How compounding works
With simple interest you earn interest only on the money you put in. With compound interest, interest is added to the balance at regular intervals, and from then on it earns interest too. Each period starts from a slightly bigger balance than the one before, so the growth speeds up over time.
How often interest is added is the compounding frequency: once a year, twice a year, quarterly, monthly, daily, or — as a mathematical limit — continuously. The more often it is added, the sooner interest starts earning interest, so the same stated rate grows money slightly faster.
That is why banks quote an annual percentage yield (APY) next to the interest rate. The APY is the rate with a year of compounding built in: 5% compounded monthly has an APY of 5.116%. Comparing APYs compares accounts on equal terms, whatever their compounding.
The compound interest formula
A = P × (1 + r ÷ m)^(m × t)
- A
- balance at the end
- P
- starting amount
- r
- annual interest rate as a decimal (5% → 0.05)
- m
- compounding periods per year: 1, 2, 4, 12 or 365
- t
- time in years (10 years 6 months → 10.5)
Continuous compounding is the limit as m grows without bound:
A = P × e^(r × t)
The effective annual rate follows from one year of compounding:
APY = (1 + r ÷ m)^m − 1
Regular deposits are added with the future value of an annuity, where i is the rate per deposit period and n the number of deposits. A deposit made at the start of each period earns one extra period of interest, so the result is multiplied by (1 + i):
FV = deposit × ((1 + i)ⁿ − 1) ÷ i
When deposits and compounding don’t line up — monthly deposits into an account that compounds quarterly or daily — the calculator converts the rate to the monthly rate that gives exactly the same yearly growth, (1 + r ÷ m)^(m ÷ 12) − 1. With yearly deposits and a final part-year, a start-of-year deposit is still made for that part-year; an end-of-year deposit is not.
Worked example: $10,000 at 5% for 10 years
- Monthly compounding: r ÷ m = 0.05 ÷ 12 = 0.00416667, and m × t = 12 × 10 = 120 periods
- Growth factor: 1.00416667¹²⁰ = 1.647009
- Balance: $10,000 × 1.647009 = $16,470.09, so interest earned is $6,470.09
- APY: 1.00416667¹² − 1 = 5.116%
The same $10,000 under each compounding frequency:
| Compounded | APY | Balance after 10 years |
|---|---|---|
| Annually | 5.000% | $16,288.95 |
| Semiannually | 5.063% | $16,386.16 |
| Quarterly | 5.095% | $16,436.19 |
| Monthly | 5.116% | $16,470.09 |
| Daily | 5.127% | $16,486.65 |
| Continuously | 5.127% | $16,487.21 |
Going from yearly to daily compounding adds $197.70 over ten years; going from daily to continuous adds only 56 cents. Simple interest on the same deposit would pay $10,000 × 5% × 10 = $5,000, which is $1,470.09 less than monthly compounding. Adding $100 at the end of every month (monthly compounding) would bring the balance to $31,998.32.
The Rule of 72 is an estimate
Divide 72 by the annual rate in percent and you get roughly how many years it takes money to double: at 5%, 72 ÷ 5 = 14.4 years. It is a mental shortcut for the exact doubling time, which is:
years to double = ln 2 ÷ ln(1 + APY)
The rule assumes yearly compounding and is most accurate for rates around 8%, where 72 ÷ 8 = 9 years and the exact answer is 9.01. At 5% compounded yearly money actually doubles in 14.2 years; compounded monthly, in 13.9 years. At low rates the rule overstates the time (at 1%: 72 years against 69.7), at high rates it understates it (at 20%: 3.6 years against 3.8). For continuous compounding the exact figure is 69.3 ÷ rate. The calculator shows both the exact doubling time and the Rule of 72 estimate.
What this leaves out
- A fixed rate. Savings rates change, and investment returns vary from year to year. For investing with regular contributions and inflation, use the investment calculator.
- Tax and inflation. Interest is usually taxable, and inflation reduces what the final balance can buy.
- Crediting. Banks compound on the balance but credit interest to the account on their own schedule and round to the cent; results can differ by a few cents.
Compounding works against you on debt as well. To see interest that does not compound, try the simple interest calculator.
Frequently asked questions
What is the difference between the interest rate and the APY?
The interest rate is the nominal yearly rate before compounding. The APY is the yearly growth after compounding is included, so it is higher whenever interest compounds more than once a year: 5% compounded monthly is an APY of 5.116%. US banks must disclose the APY on deposit accounts, which makes accounts easy to compare.
Does daily compounding make a big difference compared with monthly?
Not much. On $10,000 at 5% for 10 years, daily compounding ends at $16,486.65 and monthly at $16,470.09, a difference of $16.56. The rate itself and the time you leave the money matter far more than the compounding frequency.
What does compounded continuously mean?
It is the limit of compounding more and more often — every hour, every second, and so on. The balance is P × e^(rt), and 5% compounded continuously has an APY of 5.127%, barely above daily compounding. It is used mainly in finance theory and textbooks.
How accurate is the Rule of 72?
It is an approximation that works best for rates between about 6% and 10% compounded yearly. At 5% it says 14.4 years, while the exact answer is 14.2 years with yearly compounding and 13.9 years with monthly compounding. The calculator shows the exact figure next to the estimate.
Should deposits be at the start or the end of the month?
Choose what matches when you actually pay in. A deposit at the start of a period earns interest for that period; one at the end does not. With $100 a month on top of $10,000 at 5% for 10 years, start-of-month deposits end at $32,063.02 and end-of-month deposits at $31,998.32.
Sources
Last reviewed September 15, 2026