How compound interest actually works

"Compound interest is the eighth wonder of the world" — usually misattributed to Einstein, and worth understanding precisely instead of mystically. Here's the math, where the formula comes from, and where it stops being useful.

The intuition first

Put $1,000 in a savings account at 5% interest. After year one, you have $1,050. That's simple interest — you earned 5% of $1,000 = $50.

Now leave it for year two. With simple interest, you'd earn another $50 (5% of the original $1,000). But banks (and the universe) actually pay you 5% of your current balance, which is $1,050. So year two interest is $52.50, and you end year two with $1,102.50.

That extra $2.50 is compound interest — interest on the previously earned interest. It's a small number in year two. By year 30, it's the entire reason you bothered investing.

The formula, derived

The standard formula:

Where:

• A = final amount
• P = principal (starting amount)
• r = annual interest rate as a decimal (5% = 0.05)
• n = compounds per year (yearly = 1, monthly = 12, daily = 365)
• t = years

Where does the exponent come from? At every compounding period, you multiply by (1 + r/n) — your balance grows by one period's interest rate. After nt total periods, you've multiplied by that factor nt times, which is exactly what the exponent expresses.

Worked example

$1,000, 5% annual, compounded monthly, 10 years:

• P = 1000, r = 0.05, n = 12, t = 10
• A = 1000 × (1 + 0.05/12)^(12 × 10)
• A = 1000 × (1.004167)^120
• A = 1000 × 1.64701
• A = $1,647.01

That's the textbook answer — and if your compound-interest calculator doesn't return $1,647.01 for those inputs, it's broken.

Compounding frequency matters less than people think

The marketing talks up "compounded daily" as if it's a meaningful advantage. Let's actually run the numbers — same $1,000 at 5% for 10 years:

• Yearly (n=1): $1,628.89
• Monthly (n=12): $1,647.01
• Daily (n=365): $1,648.66
• Continuously (n=∞): $1,648.72

The gap between yearly and continuous compounding on this example is $20 over a decade — less than 1.3% of the final amount. The frequency matters when you're choosing between two products with otherwise identical rates, but it's a small effect compared to the rate itself.

The continuous compounding limit, by the way, is A = Pe^(rt) — where e is the same Euler's number from natural logs. The fact that this constant shows up here, in financial math, is the same fact that makes it show up in radioactive decay, population growth, and probability theory. Continuous proportional growth has one universal mathematical signature.

The rule of 72

How long until your money doubles? Approximation: 72 ÷ rate. At 6% it doubles in 12 years. At 8% in 9. At 12% in 6.

The rule works because ln(2) ≈ 0.693, and dividing 0.693 by the rate gives the doubling time for continuous compounding. 72 is just a slightly larger number that's easier to divide by common percentages (72 / 6 / 8 / 9 / 12 are all clean). For typical investment rates (4–10%), the rule is accurate to within a few months over decades.

Where the formula breaks

The compound interest formula assumes:

1. A constant rate. Real markets don't have constant rates. The S&P 500's "average ~10% annual return" is a long-run average; individual years range from −37% (2008) to +37% (1995). Compound returns in volatile assets are calculated with the geometric mean, which is always lower than the arithmetic mean.
2. No taxes or fees. In a taxable account, your effective rate is lower by your capital-gains rate. In retirement accounts (Roth IRA, 401k) the formula does apply more cleanly because taxation is deferred or absent.
3. No withdrawals. Adding contributions extends the formula (annuity-due / annuity-ordinary). Pulling money out — say, retirement spending — completely changes the math (see our safe withdrawal rate guide).
4. Inflation isn't accounted for. A $1,000 → $1,647 growth in 10 years sounds great, but if inflation averaged 3%, the real growth was 5% − 3% = 2% — and $1,647 in 10 years buys what $1,225 buys today. The formula gives nominal growth; for purchasing power, use the real rate (nominal − inflation) instead.

The takeaway

Compound interest isn't magic, it's just multiplication done many times. The math is:

• The exponent is everything. Doubling the time has a bigger effect than doubling the rate, because time appears in the exponent.
• Start early, even with small amounts. $5k invested at 25 outperforms $50k invested at 55, at the same return rate, over a normal retirement window.
• Watch the rate after taxes and after inflation — the headline rate isn't what you actually keep.
• Use the rule of 72 as a fast sanity check against anyone selling you growth.