Lesson Withdrawing a certain amount of money periodically from a compounded saving account

Withdrawing a certain amount of money periodically from a compounded saving account

Problem 1

Let  r = 6% = 0.06  and let  p = .  Let us move in time back from the 20-th year to the beginning.


20)  At the beginning of the 20-th year, before withdrawing, the account must have exactly W = 24000 dollars in order for the person 
     could withdraw it and complete/(zeroed) the account.

     It means that at the end of the 19-th year, after compounding, the account should have exactly W = 24000 dollars.
     In turn, it means that at the end of the 19-th year, before compounding, the account should be exactly   dollars.

     Hence, at the beginning of the 19-th year, after withdrawing of W = $24000, the account must be exactly  the same   dollars.


19)  It means that at the beginning of the 19-th year, before withdrawing, the account must be   dollars.

     Hence, at the end of the 18-th year, after compounding, the account should be the same  dollars.
     In turn, it means that at the end of the 18-th year, before compounding, the account shoud be exactly   dollars.

     Hence, at the beginning of the 18-th year, after withdrawing of W = $24000, the account must be exactly the same   dollars.


18)  It means that at the beginning of the 18-th year, before withdrawing, the account must be   dollars.

     Hence, at the end of the 17-th year, after compounding, the account should be the same   dollars.
     In turn, it means that at the end of the 17-th year, before compounding, the account should be exactly   dollars.

     Hence, at the beginning of the 18-th year, after withdrawing of W = $24000, the account must be exactly the same   dollars.


17) - 16) - 15) - . . . . . . . . - 2)


1)    It means that at the beginning of the 1-st year, before withdrawing, the account must be   +  + . . .  +  +  dollars,
      and it is exactly the starting amount X, which is under the problem question.


      Hence,  X =  +  + . . .  +  + ,  which is equal, by applying the formula of the sum of geometric progression

              X = W*(((1/p^20)-1) / ((1/p)-1)) = .


ANSWER.  The initial value X of the account to feed the person with W dollars per year for 20 years should be  X = .

         The initial value X of the account to feed the person with W dollars per year for n years should be  X = .

         With the given data  W = 24000 dollars,  r = 6% = 0.06,  p = 1 + 0.06,  n = 20 years  the initial amount at the account should be  


                   X =  = 291794.80 dollars.


Notice that the person get from the fund  $24000*20 = 480,000 dollars in all, in 20 years.

Problem 2

Use the general formula  X = .

In this case  W = $6000,  the quarterly rate is  r = 0.06/4 = 0.015,  p = 1 + 0.015 = 1.015, the number of payment periods (= the number of quarters) 
is n = 20*4 = 80, so


          X =  = 282620.60 dollars.     ANSWER


Notice that the person get from the fund  $6000*20*4 = 480,000 dollars in all, in 20 years.

Problem 3

Use the general formula  X = .

In this case  W = $2000,  the monthly rate is  r = 0.06/12 = 0.005,  p = 1 + 0.005 = 1.005, the number of payment periods (= the number of months) 
is n = 20*12 = 240, so


          X =  = 280557.40 dollars.     ANSWER


Notice that the person get from the fund  $2000*20*12 = 480,000 dollars in all, in 20 years.

Problem 4

Use the general formula  A = .


Here A is the initial amount at the account; W is the monthly withdrawn value; r is the nominal monthly percentage r = 0.1/12; 
presented as a decimal;  p = 1 + r  and n is the number of withdrawing periods (months, in this case).


In this problem,  W is the unknown;  the monthly rate is  r = 0.10/12 = 0.00833,  p = 1 + 0.00833 = 1.00833, the number of payment 
periods (= the number of months) is n = 25*12 = 300.  So


          300000 = .


The factor   is equal to 110;  therefore


           W =  = 2727 dollars  (rounded to the closest lesser dollar).


Thus you will be able to withdraw about $2727 every month during 25 years.    ANSWER

        The conclusion and the lesson to learn

    There is a saving account with initial amount of  X  dollars, compounded periodically (yearly, or quarterly, or monthly).
    A person withdraws certain constant amount  W  from the account at the beginning of each period.
    The account is compounded at the end of each period at the annual rate  r.  
    What the initial value  X  of the account should be in order for to feed the person for  n  years 
    in accordance with this scheme ? 

    a)  Use the decimal rate for the withdrawal (compounding) period  r (for the yearly period ),  or  r/4 (for the quarterly period)  or  r/12 (for the monthly period).  

        Then compute p = 1+r  and the number of the withdrawal periods m = n, or m = 4n  or  m = 12n, correspondingly;


    b)  then use the formula  X =   to determine the initial amount  X  at the account.