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: $B = P \cdot (1 + i_{\text{monthly}})^n - \frac{PMT \cdot ((1 + i_{\text{monthly}})^n - 1)}{i_{\text{monthly}}}$