Allison is 33 years old and plans to retire at age 65 with \( \$ 1,390,000 \) in her retirement account. What amount would she have to set aside now in an investment paying \( 5 \% \) annual interest if the compounding is done daily (assume 365 days in a year)? The amount to be invested now is \( \$ \square \). (Round to the nearest cent as needed.)

To find out how much Allison needs to invest now to accumulate \( \$ 1,390,000 \) by retirement age at a \( 5\% \) annual interest rate with daily compounding, we can use the formula for compound interest: \[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \] Where: - \( P \) = present value (the amount to invest now) - \( A \) = future value (\$1,390,000) - \( r \) = annual interest rate (0.05) - \( n \) = number of times that interest is compounded per year (365) - \( t \) = number of years until retirement (65 - 33 = 32) Substituting in the values: \[ P = \frac{1,390,000}{(1 + \frac{0.05}{365})^{365 \times 32}} \] Calculating this step-by-step: 1. Calculate \( r/n \): \[ \frac{0.05}{365} \approx 0.00013698630 \] 2. Calculate \( nt \): \[ 365 \times 32 = 11680 \] 3. Calculate \( (1 + \frac{r}{n})^{nt} \): \[ (1 + 0.00013698630)^{11680} \approx 4.466585 \] 4. Finally, calculate \( P \): \[ P = \frac{1,390,000}{4.466585} \approx 311,436.71 \] Allison would need to set aside approximately \( \$ 311,436.71 \) now to meet her retirement goal of \( \$ 1,390,000 \) at age 65, rounded to the nearest cent.