The Standard Formula
- A
- Final amount (principal + interest)
- P
- Principal — your initial investment or deposit
- r
- Annual interest rate as a decimal (e.g. 6% → 0.06)
- n
- Number of times interest compounds per year
- t
- Time in years
Compounding Frequency Values for n
| Frequency | n value | Interest added every… |
|---|---|---|
| Annual | 1 | 12 months |
| Semi-annual | 2 | 6 months |
| Quarterly | 4 | 3 months |
| Monthly | 12 | 1 month |
| Daily | 365 | 1 day |
| Continuous | ∞ | Every instant → use A = Pert |
Step-by-Step Worked Example
Scenario: You invest $10,000 at 6% annual interest, compounded monthly, for 10 years.
Identify your variables:
P = 10,000,r = 0.06,n = 12,t = 10Divide the rate by compounding periods:
r/n = 0.06/12 = 0.005Add 1:
1 + 0.005 = 1.005Calculate the exponent:
n × t = 12 × 10 = 120Raise to the power:
1.005^120 = 1.81940Multiply by principal:
A = 10,000 × 1.81940 = $18,193.97
Final amount: $18,193.97 — Total interest earned: $8,193.97 (81.94% return on your original $10,000)
How Changing Each Variable Affects the Outcome
Starting with the same $10,000 over 10 years at 6% monthly:
| Change | Final Amount | vs Baseline |
|---|---|---|
| Baseline: $10k, 6%, 10 yrs | $18,193.97 | — |
| Double principal ($20k) | $36,387.93 | +$18,193 |
| Rate up 2% (8%) | $22,196.40 | +$4,002 |
| Add 5 more years (15 yrs) | $24,540.94 | +$6,347 |
| Add 10 more years (20 yrs) | $33,102.04 | +$14,908 |
Time has the most dramatic effect because the exponent nt grows it exponentially — this is why starting early is the single most powerful investment decision you can make.
The Continuous Compounding Formula
When interest compounds infinitely many times per second, the formula simplifies to:
- e
- Euler's number ≈ 2.71828 (the base of natural logarithms)
- P
- Principal
- r
- Annual rate (decimal)
- t
- Time in years
Example: $10,000 at 6% for 10 years continuously compounded:
A = 10,000 × e^(0.06 × 10) = 10,000 × e^0.6 = 10,000 × 1.8221 = $18,221.19
Compare to monthly compounding ($18,193.97) — only $27.22 more. The real-world difference between daily and continuous compounding is negligible. See the full continuous compounding guide →
With Regular Monthly Contributions
When you add money every month (like to a savings account or 401k), the formula expands. First convert to an effective monthly rate m:
Then the future value with monthly contribution PMT:
Our main calculator handles this automatically for all compounding frequencies.
Compound Interest Formula in Excel & Google Sheets
You can use the built-in FV (Future Value) function instead of manual formulas:
=FV(rate/periods, periods*years, -monthly_pmt, -principal)Example: $10,000, 6% monthly for 10 years with $200/mo contributions:
=FV(0.06/12, 12*10, -200, -10000)
Try the Formula Yourself
Frequently Asked Questions
r = n × ((A/P)^(1/nt) − 1). For example, if $10,000 grew to $18,000 in 10 years with monthly compounding: r = 12 × ((18000/10000)^(1/120) − 1) ≈ 5.88%.t = ln(A/P) / (n × ln(1 + r/n)). Example: how long to double $10,000 at 6% monthly? t = ln(2) / (12 × ln(1.005)) ≈ 11.58 years. (The Rule of 72 gives a quick estimate: 72/6 = 12 years.)APY = (1 + r/n)^n − 1. A 6% rate compounded monthly gives APY = (1 + 0.06/12)^12 − 1 = 6.168%. APY is what you actually earn in a year after accounting for compounding.