The formula
- A is the amount at the end, interest included.
- P is the principal: the money you start with.
- r is the yearly interest rate as a decimal (5% = 0.05).
- n is how many times a year interest is compounded (12 for monthly, 365 for daily).
- t is the number of years.
The interest earned is A − P. Each period, the balance is multiplied by (1 + r/n), so interest from earlier periods earns interest too. That is the difference from simple interest, which only ever pays on the original P.
Worked example
$10,000 at 5% a year for 10 years, compounded monthly: r/n = 0.05 ÷ 12, n × t = 120.
You earn $6,470.09 in interest. With simple interest you would earn only $5,000 (10,000 × 0.05 × 10).
How often it compounds
More frequent compounding adds a little, but less than most people expect:
| $10,000 at 5% for 10 years | Balance |
|---|---|
| Yearly (n = 1) | $16,288.95 |
| Quarterly (n = 4) | $16,436.19 |
| Monthly (n = 12) | $16,470.09 |
| Daily (n = 365) | $16,486.65 |
| Continuously | $16,487.21 |
The rate matters far more than the compounding schedule. That is why savings accounts quote an APY, which already includes compounding: 5% compounded monthly is an APY of about 5.12%. See the APY calculator.
Continuous compounding
If interest compounded every instant, the formula becomes:
where e ≈ 2.718282. For the example above, 10,000 × e^(0.05 × 10) = $16,487.21, only 56 cents more than daily compounding.
Adding money every month
Regular deposits add a second part, the future value of a series of payments:
Here i is the rate per period (r/n) and N the number of deposits. Saving $200 a month at 7% for 30 years: i = 0.07 ÷ 12, N = 360, so FV = $243,994.20. You paid in $72,000; the other $171,994.20 is compound growth. Add the A from the first formula if you also start with a lump sum. The compound interest calculator does both at once.
Solving for the rate or the time
Rearranging gives the yearly growth rate when you know the start and end (yearly compounding):
Doubling $10,000 to $20,000 in 10 years means r = 2^(1/10) − 1 ≈ 7.18% a year. This is the same as a CAGR. To solve for time, use logarithms: t = ln(A/P) ÷ (n × ln(1 + r/n)). At 7% a year, money doubles in ln 2 ÷ ln 1.07 ≈ 10.24 years; the "rule of 72" (72 ÷ 7 ≈ 10.3) is a quick estimate of the same thing.
In a spreadsheet
In Excel or Google Sheets, =10000*(1+0.05/12)^(12*10) gives 16,470.09, and =FV(0.05/12, 120, 0, -10000) gives the same answer.
Sources: OpenStax Contemporary Mathematics 6.4: Compound Interest; College Algebra key equations (adapted from OpenStax): continuous growth; SEC Investor.gov compound interest calculator.
Every worked example on this page was checked with Calcist's own engine and an independent reference.