You want to be able to withdraw \( \$ 30,000 \) from your account each year for 20 years after you retire, You expect to retire in 15 years. If your account earns \( 9 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals? Round your answer to the nearest cent.
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To find out how much you'll need to deposit each year until retirement, we first need to determine the total amount needed at retirement. You'll be withdrawing $30,000 for 20 years, and we can use the Present Value of an Annuity formula: \[ PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \] Where: - \( PMT = 30000 \) (annual withdrawal), - \( r = 0.09 \) (annual interest rate), - \( n = 20 \) (number of withdrawals). Calculating this gives: \[ PV = 30000 \times \left(\frac{1 - (1 + 0.09)^{-20}}{0.09}\right) \approx 30000 \times 9.645 = 289,350 \] This means you'll need about $289,350 at retirement. Next, we need to find out how much to save each year for the next 15 years. We'll use the Future Value of an Annuity formula: \[ FV = PMT \times \left(\frac{(1 + r)^{n} - 1}{r}\right) \] We want \( FV \) to equal $289,350, so substituting our values, we have: \[ 289350 = PMT \times \left(\frac{(1 + 0.09)^{15} - 1}{0.09}\right) \] Calculating \( \frac{(1 + 0.09)^{15} - 1}{0.09} \approx \frac{4.412 - 1}{0.09} \approx 37.580 \). Now, we can rearrange this to solve for \( PMT \): \[ PMT = \frac{289350}{37.580} \approx 7,707.62 \] So, you'll need to deposit approximately \( \$7,707.62 \) each year for the next 15 years to achieve your retirement goals.
