You want to be able to withdraw $\$50,000$ from your account each year for $30$ years after you retire. You expect to retire in $20$ years. If your account earns $8\%$ interest, how much will you need to deposit each year until retirement to achieve your retirement goals? Enter an integer or decimal number (more..) Add Work

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Q. You want to be able to withdraw

\$50,000

from your account each year for

30

years after you retire. You expect to retire in

20

years. If your account earns

8\%

interest, how much will you need to deposit each year until retirement to achieve your retirement goals? Enter an integer or decimal number (more..) Add Work

Calculate Present Value: To solve this problem, we need to use the formula for the present value of an annuity to determine how much money needs to be in the account at the time of retirement to allow for $30$ years of $50$,$000$ withdrawals. The formula for the present value of an annuity is:$\newline$\[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]$\newline$where:$\newline$PV = present value of the annuity (the amount needed at the time of retirement)$\newline$PMT = annual payment (withdrawal amount)$\newline$r = annual interest rate (as a decimal)$\newline$n = number of years the payments are to be received$\newline$First, we need to calculate the present value of the annuity that will allow for $50$ , $000$ withdrawals for $30$ years at an $8$ % interest rate.
Calculate PV with Formula: Let's plug in the values into the formula: $\newline$ PMT = $50$,$000$$\newline$r = $8$% or $0$.$08$$\newline$n = $30$$\newline$\[ PV = 50,000 \times \left( \frac{1 - (1 + 0.08)^{-30}}{0.08} \right) \]
Calculate Annual Deposit: Now, calculate the present value (PV): $\newline$ PV = $50,000 \times \left( \frac{1 - (1 + 0.08)^{-30}}{0.08} \right) $\newline$ PV = $50,000 \times \left( \frac{1 - (1.08)^{-30}}{0.08} \right) $\newline$ PV = $50,000 \times \left( \frac{1 - 0.1003}{0.08} \right) $\newline$ PV = $50,000 \times \left( \frac{0.8997}{0.08} \right) $\newline$ PV = $50,000 \times 11.24625 $\newline$ PV = $562,312.50
Calculate PMT with Formula: The present value of $562$,$312$.$50$ is the amount needed at retirement to withdraw $50$ , $000$ annually for $30$ years. Now, we need to calculate how much to deposit each year for the next $20$ years to reach this amount, considering the $8$ % interest rate. We will use the future value of a series of annuities formula: $\newline$ $FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)$ $\newline$ We need to rearrange the formula to solve for PMT (the annual deposit): $\newline$ $PMT = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}$
Calculate PMT with Formula: The present value of $562$,$312$.$50$ is the amount needed at retirement to withdraw $50$ , $000$ annually for $30$ years. Now, we need to calculate how much to deposit each year for the next $20$ years to reach this amount, considering the $8$ % interest rate. We will use the future value of a series of annuities formula: $\newline$ $FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)$ $\newline$ We need to rearrange the formula to solve for PMT (the annual deposit): $\newline$ $PMT = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}$ Now, let's plug in the values to find the annual deposit (PMT): $\newline$ FV = $562$,$312$.$50$ (the amount needed at retirement)$\newline$r = $8$% or $0$.$08$$\newline$n = $20$ (years until retirement)$\newline$\[ PMT = \frac{$562,312.50}{\left( \frac{(1 + 0.08)^{20} - 1}{0.08} \right)} \]
Calculate PMT with Formula: The present value of $562$ , $312$ . $50$ is the amount needed at retirement to withdraw $50$,$000$ annually for $30$ years. Now, we need to calculate how much to deposit each year for the next $20$ years to reach this amount, considering the $8$% interest rate. We will use the future value of a series of annuities formula:$\newline$\[ FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \]$\newline$We need to rearrange the formula to solve for PMT (the annual deposit):$\newline$\[ PMT = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)} \]Now, let's plug in the values to find the annual deposit (PMT):$\newline$FV = $562$ , $312$ . $50$ (the amount needed at retirement) $\newline$ r = $8$ % or $0$ . $08$ $\newline$ n = $20$ (years until retirement) $\newline$ PMT = \frac{$562,312.50}{\left( \frac{(1 + 0.08)^{20} - 1}{0.08} \right)} Calculate the annual deposit (PMT): $\newline$ PMT = \frac{$562,312.50}{\left( \frac{(1.08)^{20} - 1}{0.08} \right)} $\newline$ PMT = \frac{$562,312.50}{\left( \frac{4.6604 - 1}{0.08} \right)} $\newline$ PMT = \frac{$562,312.50}{\left( \frac{3.6604}{0.08} \right)} $\newline$ PMT = \frac{$562,312.50}{45.755} $\newline$ PMT = $12,287.71