Compound Interest Explained: The Formula, the Rule of 72, and Worked Examples You Can Check

How compound interest actually works, why the Rule of 72 works, how compounding frequency changes the result, and the math behind "start early" — with numbers you can reproduce on any calculator.

Simple interest pays you on the principal. Compound interest pays you on the principal and on the interest you already earned. Over one year the difference is trivial. Over thirty years it is the difference between a modest sum and a retirement.

The formula

A = P × (1 + r/n)^(n·t)

A = final amount        r = annual rate (decimal)
P = principal           n = compounding periods per year
t = years

Example: $10 000 at 7% for 30 years, compounded annually (n = 1):

A = 10 000 × 1.07^30 = 10 000 × 7.612 = $76 123

Simple interest over the same period would give 10 000 × (1 + 0.07 × 30) = $31 000. Compounding contributed the other $45 000 — more than the principal and the simple interest combined.

Why the Rule of 72 works

The Rule of 72 says: years to double ≈ 72 ÷ annual rate (in %). At 8% money doubles in about 9 years; at 6%, 12 years; at 3%, 24 years.

The derivation is short. Doubling means (1 + r)ᵗ = 2, so t = ln 2 / ln(1 + r). For small r, ln(1 + r) ≈ r, giving t ≈ 0.693 / r ≈ 69.3 / (rate in %). The number 72 is chosen instead of 69.3 because it has more divisors (2, 3, 4, 6, 8, 9, 12) and because it corrects slightly for the ln(1 + r) ≈ r approximation at typical rates of 6–10%.

It runs in reverse too: if your index fund tripled in 15 years, the annualized return was about ln 3 / 15 ≈ 7.3%. And it works on your enemies: at 3% inflation, purchasing power halves every 24 years; at 18% credit-card APR, debt doubles every 4.

Compounding frequency: less dramatic than advertised

$10 000 at 6% for 10 years:

Going from annual to daily compounding adds about 1.7% over a decade. The rate and the time horizon matter enormously; the compounding frequency is a rounding error. When a bank advertises "daily compounding", it is marketing, not math. Compare the APY (effective annual rate), which already includes the frequency: APY = (1 + r/n)ⁿ − 1.

Regular contributions: the future value of an annuity

Most people don't invest a lump sum; they invest monthly. The formula for contributing PMT at the end of each period:

FV = PMT × [ ((1 + i)^N − 1) / i ]      i = periodic rate, N = number of periods

Example: $500/month at 7%/yr (i = 0.07/12 = 0.005833) for 30 years (N = 360):

FV = 500 × [ (1.005833^360 − 1) / 0.005833 ]
   = 500 × [ (8.116 − 1) / 0.005833 ]
   = 500 × 1 219.97 ≈ $610 000

Total contributed: $180 000. Growth: $430 000. Roughly 70% of the final balance is compound growth, not money you put in.

Why "start early" is a math statement, not a motivational poster

Two investors, same 7% return, same $500/month:

A invested one-third as much and finished ahead, because A's money had 30 extra years of doubling (about 3 doublings at 7%: ×8). The single most valuable input to the formula is t, and it is the only one you cannot buy later.

Checklist for using the formula on real products

• Use the net rate: subtract fund fees (an ETF at 0.03% vs a fund at 1% is 0.97%/yr, which over 30 years is a 25% difference in final balance).
• Use a real rate if you want today's purchasing power: (1 + nominal)/(1 + inflation) − 1.
• Match periods: monthly contributions need a monthly rate and monthly N.
• Returns are not constant; the formula gives an expected path, not a guarantee. Sequence matters when you are withdrawing.