The management of Gibraltar Brokerage Services anticipates a capital expenditure of $20,000 in 4 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 4%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

To solve this problem, we need to determine the quarterly installment Gibraltar Brokerage Services should deposit into a sinking fund to accumulate $20,000 in 4 years. The fund earns interest at an annual rate of 4% compounded quarterly.

### Step-by-Step Solution:

1. **Identify the variables:**
   - Future value (FV): $20,000 (the amount needed in the sinking fund)
   - Interest rate (annual): 4%
   - Compounding frequency: quarterly
   - Number of years (t): 4 years
   - Quarterly installment (PMT): this is what we are solving for
   - Number of compounding periods per year (n): 4 (quarterly compounding)

2. **Convert the annual interest rate to a quarterly interest rate:**
   \[
   r = \frac{4\%}{4} = \frac{0.04}{4} = 0.01
   \]
   So, the quarterly interest rate $r = 0.01$.

3. **Calculate the total number of payments:**
   \[
   N = 4 \times 4 = 16
   \]

4. **Use the formula for the future value of an ordinary annuity to find the quarterly installment $PMT$:**
   \[
   FV = PMT \times \frac{(1 + r)^N - 1}{r}
   \]
   Rearranging to solve for $PMT$:
   \[
   PMT = \frac{FV \times r}{(1 + r)^N - 1}
   \]

5. **Plug in the known values:**
   \[
   PMT = \frac{20,000 \times 0.01}{(1 + 0.01)^{16} - 1}
   \]

6. **Calculate the values step-by-step:**

   - Compute $(1 + r)^N$:
   \[
   (1 + 0.01)^{16}
   \]

   - Then compute the entire expression for $PMT$.

Let's calculate $PMT$:

\[
PMT = \frac{20,000 \times 0.01}{(1.01)^{16} - 1}
\]

Now, let's compute this value.The size of each quarterly installment that Gibraltar Brokerage Services should deposit in the sinking fund is approximately **$1,158.89**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the quarterly installment change if the interest rate was 5% compounded quarterly?
2. What if the sinking fund needed to be ready in 5 years instead of 4? How would this affect the installment size?
3. How much total interest will be earned in the fund over the 4 years?
4. What is the impact of compounding frequency on the installment size?
5. If the fund earns interest compounded monthly instead of quarterly, what would be the new quarterly installment?

**Tip:** Regularly reviewing and adjusting installment amounts for a sinking fund can help ensure the target amount is met within the desired timeframe.

Learn how to calculate the quarterly installment for a sinking fund using compound interest principles. Gibraltar Brokerage Services needs to accumulate $20,000 in 4 years with a quarterly interest rate of 4%. Get a step-by-step solution and explore related questions.

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