Money
Compound interest
Interest earns interest. Slide the years and watch the curve bend up — slowly at first, then faster.
Save $5 a day from age 20, get 7% a year, and at retirement you're a millionaire. Save $5 a day from age 40 and you're not. Compound interest is the cheat code most people learn 20 years too late.
Compound interest earns interest on previously earned interest. It starts slow but accelerates — Einstein supposedly called it the eighth wonder of the world.
Pensions, mortgages, credit-card debt, savings, inflation, business growth — everything that compounds over years grows or shrinks exponentially.
A = final amount, P = principal, r = rate per period, t = periods
$1,000 at 5% compounded yearly, 3 years.
A = 1000 × (1.05)³ = $1,157.63. Compare to $1,150 with simple interest — small difference now, big over decades.
$1,000 at 6% compounded yearly. How much after 30 years?
Rule of 72 — divide 72 by the interest rate to estimate years to double. At 6%, money roughly doubles every 12 years.
Compounding frequency matters. The same '12% APR' compounded monthly grows faster than annually. Always check whether the rate is *nominal* or *effective* before comparing.
- A = P(1 + r)ᵗ — exponential growth, not linear.
- More frequent compounding = slightly faster growth, with diminishing returns.
- Rule of 72: time to double ≈ 72 / (rate as percent). The cleanest mental shortcut in finance.