Determining the size of quarterly installments

Question

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $20,000 in 2 years for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of 5%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund. (Round your answer to the nearest cent.)

Answer 1

This problem is a future value problem, except in this case, instead of wanting to find out how much money the fund will hold given a periodic investment over some time period, the investment required to have a certain amount of money after the time period is what is needed (people will use this for retirement planning - if I want to retire with $X, then I need to invest this amount each paycheck).

So the equation is the future value equation:

FV = C [ (1 + i )ⁿ - 1 ] ⁄ i rearranged to solve for C: C = ( FV * i ) ⁄ [ (1 + i )ⁿ - 1 ]

FV = how much money is needed at the end of the time period

r = annualized interest rate (in decimal form)

p = # time periods per year

i = r ⁄ p = period interest rate (in decimal form)

n = # of periods = [# of years] * p

C = the amount paid per time period

For this problem,

FV = $20,000 (the amount needed)

r = 5% = 0.05

p = 4

i = 0.05 ⁄ 4 = 0.0125

n = 2 * 4 = 8

C = [($20,000) * ( 0.0125)] ⁄ [ (1.0125 )⁸ - 1 ] = $2392.66

(if this is worked in reverse, the FV = $19,999.98)