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A small company wishes to set up a fund that can be used for technology purchases over the next 6 years. their forecast is for $18,000 to be needed at the end of year 1, decreasing by $2,000 each year thereafter. the fund earns 9% per year. how much money must be deposited to the fund at the end of year 0 to just deplete the fund after the last withdrawal? $ round entry to the nearest dollar. the tolerance is ±4.

Respuesta :

mathmate
Cash flow diagram:

? k
|
|
|
|
|
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|    1   2  3   4   5   6
|__ __ __ __ __ __ 
     |    |    |    |    |    |
     |    |    |    |    |    |
     |    |    |    |    |    |
     |    |    |    |    |    |
     |    |    |    |    |   8k
     |    |    |    |   10k
     |    |    |   12k
     |    |   14k
     |   16k
   18k

We need the amount x deposited at the end of year 0 (or beginning of year 1) to provide exactly the cash flow shown.
To solve the problem, we can replace the problem by
A. an annual cash-out of A=$18000 for n=6 years, added onto
B. a cash-in gradient of G=$2000 annually starting at the end of year 2.
with a 9% annual rate.

A. Present value of $18000 annual cash-out
[tex]Pa=\frac{A((1+i)^n-1)}{i(1+i)^n}[/tex]
[tex]=\frac{-18000((1+.09)^6-1)}{.09(1+.09)^6}[/tex]
[tex]=-$80746.53[/tex]

B. Present value of G=$2000 annual gradient
[tex]Pb=\frac{G((1+i)^n-in-1)}{i^2(1+i)^n}[/tex]
[tex]=\frac{2000((1+.09)^6-.09(6)-1)}{.09^2(1+.09)^6}[/tex]
[tex]=$20184.77[/tex]

Total present value of expenses = -$80746.53+$20184.77=-60561.77
Amount needed at the end of year zero = -(-60561.77)= $60561.77

Answer: amount needed to be deposited at the end of year zero is $60561.77 (to the nearest cent).

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