A person purchased a ​$129,990 home 10 years ago by paying 55​% down and signing a​ 30-year mortgage at 7.8​% compounded monthly. Interest rates have dropped and the owner wants to refinance the unpaid balance by signing a new 2020​-year mortgage a 4.2% compounded monthly. How much interest will refinancing​ save?

Let's break down the problem into manageable steps:

### 1. **Determine the initial loan amount:**

The person paid 55% down, so they borrowed the remaining 45% of the $129,990 home price.

\[
\text{Loan amount} = 129,990 \times (1 - 0.55) = 129,990 \times 0.45 = 58,495.5
\]

### 2. **Find the monthly payment for the original loan:**

The original loan was for 30 years at 7.8% annual interest, compounded monthly. We'll use the formula for the monthly payment on a mortgage:

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

Where:
- $ M $ = monthly payment
- $ P $ = loan amount = $58,495.50
- $ r $ = monthly interest rate = $\frac{7.8\%}{12} = 0.0065$
- $ n $ = total number of payments = $30 \times 12 = 360$

Substitute these values into the formula to calculate the original monthly payment.

### 3. **Calculate the remaining balance after 10 years:**

After 10 years (120 months), we need to calculate the remaining balance of the loan using the amortization formula. We'll use the formula for the remaining balance:

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

Where:
- $ B $ = remaining balance
- $ P $ = original loan amount = $58,495.50
- $ r $ = monthly interest rate = 0.0065
- $ n $ = total number of payments = 360
- $ p $ = number of payments made = 120 (since 10 years have passed)

We'll calculate the remaining balance after 120 months.

### 4. **Find the new monthly payment after refinancing:**

The owner refinances the unpaid balance at 4.2% interest for 20 years. Using the same mortgage formula as before, but now with:
- The remaining balance as the new loan amount
- Monthly interest rate = $\frac{4.2\%}{12} = 0.0035$
- Total number of payments = $20 \times 12 = 240$

We calculate the new monthly payment.

### 5. **Calculate total payments before and after refinancing:**

- **Total payments without refinancing**: Calculate how much the person would have paid over the remaining 20 years under the original loan.
- **Total payments with refinancing**: Calculate the total payments after refinancing for 20 years at the new rate.

### 6. **Find the interest saved:**

\[
\text{Interest saved} = \text{Total payments without refinancing} - \text{Total payments with refinancing}
\]

Let me now calculate these steps for you.Here are the calculations:

1. **Original monthly payment**: \$421.09
2. **Remaining balance after 10 years**: \$51,101.14
3. **New monthly payment after refinancing**: \$315.07
4. **Total payments over the remaining 20 years without refinancing**: \$101,062.04
5. **Total payments over 20 years with refinancing**: \$75,617.93

### **Interest saved by refinancing**: \$25,444.11

Would you like more details on any part of the process? Here are some related questions:

1. What if the initial down payment percentage was different?
2. How would changing the new mortgage duration affect savings?
3. How does the compounding frequency affect the interest savings?
4. How do varying interest rates impact long-term savings?
5. What happens if the refinancing happens later than 10 years?

**Tip**: Refinancing makes sense when the interest rate reduction outweighs the costs of refinancing, and the savings over time are significant.

A person purchased a $129,990 home 10 years ago by paying 55% down and signing a 30-year mortgage at 7.8% compounded monthly. Interest rates have dropped, and the owner wants to refinance the unpaid balance by signing a new 20-year mortgage at 4.2% compounded monthly. How much interest will refinancing save?

Learn how to calculate interest savings when refinancing a mortgage with a detailed step-by-step solution. This problem covers mortgage amortization, loan refinancing, and compound interest. Includes formulas and savings calculations for refinancing a 30-year mortgage at 7.8% to a 20-year mortgage at 4.2%.

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