A client enters into a loan agreement to borrow R2 800 000 at an annual interest rate of 11.5%, compounded monthly. The loan is repayable in equal payments over 20 years, payable at the end of each month. If the interest rate changes to 9.0% after 5 years, calculate the new monthly payment for the remaining period.

To calculate the new monthly payment after the interest rate changes, we can follow these steps:

1. **Calculate the initial monthly payment for the first 5 years at 11.5% compounded monthly.**
2. **Determine the outstanding loan balance after 5 years.**
3. **Calculate the new monthly payment based on the remaining balance, adjusted interest rate, and remaining term.**

### Step 1: Calculate the Initial Monthly Payment
Given:
- Loan amount, \( P = R2,800,000 \)
- Annual interest rate, \( r = 11.5\% \) compounded monthly
- Loan term, \( n = 20 \) years, or 240 months

The monthly interest rate is:
\[
\text{Monthly interest rate} = \frac{11.5\%}{12} = \frac{0.115}{12} \approx 0.00958333
\]

Using the formula for a fixed monthly payment on an amortizing loan:
\[
M = P \cdot \frac{i(1 + i)^n}{(1 + i)^n - 1}
\]
where:
- \( M \) is the monthly payment,
- \( i \) is the monthly interest rate, and
- \( n \) is the total number of payments (240).

Substituting the values:
\[
M = 2,800,000 \cdot \frac{0.00958333 \cdot (1 + 0.00958333)^{240}}{(1 + 0.00958333)^{240} - 1}
\]

Calculating \( M \) will give us the initial monthly payment.

### Step 2: Calculate the Outstanding Balance After 5 Years
After 5 years (60 months), we use the loan balance formula:
\[
B = P \cdot \frac{(1 + i)^n - (1 + i)^p}{(1 + i)^n - 1}
\]
where:
- \( B \) is the remaining balance,
- \( p \) is the number of payments made (60).

Substituting the values will give the outstanding balance after 5 years.

### Step 3: Calculate the New Monthly Payment for the Remaining Term
Now, with the new interest rate of 9.0% compounded monthly, and the remaining loan balance from Step 2:

- New monthly interest rate: \( \frac{9.0\%}{12} = \frac{0.09}{12} = 0.0075 \)
- Remaining term: 15 years or 180 months

The new monthly payment \( M_{\text{new}} \) is calculated as:
\[
M_{\text{new}} = B \cdot \frac{0.0075 \cdot (1 + 0.0075)^{180}}{(1 + 0.0075)^{180} - 1}
\]

This value, \( M_{\text{new}} \), will be the revised monthly payment after the interest rate changes to 9.0%.

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### Related Questions:
1. How does an interest rate change impact the total interest paid over the loan term?
2. What would be the effect if the interest rate decreased to 7% instead of 9%?
3. How would the calculations differ if the interest was compounded annually instead of monthly?
4. What if the loan term was shortened to 15 years instead of 20 years?
5. How can early repayments affect the outstanding balance and future monthly payments?

### Tip:
When handling loans, always verify the compounding frequency, as it significantly impacts the monthly payment and total cost.

Learn how to calculate the new monthly payment for a R2 800 000 loan over 20 years when the interest rate changes from 11.5% to 9.0% after 5 years. This example uses amortization and compound interest formulas to find the new payment for the remaining term.

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