Compound interest explained: formula, APY vs APR and the Rule of 72
8 min read · Updated 23 October 2026
What is compound interest? Compound interest is interest paid on your original money and on the interest it has already earned. Each period, the interest is added to the balance, and the next period's interest is figured on the larger amount. The formula is A = P(1 + r/n)^(nt). For example, $10,000 at 5 percent compounded annually for 10 years grows to $16,288.95. You can try your own numbers in the compound interest calculator.
Compound interest works for you when you save and against you when you borrow. It is also the reason time matters so much in saving: the same deposit grows faster the longer it sits. This guide explains the formula, shows how the compounding frequency changes the result, clears up APY and APR, and teaches a quick mental shortcut, the Rule of 72. All examples use US dollars, and every number can be checked with a basic calculator.
Simple vs compound interest
With simple interest, you earn interest only on the original amount. $1,000 at 5 percent simple interest earns $50 every year, forever. After 10 years you have $1,500.
With compound interest, each year's interest is added to the balance. In year 1, you earn $50 and end with $1,050. In year 2 you earn 5 percent of $1,050, which is $52.50. In year 3 you earn 5 percent of $1,102.50, and so on. After 10 years you have $1,628.89. The extra $128.89 is interest on interest.
The gap looks small at first and gets wider every year.
The formula
A = P x (1 + r/n)^(n x t)
- A is the final amount.
- P is the principal, the starting amount.
- r is the annual interest rate as a decimal (5 percent is 0.05).
- n is how many times per year interest is compounded.
- t is the number of years.
The interest earned is A minus P.
Here is the formula in action with $10,000 at 5 percent for 10 years, compounded once a year:
- r/n = 0.05 / 1 = 0.05
- n x t = 1 x 10 = 10
- (1.05)^10 = 1.62889
- A = 10,000 x 1.62889 = $16,288.95
The calculator uses this same formula. The interest is credited at the end of each compounding period, and balances are kept at full precision with rounding only for display.
Monthly vs daily compounding
Compounding more often gives a slightly higher result, because interest starts earning interest sooner. Take the same $10,000 at 5 percent for 10 years and change only n:
| Compounding | n | Final amount | Interest earned |
|---|---|---|---|
| Annually | 1 | $16,288.95 | $6,288.95 |
| Quarterly | 4 | $16,436.19 | $6,436.19 |
| Monthly | 12 | $16,470.09 | $6,470.09 |
| Daily | 365 | $16,486.65 | $6,486.65 |
Going from annual to daily adds about $198 over ten years on $10,000. That is real money but not dramatic. Most of the growth comes from the rate and the time, not from the compounding frequency. Do not choose an account only because it compounds daily. Compare the APY instead.
APY vs APR
These two letters cause a lot of confusion.
APR is the annual percentage rate. It is a yearly rate that, for loans, is meant to express the cost of borrowing. For deposit accounts, a bank's stated or nominal rate does not include the effect of compounding.
APY is the annual percentage yield. It is the rate you actually earn over a year after compounding is included. US rules for deposit accounts, known as Regulation DD under the Truth in Savings Act, make banks disclose an APY so that you can compare accounts fairly.
The common formula for APY is:
APY = (1 + r/n)^n - 1
For a 5 percent nominal rate:
| Compounding | APY |
|---|---|
| Annually | 5.000% |
| Monthly | 5.116% |
| Daily | 5.127% |
So an account paying 5 percent compounded monthly has an APY of about 5.12 percent. Two accounts can both advertise "5 percent" and differ slightly in what they pay. The APY lets you compare them. When you compare savings products, look at APY. When you compare loans, look at APR and also at the fees.
Adding regular deposits
Most people save by adding money regularly. The calculator handles contributions weekly to yearly, at the start or end of each period.
Here is a longer example. You invest $500 at the end of every month. The account earns 7 percent a year compounded monthly. This is an assumption for the example, not a forecast.
| Years | You put in | Balance | Interest earned |
|---|---|---|---|
| 20 | $120,000 | $260,463.33 | $140,463.33 |
| 30 | $180,000 | $609,985.50 | $429,985.50 |
Look at the second row. Going from 20 to 30 years adds only $60,000 of your own money, but the balance grows by about $349,500. The last ten years earn more than the first twenty put together. That is compounding at work, and it explains why starting early matters so much.
