The Compound Interest Formula, Explained
A = P(1 + r/n)^(nt), broken down variable by variable with worked examples.
The compound interest formula is A = P(1 + r/n)^(nt). It calculates how a principal P grows at annual rate r, compounded n times per year, over t years. On $1,000 at 8% for 10 years, annual compounding gives $2,158.92 while daily compounding gives $2,225.35 — compounding frequency matters, but time and rate matter more.
The formula
A = P(1 + r/n)nt
Each variable means something specific:
| Variable | Meaning |
|---|---|
| A | Final amount after interest |
| P | Principal — the starting amount |
| r | Annual interest rate, as a decimal (8% → 0.08) |
| n | Compounding periods per year (12 = monthly, 365 = daily) |
| t | Number of years |
Worked example
Let's apply it: $1,000 at 8% annual return for 10 years, with no contributions. The only change is how often interest is credited:
| Compounding frequency | n | Result |
|---|---|---|
| Annual | 1 | $2,158.92 |
| Semi-annual | 2 | $2,191.12 |
| Quarterly | 4 | $2,208.04 |
| Monthly | 12 | $2,219.64 |
| Daily | 365 | $2,225.35 |
| Continuous | ∞ | $2,225.54 |
Annual compounding grows $1,000 to $2,158.92; daily compounding reaches $2,225.35 — a difference of about $66. Compounding frequency is real, but it's a small effect next to rate and time.
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Why time is the real lever
Because time sits in the exponent (nt), it doesn't add to growth — it multiplies it. Doubling the time doesn't double the result; it roughly squares the growth factor. That's why starting earlier matters more than trying to pick a higher rate.
For example, $1,000 at 8% for 10 years is $2,158.92, but for 20 years it's $4,660.96 — more than double, because the growth compounds on itself.
Continuous compounding
As n grows toward infinity, the formula becomes A = Pert. It's the mathematical ceiling of compounding — the most growth mathematically possible at a given rate and time. For real accounts that credit daily or monthly, it's a few dollars' difference on modest sums, so it's mostly useful as a clean upper bound and a shortcut in higher math.
Risk & limitations
The formula assumes a constant rate, but real investment returns fluctuate year to year. Inflation, taxes, and fees are excluded and reduce real results. More frequent compounding helps, but it does not change risk. This is educational math, not a prediction or advice.