Question

What are the semi-annual payments on a 20-year loan of $40,000 at 10.75% compounded semi-annually?

Answer 1

To find the semi-annual payments on a loan, we can use the formula for the annuity payment:

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

Where:
- \(PMT\) is the payment amount per period.
- \(P\) is the principal amount (the loan amount).
- \(r\) is the interest rate per period.
- \(n\) is the total number of payments.

### Given:
- Principal amount \(P = 40,000\)
- Annual interest rate \(= 10.75\%\)
- Compounding frequency = semi-annual (2 times a year)
- Loan term = 20 years, so total periods \(n = 20 \times 2 = 40\)

### Step 1: Calculate the interest rate per period
The semi-annual interest rate \(r\) can be calculated as:

\[
r = \frac{\text{Annual Interest Rate}}{\text{Number of periods per year}} = \frac{10.75\%}{2} = \frac{10.75}{100 \times 2} = 0.05375
\]

### Step 2: Calculate the total number of payments
\[
n = 20 \times 2 = 40
\]

### Step 3: Substitute values into the annuity payment formula
Now we can use these values in the payment formula:

\[
PMT = \frac{40,000 \times 0.05375}{1 - (1 + 0.05375)^{-40}}
\]

### Step 4: Calculate \((1 + r)^{-n}\)
Calculate \((1 + 0.05375)^{-40}\):

\[
(1 + 0.05375)^{-40} \approx 0.188586
\]

### Step 5: Now calculate \(PMT\)
Substituting back:

\[
PMT = \frac{40,000 \times 0.05375}{1 - 0.188586}
\]

Calculate the denominator:

\[
1 - 0.188586 \approx 0.811414
\]

Now calculate \(PMT\):

\[
PMT = \frac{40,000 \times 0.05375}{0.811414} \approx \frac{2150}{0.811414} \approx 2658.47
\]

### Result
The semi-annual payments on a 20-year loan of $40,000 at an annual interest rate of 10.75% compounded semi-annually is approximately **$2,658.47**.