The management of Gibraltar Brokerage Services anticipates a capital expenditure of $23,000 in 4 years for the purpose of purchasing new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of 11%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.
To calculate the size of each quarterly installment, we need to determine the future value of the sinking fund.
The future value formula for a sinking fund is given by:
FV = P * ((1 + r/n)^(n*t) - 1) / (r/n)
Where:
FV = Future value of the sinking fund
P = Amount deposited each period
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case, we know:
FV = $23,000 (the capital expenditure)
r = 11% = 0.11 (interest rate)
n = 4 (compounding periods per year)
t = 4 (number of years)
Let's plug in these values and solve for P:
$23,000 = P * ((1 + 0.11/4)^(4*4) - 1) / (0.11/4)
Simplifying the equation, we have:
$23,000 = P * (1.0275^16 - 1) / 0.0275
To solve for P, we'll isolate it on one side of the equation:
P = ($23,000 * 0.0275) / (1.0275^16 - 1)
Calculating the right-hand side of the equation gives us:
P ≈ $23,000 * 0.0275 / 0.529
P ≈ $1,192.45
Therefore, each quarterly installment that should be deposited in the fund is approximately $1,192.45.