The Compound Interest Formula
Compound interest is interest on your interest: each period, the interest earned is added to the balance and starts earning interest itself. The formula for a single deposit is:
A = P × (1 + r/n)n×t
- A = the final amount
- P = the principal (starting amount)
- r = annual interest rate as a decimal (7% = 0.07)
- n = times interest compounds per year (1 annually, 12 monthly, 365 daily)
- t = time in years
The compound interest earned is A − P. Compare simple interest, A = P(1 + r×t), which pays interest only on the original principal: $10,000 at 7% simple interest for 10 years grows to $17,000, while compounding annually grows it to $19,671.51.
Worked Example: $10,000 at 7% for 10 Years
With monthly compounding, r/n = 0.07 ÷ 12 and n×t = 12 × 10 = 120:
A = 10,000 × (1 + 0.07/12)120
A = 10,000 × (1.0058333)120 = 10,000 × 2.009661
A = $20,096.61, so the compound interest is $10,096.61
How often interest compounds changes the result, though less than most people expect:
| Compounding | n | Balance after 10 years | Interest earned |
|---|---|---|---|
| Annually | 1 | $19,671.51 | $9,671.51 |
| Semiannually | 2 | $19,897.89 | $9,897.89 |
| Quarterly | 4 | $20,015.97 | $10,015.97 |
| Monthly | 12 | $20,096.61 | $10,096.61 |
| Daily | 365 | $20,136.18 | $10,136.18 |
| Continuously | ∞ | $20,137.53 | $10,137.53 |
Going from annual to daily compounding adds $465 over ten years. The rate and the time matter far more than the frequency.
Compound Interest With Monthly Contributions
Most people add money regularly. The future value of a starting balance plus a fixed monthly contribution combines the compound interest formula with the future value of an annuity:
FV = P(1 + i)N + PMT × [(1 + i)N − 1] / i
Here i is the monthly rate (r ÷ 12), N is the number of months and PMT is the monthly contribution, assumed to be made at the end of each month. Starting with $10,000, adding $200 a month at 7% for 20 years:
i = 0.07 ÷ 12 = 0.0058333, N = 240, (1 + i)N = 4.038739
Starting balance: 10,000 × 4.038739 = $40,387.39
Contributions: 200 × (4.038739 − 1) ÷ 0.0058333 = $104,185.33
FV = $144,572.72
You put in $58,000 ($10,000 plus 240 × $200), so $86,573 is interest. If you contribute at the start of each month instead, multiply the contribution part by (1 + i): the total becomes $145,180.47.
Run the Formula With Your Numbers
The calculator uses the monthly-contribution formula above and shows the balance year by year. It starts with the example: $10,000, $200 a month, 7%, 20 years.
Continuous Compounding: A = Pe^rt
If interest compounded infinitely often, the formula becomes:
A = P × er×t
where e ≈ 2.71828. $5,000 at 5% compounded continuously for 8 years grows to 5,000 × e0.4 = $7,459.12. Continuous compounding is the upper limit for a given rate: at 7%, the effective annual yield is 7.251% versus 7.250% with daily compounding. It appears mostly in finance courses and in pricing models for bonds and options.
How Long Does Money Take to Double? The Rule of 72
The exact doubling time with annual compounding is t = ln(2) ÷ ln(1 + r). The Rule of 72 approximates it as 72 ÷ the rate in percent:
| Annual return | Rule of 72 | Exact |
|---|---|---|
| 2% | 36.0 years | 35.0 years |
| 3% | 24.0 years | 23.4 years |
| 4% | 18.0 years | 17.7 years |
| 6% | 12.0 years | 11.9 years |
| 7% | 10.3 years | 10.2 years |
| 8% | 9.0 years | 9.0 years |
| 10% | 7.2 years | 7.3 years |
| 12% | 6.0 years | 6.1 years |
The rule is most accurate between about 6% and 10%. For low rates, 70 divided by the rate is closer; for continuous compounding, use 69.3.
Solving the Formula for Rate or Time
Rearranging A = P(1 + r)t answers the two other common questions.
What rate did my money earn?
r = (A / P)1/t − 1
An investment that grew from $10,000 to $16,000 in 6 years earned (1.6)1/6 − 1 = 8.15% a year. This is the compound annual growth rate (CAGR).
How long until I reach a target?
t = ln(A / P) / ln(1 + r)
To grow $10,000 into $25,000 at 6% a year takes ln(2.5) ÷ ln(1.06) = 15.7 years.
Compound Interest in Excel and Google Sheets
| What you want | Formula | Result |
|---|---|---|
| Single deposit, monthly compounding | =10000*(1+0.07/12)^(12*10) | $20,096.61 |
| With $200 monthly contributions | =FV(0.07/12, 240, -200, -10000) | $144,572.72 |
| Contributions at the start of each month | =FV(0.07/12, 240, -200, -10000, 1) | $145,180.47 |
| Annual growth rate (CAGR) | =RATE(6, 0, -10000, 16000) | 8.15% |
| Years to reach a target | =NPER(0.06, 0, -10000, 25000) | 15.73 |
| Effective annual rate of 7% compounded monthly | =EFFECT(0.07, 12) | 7.229% |
Money you put in is entered as a negative number, so FV returns a positive balance.
Adjusting Compound Interest for Inflation
A balance that grows at 7% while prices rise 3% a year gains much less purchasing power. The real return is:
real rate = (1 + nominal rate) / (1 + inflation) − 1
At 7% nominal and 3% inflation, the real rate is 3.88%. Run the compound interest formula with that real rate to see a future balance in today's dollars. Savings that earn less than inflation lose value even as the balance rises; our average CD rates report tracks how bank deposits compare with inflation each month.