Dave takes out a 26-year mortgage of 270000 dollars for his new house. Dave gets an interest rate of 15.6 percent compounded monthly. He agrees to make equal monthly payments, the first coming in one month. After making the 66th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 400 dollars, and to get a better interest rate. In particular, he negotiates a new rate of 8.4 percent compounded monthly, and agrees to make equal monthly payments (each 400 dollars less than his original payments) for as long as necessary, followed by a single smaller payment. How large will Dave's final loan payment be?

Let's break down the problem into the following steps to find out how large Dave's final loan payment will be after refinancing:

### Step 1: Understanding the mortgage details before refinancing
- **Initial loan amount**: $ P = 270,000 $ dollars
- **Mortgage term**: $ t = 26 $ years
- **Interest rate**: 15.6% compounded monthly, so the monthly interest rate is:
  \[
  r = \frac{15.6\%}{12} = 0.013 \, \text{(or 1.3% per month)}
  \]
- **Number of monthly payments**: $ n = 26 \times 12 = 312 $ payments.

### Step 2: Calculating the original monthly payment
To find the monthly payment $ M $, we use the formula for a fixed-rate mortgage:
\[
M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Substituting the known values, we can calculate the monthly payment $ M $.

### Step 3: Calculate the remaining loan after 66 payments
After Dave has made 66 payments, the remaining balance on the loan can be found using the remaining balance formula for an amortized loan:
\[
B_{66} = P \cdot \frac{(1 + r)^n - (1 + r)^{66}}{(1 + r)^n - 1}
\]
This will give us the remaining principal on the loan after 66 payments.

### Step 4: Refinancing the loan
Once Dave refinances, the new loan terms are:
- **New interest rate**: 8.4% compounded monthly, so the new monthly interest rate is:
  \[
  r' = \frac{8.4\%}{12} = 0.007 \, \text{(or 0.7% per month)}
  \]
- **New monthly payment**: $ M' = M - 400 $ dollars.
- The loan is paid off in monthly installments until a single smaller payment is made at the end.

### Step 5: Finding the final loan payment
The structure of this loan indicates that Dave will make equal payments until one final smaller payment is required to pay off the remaining balance. To calculate this final payment, we need to compute the monthly payments and then solve for the last payment.

Let me first calculate the values step-by-step for you.Here are the results from each step:

1. **Original monthly payment**: Dave's original monthly payment was approximately **$3,573.53**.
2. **Remaining loan balance after 66 payments**: After making 66 payments, the remaining loan balance was approximately **$263,425.40**.
3. **New monthly payment after refinancing**: After refinancing, Dave's new monthly payment, reduced by $400, is approximately **$3,173.53**.

However, the calculated final payment appears negative, which suggests there was an issue in the calculation related to how the payments are structured or interpreted. I'll re-evaluate the approach for the final payment to ensure precision.

Let me check and adjust the final payment computation for accuracy.It seems there is still an issue with the final payment calculation, as it's returning an incorrect (negative) result. I will correct the computation process to ensure that Dave's final loan payment after refinancing is correctly determined.

Allow me a moment to resolve this and provide the correct final payment.The calculations for the final payment continue to yield unexpected results, suggesting that the approach to amortization might need a further review, especially when considering a fixed term followed by a final lump sum.

Let me address the issue and provide an accurate solution. Meanwhile, feel free to ask more detailed questions or clarifications!

This problem involves calculating the final payment on a mortgage after refinancing. Dave refinances his 26-year mortgage of $270,000 with a new interest rate of 8.4% after 66 payments. Learn how to calculate the original monthly payments, the remaining balance, and the new reduced payments. Suitable for students with knowledge of amortization and compound interest.

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