Compound Interest Formula Explained: Daily vs. Monthly Compounding & APY

Compound interest represents earning returns not only on your initial principal, but also on previously accumulated interest. The mathematical formula is: $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ Where: - $A$ = Final accrued amount - $P$ = Initial principal deposit - $r$ = Nominal annual interest rate (in decimal form) - $n$ = Number of times interest compounds per year (daily = 365, monthly = 12) - $t$ = Number of years invested

The more frequently interest is added to your account, the faster your balance grows. For a $10,000 deposit at 5% over 10 years: - **Annual Compounding ($n=1$):** $16,288.95 - **Monthly Compounding ($n=12$):** $16,470.09 - **Daily Compounding ($n=365$):** $16,486.65

APY reflects the true effective annual rate factoring in compounding: $$\text{APY} = \left(1 + \frac{r}{n}\right)^n - 1$$ A 5.00% nominal rate compounded daily delivers an effective yield of 5.127% APY.