$3.($a$)$ You are saving for retirement, and you can afford to save $$18,000$ every year, starting one year from today. If you invest for $30$ years earning an average return of $3\mathrm{.}8\%$ per year, how much will you have saved for your retirement? Hint, this is the FV of an annuity. You may want to solve parts $($c$)$ and $($e$),$ then come back and solve $($b$)$ and $($d$),$ which are annuities due. $$

$($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 $$72,000$ per year $($starting one year from your retirement$)$ for a total of $25$ years during your retirement. How much would you need in your account when you retire to make this work assuming an annual interest rate of $3\mathrm{.}8\%?$ Hints: This is the PV of a regular annuity and remember that with this type of problem, you are withdrawing a set amount every year and at the end of the $25$ years, your account balance is zero. $$

$($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$$s assume that you want to have $$1,800,000$ in your retirement account at the end of $30$ years. You have now decided to deposit funds at the end of every month for $30$ years. The interest rate is still $3\mathrm{.}8\%$ per year. How much must you deposit each month to reach your goal in $30$ years?

$4.$ Compare the results you got in part $3$a for future value of a "regular" annuity compared to the value you got for the annuity due $($part $3$b$).$ Now compare the PV of the regular annuity in part $(3$c$)$ to the PV of an annuity due in part $(3$d$).$ What is the relationship that you see? Using the time value of money concepts, you have learned so far, why does this relationship $($FV of regular annuity vs$.$ annuity due and PV of regular annuity vs$.$ annuity due$)$ occur?