Q$2.$ Today is John Tardy's ${45}^{th}$ birthday and, despite his finance professor's advice, he

hasn't started to save for retirement. John can't believe time has passed so quickly. John

wants to retire on his ${65}^{th}$ birthday; mortality tables indicate his life expectancy is $85.$

Therefore, he will need a retirement fund that will sustain him for $20$ years, or $240$

months. John intends to completely deplete this fund by age $85,$ leaving nothing for his

heirs.

a$)$ What amount of money must be in the retirement fund on John's ${65}^{th}$ birthday if he

withdrew $$7000$ then and every subsequent month? John will withdraw the last payment

one month before his ${85}^{th}$ birthday for a total of $240$ payments $($Annuity Due$).$ The rate of

return on the retirement fund during this period is $9\%$ per year.

b$)$ How much money must John save each month $-$ beginning one month from today

$($Annuity$),$ his ${45}^{th}$ birthday $-$ to accumulate the necessary funds? The rate of return

during this period is $10\%$ per year.

c$)$ John would like to start saving right now, of course, but his $18-$year$-$old daughter is

about to start college at an exclusive eastern school, and he hasn't saved for her tuition

either. Suppose John can pay his daughter's expenses out of his current income but can't

start saving for his retirement until she graduates $($she is almost certain to take four years

to finish$).$ How much money must John save each month, beginning one month after his

$4^{th}$ birthday to accumulate the necessary funds.

$(20$ points$)$