To calculate the monthly mortgage payment for a loan of $300,000 at an 8% APR with semi-annual compounding for 15 years, we need to break the problem into steps and apply the right formula. Here's the process:

Step 1: Convert the nominal interest rate to the effective semi-annual rate

Since the loan compounds semi-annually (twice a year), the interest rate for each semi-annual period is:

$r_{\text{semi-annual}} = \frac{8\%}{2} = 4\% = 0.04 \text{ per period}$

Step 2: Find the number of total semi-annual periods

There are two compounding periods per year, and the loan is for 15 years. Therefore, the number of semi-annual periods is:

$n_{\text{semi-annual}} = 15 \times 2 = 30 \text{ periods}$

Step 3: Calculate the equivalent monthly interest rate

To calculate monthly payments, we need to find an equivalent monthly interest rate. The formula to convert the semi-annual rate to a monthly rate is based on compounding:

$r_{\text{monthly}} = \left(1 + r_{\text{semi-annual}}\right)^{\frac{1}{6}} - 1$

Substitute $r_{\text{semi-annual}} = 0.04$:

$r_{\text{monthly}} = \left(1 + 0.04\right)^{\frac{1}{6}} - 1$

Step 4: Use the monthly payment formula for a fixed-rate loan

The formula for calculating the monthly mortgage payment is:

$M = P \times \frac{r_{\text{monthly}} \times (1 + r_{\text{monthly}})^{n_{\text{months}}}}{(1 + r_{\text{monthly}})^{n_{\text{months}}} - 1}$

Where:

• $M$ is the monthly payment,
• $P = 300,000$ is the loan principal,
• $r_{\text{monthly}}$ is the monthly interest rate,
• $n_{\text{months}} = 15 \times 12 = 180$ is the total number of months.

Once we calculate $r_{\text{monthly}}$, we can plug it into the equation to find the monthly payment.

Let me calculate this for you now.The monthly payment for a loan of $300,000 at an 8% APR with semi-annual compounding for 15 years is approximately $2,844.46.

Would you like more details on how this was calculated, or do you have any other questions?

Here are 5 related questions to explore:

1. How does changing the loan term affect the monthly payment?
2. What would the monthly payment be if the interest rate were 7% instead of 8%?
3. How is the total interest paid over the life of the loan calculated?
4. What is the difference between nominal and effective interest rates?
5. How does semi-annual compounding differ from monthly compounding in loans?

Tip: When taking a loan, knowing the total interest you'll pay over time helps you understand the full cost, not just the monthly payments.