How compound interest is calculated, step by step

The compound interest calculator takes a starting amount, a monthly contribution, a number of years and a yearly return, and tells you the balance at the end. This page shows exactly what happens in between, so you can check the numbers yourself.

The formula

Write the yearly return as a decimal (10.7% is 0.107), call it r, and call the number of compounding periods in a year n. Then after k years:

balance = P x (1 + r/n)^(k x n) + c x ((1 + r/n)^(k x n) - 1) / (r/n)

P = starting amount
c = contribution per period = monthly contribution x 12 / n
n = 1 (yearly), 2 (twice a year), 4 (quarterly), 12 (monthly) or 365 (daily)

The first half is the starting amount growing on its own. The second half is every contribution, each growing from the moment it went in until the end. When the return is exactly zero the formula is simply the starting amount plus everything paid in.

What the compounding setting changes

The compounding setting decides how often growth is added to the balance and starts growing itself. Yearly is the usual assumption for stock market averages, which are quoted as a return per year. Savings accounts often compound monthly or daily. More frequent compounding gives a slightly higher result for the same yearly rate, because growth starts earning sooner.

How monthly contributions are counted

Contributions are pooled for each compounding period and added at the end of it. With yearly compounding, a year's twelve deposits earn nothing until the year ends. With monthly compounding, each deposit starts earning the next month. With daily compounding, the monthly amount is spread evenly over 365 days.

A worked example makes the difference visible. Save 100 a month for one year at 12%, starting from nothing:

• Yearly compounding: 1,200.00. The deposits were pooled and had not compounded yet.
• Monthly compounding: 1,268.25. Each deposit earned 1% a month from the month after it went in.
• Daily compounding: 1,274.75. The same money, spread over 365 days, earning every day.

The low and high lines

Nobody knows the future return, so the calculator also runs the same formula at a lower and a higher rate. The give-or-take field sets how far: with an expected 10.7% and 2 in that field, the low line uses 8.7% and the high line 12.7%. Set it to 0 to see one line only.

Today's money

Prices rise every year, so a future balance buys less than the same amount today. The calculator takes the inflation rate off the final balance for every year that passed:

today's money = balance / (1 + inflation)^k

At 3% inflation a year, 1,000,000 in thirty years buys what about 412,000 buys today. Leave the inflation field at 0 and the calculator shows the plain balance only.

Goal mode: how much to save a month

The balance formula is linear in the monthly contribution: doubling the monthly amount doubles the part of the balance the contributions produced. So the monthly amount that lands exactly on a target is one division away: the target minus what the starting amount grows to on its own, divided by what a contribution of 1 a month grows to over the same years. The calculator repeats this at the low and high rates too.

Where the preset return figures come from

The presets are long-run averages, each with its period and source. Past averages are not forecasts.

What the calculator leaves out

• Taxes and fees. Returns are before any tax or fund cost.
• Changing contributions. The monthly amount stays the same every month.
• Real market paths. Markets do not return the same percentage every year. The projection is arithmetic on the assumptions you enter, not a forecast.

Ready to try it with your own numbers? Open the compound interest calculator. Every input travels in the link, so you can share a worked example.