Question

Lady Gaga is 30 and already worried about her future. She wants to make sure that she’ll be able to keep up with the life standard she got used to – at the end of the day, she was born this way and wants to die this way, too. She has couple of goals that she wants to achieve after she retires. First, she wants to be able to withdraw $150,000 each month to cover her clothing and make-up expenses for 15 years after she stops singing and retires at the age of 65. Second, she wants to be able to donate $3,000,000 to St. Jude Children’s Hospital at the age of 75. Lastly, the year she retires, she also wants to buy a house in Honolulu, HI that costs $7,500,000 today, with the price being estimated to increase by 1% each year.

a. If she can earn 15% compounded monthly on her retirement account, how much does she need to deposit into her account each month until retirement to achieve her goals?

b. What if she decides to save $40,000 each month for the first 10 years, then not to save for 15 years (but the money still stays in the account), and to go back to saving and investing for the last 10 years before retirement – how much would she then need to save every month for the last 10 years to achieve the same goals?

Answer 1

Lets compute the PV of goals at the time of retirement at age 65
PV of donation=	3000000/(1+15%)^10
	741,554.12
FV of house property at age 65=	7500000*(1+1%)^35
	10,624,520.67
PV of monthly withdrawal at age 65=
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
Monthly withdrawal	150000
Interest	15%
Time	15
Effective interest rate	(1+15%/12)^12)-1)
Effective interest rate	16.075%
PV	=150000*12*(((1-(1 + 16.075%) ^- 15)) / 15%)
PV	10,717,446.45
Total funds required in retirement account at age 65
Donation	741,554.12
House	10,624,520.67
Monthly withdrawal	10,717,446.45
Total funds required	22,083,521.24
Lets compute the monthly deposit required
FV of annuity
The formula for the future value of an ordinary annuity, as opposed to an annuity due, is as follows:
P = PMT x ((((1 + r) ^ n) - 1) / i)
Where:
P = the future 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
Interest nominal	15%
Effective interest	16.075%
Time	35 years
Future value required	22,083,521.24
22,083,521.24	=PMT*((((1 + 16.075%) ^ 35) - 1) / 15%)
22,083,521.24	=PMT*1222.93
Monthly deposit required=	22083521.24/1222.93
Annual deposit required=	18,057.88
Monthly deposit required=	1,504.82
Solution B
First deposit 40000 per month for first 10 years
FV of this deposit at age 40
FV=	PMT*((((1 + 16.075%) ^ 10) - 1) / 15%)
FV=	40000*12*((((1 + 16.075%) ^ 10) - 1) / 15%)
FV=	11,008,129.33
Now this amount remains invested for next 25 years
FV of this amount	11008129.33*(1+15%)^25
FV of this amount	362,376,088
So the amount is already saved enough to fund the retirement requirement so she need not save any thing in her last 10 years