Real returns are not smooth. Investments can lose value, and a constant rate is a simplification. The calculator shows a constant rate, so treat the output as a "what if" and not a prediction. Taxes and account fees are not deducted.
Does deposit timing matter?
A little. A deposit made at the start of a period earns interest for that whole period. One made at the end earns nothing until the next period. So start-of-period deposits end slightly higher. The calculator lets you choose either.
The Rule of 72
The Rule of 72 is a mental shortcut. To estimate how many years it takes money to double, divide 72 by the annual rate as a whole number.
| Annual rate | 72 divided by rate | Exact answer |
|---|---|---|
| 3% | 24 years | 23.4 years |
| 6% | 12 years | 11.9 years |
| 8% | 9 years | 9.0 years |
| 12% | 6 years | 6.1 years |
The rule is most accurate near 8 percent and drifts a little at very low or very high rates. It works for growth that compounds once a year. You can also use it in reverse: if you want to double your money in 9 years, you need about 72 / 9 = 8 percent.
It also works for costs. If prices rise 3 percent a year, what you buy with a dollar is cut roughly in half in about 24 years.
Compound interest on debt
The same math that grows savings grows debt. If you owe money and do not pay the interest, it is added to the balance and then earns interest itself.
For a regular installment loan, payments cover the interest first, and the rest reduces the balance. For example, a $25,000 car loan at 7 percent for 60 months has a payment of $495.03. Total interest is about $4,702. The loan calculator shows each month's split between interest and principal. Our guide to how loan amortization works explains why early payments are mostly interest, and what extra payments save.
Credit cards are different, because many issuers compound interest daily on the balance you carry. Check the terms on your own card.
A worked example with a savings goal
You want $12,000 in 24 months for a down payment on a used car.
- With no interest, you need $12,000 / 24 = $500 a month.
- If your savings account earns 4 percent a year, with interest credited monthly and deposits made at the end of each month, the savings goal calculator gives about $481.10 a month.
The interest does about $19 a month of the work. That is not much over two years. The same 4 percent matters far more over 20 years. This is the practical lesson: for short goals, the amount you deposit does most of the work. For long goals, time and the rate do.
Common mistakes
- Comparing APR to APY. They are not the same number. Compare APY with APY, and APR with APR.
- Using whole-number rates in the formula. Use 0.05, not 5.
- Forgetting the nt exponent. Compounding twelve times a year for ten years is an exponent of 120, not 12.
- Ignoring inflation. A balance of $16,289 in ten years buys less than $16,289 does today. The calculator can show a result in today's money when you enter an inflation rate.
- Ignoring taxes and fees. Neither is deducted in a basic calculation.
- Assuming the rate will stay the same. Savings accounts and investments differ. A constant rate is an assumption.
- Using the Rule of 72 for tiny or huge rates. It gets less accurate at the extremes.
- Believing compounding frequency is the big lever. Rate, time and deposits matter more.
This guide is general education, not personal financial advice.
FAQ
What is the compound interest formula? A = P(1 + r/n)^(nt), where P is the starting amount, r the annual rate as a decimal, n the number of compounding periods per year and t the number of years.
Is monthly or daily compounding better? Daily is slightly higher, but the difference is small. $10,000 at 5 percent for 10 years ends at $16,470.09 with monthly compounding and $16,486.65 with daily.
What is the difference between APY and interest rate? The interest rate is the nominal yearly rate. The APY includes compounding, so it is a little higher whenever interest is paid more often than once a year.
How long does it take to double my money? Divide 72 by the annual rate. At 6 percent, about 12 years. The exact figure is 11.9 years.
How do I calculate compound interest in a spreadsheet? Use the formula with cell references: principal x (1 + rate / periods) ^ (periods x years).
Does compound interest work on loans too? Yes. Interest unpaid on a loan can be added to what you owe. With regular installment payments, the schedule shows how much of each payment is interest.