Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. The compound interest formula is A = P(1 + r/n)^(nt), where P is principal, r is annual nominal interest rate, n is compounding frequency per year, t is time in years, and A is the final amount.
The compounding frequency n determines how often interest is added back to the principal. Annual compounding (n = 1) adds interest once a year. Monthly compounding (n = 12) adds interest 12 times a year. Daily compounding (n = 365) adds interest every day. Higher compounding frequencies result in slightly higher effective annual yields (APY) for the same nominal APR.
Comparing compounding frequencies with a concrete example
Suppose you deposit $10,000 at a 6% nominal annual rate for 10 years. With annual compounding (n = 1), A = $10,000(1 + 0.06/1)^(10) = $17,908.48. With monthly compounding (n = 12), A = $10,000(1 + 0.06/12)^(120) = $18,193.97, yielding $285.49 more.
With daily compounding (n = 365), A = $10,000(1 + 0.06/365)^(3650) = $18,220.30. Notice that moving from annual to monthly compounding gained $285.49, but moving from monthly to daily compounding only gained an extra $26.33. As compounding frequency approaches infinity (continuous compounding: A = Pe^(rt)), the gains diminish rapidly toward an absolute mathematical limit.
APR vs APY: why banks use both
Annual Percentage Rate (APR) is the simple nominal rate without compounding. Annual Percentage Yield (APY) is the effective rate after accounting for compounding: APY = (1 + r/n)^n - 1. A 6% APR compounded monthly yields an APY of (1 + 0.06/12)^12 - 1 = 6.168%.
Financial institutions often advertise APY on savings accounts (because 6.17% APY sounds better than 6.00% APR) and APR on loans (because 6.00% APR sounds lower than 6.17% APY). Understanding the relationship lets you compare products on equal footing regardless of how they are marketed.