The formula
Compound interest grows money in two places at once: the amount you start with grows, and every monthly contribution grows too — and the growth itself grows. The formula has two parts, one for each place. The starting amount grows to P times (1 + i) to the power N, where P is the starting amount, i is the monthly rate, and N is the number of months. The monthly contributions grow to PMT times ((1 + i) to the power N minus 1) divided by i, where PMT is the monthly contribution. Add the two parts and you have the future value. The monthly rate is the annual rate divided by 12, and the months are the years times 12. Contributions go in at month-end.
Step 1 — Write down your starting amount
Start with the lump sum you already have. In our worked example that is $10,000. If you are starting from nothing, write down $0.
Step 2 — Write down your monthly contribution
Next, the amount you add every month. Our example uses $200 a month, added at month-end. Over 20 years that is 240 contributions, which is $48,000 of your own money on top of the starting amount. If your contribution changes over time, use the number you can actually hold steady.
Step 3 — Turn the annual rate into a monthly rate and count the months
The formula compounds monthly, so the annual rate has to become a monthly one: divide it by 12. At 7% a year the monthly rate is 7% divided by 12. Then count the months: 20 years times 12 is 240 months. These two numbers are the engine of the whole calculation.
Step 4 — Grow the starting amount
Raise (1 + the monthly rate) to the power of the months, then multiply by the starting amount. At 7% for 20 years that is $10,000 times (1 + 7% divided by 12) to the power 240, which is $40,387.39. Most of that growth happens late, because the growth itself is growing.
Step 5 — Grow the monthly contributions
The contributions need their own calculation because each month's deposit has a different amount of time to grow. The formula is the monthly contribution times ((1 + the monthly rate) to the power of the months minus 1) divided by the monthly rate. At $200 a month, 7%, and 20 years that is $104,185.33 — more than double the $48,000 you actually put in as contributions.
Step 6 — Add the two parts and split the ending
Add the two grown parts: $40,387.39 plus $104,185.33 is a future value of $144,572.72. Then split the ending into what you put in and what compounding earned. What you put in is the starting amount plus the contributions: $10,000 plus $200 times 240 months is $58,000.00. What compounding earned is the future value minus what you put in: $144,572.72 minus $58,000.00 is $86,572.72. The interest earned is bigger than the money you put in — that is the whole point of compounding.
The worked example
Setup: $10,000 starting amount, $200 a month, 7% annual rate, 20 years. Line 1: the monthly rate is 7% divided by 12 and the months are 20 times 12, which is 240. Line 2: the starting amount grows to $10,000 times (1 + 7% divided by 12) to the power 240, which is $40,387.39. Line 3: the contributions grow to $200 times ((1 + 7% divided by 12) to the power 240 minus 1) divided by (7% divided by 12), which is $104,185.33. Line 4: the future value is $40,387.39 plus $104,185.33, which is $144,572.72; the total contributions are $10,000 plus $200 times 240, which is $58,000.00; and the interest earned is $144,572.72 minus $58,000.00, which is $86,572.72. Check it against the calculator on the compound interest page and you will get the same numbers.
How the numbers change
Two changes show what matters most. First, give it ten more years: at 30 years instead of 20, the same $10,000 and $200 a month at 7% grow to $325,159.17 — more than double the 20-year ending. Time is the biggest lever in the whole calculation, because compounding accelerates. Second, raise the rate: at 10% instead of 7% for the same 20 years, the ending is $225,154.51. At 10 years the ending is only $54,713.57, because the growth has not had time to start growing on itself.
Working backwards
Sometimes the question runs the other way: you know the target and want the input. To reach a $500,000.00 future value in 20 years at 7%, you would need $882.30 a month — the exact figure is 882.2981, and at the displayed $882.30 the future value is $500,000.98, which is $0.98 over the target because of rounding. And if the target is time rather than money: 27 years is the first whole-year horizon past $250,000 — at 27 years the future value is $257,252.47, while at 26 years it is only $237,597.98. There is no exact in-between, because years come in whole years. Both are worked out in full on the calculator page.
What the calculation leaves out
The formula assumes one flat return every month, and real returns are never flat — some years are up, some are down, and the average hides both. It ignores taxes, which take a cut of the growth; fees, which take a cut every year; and inflation, which shrinks what the ending can buy. It also assumes the contributions never change and nothing is ever withdrawn. Use it to compare plans and see which lever moves the ending most, then check a full plan, with taxes and fees included, before you act on it.
What to do next
Run your own numbers on compound interest calculator: put in your starting amount, your monthly contribution, your rate, and your years, and it will give you the future value, the contributions, and the interest earned — and it will work backwards from any target you set. If you want the one-lump-sum version without contributions, future value calculator grows a single amount once. And if you want the doubling shortcut, rule of 72 chart tells you how long any rate takes to double your money.