Lesson Calculating a retirement plan

Calculating a retirement plan

Problem 1

First, let's determine how much money X should be accumulated on the account during 25 years when you make your annual deposits,
in order for to have enough to withdraw $45,000 each year for 15 years of your retirement.


Withdrawing $45,000 each year, your account (the remaining money) still earns 4% per annum compounded at the end of each year.   
So, during this period of 15 years, the annuity works as a sinking fund.


If you withdraw at the beginning of each year for living, the formula for the starting value of sinking fund is

    X = ,     (1)


where W is the annual withdraw amount; r is the annual rate of compounding as the decimal,
and n is the number of withdrawals.


In this problem,  W = $45,000,  r = 0.04;  n = 15.  Substitute these values into the formula (1) and calculate X

    X =  = $520340.53.     (2)


So, after 25 years of accumulating money, the fund should have $520340.53.


Thus, the first step of calculations is complete.

Now we are in position to determine how much should be deposited each year during 25 years when you make annual deposits.


I will assume that the annual deposits are made at the end of each of 25 years.


Then the fund works as a standard Ordinary Annuity saving plan.
So, we use this formula for the future value

    FV = ,     (3)


where D is the annual deposit amount; r is the annual rate of compounding as the decimal,
and m is the number of deposits.


In this problem,  FV = $500327.43 as we determined above;  D is the unknown value of the deposit to find from the equation; 
r = 0.04;  m = 25.  Substitute these values into the formula (3) 

    520340.53 = .     (4)


The multiplier    is  41.64591,  


It implies from equation (4)  that  D =  = $12494.40.


It is your ANSWER:  During 25 years of the accumulating period you should deposit $12494.40 at the end of each year to your account.

                    Then you will be able to withdraw $45000 at the beginning of every year during 15 years.


For clarity, notice that in 25 years, you deposit in the bank 25*12494.40 = 312360 dollars.


During following 15 years, you withdraw from the bank 15*45000 = 675000  dollars.


The difference 675000 - 312360 = 362640 dollars is the interest, which your account earned 
during 25 + 15 = 40 years.