A loan of R15 080,00 is being repaid by 14 half-yearly payments, with the first payment made one half year after the loan is taken out. It is paid in a period of 7 years. The interest rate is 9% per year, compounded half-yearly. The size of each of the half-yearly payments is R1475,13. the outstanding principle after two years is

To determine the outstanding principal after two years, we'll break the problem into steps:

### Given:
- **Loan amount (Principal, \( P_0 \))**: R15,080.00
- **Number of payments**: 14 half-yearly payments
- **Payment amount**: R1,475.13 per half-year
- **Interest rate**: 9% per annum, compounded half-yearly
- **Time period after which we need to find the outstanding principal**: 2 years (which is 4 half-yearly periods)

### Steps to Calculate the Outstanding Principal:
1. **Convert the annual interest rate to a half-yearly interest rate:**
   \[
   i = \frac{9\%}{2} = \frac{0.09}{2} = 0.045 \text{ per half-year}
   \]

2. **Total number of payments made in 2 years:**
   Since payments are made every half-year, in 2 years, the number of payments made will be:
   \[
   \text{Number of payments} = 2 \times 2 = 4 \text{ payments}
   \]

3. **Outstanding principal calculation:**
   The outstanding principal after 4 payments is calculated using the present value of the remaining payments. If the original loan has \( n \) periods and payments, and \( k \) payments have been made, the outstanding principal is the present value of the remaining \( n - k \) payments:
   
   \[
   P_k = R \times \left[\frac{1 - (1 + i)^{-(n-k)}}{i}\right]
   \]
   where:
   - \( P_k \) is the outstanding principal after \( k \) payments,
   - \( R \) is the half-yearly payment,
   - \( i \) is the half-yearly interest rate,
   - \( n \) is the total number of payments,
   - \( k \) is the number of payments made.

Let's calculate the outstanding principal after 2 years.The outstanding principal after 2 years, or 4 half-yearly payments, is approximately **R11,672.29**.

Would you like any further details or have any questions regarding this calculation?

Here are some related questions that might interest you:
1. How would the outstanding principal change if the interest rate were different?
2. How is the formula for calculating the outstanding principal derived?
3. What would the outstanding principal be after 5 years?
4. How much total interest will be paid by the time the loan is fully repaid?
5. How does the timing of payments (beginning vs. end of the period) affect the outstanding principal?
6. What happens if the payments are increased or decreased?
7. How does this loan compare to one with monthly payments instead of half-yearly payments?
8. What is the total amount paid over the life of the loan?

**Tip:** Understanding how to calculate the outstanding principal at any point during a loan's term is crucial for planning refinancing options or early repayments.

Learn how to calculate the outstanding principal of a loan after two years with half-yearly payments. This problem involves finance concepts such as interest calculation, present value, and annuity payments. The loan details include a principal of R15,080.00, 14 half-yearly payments of R1,475.13 each, and an interest rate of 9% per year, compounded half-yearly.

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