You have a choice between a​ 30-year fixed rate loan at 3.5​% and an adjustable rate mortgage​ (ARM) with a first year rate of 2​%. Neglecting compounding and changes in​ principal, estimate your monthly savings with the ARM during the first year on a ​$175 comma 000 loan. Suppose that the ARM rate rises to 11.5​% at the start of the third year. Approximately how much extra will you then be paying over what you would have paid if you had taken the fixed rate​ loan?
What is the approximate monthly savings with the ARM during the first​ year?
​$___. (Round to the nearest dollar as​ needed.)

Let's start by calculating the monthly payments for both the 30-year fixed-rate loan and the adjustable-rate mortgage (ARM) during the first year.

### 1. **Monthly Payment Formula:**

The formula to calculate monthly mortgage payments is:

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

Where:
- $ M $ is the monthly payment
- $ P $ is the loan principal (in this case, $175,000)
- $ r $ is the monthly interest rate (annual rate divided by 12)
- $ n $ is the number of total payments (loan term in months, 30 years = 360 months)

---

### 2. **Fixed-Rate Loan:**

For the 30-year fixed-rate loan:
- Loan amount $ P = 175,000 $
- Annual interest rate = 3.5%
- Monthly interest rate $ r = \frac{3.5\%}{12} = 0.0029167 $
- Loan term $ n = 30 \times 12 = 360 $

Now, let me calculate the monthly payment for this fixed-rate loan.The monthly payment for the 30-year fixed-rate loan is approximately \$786.

---

### 3. **Adjustable-Rate Mortgage (ARM) - First Year:**

For the ARM during the first year:
- Loan amount $ P = 175,000 $
- Annual interest rate = 2%
- Monthly interest rate $ r = \frac{2\%}{12} = 0.0016667 $
- Loan term $ n = 30 \times 12 = 360 $

Now, I'll calculate the monthly payment for the ARM during the first year.The monthly payment for the adjustable-rate mortgage (ARM) during the first year is approximately \$647.

---

### 4. **Monthly Savings in the First Year:**

To find the monthly savings with the ARM during the first year, we subtract the ARM payment from the fixed-rate payment:

\[
\text{Monthly Savings} = 786 - 647 = 139
\]

Thus, the approximate monthly savings with the ARM during the first year is **\$139**.

---

### 5. **ARM Payment in the Third Year (11.5% Interest Rate):**

Now, let's calculate the new monthly payment when the ARM rate rises to 11.5% at the start of the third year.

For the third year:
- Annual interest rate = 11.5%
- Monthly interest rate $ r = \frac{11.5\%}{12} = 0.0095833 $

Let's calculate the new monthly payment with this rate.The monthly payment for the adjustable-rate mortgage (ARM) when the rate rises to 11.5% at the start of the third year is approximately **\$1,733**.

---

### 6. **Additional Monthly Payment in the Third Year:**

To find the additional amount you'll be paying in the third year compared to the fixed-rate loan, we subtract the fixed-rate payment from the new ARM payment:

\[
\text{Extra Payment} = 1,733 - 786 = 947
\]

Thus, you would be paying approximately **\$947 extra** per month starting in the third year compared to what you would have paid with the fixed-rate loan.

---

Would you like more details or further clarification on any part of the calculation?

Here are 5 related questions to explore further:
1. How does refinancing affect long-term savings with variable-rate loans?
2. What are the risks associated with adjustable-rate mortgages (ARMs)?
3. How is the amortization schedule different for fixed and adjustable-rate loans?
4. How does inflation impact the decision between a fixed-rate and ARM?
5. Can prepaying on an ARM reduce the risk of future rate hikes?

**Tip:** Always consider the potential long-term increases in interest rates when opting for an adjustable-rate mortgage, as initial savings might be offset by higher payments later.

You have a choice between a​ 30-year fixed rate loan at 3.5​% and an adjustable rate mortgage​ (ARM) with a first year rate of 2​%. Neglecting compounding and changes in​ principal, estimate your monthly savings with the ARM during the first year on a ​$175,000 loan. Suppose that the ARM rate rises to 11.5​% at the start of the third year. Approximately how much extra will you then be paying over what you would have paid if you had taken the fixed rate​ loan? What is the approximate monthly savings with the ARM during the first​ year? ​$___. (Round to the nearest dollar as​ needed.)

Estimate monthly savings for a 30-year $175,000 loan with a fixed rate of 3.5% vs an ARM starting at 2%. Learn how interest rate increases to 11.5% impact your monthly payments in the third year. Suitable for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation