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 the deposit-style formula to $1,000 at a fixed 8% annual rate for 10 years, with no contributions. The only change is how often interest is credited. For investments such as stocks or ETFs, actual returns are not fixed interest rates, so this is a mathematical illustration rather than a market forecast.
| 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.
Try the calculator with your own numbers →
Why time is the real lever
Because time sits in the exponent (nt), extending the horizon can increase the effect of compounding substantially when the rate is held constant. The exact relationship depends on the rate and compounding convention; a real investment does not follow a constant-rate path. 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.
Note: The formula is standard compound-interest math. Long-run return context: NYU Stern (Damodaran) provides long-run return context. Educational projection only.
Frequently asked questions
What is the compound interest formula?
What does n mean in the compound interest formula?
How do you calculate monthly compound interest?
What is continuous compounding?
Related calculators
Last reviewed: September 2026.
Source & scope
The formula A = P(1 + r/n)nt describes discrete compounding at a fixed nominal rate. Investor.gov uses the same core inputs—principal, estimated annual rate, time and compounding frequency—in its educational compound-interest calculator. Market investments do not provide a fixed rate, so use this formula as a scenario model rather than a promise of returns.