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.

To determine the annual percentage yield (APY) for \( \$400 \) invested at an interest rate of \( 3.23\% \) compounded daily over 20 years, you can use the formula for APY, which is given by: \[ APY = \left(1 + \frac{r}{n}\right)^{n \cdot t} - 1 \] where \( r \) is the nominal interest rate (decimal), \( n \) is the number of compounding periods per year, and \( t \) is the number of years. Here, \( r = 0.0323 \) (since \( 3.23\% = \frac{3.23}{100} \)), \( n = 365 \), and \( t = 20 \). Substituting these values into the formula: \[ APY = \left(1 + \frac{0.0323}{365}\right)^{365 \cdot 20} - 1 \] Calculating it step-by-step: 1. Calculate \( \frac{0.0323}{365} \approx 0.00008884 \). 2. Add \( 1 \) to this value: \( 1 + 0.00008884 \approx 1.00008884 \). 3. Calculate \( 365 \cdot 20 = 7300 \). 4. Raise \( 1.00008884 \) to the power of \( 7300 \): \[ (1.00008884)^{7300} \approx 1.83345 \] 5. Subtract \( 1 \): \( 1.83345 - 1 = 0.83345 \). Finally, convert this value to a percentage by multiplying by \( 100 \): \[ APY \approx 83.35\% \] So, the annual percentage yield for this investment is approximately \( 83.35\% \).