The formula and what each letter means
The standard compound interest formula gives the balance after interest has been added to itself again and again. It has four inputs. Once you know what each one stands for, the rest is careful button pressing.
A is the amount you end with. P is the principal, the money you start with. r is the yearly interest rate written as a decimal, so 5% becomes 0.05. n is how many times a year interest is added, such as 12 for monthly or 365 for daily. t is the number of years.
Formula notes
A = P x (1 + r / n)^(n x t)Interest earned = A - P
Working it by hand
Say you put $10,000 in an account paying 5% compounded monthly and leave it for 10 years. Divide the rate by the periods: 0.05 / 12 = 0.0041667. Add 1 to get 1.0041667. Multiply the periods by the years: 12 x 10 = 120.
Now raise 1.0041667 to the 120th power. You get about 1.647009. Multiply by $10,000 and the balance is $16,470.09. Subtract the $10,000 you started with and you've earned $6,470.09 in interest.
Simple interest on the same money would pay 5% of $10,000 each year, or $5,000 over the decade. The extra $1,470.09 is interest earned on earlier interest. That gap widens every year you wait.
What changes when n goes up
More compounding periods mean interest starts earning sooner, so the balance rises a bit. The effect is smaller than most people expect. Run the same $10,000 at 5% for 10 years with daily compounding and you get $16,486.65, just $16.56 more than monthly.
That's why APY exists. It folds the compounding schedule into one yearly figure so you can compare accounts. 5% compounded monthly is a 5.12% APY. 5% compounded daily is 5.13%.
Formula notes
APY = (1 + r / n)^n - 1
Adding regular deposits
The basic formula only handles one deposit. Monthly deposits each compound for a different length of time. So you need a second formula for the deposits, then you add the two results together.
Take $5,000 up front plus $200 at the end of each month at 7% for 10 years. The lump sum grows to $10,048.31. The deposits grow to $34,616.96. The total is $44,665.27. If the deposits land at the start of each month instead, multiply the deposit part by (1 + i) and the total becomes $44,867.20.
Formula notes
FV of deposits = PMT x ((1 + i)^N - 1) / ii = r / 12 and N = number of monthly deposits
Mistakes that throw the answer off
- Using 5 instead of 0.05 for the rate. The answer explodes, which is at least easy to spot.
- Mixing up n and t. n is periods per year, t is years. The exponent is n times t, so 10 years monthly is 120.
- Compounding an APY again. An APY already includes compounding, so plug it in with n = 1.
- Rounding r / n too early. Keep at least six decimal places until the last step.
- Forgetting that deposits need their own formula. Adding $200 x 120 to the lump sum result skips all the growth on those deposits.
Solving for time or rate
You can turn the formula around. To find how long money takes to double, set A / P to 2 and solve for t with logarithms. At 5% compounded monthly, that's about 13.89 years.
Working out a monthly deposit, a target date or a mix of fees and inflation by hand gets tedious fast. Our compound interest calculator runs every month for you and shows the yearly balances, so you can check your own working against it.
Formula notes
t = ln(A / P) / (n x ln(1 + r / n))
Frequently asked questions
What does n mean in the compound interest formula?
n is the number of times interest is added each year. Use 1 for annual, 4 for quarterly, 12 for monthly and 365 for daily compounding.
Do I use 5 or 0.05 for a 5% rate?
Use 0.05. The formula needs the rate as a decimal, so divide the percentage by 100 before you start.
What is the difference between simple and compound interest?
Simple interest is paid only on the original amount. Compound interest is also paid on interest already earned. $10,000 at 5% for 10 years earns $5,000 simple, or $6,470.09 compounded monthly.
Can you use the formula for continuous compounding?
Continuous compounding uses A = P x e^(rt) instead. It gives a slightly higher balance than daily compounding, though the difference is tiny at normal savings rates.
How do you find the time it takes to double your money?
Set the ending amount to twice the start and solve t = ln(2) / (n x ln(1 + r / n)). At 5% compounded monthly, money doubles in about 13.89 years.
References
These sources support the method or guidance used for Compound Interest Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Compound Interest Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.