Compound interest pays interest not just on the original principal but on previously-earned interest as well. Each period’s interest is added to the principal, and the next period’s interest is computed on the new (larger) base. The result is exponential growth.
For a principal at periodic rate over periods:
The exponent is what makes the growth exponential — compare with simple interest, where the same calculation would give (linear in ).
The qualitative difference is dramatic over long horizons. Compounding at 10% for 30 years multiplies your money by . Simple interest at the same rate would multiply by only .
A simple compounding 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 distinction between nominal and effective rates.
In the continuous-compounding limit (interest compounded at every instant), the formula becomes , where is the continuously-compounded rate. Continuous compounding is mostly used in financial mathematics; engineering-economic problems usually compound discretely (annually, monthly, etc.).
For the link between compound interest and the simplest exponential differential equation, see Malthusian model — both have the same governing ODE .