A loan of $6,000 is to be repaid over a 2-year period through equal quarterly installments with an interest rate of 13% per year compounded quarterly. Determine the size of each installment.

To determine the size of each quarterly installment for the loan, we can use the **loan amortization formula**, which is similar to the formula for calculating the payment on an annuity.

### Given:
- Principal (loan amount) $ P = 6,000 $
- Annual interest rate $ r = 13\% = 0.13 $
- Compounding frequency = quarterly (4 times per year)
- Time $ t = 2 $ years
- Number of quarters $ n = 2 \times 4 = 8 $ (since payments are made quarterly)

### Step 1: Calculate the quarterly interest rate
Since the loan is compounded quarterly, the quarterly interest rate $ i $ is given by:
\[
i = \frac{r}{4} = \frac{0.13}{4} = 0.0325
\]

### Step 2: Use the amortization formula
The formula for the installment payment $ A $ is:
\[
A = \frac{P \cdot i}{1 - (1 + i)^{-n}}
\]
where:
- $ A $ is the size of each installment
- $ P $ is the loan amount
- $ i $ is the quarterly interest rate
- $ n $ is the total number of quarters

### Step 3: Plug in the values
\[
A = \frac{6000 \times 0.0325}{1 - (1 + 0.0325)^{-8}}
\]

Now, let’s calculate this.

The size of each quarterly installment is approximately **$863.78**.

Would you like a more detailed breakdown of the calculations, or do you have any further questions?

Here are five related questions you might find interesting:
1. How does the loan repayment schedule affect the total interest paid?
2. How would the size of the installments change with a higher or lower interest rate?
3. What would happen to the payment amount if the loan term were extended to 3 years?
4. How can early repayments reduce the total interest paid?
5. What is the impact of compounding frequency on the loan repayment?

**Tip:** When repaying a loan, making extra payments towards the principal can significantly reduce the overall interest cost.

Learn how to calculate equal quarterly installments for a $6,000 loan repaid over 2 years with an interest rate of 13% compounded quarterly. This step-by-step guide covers key concepts such as loan amortization, interest rates, and compounding. Suitable for Grades 11-12.

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