Question

When you reach retirement​ age, you would like to have enough money saved to be able to​ “pay yourself” an annual salary of

​$70​,000

per year for 20 years. To put this another​ way, your plan is to start your retirement with a large amount of money​ saved, and you will withdraw

​$70​,000

from these savings once a year for the next 20 years until all of your savings are depleted.

In the​ meantime, you are a​ 25-year-old new UIC​ graduate, and you plan on working for 40 years until you retire. To fund your retirement​goals, you plan on investing some money in the stock market. More​ specifically, at the end of each year until you​ retire, you are going to put part of your paycheck into the stock​ market; you'll put in the same dollar amount every year for the next 40 years.

You are a pretty decent stock​ investor, and you think you can make a

10​%

return on the market each year you​ invest, both until you retire and after retirement.

What is the annuity payment​ (to the nearest​ dollar) you need to put into the stock market every year for the next 40 years to fully fund your​retirement?

Write your​ answer, without a dollar sign in​ front, rounded to the nearest whole dollar. ​ (If you do this​ correctly, it might be a smaller number than​ you'd think!)

Answer 1

First we will compute the PV of retirement withdrawals and then we will compute the annual contribution requirement to accumulate this corpus.
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	P
PMT = the dollar amount of each annuity payment	70,000
r = the effective interest rate (also known as the discount rate)	10%
n = the number of periods in which payments will be made	20
PV of retirement withdrawals=	PMT x (((1-(1 + r) ^- n)) / r)
PV of retirement withdrawals=	70000*(((1-(1+10%) ^-20)) /10%)
PV of retirement withdrawals=	$595,949
FV of annuity
P = PMT x ((((1 + r) ^ n) - 1) / r)
Where:
P = the future value of an annuity stream	$595,949
PMT = the dollar amount of each annuity payment	PMT
r = the effective interest rate (also known as the discount rate)	10%
n = the number of periods in which payments will be made	40
FV of annuity=	PMT x ((((1 + r) ^ n) - 1) / r)
595949=	PMT*((((1+10%)^40)-1)/10%)
Annual contribution=	595949/((((1+10%)^40)-1)/10%)
Annual contribution=	$    1,347