$($a$)$ You are saving for retirement, and you can afford to save $$10,500$ every year, starting one year from today. If you invest for $30$ years earning an average return of $5\mathrm{.}82\%$ per year, how much will you have

saved for your retirement?

$($b$)$ How much would you have in your retirement account if you began these same $30$ annualpayments immediately?

c$)$ Now let's look at things a little differently. Suppose that once you retire, you want to be able to withdraw $$66,000$ 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 $5\mathrm{.}82\%?$ 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? $.$

$($e$)$ Changing the scenario, now let's assume that you want to have $$1,600,000$ 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 $5\mathrm{.}82\%$ per year. How much do you need to deposit each month to reach your goal in $30$ years?

Compare the results you got in part $3$ a for future value of a "regular" annuity compare these to the value you got for the annuity due $($part $3$ b $).$ Now look 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?