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 $300,000 for a term of 30 years, with interest at the rate of 6%/year compounded monthly. Currently, the interest rate for his ARM is 2.5%/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 find George's new monthly payment after his adjustable-rate mortgage (ARM) interest rate changes, we need to follow these steps:

1. **Determine the remaining balance on the loan after 5 years** using the original terms of the mortgage.
2. **Calculate the new monthly payment** based on the current interest rate and the remaining balance over the remaining term of the loan.

### Step 1: Calculate the remaining balance after 5 years

The formula for the monthly payment $ M $ of a mortgage is:

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

where:
- $ P = 300,000 $ (initial loan amount)
- $ r = \frac{6\%}{12} = 0.005 $ (monthly interest rate)
- $ n = 30 \times 12 = 360 $ (total number of payments for 30 years)

Let's calculate the monthly payment first:

\[
M = 300,000 \times \frac{0.005(1+0.005)^{360}}{(1+0.005)^{360} - 1}
\]

Now, after 5 years (or 60 payments), we calculate the remaining balance using:

\[
B = P(1+r)^n - M \left( \frac{(1+r)^n - (1+r)^t}{r} \right)
\]

where:
- $ t = 60 $ (number of payments made)

### Step 2: Calculate the new monthly payment with the new interest rate

The new balance $ B $ will now be used to find the new monthly payment $ M_{\text{new}} $ with:
- New interest rate $ r_{\text{new}} = \frac{2.5\%}{12} = 0.00208333 $
- Remaining term $ n_{\text{remaining}} = 30 \times 12 - 60 = 300 $

The new monthly payment formula becomes:

\[
M_{\text{new}} = B \frac{r_{\text{new}}(1+r_{\text{new}})^{n_{\text{remaining}}}}{(1+r_{\text{new}})^{n_{\text{remaining}}} - 1}
\]

Let’s calculate the exact values:George's new monthly payment, after the interest rate adjustment, will be approximately **$562.98**.

### Summary of the Steps:
1. The original loan of $300,000 at 6% interest had monthly payments of around $1,799.62.
2. After 5 years of payments, the remaining loan balance was recalculated.
3. With the new 2.5% interest rate, the monthly payments are recalculated to be $562.98.

Would you like more details or have any questions?

### Related Questions:
1. How does the change in interest rate affect the overall cost of the loan?
2. What would be the monthly payment if the interest rate were 3% instead of 2.5%?
3. How does making extra payments affect the remaining balance on a mortgage?
4. What is the impact of refinancing a mortgage at a lower interest rate?
5. How can a borrower calculate the break-even point when refinancing?

### Tip:
Understanding how interest rate changes affect your mortgage can help you make informed financial decisions, such as refinancing or making additional principal payments.

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 $300,000 for a term of 30 years, with interest at the rate of 6%/year compounded monthly. Currently, the interest rate for his ARM is 2.5%/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.)

Find out how to calculate the new monthly payment for George's $300,000 ARM loan after a 5-year interest rate adjustment. This problem involves understanding mortgage calculations, compound interest, and monthly payment formulas, suitable for college-level financial mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation