Compound interest pays interest on the original principal and on previously-earned interest. Each period’s interest gets added to the principal, and the next period’s interest is computed on the new (larger) base. Growth is exponential.
For a principal at periodic rate over periods:
The exponent makes the growth exponential. Compare with simple interest, where the same calculation gives , linear in .
The difference is dramatic over long horizons. Compounding at 10% for 30 years multiplies your money by . Simple interest at the same rate multiplies by only .
Worked example: invest $1,000 at 8% for 3 years.
- After year 1: 1,000 \cdot 1.08 = \1,080$.
- After year 2: 1,080 \cdot 1.08 = \1,166.40$.
- After year 3: 1,166.40 \cdot 1.08 = \1,259.71$.
Or directly: 1,000 \cdot (1.08)^3 = \1,259.71. The interest in year 1 is \80 (on the original $1,000); in year 2 it’s $86.40 (on $1,080); in year 3 it’s $93.31 (on $1,166.40). Each year’s interest is a bit bigger than the last because it’s computed on a slightly larger base.
In the Time value of money framework, “interest rate” almost always means compound interest rate. Every standard formula (present worth, future worth, annuity factors, gradient factors) assumes compounding. The compounding frequency, how often per year interest gets added back to the principal, is a separate parameter, captured by the nominal vs effective rates distinction.
In the continuous-compounding limit (interest compounded at every instant), the formula becomes , where is the continuously-compounded rate. Continuous compounding shows up mostly in financial mathematics; engineering-economic problems usually compound discretely (annually, monthly, etc.).
Compound interest and the Malthusian model share the same governing ODE .