Five years ago, you took out a 5/1 adjustable rate mortgage and the 5-year fixed rate period has just expired. The loan was originally for $304,000 with 360 payments at 4.2% APR compounded monthly. a . Now that you have made 60 payments, what is the remaining balance on the loan? b. If the interest rate increases by 0.8%, to 5.0% APR, compounded monthly, what will be your new payments?

### Part (a): Finding the Remaining Balance After 60 Payments

1. **Loan Details:**
   - **Loan amount (P)**: $304,000
   - **Annual interest rate (APR)**: 4.2% = 0.042 (compounded monthly)
   - **Number of total payments (N)**: 360 months (30 years)
   - **Monthly payment period so far (n)**: 60 months (5 years)

#### Step 1: Calculate the Monthly Payment
The formula to calculate the monthly mortgage payment (M) for a fixed-rate loan is:

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

Where:
- $ P = 304,000 $ (principal loan amount),
- $ r = \frac{0.042}{12} = 0.0035 $ (monthly interest rate),
- $ N = 360 $ (total number of payments).

#### Step 2: Calculate the Remaining Balance
The remaining balance after $ n = 60 $ payments can be calculated using the formula for the remaining balance $ B_n $:

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

I’ll calculate both the monthly payment and the remaining balance.#### Part (a) Results:
- The **monthly payment** during the first five years (fixed-rate period) was **$1,486.61**.
- After 60 payments, the **remaining balance** on the loan is **$275,838.83**.

---

### Part (b): New Payments with Increased Interest Rate

The interest rate has increased to 5.0% APR compounded monthly. Let's recalculate the new monthly payment based on the remaining balance of $275,838.83 with the updated interest rate of 5.0%.

1. **New interest rate**: 5.0% APR = 0.05 annual rate = $\frac{0.05}{12} = 0.004167$ monthly interest rate.
2. **Remaining number of payments (N)**: 300 months (since 60 payments have already been made out of 360).

We'll use the same monthly payment formula, but now the principal is the remaining balance and the interest rate is 5.0%. Let me calculate the new monthly payment.#### Part (b) Results:
With the interest rate increasing to 5.0%, the **new monthly payment** will be **$1,612.53**.

---

### Summary:
1. The remaining balance after 60 payments is **$275,838.83**.
2. The new monthly payment after the interest rate increase to 5.0% will be **$1,612.53**.

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Do you have any questions or would you like further details on the calculations?

Here are 5 related questions to consider:
1. How would the payments change if the interest rate were to increase further?
2. What is the total amount of interest paid over the life of the loan?
3. How does the term of the loan affect the amount paid in interest?
4. What is the impact of making extra payments on the mortgage balance?
5. How would switching to a fixed-rate mortgage affect future payments?

**Tip:** If interest rates rise significantly, refinancing to a fixed-rate mortgage can help lock in a lower rate and protect against future rate hikes.

Five years ago, you took out a 5/1 adjustable rate mortgage and the 5-year fixed rate period has just expired. The loan was originally for $304,000 with 360 payments at 4.2% APR compounded monthly. a. Now that you have made 60 payments, what is the remaining balance on the loan? b. If the interest rate increases by 0.8%, to 5.0% APR, compounded monthly, what will be your new payments?

Discover how to calculate the remaining balance on a $304,000 adjustable-rate mortgage after 60 payments at 4.2% APR. Learn how an interest rate increase to 5.0% affects the new monthly payments. This problem explores mortgage amortization, compound interest, and loan balance calculations, suitable for students in Grades 11-12 or college.

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