Lesson Regular withdrawals from an annuity payout account

Regular withdrawals from an annuity payout account

Problem 1

Since withdrawals are made at the beginning of each month, use the general formula  for an Annuity Due sinking fund  

          A = .


Here A is the initial amount at the account; W is the monthly withdrawn value at the beginning of each month; 
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 = 0.05/12 = 0.004166667,  
p = 1 + 0.004166667 = 1.004166667, the number of payment periods (= the number of months) is n = 25*12 = 300.  So


          500000 = .


The factor   is equal to 171.7727904;  therefore


           W =  = 2910.82 dollars.


Thus you will be able to withdraw about $2910.82 every month (rounded to the closest lesser cent) during 25 years.

Problem 2

For such sinking fund, the formula which connects the starting amount X and the quarterly 
payment Q is 

    X = .    (1)


In this formula, Q is  the regular withdrawal per quarter;  the factual quarterly compounding rate 
is  r = ,  p = , and the number of payment periods  is n = 10 years * 4 quarters = 40. So


    200000 = .    (2)


The unknown is the value of quarterly payments Q.
In this formula, we can calculate the factor (multiplier)

     = 29.11257449.


Then  from formula (2)  Q =  = 6869.883668.


We round it to the closest cent and get the


ANSWER.  The quarterly payment out is 6869.88 dollars.

Problem 3

This problem is about the starting amount of a sinking fund.

The fund should provide the payments of $12,000 at the beginning of each quarter during 30 years.
The fund is compounded quarterly at the annual interest rate of 5.75%.


The general formula  to calculate the starting amount of the account is

    X = .


In this formula, W is  the regular withdrawal per quarter, W = $12000;  
the factual quarterly compounding rate is  r = 0.0575/4,  p = 1 + r,
and the number of payment periods  is n = 30 years * 4 quarters = 120. So


          X =  = 694044.48

 dollars.     It is the  ANSWER  to the problem's question (a).



The answer to question (b) is   = 5783.70 dollars.

    It is the averaged available uniform withdrawal from $694044.48 at each quarter 
    during 30 years - if compounding does not work.



The answer to question (c) is 30*4*12000 = 1,440,000 dollars.

    It is the real total amount that the family withdraws/receives from this sinking fund during 30 years.



The answer to question (d) is the difference 

    $1,440,000 - $694,044.48 = $745955.52.


    It is the money that the sinking fund earns/accumulates in 30 years due to compounding.

Problem 4

Similar account at the end of fifteen years would be able to provide payments of $75,000 
at the beginning of each year for 5 (five) years.


So, the question is: find the amount, which, when invested at 8% annual interest rate,
will provide payment (withdrawing) $75,000 at the beginning of each year for five years.


    +----------------------------------------------+
    |    It is a standard question/problem about   |
    |      a starting amount of a sinking fund.    |
    +----------------------------------------------+


The general formula  to calculate the starting amount at the account is

    X = .


In this formula, W is  the regular annual withdrawal, W = $75000;  the factual annual compounding rate 
is  r = 0.08,  p = 1 + 0.08 = 1.08, and the number of payment periods  is n = 5 years. So


          X =  = 323409.52 dollars.     


It is the  ANSWER  to the problem's question.

Problem 5

The sough purchase price is the sum of two amounts.

First amount is the starting amount of an annuity that provides paying (from annuity to you) $200 per month for 10 years.

Second amount is the starting amount which provides a lump payment (from the annuity to you) of $2000 in 10 years from now.


To find first amount, use the formula  X = .


In this case,  W = $200 is the monthly withdrawal;  
r is the effective monthly compounding rate  r = 0.065/12;  
p = 1 + 0.065/12; 
n is the number of withdrawals (the same as the number of months, n = 10*12 = 120).


So, 

    X =  = 17613.70 dollars for the first amount.


To find the value of the second amount, Y, use this equation

    2000 = .


From this equation,  Y =  = 1045.924586, or 1045.92 dollars (rounded).


Thus the  ANSWER  to the problem's question is this sum

    X + Y = 17613.70 + 1045.92 = 18659.62 dollars.