The management of Gibraltar Brokerage Services anticipates a capital expenditure of $24,000 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 2%/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.)

To determine the size of each quarterly installment for a sinking fund, we use the future value formula of an ordinary annuity because a sinking fund is essentially a series of equal payments made into an account that earns compound interest to reach a specific future value.

### Formula for Future Value of an Ordinary Annuity

The future value $FV$ of an ordinary annuity is given by:

\[
FV = P \times \frac{(1 + i)^n - 1}{i}
\]

where:
- $FV$ is the future value of the sinking fund (amount needed).
- $P$ is the size of each installment.
- $i$ is the interest rate per period.
- $n$ is the total number of periods.

### Given Information

- $FV = 24,000$ (future expenditure needed for computers)
- Annual interest rate $r = 2\%$ compounded quarterly
- Time period $t = 2$ years
- Compounding periods per year $m = 4$ (quarterly)

### Step-by-Step Calculation

1. **Determine the quarterly interest rate $i$:**

\[
i = \frac{r}{m} = \frac{2\%}{4} = \frac{0.02}{4} = 0.005
\]

2. **Calculate the total number of installments $n$:**

\[
n = m \times t = 4 \times 2 = 8
\]

3. **Rearrange the formula to solve for $P$ (the installment amount):**

\[
P = \frac{FV \times i}{(1 + i)^n - 1}
\]

4. **Substitute the values and calculate $P$:**

\[
P = \frac{24,000 \times 0.005}{(1 + 0.005)^8 - 1}
\]

Let's calculate the size of each quarterly installment.The size of each quarterly installment that should be deposited in the sinking fund is approximately **$2,947.89**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the installment size change if the interest rate were 3% compounded quarterly?
2. What would be the impact on the installment size if the fund was set up for 3 years instead of 2 years?
3. How does compounding frequency (e.g., monthly vs. quarterly) affect the size of each payment?
4. What is the total amount deposited into the sinking fund after 2 years?
5. How does the future value formula differ for a sinking fund compared to a retirement fund?

**Tip:** Sinking funds are often used to set aside money for large future expenditures, so understanding how different factors affect the installment size is crucial for financial planning.

Learn how to calculate the size of each quarterly installment for a sinking fund to finance a $24,000 expenditure on computers in 2 years. Understand the future value formula of an ordinary annuity with 2% interest compounded quarterly. Approximate installment: $2,947.89.

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