Compound Interest, Explained
The compound interest formula A = P(1 + r/n)^(nt) and how compound interest works, with real examples, compounding frequency, deposits, APY and the rule of 72.
By Udit Agarwal · Updated October 17, 2026
Compound interest is interest calculated on the original amount and on the interest already added to it, so a balance grows by a larger amount each period. The formula is A = P(1 + r/n)^(nt): $1,000 at 5% a year, compounded monthly, grows to $1,647.01 in 10 years, against $1,500 with simple interest. The gap is small at first and widens every year, which is why the length of time has such a large effect.
Simple interest is paid on the original principal only: interest = P × r × t. $1,000 at 5% earns $50 a year, every year, so after 10 years it has earned $500. Compound interest adds each period's interest to the balance, and the next period's interest is calculated on the larger amount. In year two below, the extra $2.50 is 5% interest on the first year's $50 of interest.
| After | Simple interest | Compounded yearly | Difference |
|---|---|---|---|
| 1 year | $1,050.00 | $1,050.00 | $0.00 |
| 2 years | $1,100.00 | $1,102.50 | $2.50 |
| 3 years | $1,150.00 | $1,157.63 | $7.63 |
| 5 years | $1,250.00 | $1,276.28 | $26.28 |
| 10 years | $1,500.00 | $1,628.89 | $128.89 |
| 20 years | $2,000.00 | $2,653.30 | $653.30 |
| 30 years | $2,500.00 | $4,321.94 | $1,821.94 |
Simple interest grows in a straight line, while compound interest curves upward. The Simple Interest Calculator covers the first column, including terms in months or days.
A = P(1 + r/n)^(nt)
The interest earned is A − P. For $1,000 at 5% compounded monthly for 10 years:
| Step | Calculation | Result |
|---|---|---|
| Rate per period | 0.05 ÷ 12 | 0.0041666… |
| Number of periods | 12 × 10 | 120 |
| Growth factor | (1 + 0.05 ÷ 12)^120 | 1.6470095 |
| Final balance | 1,000 × 1.6470095 | $1,647.01 |
| Interest earned | 1,647.01 − 1,000 | $647.01 |
The rate and the time must be converted together: divide the annual rate by n and multiply the years by n. Mixing a monthly rate with a count of years is a common slip. Keep the rate per period unrounded on a calculator, because rounding it early can move the answer by a cent or more.
More frequent compounding adds interest to the balance sooner, so it starts earning sooner. For $10,000 at 5% over 10 years:
| Compounding | n | Final balance | Interest |
|---|---|---|---|
| Annually | 1 | $16,288.95 | $6,288.95 |
| Semi-annually | 2 | $16,386.16 | $6,386.16 |
| Quarterly | 4 | $16,436.19 | $6,436.19 |
| Monthly | 12 | $16,470.09 | $6,470.09 |
| Daily | 365 | $16,486.65 | $6,486.65 |
The gains shrink quickly. Moving from annual to monthly compounding adds $181.14, while moving from monthly to daily adds only $16.56. The mathematical limit, continuous compounding, uses A = Pe^(rt) and gives $16,487.21, 56 cents more than daily. The rate matters far more: the same $10,000 at 6% compounded annually reaches $17,908.48, which is $1,619.53 more than at 5%.
A nominal annual rate, often labelled APR, leaves out the effect of compounding. The annual percentage yield (APY) includes it: it is the actual growth over one year, APY = (1 + r/n)^n − 1.
| 5% compounded | APY |
|---|---|
| Annually | 5.000% |
| Quarterly | 5.095% |
| Monthly | 5.116% |
| Daily | 5.127% |
Because APY already includes compounding, it puts savings accounts with different compounding on the same footing. In the US, Regulation DD, which implements the Truth in Savings Act, sets out how banks calculate the APY they disclose on deposit accounts. The CD Calculator reports the APY of a certificate of deposit next to its maturity value. Loan APRs follow different rules and include certain fees, which the mortgage guide covers.
To model an account that quotes only an APY, enter the APY as the rate and choose annual compounding. A 5.1162% APY entered that way grows $10,000 to $16,470.11 in 10 years, within a few cents of 5% compounded monthly.
