You want to be able to withdraw \( \$ 35,000 \) from your account each year for 30 years after you retire. You expect to retire in 20 years. If your account earns \( 7 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals?
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To find out how much you will need to deposit each year until retirement, we first need to calculate the total amount needed at retirement to withdraw \( \$ 35,000 \) annually for 30 years at an interest rate of \( 7\% \). The total amount \( S \) required at retirement can be found using the formula for the present value of an annuity: \[ S = \frac{C \times (1 - (1 + r)^{-n})}{r} \] where \( C = 35,000 \), \( r = 0.07 \), and \( n = 30 \). Plugging in the values: \[ S = \frac{35,000 \times (1 - (1 + 0.07)^{-30})}{0.07} \approx \frac{35,000 \times 13.385}{0.07} \approx 6,604,250 \] Now, we need to determine how much to deposit each year for the next 20 years to reach this amount. Using the future value of a series formula: \[ FV = PMT \times \frac{(1 + r)^{n} - 1}{r} \] We rearrange it to solve for \( PMT \): \[ PMT = \frac{S \times r}{(1 + r)^{n} - 1} \] Substituting our values into the formula: \[ PMT = \frac{6,604,250 \times 0.07}{(1 + 0.07)^{20} - 1} \approx \frac{462,297.5}{3.86968} \approx 119,607.83 \] So you will need to deposit approximately \( \$ 119,607.83 \) each year until retirement to achieve your goal.
