Lesson One level more complicated problems on sinking funds

One level more complicated problems on sinking funds

Problem 1

It is a non-traditional sinking fund.

The complication is that the outpayments are made annually, while compounding are made quarterly.


So, we should construct an equivalent model, which will smoothly combine/treat these features.


An account with the annual rate of 8% compounded quarterly works as the account with effective 
quarterly rate r = 0.08/4; hence, its the effective annual growth coefficient is

    t =  =  = 1.08243216,  or an effective annual rate q = 0.08243216.


Now this given sinking fund is equivalent to the ordinary annuity sinking fund 

with annual down payments of $20,000 and with effective annual rate q = 0.071859031 compounded yearly.


Therefore, we can apply the standard formula for the starting value A of such sinking fund

   A = ,

where A is the starting value of a sinking fund;  W is the annual withdrawal; 
n is the number of years;  r is the annual rate (expressed as a decimal);  and p = 1+r.


In our problem 

   A  =  = 132741.78  (rounded).


ANSWER.  The starting value of this sinking fund is  $132741.78.

Problem 2

In other terms, this investment fund is a sinking fund, which provides regular payouts
annually and is compounded quarterly at the nominal compounding rate of 6% per year.
The fund works in this mode during 10 years. After 10 years, the fund is empty.


It is a non-traditional sinking fund.

The complication is that the payouts are made annually, while compounding are made quarterly.


So, we should construct an equivalent scheme, which will smoothly combine/treat these features.


An account with 6% interest rate compounded quarterly works is equivalent to the account 
compounded annually with the effective rate of 

    t =  =  =  = 0.061363551.


    +---------------------------------------------------------------------------+
    |   For such a scheme, the outpayments are synchronized with compounding,   |
    |              so we have a classic regular sinking fund.                   |
    +---------------------------------------------------------------------------+


Now we can apply a classic standard formula for sinking account, which connects the starting amount 
of the fund  A  with the annual outpayments W 

    A = .    (1)


In this formula, A is the starting amount of the sinking fund; W is the regular annual outpayment 
values; the effective annual compounding rate is  

    t =  =  =  = 0.061363551, 

p = 1 + t = 1 + 0.061363551 = 1.061363551,  and the number of payment periods  is n = 10 years. 

So, the equation (1) takes the form


    200000 = .    (2)


The unknown is the value of annual payments W.


In this formula, we can calculate the factor (multiplier) separately

     = 7.312772366.


Then  from formula (2)  W =  = 27349.41.


We round it to the closest cent and get the


ANSWER.  The annual outpayment is 27349.41 dollars.