Visualization for problem $3$

Investing on a regular basis for $30$ years

In retirement, withdrawing funds for $25$ yrs

$3.($a$)$ You are saving for retirement, and you can afford to save $$9,750$ every year, starting one year from today. If you invest for $30$ years earning an average return of $5\mathrm{.}64\%$ per year, how much will you have saved for your retirement? Hint, this is the FV of an annuity. You may want to solve parts $($c$)$ and $($e$),$ then come back and solve $($b$)$ and $($d$),$ which are annuities due.

$($b$)$ How much would you have in your retirement account if you began these same $30$ annual payments immediately? Hint: This is now the FV of an annuity due.

c$)$ Now let's look at things a little differently. Suppose that once you retire, you want to be able to withdraw $$68,000$ per year $($starting one year from your retirement$)$ for a total of $25$ years during your retirement. How much would you need to have in your account when you retire to make this work assuming an annual interest rate of $5\mathrm{.}64\%?$ Hints: This is the $PV$ of an annuity and remember that with this type of problem, you are withdrawing a set amount every year and at the end of the $25$ years, your account has zero dollars in it$.$

$($d$)$ How much would you need to have in your retirement account if you began these same $25$ annual withdrawals immediately? Hint: This is now the PV of an annuity due. Reset your calculator to "END" of period payments when you have finished the annuity due problems.

$($e$)$ Changing the scenario, now let's assume that you want to have $$1,460,000$ in your retirement account at the end of $30$ years $($this is the goal that WSJ reports as what "Americans think they need for retirement" as of late $2023).$ You have now decided that you will deposit funds at the end of every month for $30$ years. The interest rate is still $5\mathrm{.}64\%$ per year. How much do you need to deposit each month in order to reach your goal in $30$ years?