The Math Behind Compound Interest

Simple interest: the baseline

Simple interest is calculated only on the original principal:

A = P × (1 + r × t)

$10,000 at 8% simple interest for 20 years:
A = $10,000 × (1 + 0.08 × 20) = $26,000

Simple interest grows linearly. Every year, the same $800 in interest is earned, regardless of the accumulated total. After 20 years you have $26,000. This is how some short-term loans and bonds work.

Compound interest: interest on interest

A = P × (1 + r/n)^(n×t)

Where:
  P = principal
  r = annual rate (decimal)
  n = compounding periods per year
  t = years

$10,000 at 8% compounded annually for 20 years:
A = $10,000 × (1 + 0.08)^20 = $46,610

That's $46,610 versus $26,000 from simple interest — $20,610 more, earned from interest compounding on itself. Each year's interest is added to the principal, so the next year's interest is calculated on a larger base.

Year-by-year breakdown: seeing the curve

$10,000 at 8% annual compound interest:

Year  Balance     Interest earned this year
1     $10,800     $800
5     $14,693     $1,089
10    $21,589     $1,599
15    $31,722     $2,348
20    $46,610     $3,449
30    $100,627    $7,451
40    $217,245    $16,093

Notice that the interest earned per year grows every year — from $800 in year 1 to $16,093 in year 40. The balance hasn't changed (same 8% rate) but the base is now $200,000 instead of $10,000. This is the exponential curve — slow at first, dramatic later.

Adding regular contributions: the power multiplier

Most real investors don't invest a lump sum and wait — they add to their investment regularly. The future value of an annuity (regular equal payments) is:

FV = PMT × [(1 + r)^n - 1] / r

$500/month at 8% annual return over 30 years:
r (monthly) = 0.08/12 = 0.00667
n = 360 months
FV = $500 × [(1.00667)^360 - 1] / 0.00667
FV ≈ $745,180

Total contributed: $180,000
Investment growth: $565,180

You contribute $180,000 and end up with $745,000. More than three-quarters of the final balance is compound growth — interest earned on interest. The contributions are the seed; compounding does the rest.

The Rule of 72: mental math for doubling

Divide 72 by the annual interest rate to estimate how long it takes to double your money: at 6% it doubles in 12 years; at 8% in 9 years; at 12% in 6 years. The rule is an approximation — the actual formula is 69.3/r — but it's accurate enough for most planning purposes in the typical 2-15% range. It also works in reverse: if you need your money to double in 8 years, you need approximately 72/8 = 9% annual return.

Why the rate matters less than you think

It's tempting to chase higher returns, but time in the market has a bigger impact than a few percentage points of extra return. $10,000 at 6% for 40 years grows to $102,857. $10,000 at 8% for 30 years grows to $100,627. Despite a lower rate over a longer period, the outcome is nearly identical — because the time advantage offsets the rate disadvantage. The practical implication: start early, stay consistent, don't pause contributions during market downturns. The mathematical damage from stopping contributions for even 2-3 years is larger than most people expect.