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?

To solve this problem, we need to follow these steps in order to find Dave's final loan payment after he refinances his mortgage. The process involves multiple steps involving annuity formulas for mortgage calculations before and after refinancing.

### Step 1: Calculate Dave's original monthly payment

Dave took out a mortgage of $ P_0 = 270,000 $ dollars at an interest rate of 15.6% compounded monthly. The monthly interest rate is:

\[
r_1 = \frac{15.6\%}{12} = \frac{0.156}{12} = 0.013
\]

The mortgage term is 26 years, which means the total number of payments is:

\[
n_1 = 26 \times 12 = 312 \text{ months}
\]

Using the annuity formula for calculating monthly payments $ M_1 $, we have:

\[
M_1 = \frac{P_0 \cdot r_1}{1 - (1 + r_1)^{-n_1}}
\]

Substitute the values into the formula:

\[
M_1 = \frac{270,000 \cdot 0.013}{1 - (1 + 0.013)^{-312}}
\]

### Step 2: Determine the outstanding balance after 66 payments

After Dave makes 66 payments, he has reduced his mortgage balance. The outstanding balance after $ p = 66 $ payments can be calculated using the formula for the remaining balance $ B_{66} $:

\[
B_{66} = M_1 \cdot \frac{1 - (1 + r_1)^{-(n_1 - p)}}{r_1}
\]

This will give the remaining mortgage balance after the 66th payment.

### Step 3: Refinancing the loan

Dave refinances the remaining balance $ B_{66} $ with a new interest rate of 8.4% compounded monthly. The new monthly interest rate is:

\[
r_2 = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007
\]

His new monthly payments $ M_2 $ are $400 less than his original monthly payment $ M_1 $:

\[
M_2 = M_1 - 400
\]

Dave will now pay off the remaining balance over time. The number of monthly payments required, say $ n_2 $, can be calculated using the annuity formula for the new loan until the final payment is reached. However, Dave is required to make a final lump-sum payment after making all the equal monthly payments. The size of this lump-sum payment can be found by calculating the remaining balance after $ n_2 $ payments and subtracting the monthly payments.

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I'll proceed with the calculation to find $ M_1 $, $ B_{66} $, and the final payment. Let me compute this.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the manual calculations or answer any related questions if you'd like. Let me know how you'd like to proceed!

Here are a few questions to consider next:

1. Do you need further clarification on how to calculate the monthly mortgage payment?
2. Would you like me to explain the remaining balance formula in detail?
3. Are you interested in understanding how the refinancing process affects loan payments?
4. Should I assist you in setting up the equations for manual calculation?
5. Do you want more details on how interest rates impact mortgage duration?

**Tip**: When refinancing a loan, always calculate the remaining balance first to ensure the new monthly payments align with your budget.

This problem guides you through calculating the final loan payment after refinancing a 270000 mortgage with an interest rate change from 15.6% to 8.4%, compounded monthly. Learn to apply the annuity formula and understand how refinancing impacts monthly payments and the remaining balance.

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