Question

You are trying to decide how much money you will need at retirement. You expect to retire at age 65. You hope to travel extensively while you are healthy enough. To finance your travels (and cover your basic living expenses) you think you will need income of $241,000 per year (at the end of each year). You will make your first withdrawal on your 66th birthday. You expect to stay healthy enough for travel for the first 19 years after retirement and thus make a withdrawal of $241,000 at the end of each of those 19 years. Once your active travel years are over you will settle down into a retirement home. During your retirement home years you will only need $225,000 in income per year. You expect to live in the retirement home for 12 years. You will make the first retirement home withdrawal 20 years after retirement and all subsequent withdrawals will continue to be at year-end.

If retirement savings will earn a return of 4.5%, then how much will you need at retirement to fund these planned withdrawals? How much will you need at retirement to fund these planned withdrawals? $ (Round to the nearest cent.)

Answer 1

To get the requisite retirement fund value, we need to calculate the PV of 2 annuities.
1 annuity of annual withdrawal of $ 241,000 for 19 years
2nd annuity of annual withdrawal of $ 225,000 for the next 12 years.
All these present values are computed at T0 i.e. at age 65
PV of annuity
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	To be computed
PMT = the dollar amount of each annuity payment	$    241,000.00
r = the effective interest rate (also known as the discount rate)	4.50%
n = the number of periods in which payments will be made	19
PV of first annuity=	PMT x (((1-(1 + r) ^- n)) / r)
PV of first annuity=	241000* (((1-(1 + 4.5%) ^- 19)) / 4.5%)
PV of first annuity=	$ 3,034,983.76
PV of the second annuity at t19=	225000* (((1-(1 + 4.5%) ^- 12)) / 4.5%)
PV of the second annuity at t19=	$ 2,051,680.68
PV of the second annuity at t0=	2051680.68/(1+4.5%)^19
PV of the second annuity at t0=	$    888,996.91
So total corpus required	3034983.76+888996.91
So total corpus required	$ 3,923,980.66