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 ​$200 comma 000 loan. Suppose that the ARM rate rises to 9​% at the start of the third year. Approximately how much extra will you then be paying over what you would have paid if you had taken the fixed rate​ loan?
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Part 1
What is the approximate monthly savings with the ARM during the first​ year?
​$
  
enter your response here ​(Round to the nearest dollar as​ needed.)

### Part 1: Calculating Monthly Payments for the Fixed-Rate Loan

The monthly payment for a fixed-rate loan can be calculated using the formula for a fixed-rate mortgage:

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

Where:
- $ M $ is the monthly payment,
- $ P $ is the loan amount ($200,000),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the number of payments (for a 30-year loan, $ n = 30 \times 12 = 360 $).

For the fixed-rate loan:
- The annual interest rate is 3.5%, so the monthly rate $ r = \frac{3.5\%}{12} = 0.002917 $,
- The loan term is 30 years (360 months).

Substitute these values into the formula to calculate the monthly payment for the fixed-rate loan.

### Part 2: Calculating Monthly Payments for the ARM (Adjustable Rate Mortgage)

For the ARM:
- The first-year interest rate is 2%, so the monthly rate $ r = \frac{2\%}{12} = 0.001667 $.
- The loan term is still 30 years (360 months).

Substitute these values into the formula to calculate the monthly payment for the first year of the ARM.

### Part 3: Calculating Monthly Savings for the First Year

Once we have the monthly payments for both the fixed-rate loan and the ARM, the monthly savings during the first year is simply the difference between these two values:

\[
\text{Monthly Savings} = \text{Fixed Rate Payment} - \text{ARM Payment}
\]

### Part 4: Impact of the Rate Increase in Year 3

If the ARM rate rises to 9% at the start of the third year, the new monthly payment can be calculated using the same formula, but with the new interest rate:

\[
r = \frac{9\%}{12} = 0.0075
\]

This will give us the new monthly payment for the ARM starting in year 3. To find how much extra you will be paying compared to the fixed-rate loan, we will calculate the difference between the new ARM payment and the fixed-rate payment.

Let's calculate everything now.### Part 1: Monthly Savings During the First Year
- The monthly payment for the fixed-rate loan is approximately **$898**.
- The monthly payment for the ARM in the first year is approximately **$739**.
- Therefore, the monthly savings during the first year is approximately **$159**.

### Part 2: Extra Payment in Year 3
- If the ARM rate rises to 9% at the start of the third year, the new monthly payment will be approximately **$1609**.
- Compared to the fixed-rate loan payment of **$898**, this results in an extra payment of approximately **$711** per month starting in year 3.

Would you like to explore more details or have any further questions?

Here are 5 related questions:
1. How does the loan term affect monthly payments?
2. What happens to your total interest paid if you choose an ARM?
3. What factors influence the future rate of an ARM?
4. How can refinancing help when ARM rates rise?
5. What other types of mortgage plans are available?

**Tip:** Consider how long you plan to stay in your home before choosing between an ARM and a fixed-rate mortgage.

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 $200,000 loan. Suppose that the ARM rate rises to 9% 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?

Explore the monthly savings of an ARM with a first-year rate of 2% compared to a 30-year fixed-rate loan at 3.5%. Learn how rising rates impact your mortgage payments in the third year for a $200,000 loan. Suitable for college-level finance or mathematics students.

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