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

View step-by-step help

Home

Math Problems

Grade 6

Debit cards and credit cards

Full solution

Q. You want to be able to withdraw

\$ 50,000

from your account each year for

30

years after you retire.

\newline

You expect to retire in

20

years.

\newline

If your account earns

8 \%

interest, how much will you need to deposit each year until retirement to achieve your retirement goals?

\newline

Enter an inteser or decimal number [more..]

\newline

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 the $\$50,000$ annual withdrawals. The formula for the present value of an annuity is: $\newline$ $PVA = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)$ $\newline$ where $PVA$ is the present value of the annuity, $PMT$ is the annual payment ( $\$50,000$ ), $r$ is the annual interest rate ( $8\%$ or $0.08$ ), and $n$ is the number of years the payments are to be received ( $30$ years). $\newline$ First, we calculate the present value of the annuity.
Substitute Values in Formula: We plug the values into the formula: $\newline$ $PVA = \$50,000 \times \left[\frac{1 - (1 + 0.08)^{-30}}{0.08}\right]$ $\newline$ Now we calculate the value inside the brackets.
Calculate Value Inside Brackets: Calculate $(1 + 0.08)^{-30}$ : $\newline$ $(1 + 0.08)^{-30} \approx 0.09938$ $\newline$ Now we substitute this value back into the formula.
Continue Calculation: We continue with the calculation: $\newline$ $PVA = \$50,000 \times [(1 - 0.09938) / 0.08]$ $\newline$ $PVA = \$50,000 \times [0.90062 / 0.08]$ $\newline$ $PVA = \$50,000 \times 11.25775$ $\newline$ Now we calculate the present value of the annuity.
Calculate Present Value: We multiply $\$50,000$ by $11.25775$ : $\newline$ $PVA = \$50,000 \times 11.25775 \approx \$562,887.50$ $\newline$ This is the amount that needs to be in the account at the time of retirement.
Calculate Future Value: Next, we need to calculate how much needs to be deposited each year for the next $20$ years to reach the present value of $\$562,887.50$ . We will use the future value of a series formula: $\newline$ $FV = PMT \times \left(\frac{(1 + r)^n - 1}{r}\right)$ $\newline$ where $FV$ is the future value we want to achieve ( $\$562,887.50$ ), $PMT$ is the annual payment we need to find, $r$ is the annual interest rate ( $8\%$ or $0.08$ ), and $n$ is the number of years until retirement ( $20$ years). $\newline$ We need to rearrange the formula to solve for $PMT$ .
Rearrange Formula for PMT: Rearrange the formula to solve for PMT: $\newline$ $PMT = \frac{FV}{\left(\left(1 + r\right)^n - 1\right) / r}$ $\newline$ Now we substitute the values into the formula.
Substitute Values in Formula: Calculate $(1 + 0.08)^{20}$ : $\newline$ $(1 + 0.08)^{20} \approx 4.66096$ $\newline$ Now we substitute this value back into the formula.
Calculate Value Inside Brackets: We continue with the calculation: $\newline$ $PMT = \$562,887.50 / [(4.66096 - 1) / 0.08]$ $\newline$ $PMT = \$562,887.50 / [3.66096 / 0.08]$ $\newline$ $PMT = \$562,887.50 / 45.762$ $\newline$ Now we calculate the annual deposit.
Continue Calculation: We divide $\$562,887.50$ by $45.762$ : $\newline$ $\text{PMT} \approx \$12,300.47$ $\newline$ This is the amount that needs to be deposited each year until retirement.