Question

Ms. Patricia Sullivan plans to create a fund from her lottery winnings to meet three objectives. First, she wants to create a fund so that her mother can withdraw $20,000 per month for the remainder of her expected life of 20 years. Second, she wants to pay the down payment for her brother to buy a house upon graduation from college four years from now. She expects that he will need $100,000 for down payment at that time. Finally, she wants to retire after 15 years and be able to withdraw $30,000 per month starting a month from her retirement. She expects to live for 30 years after retirement. All monies earn 8 percent compounded monthly and all cash flows occur at the end of the relevant period. (Answers are rounded.) How much money does she need to invest today to meet her first, 2nd and 3rd objective? Thank you. Emmy

Answer 1

First objective
Annual payment	20,000 PM	$      240,000
Payment time in years		20
One time payment		-
Payment time		-
Interest	8% compounded monthly
Effective Interest	((1+8%/12)^12)-1)
Effective Interest	8.30%
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
i=nominal Interest rate
n = the number of periods in which payments will be made
PV of 1st objective=	PMT x (((1-(1 + r) ^- n)) / i)
PV of 1st objective=	240000*(((1-(1 + 8.30%) ^- 20)) / 8%)
PV of 1st objective=	$                         2,391,091
Second objective
Annual payment		$                 -
Payment time in years		-
One time payment		$      100,000
Payment time		4
Interest	8% compounded monthly
Effective Interest	((1+8%/12)^12)-1)
Effective Interest	8.30%
PV of 2nd objective=	Payment/(1+r)^time
PV of 2nd objective=	100000/(1+8.30%)^4
PV of 2nd objective=	$                         72,691.93
3rd objective
Annual payment	30,000 PM	$      360,000
Payment time in years		30
Payment starting year		T15
Interest	8% compounded monthly
Effective Interest	((1+8%/12)^12)-1)
Effective Interest	8.30%
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
i=nominal Interest rate
n = the number of periods in which payments will be made
PV of 3rd objective at T15=	PMT x (((1-(1 + r) ^- n)) / i)
PV of 3rd objective at T15=	360000*(((1-(1 + 8.30%) ^- 30)) / 8%)
PV of 3rd objective at T15=	$                    4,088,510.45
PV of 3rd objective at T0=	4088510.45/(1+8.30%)^15
PV of 3rd objective at T0=	$                    1,236,340.97
Conclusion
PV of 1st objective	$                         2,391,091
PV of 2nd objective	$                               72,692
PV of 3rd objective	$                         1,236,341
Total amount required	$                         3,700,124