$($a$)$ You are saving for retirement, and you can afford to save $$12,600$ every year, starting one year from today. If you invest for $30$ years earning an average return of $6\mathrm{.}24\%$ 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.

$N=30,\frac{\text{I}}{Y}=6\mathrm{.}24,PMT=0,PV=-12,600,FV=77,448\mathrm{.}39$

$($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 $$78,000$ 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 $6\mathrm{.}24\%?$ 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$.$