The management of Gibraltar Brokerage Services anticipates a capital expenditure of $27,000 in 3 yr 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 9%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.
To determine the size of each quarterly installment that should be deposited in the sinking fund, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r/n)^(n*t) - 1] / (r/n)
Where:
FV = Future value (total amount needed)
P = Quarterly installment (amount to be deposited)
r = Annual interest rate (9%)
n = Number of compounding periods per year (quarterly, so n = 4)
t = Number of years (3)
We are given that the future value (FV) is $27,000, so we can plug in the values and solve for P:
$27,000 = P * [(1 + 0.09/4)^(4*3) - 1] / (0.09/4)
Simplifying the equation:
27,000 = P * [(1.0225)^(12) - 1] / 0.0225
Now, let's solve for P:
27,000 * 0.0225 = P * [(1.0225)^(12) - 1]
607.5 = P * (1.297032313 - 1)
607.5 = P * 0.297032313
P = 607.5 / 0.297032313
P ≈ $2046.25
Therefore, each quarterly installment that should be deposited in the sinking fund is approximately $2046.25.
To determine the size of each quarterly installment for the sinking fund, we can use the formula for the future value of an ordinary annuity:
FV = P [(1 + r/n)^(nt) - 1] / (r/n)
where:
FV = Future value of the sinking fund (equal to the capital expenditure)
P = Quarterly installment
r = Annual interest rate (9% = 0.09)
n = Number of times interest is compounded per year (quarterly = 4)
t = Number of years (3)
In this case, the future value of the sinking fund (FV) is equal to the capital expenditure of $27,000. We need to solve for the quarterly installment (P).
Let's plug in the values into the formula and solve for P:
27,000 = P [(1 + 0.09/4)^(4*3) - 1] / (0.09/4)
Simplifying the equation:
27,000 = P [1.0225^(12) - 1] / (0.0225)
Now, we can calculate the quarterly installment:
P = (27,000 * 0.0225) / [1.0225^(12) - 1]
P ≈ $867.32 (rounded to the nearest cent)
Therefore, each quarterly installment that should be deposited in the sinking fund is approximately $867.32.
27,000 in 3yr, 9%/yr compounded quarterly
R = Ai/(1 + i)^n - 1
R = Payment
A = Total needed = 27,000
n = Number of payments = 4 * 3 = 12
i = Interest rate = 0.09/4 = 0.0225
R = 27000(0.0225)/((1 + 0.0225)^12 - 1)
R = 607.50/((1.0225)^12 - 1)
R = 607.50/(1.30605 - 1)
R = 607.50/0.30605
R = 1984.97