Dave takes out a 28-year mortgage of 280000 dollars for his new house. Dave gets an interest rate of 14.4 percent compounded monthly. He agrees to make equal monthly payments, the first coming in one month. After making the 65th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 500 dollars, and to get a better interest rate. In particular, he negotiates a new rate of 7.2 percent compounded monthly, and agrees to make equal monthly payments (each 500 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?

To find the size of Dave's final loan payment, we need to follow a structured approach based on mortgage and refinancing calculations. We'll break the problem into parts:

### 1. **Determine the original monthly payment**
The original mortgage has:
- Principal ($P_0$) = \$280,000
- Annual interest rate = 14.4%, compounded monthly
- Mortgage term = 28 years

First, convert the annual interest rate to a monthly interest rate:
\[
r = \frac{14.4\%}{12} = 0.012 \text{ (monthly interest rate)}
\]

The number of payments for a 28-year mortgage is:
\[
n = 28 \times 12 = 336 \text{ payments}
\]

Using the formula for the monthly payment ($M$) of a mortgage:
\[
M = \frac{P_0 \cdot r}{1 - (1 + r)^{-n}}
\]

Substitute the values and calculate $M$.

### 2. **Calculate the loan balance after 65 payments**
After making 65 payments, the balance of the loan ($P_{65}$) can be found using the formula for the remaining balance of a mortgage:
\[
P_{65} = M \times \frac{1 - (1 + r)^{-(n - 65)}}{r}
\]
This will give us the remaining principal after 65 payments.

### 3. **Refinance the remaining balance**
Dave refinances the remaining balance $P_{65}$ with a new interest rate of 7.2%, compounded monthly. The new monthly interest rate is:
\[
r' = \frac{7.2\%}{12} = 0.006
\]

The new monthly payment is reduced by \$500, so the new monthly payment is:
\[
M' = M - 500
\]

Dave makes monthly payments until the loan is almost paid off, but there is a final smaller payment after that period. To calculate the size of that final payment, we need to first compute the total number of new payments Dave will make.

### 4. **Calculate the final loan payment**
The final loan payment occurs when the remaining balance is small. We can use a mortgage amortization approach to find the size of that final payment.

Let's compute everything step by step.1. **Original monthly payment:** \$3,422.18
2. **Remaining balance after 65 payments:** \$273,930.37
3. **New monthly payment after refinancing (reduced by \$500):** \$2,922.18

Now, we can proceed to calculate the size of the final loan payment. After refinancing, Dave continues making payments of \$2,922.18 at the new interest rate of 7.2%. We'll compute the final loan payment by determining the balance after a set number of monthly payments until only a small amount remains. Let's proceed with this calculation.After refinancing, Dave will make **138 more monthly payments** of \$2,922.18, followed by a **final payment of \$509.29**.

Would you like further clarification or details on any part of this process?

Here are 5 follow-up questions that could deepen your understanding:
1. How does the remaining balance of a mortgage change over time with different interest rates?
2. What is the impact of the compounding frequency on the total interest paid?
3. How is the monthly payment affected by extending or shortening the mortgage term?
4. What are the advantages of refinancing a mortgage to a lower interest rate?
5. How does the loan amortization schedule work in a typical mortgage?

**Tip:** Always check how much interest you'll save when refinancing, especially if you're reducing the loan term or getting a better interest rate.

Learn how to calculate the final loan payment for a mortgage after refinancing. This detailed step-by-step solution covers key concepts like compound interest, amortization, and refinancing. Suitable for undergraduate students studying finance or mathematics.

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