When you reach retirement​ age

When you reach retirement​ age, you would like to have enoughmoney 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 tostart your retirement with a large amount of money​ saved, and youwill withdraw

​$70​,000

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

In the​ meantime, you are a​ 25-year-old new UIC​ graduate, andyou plan on working for 40 years until you retire. To fund yourretirement​goals, you plan on investing some money in the stockmarket. 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 thenext 40 years.

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

10​%

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

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

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

Answer:

First we will compute the PV ofretirement withdrawals and then we will compute the annualcontribution requirement to accumulate this corpus.
PV of annuity for making pthlypayment
P = PMT x (((1-(1 + r) ^- n)) /r)
Where:
P = the present value of an annuitystream P
PMT = the dollar amount of eachannuity payment          70,000
r = the effective interest rate (alsoknown as the discount rate) 10%
n = the number of periods in whichpayments will be made 20
PV of retirementwithdrawals= PMT x (((1-(1 + r) ^- n)) /r)
PV of retirementwithdrawals= 70000*(((1-(1+10%) ^-20))/10%)
PV of retirementwithdrawals= $595,949
FV of annuity
P = PMT x ((((1 + r) ^ n) – 1) /r)
Where:
P = the future value of an annuitystream $595,949
PMT = the dollar amount of eachannuity payment PMT
r = the effective interest rate (alsoknown as the discount rate) 10%
n = the number of periods in whichpayments 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

 
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