Most savings grow from deposits as well as interest. When a payment PMT is added at the end of each compounding period, the balance after t years is:
FV = P(1 + r/n)^(nt) + PMT × ((1 + r/n)^(nt) − 1) ÷ (r/n)
The second part is the future value of a series of equal payments. Take $1,000 to start plus $100 a month at 5%, compounded monthly, for 10 years:
Timing matters. Deposits made at the start of each month earn one more month of interest, which would give $17,239.94 here. When deposits and compounding run at different frequencies, such as monthly deposits into an account that compounds daily, the annual rate is first turned into an equivalent rate per deposit period: i = (1 + r/n)^(n/f) − 1, where f is the number of deposits a year. With daily compounding, the same $1,000 and $100 a month reach $17,185.28. That conversion is how the ArrayKit calculator handles the mismatch.
Starting with $5,000 and adding $200 a month at a constant 6%, compounded monthly (a simplification, since real returns and savings rates change over time):
| After | Balance | Paid in | Interest earned | Interest share |
|---|---|---|---|---|
| 5 years | $20,698.26 | $17,000.00 | $3,698.26 | 18% |
| 10 years | $41,872.85 | $29,000.00 | $12,872.85 | 31% |
| 20 years | $108,959.20 | $53,000.00 | $55,959.20 | 51% |
| 30 years | $231,015.88 | $77,000.00 | $154,015.88 | 67% |
The money paid in grows in a straight line, $2,400 a year. The interest accelerates: in year 20 it overtakes the total paid in, and by year 30 it makes up two-thirds of the balance.
To estimate how long money takes to double, divide 72 by the annual rate in percent. At 6%, 72 ÷ 6 = 12 years. The exact answer with annual compounding is ln 2 ÷ ln 1.06 = 11.9 years.
| Rate | Rule of 72 | Exact, compounded yearly |
|---|---|---|
| 2% | 36 years | 35.0 years |
| 4% | 18 years | 17.7 years |
| 6% | 12 years | 11.9 years |
| 8% | 9 years | 9.0 years |
| 10% | 7.2 years | 7.3 years |
| 12% | 6 years | 6.1 years |
| 24% | 3 years | 3.2 years |
The rule is an approximation. It is closest for rates of about 6% to 10% and drifts at the extremes. For continuous compounding the exact constant is 69.3 (100 × ln 2); 72 is popular partly because it divides evenly by 2, 3, 4, 6, 8, 9 and 12. The rule works for anything that grows at a steady rate: at 3% inflation, prices double in roughly 24 years (23.4 exactly), and the Inflation Calculator projects the effect on buying power.
The Compound Interest Calculator uses the formulas above. Enter a starting principal, an annual rate and a number of years (decimals work), then choose annual, semi-annual, quarterly, monthly or daily compounding; daily uses n = 365. You can add a regular contribution made monthly, quarterly or annually, applied at the end of each period. It shows the final balance, the total of your regular contributions (not counting the starting principal), the total interest, and a year-by-year balance table that shows the last 12 years of longer terms. A summary can be copied as text.
It assumes one fixed rate for the whole period and no withdrawals, fees, taxes or inflation, and the starting principal must be greater than zero. The figures are calculated in your browser and are not uploaded. The results are estimates, not financial advice.
What is the formula for compound interest? 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. The interest earned is A − P. With a deposit PMT at the end of each period, add PMT × ((1 + r/n)^(nt) − 1) ÷ (r/n).
How does compound interest work, in simple terms? Each period's interest is added to the balance, and the next period's interest is worked out on that larger balance, so interest earns interest. The effect is small at first, but after 30 years at 5% compounded yearly, $1,000 becomes $4,321.94, against $2,500 with simple interest.
How much will $10,000 grow in 10 years at 5%? It reaches $16,288.95 with annual compounding and $16,470.09 compounded monthly, before taxes or fees. Regular deposits can change the result far more than the compounding frequency does.
Does daily compounding make a big difference? Not compared with monthly. On $10,000 at 5% for 10 years, daily compounding earns $16.56 more than monthly. The rate and the number of years matter much more.
What is the difference between APR and APY? For savings, the APR is the nominal yearly rate before compounding and the APY is the growth after a year of compounding, (1 + r/n)^n − 1. A 5% rate compounded monthly is an APY of 5.116%. For loans, the APR has a legal definition that also includes certain fees.
How accurate is the rule of 72? For rates between about 4% and 12%, it lands within about four months of the exact doubling time with yearly compounding. At 6% it gives 12 years against an exact 11.9. At very low or very high rates the error grows.
Does compound interest apply to debt? Yes, and it works against the borrower: interest that is added to a balance is then charged interest itself. Credit cards are the familiar example, and the Credit Card Payoff Calculator shows how long a balance takes to clear at a given payment.
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