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Compounding — Definition, Formula & Examples

Compounding is the process where earned interest is added to the principal so that, in the next period, interest is calculated on the new, larger balance. The more frequently compounding occurs, the faster the total amount grows.

Compounding refers to the iterative application of an interest rate to a principal sum plus all previously accumulated interest. If interest is compounded nn times per year at an annual rate rr, the accumulated amount after tt years is determined by scaling the principal by the factor (1+rn)nt\left(1 + \frac{r}{n}\right)^{nt}.

Key Formula

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}
Where:
  • AA = Future value (total amount after compounding)
  • PP = Principal (initial amount invested or borrowed)
  • rr = Annual interest rate (as a decimal)
  • nn = Number of times interest is compounded per year
  • tt = Time in years

How It Works

Each compounding period, the bank (or investment) calculates interest on your current balance—not just the original deposit. That new interest gets added to the balance, so the next period's interest is slightly larger. Over many periods, this snowball effect causes exponential growth. Compounding can happen annually, semiannually, quarterly, monthly, or even continuously, and more frequent compounding produces a larger final amount for the same nominal rate.

Worked Example

Problem: You deposit $1,000 in a savings account that pays 6% annual interest, compounded monthly. How much will you have after 5 years?
Identify variables: Principal P=1000P = 1000, annual rate r=0.06r = 0.06, compounding frequency n=12n = 12 (monthly), and time t=5t = 5 years.
Substitute into the formula: Plug the values into the compound interest formula.
A=1000(1+0.0612)12×5=1000(1.005)60A = 1000\left(1 + \frac{0.06}{12}\right)^{12 \times 5} = 1000\left(1.005\right)^{60}
Evaluate: Compute (1.005)60≈1.34885(1.005)^{60} \approx 1.34885.
A≈1000×1.34885=1348.85A \approx 1000 \times 1.34885 = 1348.85
Answer: After 5 years, the account holds approximately $1,348.85. Of that, $348.85 is earned interest—compared to only $300 you would earn with simple interest at the same rate.

Why It Matters

Understanding compounding is essential for evaluating loans, mortgages, and retirement savings. In personal finance, even a small difference in compounding frequency can mean thousands of dollars over decades. The concept also appears in AP math courses and standardized tests whenever exponential growth models are tested.

Common Mistakes

Mistake: Using the annual rate directly instead of dividing by the number of compounding periods.
Correction: Always divide the annual rate rr by nn and multiply the exponent by nn. For monthly compounding at 6%, the periodic rate is 0.06/12=0.0050.06/12 = 0.005, not 0.060.06.