Lesson Accumulating money via ordinary annuity and spending simultaneously via sinking fund

Accumulating money via ordinary annuity and spending simultaneously via sinking fund

Problem 1

Below is the time scale to this problem, showing years, contributions and withdrawals.


contributions  C1    C2    C3    C4    C5    C6    C7    C8    C9    C10
                 
years           |--1--|--2--|--3--|--4--|--5--|--6--|--7--|--8--|--9--|--10-|

withdrawals                                  W1    W2    W3    W4    W5    


Let say that every contribution is made September, 1, each year;
every compounding is made August, 31, each year, 
and every withdrawal is made September, 1, each year.


Then in first five years we have annuity due saving plan with annual deposits X dollars.
So, we can write expression for the future value of the account after 5 years (at the end of the 5-th year)


    FV(5) =  = .    (1)


Starting from the beginning of the 6-th year, we have contributions of X dollars and withdrawals of 36500 dollars
at the same day. So, we can interpret it as if we have withdrawals only of the size  (35600-X)  dollars
at the beginning of each year starting from 6-th year .


So, on years from 6-th to 10-th, we have, actually, a sinking fund, which has the starting amount FV(5) and
five withdrawals of  (35600-X) dollars each, compounded annually at 3.4%


    FV(5) =  = .   (2)


Together with expression (1), it gives us this equation


     = .    (3)


Cancel factors 1.034 in both sides, and cancel 0.34 in denominators.  You will get then


     = .    (4)


From this point, there are two ways to proceed.


One way is to calculate coefficients in each side directly.  You will get


    X*0.181959767 = (36500-X)*0.153947513.


Simplify and find X

    0.181959767*X = 36500*0.153947513 - 0.153947513*X

    0.181959767*X + 0.153947513*X = 36500*0.153947513

    (0.181959767 + 0.153947513)*X = 36500*0.153947513

    X =  = 16728.08.


Thus, the regular annual deposit should be  16,728.08 dollars.   


Another way is to notice that equation (4) is equivalent to

    
    1.034^5*X = 36500 - X,

    X =  =  = 16728.08  dollars,


and you get the same answer as above.
    

ANSWER.  The regular annual deposit should be  16,728.08 dollars (ten equal installments).