Compound interest: the formula and a £100-a-month example
Compound interest is not complicated and it is not a trick. It is one sentence: the interest you earned last period earns interest this period. Everything else on this page follows from that sentence, including the part at the end where the interest is larger than everything you paid in.
I am going to use 4.5% AER throughout. It is an illustrative rate, chosen because it is a plausible round-ish figure, and nothing in the method depends on it. Substitute your own and every step still works. If the word “rate” itself needs a foundation, the Bank of England’s explainer on what interest rates are is the shortest honest account of it.
Start with one pound, then one thousand
Put £1,000 into an account paying 4.5% a year.
After one year you have £1,000 plus 4.5% of £1,000, which is £1,045.00.
After two years you earn 4.5% on £1,045.00, not on £1,000. That is £47.02 rather than £45.00, giving £1,092.02.
After three years you earn 4.5% on £1,092.02, giving £1,141.17.
The whole of compound interest is in the gap between £45.00 and £47.02. That is £2.02, and it looks like nothing. Over thirty years the same £1,000 becomes £3,745.32, and the account has been given no further money at any point.
Writing what we just did as a formula:
A = P x (1 + r)^n
where P is what you put in, r is the rate as a decimal, n is the number of years, and A is what you end up with. Check it: 1000 x 1.045^30 = 3,745.32.
Now the version that matches how people actually save
Almost nobody deposits a lump sum and stops. They pay in monthly. That needs two adjustments.
First, convert the annual rate into a monthly one. Not by dividing by twelve. Dividing by twelve would ignore the compounding within the year, which is the entire subject. The monthly rate that compounds twelve times into 4.5% is:
i = (1 + r)^(1/12) – 1 = (1.045)^(1/12) – 1 = 0.00367481
That is 0.3675% a month. Multiply it by twelve and you get 4.41%, which is less than 4.5%, and the missing 0.09 percentage points is exactly the compounding.
Second, add up a stream of payments rather than one. Each £100 you pay in sits in the account for a different length of time. The first payment compounds for 359 months, the last for none. Summing that series gives the future value of a regular payment:
FV = PMT x [ ((1 + i)^n – 1) / i ]
where PMT is the monthly payment, i is the monthly rate, and n is the number of months.
The worked example: £100 a month
Put £100 in at the end of every month, at 4.5% AER, and here is what happens.
| After | Paid in | Value | Of which interest | Interest as a share |
|---|---|---|---|---|
| 1 year | £1,200 | £1,224.55 | £24.55 | 2.0% |
| 5 years | £6,000 | £6,699.18 | £699.18 | 10.4% |
| 10 years | £12,000 | £15,047.57 | £3,047.57 | 20.3% |
| 15 years | £18,000 | £25,451.18 | £7,451.18 | 29.3% |
| 20 years | £24,000 | £38,415.98 | £14,415.98 | 37.5% |
| 25 years | £30,000 | £54,572.48 | £24,572.48 | 45.0% |
| 30 years | £36,000 | £74,706.41 | £38,706.41 | 51.8% |
Let us check the thirty-year row by hand, because a formula you have not verified once is a formula you do not own.
n = 360 months. (1.00367481)^360 = 3.745318. Subtract 1 to get 2.745318. Divide by 0.00367481 to get 747.064. Multiply by £100: £74,706.41.
The two things the table is actually telling you
First: the interest overtakes the contributions. After one year interest is 2.0% of the balance. After thirty it is 51.8%. Somewhere in the late twenties the account crosses the line where more than half of it is money nobody paid in.
Second: almost all of that is the compounding, not the interest rate. Suppose the interest never compounded, so each £100 simply earned 4.5% a year for however long it sat there, with no interest on interest. The same 360 payments would reach £60,232.50.
The difference is £14,473.91. That is what “interest on interest” is worth over thirty years at 4.5%, and it is the only number on this page that is unique to compounding.
What thirty years of inflation does to it
This is where a lot of compound interest teaching stops, and stopping here is misleading.
£74,706.41 in thirty years is not £74,706.41 in today’s money. To find what it is worth you run the same formula backwards:
Real value = A / (1 + inflation)^n
The Bank of England’s inflation target is 2%. At 2% for thirty years, £74,706.41 is worth £41,243.23 in today’s money. At 2.9%, which is where the ONS put CPI in the 12 months to July 2026, it is worth £31,688.13.
Those are very different numbers and the difference is 0.9 percentage points of inflation. A rate only means something next to the inflation rate beside it, which is the subject of the next lesson in this section.
The rule of 72, and when it fails
A useful shortcut: divide 72 by the interest rate to get the approximate years to double. At 4.5%, that is 72 / 4.5 = 16 years.
The exact answer is ln(2) / ln(1.045) = 15.75 years. The rule is out by three months, which is fine.
It stays accurate for rates roughly between 4% and 12%. At 1% it tells you 72 years when the answer is 69.7. At 30% it tells you 2.4 when the answer is 2.64. Know the range it works in.
Why starting earlier beats paying more
Run the same £100 a month, but start five years later and stop at the same point. You pay in £30,000 instead of £36,000 and you finish with £54,572.48 instead of £74,706.41.
The five years cost £20,133.93, and only £6,000 of that is money you did not pay in. The other £14,133.93 is growth those five years of payments would have earned.
This is the one genuinely actionable thing in the mathematics of saving, and it is why the first years feel so pointless. In year one the interest is £24.55. It is meant to look like that.
None of which says you should be saving £100 a month, or that you can. That is a question about your own budget rather than about arithmetic, and MoneyHelper’s budget planner is a better place to answer it than this page.
Summary box
- Compound interest is interest earning interest. One sentence, and every formula here follows from it.
- Lump sum: A = P(1 + r)^n. £1,000 at 4.5% becomes £3,745.32 in thirty years.
- Convert an annual rate to a monthly one with i = (1 + r)^(1/12) – 1, never by dividing by twelve.
- Regular saving: FV = PMT x [((1 + i)^n – 1) / i]. £100 a month at 4.5% for 30 years is £74,706.41.
- £36,000 of that was paid in and £38,706.41 is interest, so interest is 51.8% of the final figure.
- Compounding specifically, as opposed to simple interest, is worth £14,473.91 of it.
- Always deflate the answer. At 2% inflation, £74,706.41 is £41,243.23 in today’s money.
- Rule of 72 for a quick doubling time. Reliable roughly between 4% and 12%.
- Five years of delay costs £20,133.93, of which only £6,000 is contributions and £14,133.93 is forgone growth.
Frequently asked questions
Why not just divide the annual rate by twelve? Because that ignores compounding inside the year. Dividing 4.5% by twelve and compounding it monthly produces 4.59% a year, not 4.5%. The twelfth-root method is the one that returns the rate you were quoted.
Does it matter whether I pay in at the start or the end of the month? Slightly. Every figure here assumes end of month. Paying at the start gives each payment one extra month of growth, which raises the thirty-year total by about 0.37%.
Is 4.5% a realistic rate? It is used here as an illustration and nothing in the method depends on it. Put your own rate into the same formulas and the arithmetic is identical.
What is AER? Annual Equivalent Rate: the rate expressed as if interest were paid once a year, so that accounts paying monthly, quarterly or annually can be compared on one number. It is the figure to convert from, which is what the twelfth root above does.
Updated on 11 September 2026.
Sources
- Bank of England, what is inflation
- Bank of England, what are interest rates
- ONS, consumer price inflation, published 19 August 2026
- MoneyHelper, Money and Pensions Service
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