A client enters into a loan agreement to borrow R1 350 000 at an annual interest rate of 8.4%, compounded monthly. The loan is repayable in equal payments over 8 years, payable at the end of each month. Calculate the value of each instalment.


To calculate the monthly installment for this loan, we can use the formula for the monthly payment of a loan with compound interest:

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

where:
- \( P \) is the principal loan amount,
- \( i \) is the monthly interest rate,
- \( n \) is the total number of payments,
- \( PMT \) is the monthly payment.

### Step 1: Identify the known values
- **Principal** \( P = R1,350,000 \)
- **Annual interest rate** \( = 8.4\% \)
- **Monthly interest rate** \( i = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007 \)
- **Loan term** \( = 8 \text{ years} \)
- **Total number of payments** \( n = 8 \times 12 = 96 \text{ months} \)

### Step 2: Substitute the values into the formula

\[
PMT = \frac{1,350,000 \times 0.007}{1 - (1 + 0.007)^{-96}}
\]

### Step 3: Calculate the monthly installment

Breaking it down:

1. \( (1 + 0.007)^{96} \approx 1.7474 \)
2. \( (1 + 0.007)^{-96} \approx \frac{1}{1.7474} \approx 0.5723 \)
3. \( 1 - 0.5723 \approx 0.4277 \)
4. \( \frac{1,350,000 \times 0.007}{0.4277} \approx R22,085.84 \)

So, the value of each monthly installment is approximately **R22,085.84**.

Would you like further details or explanations? 

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Here are five related questions for further exploration:
1. How does changing the loan term to 10 years affect the installment?
2. What would the installment be if the interest rate were 10% instead of 8.4%?
3. How much interest is paid over the entire loan period?
4. How would switching to quarterly compounding impact the monthly installment?
5. What would be the effect of making extra payments on the loan?

**Tip:** When taking a loan, understanding the breakdown of interest and principal in each payment can help you plan for faster repayment.

A client enters into a loan agreement to borrow R1 350 000 at an annual interest rate of 8.4%, compounded monthly. The loan is repayable in equal payments over 8 years, payable at the end of each month. Calculate the value of each instalment.

Find the monthly instalment for a R1,350,000 loan at an 8.4% annual interest rate, compounded monthly, repayable over 8 years. This solution covers compound interest calculations and annuity formulas, providing detailed steps suitable for high school students in Grades 10-12.

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