should be able to calculate simple interest and use powers. See Interest and Percentages.

The idea

With simple interest, you earn the same amount every year. With compound interest, interest is added to your balance, and next year you earn interest on that larger balance. You earn interest on your interest.

Each year the balance is multiplied by the same factor. At $5\%$ interest, that factor is $1.05$ . After $t$ years:

$A = P(1 + r)^t$

Because the growth is multiplicative, compound interest is an example of exponential growth. Slow at first, it accelerates over time.

Simple and compound interest on £1,000 at 10%. Compound interest pulls ahead because interest earns interest.

Compounding frequency

Interest can be added yearly, monthly or even daily. If the annual rate $r$ is compounded $n$ times a year, each period uses rate $\frac{r}{n}$ and there are $nt$ periods:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

More frequent compounding gives slightly more growth. As $n$ grows without limit, the formula approaches $Pe^{rt}$ , where $e \approx 2.718$ is the famous constant from calculus.

£1,000 growing at 10% a year. The line gets steeper because each year's interest is bigger than the last.

Worked examples

Example 1: Annual compounding

£1,000 is invested at $5\%$ compound interest per year. What is it worth after $3$ years?

The multiplier is $1.05$ .
$A = 1000 \times 1.05^3 = 1000 \times 1.157625 = 1157.63$ .
Compare: simple interest would give $1000 + 3 \times 50 = £1,150$ .

Answer: £1,157.63, which is £7.63 more than simple interest.

Year	Start	Interest (5%)	End
1	£1,000.00	£50.00	£1,050.00
2	£1,050.00	£52.50	£1,102.50
3	£1,102.50	£55.13	£1,157.63

Example 2: Monthly compounding

£5,000 is saved at $4.8\%$ per year compounded monthly for $10$ years. Find the final amount.

$r = 0.048$ , $n = 12$ , so the monthly rate is $0.048 \div 12 = 0.004$ .
Number of periods: $12 \times 10 = 120$ .
$A = 5000 \times 1.004^{120} \approx 5000 \times 1.6150 \approx 8075$ .

Answer: about £8,075

Example 3: How long to double?

Roughly how long does money take to double at $6\%$ a year? Check with the formula.

Rule of 72: $72 \div 6 = 12$ years.
Check: $1.06^{12} \approx 2.012$ . ✓

Answer: about $12$ years.

Example 4: Depreciation

A £20,000 car loses $15\%$ of its value each year. What is it worth after $4$ years?

Losing $15\%$ means keeping $85\%$ , a multiplier of $0.85$ .
$20\,000 \times 0.85^4 \approx 20\,000 \times 0.5220 = 10\,440$ .

Answer: about £10,440. Compounding works downwards too.

Compounding also works against borrowers. Unpaid credit card balances compound, which is why small debts can grow surprisingly quickly.

Why it matters

Compound interest is why starting to save early matters so much: money invested at 20 has decades longer to snowball than money invested at 40. It explains how pensions grow, how inflation erodes the value of cash, why population and viral growth can explode, and why long-term debt can cost far more than the original loan. Try different rates and terms with our compound interest calculator.

This lesson is educational and is not financial advice.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.