If you borrow $300,000 for a house at an 8% APR with semi-annual compounding for 15 years, what is your monthly payment?

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.

Find the monthly mortgage payment for a $300,000 loan with an 8% APR and semi-annual compounding over 15 years. Learn about compound interest, loan amortization, and key formulas used in mortgage calculations. Suitable for finance and business students.

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