George secured an adjustable-rate mortgage (ARM) loan to help finance the purchase of his home 5 years ago. The amount of the loan was $400,000 for a term of 30 years, with interest at the rate of 5%/year compounded monthly. Currently, the interest rate for his ARM is 4.5%/year compounded monthly, and George's monthly payments are due to be reset. What will be the new monthly payment? (Round your answer to the nearest cent.)

To determine George's new monthly payment after the interest rate reset on his adjustable-rate mortgage (ARM), we can follow these steps:

### 1. **Determine the remaining loan balance:**

First, we calculate the remaining loan balance after 5 years at the initial interest rate of 5% compounded monthly.

The formula to calculate the remaining balance of a loan is:

\[
B = P \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - \left(1 + \frac{r}{n}\right)^{n \times t_{\text{paid}}}}{\left(1 + \frac{r}{n}\right)^{nt} - 1}
\]

Where:
- $ B $ = Remaining loan balance
- $ P $ = Original loan amount = $400,000
- $ r $ = Annual interest rate (as a decimal) = 0.05
- $ n $ = Number of compounding periods per year = 12 (monthly compounding)
- $ t $ = Total loan term in years = 30
- $ t_{\text{paid}} $ = Time already paid in years = 5

### 2. **Calculate the remaining balance:**

After calculating the remaining balance, we need to use this balance to determine the new monthly payment based on the new interest rate of 4.5%.

The formula for the monthly payment on a mortgage is:

\[
M = B \times \frac{\left(\frac{r_{\text{new}}}{n}\right) \times \left(1 + \frac{r_{\text{new}}}{n}\right)^{n \times t_{\text{remaining}}}}{\left(1 + \frac{r_{\text{new}}}{n}\right)^{n \times t_{\text{remaining}}} - 1}
\]

Where:
- $ M $ = New monthly payment
- $ B $ = Remaining balance after 5 years
- $ r_{\text{new}} $ = New annual interest rate (as a decimal) = 0.045
- $ t_{\text{remaining}} $ = Remaining loan term in years = 25

Let's go ahead and calculate the new monthly payment.The new monthly payment for George's adjustable-rate mortgage (ARM) will be approximately **$2,041.66** after the interest rate is reset to 4.5% compounded monthly.

Would you like more details or have any other questions? 

Here are 5 related questions:
1. How does an adjustable-rate mortgage (ARM) differ from a fixed-rate mortgage?
2. What factors influence the reset of interest rates in an ARM?
3. How would an increase in the loan term affect the monthly payment amount?
4. What are the potential risks associated with adjustable-rate mortgages?
5. How can refinancing impact the overall cost of a mortgage?

**Tip:** Keeping an eye on the interest rate trends can help you plan for possible changes in your mortgage payments.

Learn how to calculate the new monthly payment for George's $400,000 adjustable-rate mortgage (ARM) after the interest rate is reset from 5% to 4.5%. This problem covers compound interest, loan amortization, and monthly payment calculation. Suitable for finance students.

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