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 solve this problem, we’ll go through the following steps:

1. **Calculate the initial monthly payment for the loan with a 11.5% interest rate for the first 5 years** (20 years total term).
2. **Determine the loan balance after 5 years** with the initial interest rate.
3. **Calculate the new monthly payment** required to pay off the remaining balance over the remaining 15 years, using the new interest rate of 9.0%.

Let’s work through these steps.

### Step 1: Calculate the Initial Monthly Payment (11.5% Interest)

For a loan with principal \( P = R2,800,000 \), an annual interest rate \( i = 11.5\% \) compounded monthly, and a term of \( n = 20 \) years (or 240 months), we use the monthly interest rate in our calculation.

1. **Monthly interest rate (11.5% annual):**
   \[
   i_{\text{monthly}} = \frac{11.5\%}{12} = \frac{0.115}{12} = 0.0095833
   \]

2. **Calculate the initial monthly payment** using the formula for a fixed-rate annuity:
   \[
   PMT = \frac{P \cdot i_{\text{monthly}}}{1 - (1 + i_{\text{monthly}})^{-n}}
   \]
   where:
   - \( P = 2,800,000 \)
   - \( i_{\text{monthly}} = 0.0095833 \)
   - \( n = 240 \)

   Plugging in these values:
   \[
   PMT = \frac{2,800,000 \times 0.0095833}{1 - (1 + 0.0095833)^{-240}}
   \]

   Calculating this, we get the initial monthly payment.

### Step 2: Calculate the Balance After 5 Years (11.5% Interest)

After 5 years (60 months), we need to find the remaining balance. We use the loan balance formula after a certain number of payments have been made:
   \[
   B = P \cdot (1 + i_{\text{monthly}})^n - \frac{PMT \cdot ((1 + i_{\text{monthly}})^n - 1)}{i_{\text{monthly}}}
   \]
  

Calculate the new monthly payment for a R2,800,000 loan after an interest rate change from 11.5% to 9.0%. Learn how to adjust monthly payments when interest rates change, covering loan amortization and annuity formulas, suitable for advanced high school or college-level students.

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