Determine the annual percentage yield, or the effective interest rate, for \( \$ 200 \) invested at \( 5.44 \% \) over 10 years compounded daily. Round your answer to the nearest hundredth of a percent, if necessary. Formulas Simple Interest \[ \begin{array}{l} I=P r t \\ A=P(1+r t) \\ A=P\left(1+\frac{r}{n}\right)^{n t} \end{array} \] Future Value for Simple Interest Future Value for Compound Interest Future Value for Continuous Compound Interest \[ A=P e^{r} \] Annual Percentage Yield (APY) \[ \mathrm{APY}=\left(1+\frac{r}{n}\right)^{n}-1 \] Answer

To determine the annual percentage yield (APY) for an investment of \( \$200 \) at an interest rate of \( 5.44\% \) compounded daily over 10 years, we can use the APY formula: 1. First, convert the percentage rate to decimal form: \( r = 0.0544 \). 2. The number of compounding periods per year for daily compounding is \( n = 365 \). Now plug these values into the APY formula: \[ \mathrm{APY} = \left(1 + \frac{r}{n}\right)^{n} - 1 = \left(1 + \frac{0.0544}{365}\right)^{365} - 1. \] Calculating this gives: \[ \mathrm{APY} \approx \left(1 + 0.00014986\right)^{365} - 1 \approx 0.0560 \text{ or } 5.60\% \] Rounded to the nearest hundredth of a percent, the APY is approximately \( 5.60\% \). --- When considering how compounding works, think of it like a snowball effect: the interest you earn begins to earn interest itself! This is why understanding compounding – especially with daily compounding – can significantly boost your investment returns over time. Additionally, common mistakes to avoid include forgetting to adjust the interest rate for the compounding frequency and not accounting for the total investment duration properly. Make sure you’re consistent with your units, and your calculations will yield the sweetest results!