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 $250,000 for a term of 30 years, with interest at the rate of 9%/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 calculate George's new monthly payment, we need to follow these steps:

1. **Determine the remaining balance on the loan after 5 years.**
2. **Calculate the new monthly payment based on the current interest rate of 4.5%.**

### Step 1: Determine the Remaining Balance on the Loan after 5 Years

The loan was originally for $ P = \$250,000 $ at an interest rate of 9% per year, compounded monthly, over 30 years.

- **Monthly interest rate**: 
\[
r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075
\]
- **Number of payments over 30 years**:
\[
n = 30 \times 12 = 360
\]
- **Initial monthly payment** can be calculated using the formula for a fixed-rate mortgage:
\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

Substituting the values:
\[
M = 250{,}000 \times \frac{0.0075(1 + 0.0075)^{360}}{(1 + 0.0075)^{360} - 1}
\]

Let's compute this:

\[
M \approx 250{,}000 \times \frac{0.0075(1.0075)^{360}}{(1.0075)^{360} - 1}
\]
\[
M \approx 250{,}000 \times \frac{0.0075 \times 12.152}{12.152 - 1}
\]
\[
M \approx 250{,}000 \times \frac{0.09114}{11.152}
\]
\[
M \approx 250{,}000 \times 0.008172
\]
\[
M \approx 2{,}011.68
\]

### Step 2: Calculate the Remaining Balance after 5 Years

The number of payments George has made after 5 years:
\[
n_{5} = 5 \times 12 = 60
\]
The remaining balance can be calculated using the balance formula:
\[
B = P \times \frac{(1 + r)^n - (1 + r)^{n_{5}}}{(1 + r)^n - 1}
\]
Substituting:
\[
B = 250{,}000 \times \frac{(1.0075)^{360} - (1.0075)^{60}}{(1.0075)^{360} - 1}
\]

Let's calculate this balance:

\[
B \approx 250{,}000 \times \frac{12.152 - 1.488}{12.152 - 1}
\]
\[
B \approx 250{,}000 \times \frac{10.664}{11.152}
\]
\[
B \approx 250{,}000 \times 0.9564
\]
\[
B \approx 239{,}100.00
\]

### Step 3: Calculate the New Monthly Payment

Now, using the new interest rate of 4.5% per year compounded monthly:

- **New monthly interest rate**:
\[
r_{\text{new}} = \frac{4.5\%}{12} = \frac{0.045}{12} = 0.00375
\]
- **Remaining term**: 
\[
n_{\text{remaining}} = 30 \times 12 - 60 = 300
\]

The new monthly payment $ M_{\text{new}} $ can be calculated using the formula:
\[
M_{\text{new}} = B \times \frac{r_{\text{new}}(1 + r_{\text{new}})^{n_{\text{remaining}}}}{(1 + r_{\text{new}})^{n_{\text{remaining}}} - 1}
\]

Substituting:
\[
M_{\text{new}} \approx 239{,}100 \times \frac{0.00375(1.00375)^{300}}{(1.00375)^{300} - 1}
\]

Calculating this:

\[
M_{\text{new}} \approx 239{,}100 \times \frac{0.00375 \times 3.243}{3.243 - 1}
\]
\[
M_{\text{new}} \approx 239{,}100 \times \frac{0.01216}{2.243}
\]
\[
M_{\text{new}} \approx 239{,}100 \times 0.00542
\]
\[
M_{\text{new}} \approx 1{,}295.94
\]

Thus, George's new monthly payment will be **$1,295.94**.

Would you like to ask about the derivation or the formulas used? Here are some related questions:

1. How does an ARM loan differ from a fixed-rate mortgage?
2. Why does the interest rate impact the monthly payments significantly?
3. What are the implications of refinancing a mortgage?
4. How does the term of a mortgage affect the monthly payments?
5. What other factors can influence the cost of a mortgage?

**Tip:** Regularly reviewing your mortgage terms and considering refinancing when rates drop can save you money over time.

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 $250,000 for a term of 30 years, with interest at the rate of 9%/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.)

Learn how to calculate the new monthly payment for George's adjustable-rate mortgage (ARM) after 5 years, considering an interest rate drop from 9% to 4.5%. This solution covers concepts of loan amortization, compounded interest, and mortgage payment formulas. Suitable for advanced high school or college-level students.

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