Question 7 of 15, Step 1 of 1 Correct Incorrect Determine the annual percentage yield, or the effective interest rate, for \( \$ 400 \) invested at \( 3.23 \% \) over 20 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 \[ \begin{array}{l} A=P^{n} \\ A P Y=\left(1+\frac{r}{n}\right)^{n}-1 \end{array} \] Annual Percentage Yield (APY) Answer Keypad Keyboard Shortcuts

To determine the annual percentage yield (APY) for $400 invested at 3.23% over 20 years compounded daily, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the future value of the investment - \( P \) is the principal amount ($400) - \( r \) is the annual interest rate (3.23% or 0.0323) - \( n \) is the number of times the interest is compounded per year (365 for daily compounding) - \( t \) is the time in years (20 years) Substitute the given values into the formula: \[ A = 400 \left(1 + \frac{0.0323}{365}\right)^{365 \times 20} \] Now, we can calculate the future value of the investment using the formula for compound interest. Calculate the value by following steps: - step0: Calculate: \(400\left(1+\frac{0.0323}{365}\right)^{365}\times 20\) - step1: Divide the terms: \(400\left(1+\frac{323}{3650000}\right)^{365}\times 20\) - step2: Add the numbers: \(400\left(\frac{3650323}{3650000}\right)^{365}\times 20\) - step3: Multiply the terms: \(8000\left(\frac{3650323}{3650000}\right)^{365}\) - step4: Simplify: \(8000\times \frac{3650323^{365}}{3650000^{365}}\) - step5: Rewrite the expression: \(2000\times 4\times \frac{3650323^{365}}{3650000^{365}}\) - step6: Rewrite the expression: \(2000\times 4\times \frac{3650323^{365}}{2000^{365}\times 1825^{365}}\) - step7: Reduce the numbers: \(4\times \frac{3650323^{365}}{2000^{364}\times 1825^{365}}\) - step8: Rewrite the expression: \(4\times \frac{3650323^{365}}{16^{364}\times 125^{364}\times 1825^{365}}\) - step9: Rewrite the expression: \(4\times \frac{3650323^{365}}{4^{728}\times 125^{364}\times 1825^{365}}\) - step10: Reduce the numbers: \(1\times \frac{3650323^{365}}{4^{727}\times 125^{364}\times 1825^{365}}\) - step11: Multiply the fractions: \(\frac{3650323^{365}}{4^{727}\times 125^{364}\times 1825^{365}}\) The future value of the investment is approximately $8262.61. To find the annual percentage yield (APY), we can use the formula: \[ APY = \left(1 + \frac{r}{n}\right)^{n} - 1 \] Substitute the given values into the formula: \[ APY = \left(1 + \frac{0.0323}{365}\right)^{365} - 1 \] Now, we can calculate the APY. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.0323}{365}\right)^{365}-1\) - step1: Divide the terms: \(\left(1+\frac{323}{3650000}\right)^{365}-1\) - step2: Add the numbers: \(\left(\frac{3650323}{3650000}\right)^{365}-1\) - step3: Rewrite the expression: \(\frac{3650323^{365}}{3650000^{365}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{3650323^{365}}{3650000^{365}}-\frac{3650000^{365}}{3650000^{365}}\) - step5: Transform the expression: \(\frac{3650323^{365}-3650000^{365}}{3650000^{365}}\) The annual percentage yield (APY) is approximately 3.28%. Therefore, the annual percentage yield for $400 invested at 3.23% over 20 years compounded daily is approximately 3.28%.