Dave takes out a 21-year mortgage of 250000 dollars for his new house. Dave gets an interest rate of 13.2 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 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?

We can approach this problem step by step, using the concept of mortgage amortization and refinancing.

### Step 1: Calculate Dave's Original Monthly Payment

First, we calculate Dave's original monthly payment under the terms of his 21-year mortgage.

- Principal $ P = 250,000 $ dollars
- Annual interest rate $ r = 13.2\% $, so the monthly interest rate $ i = \frac{13.2\%}{12} = 0.011 $
- Number of total monthly payments $ n = 21 \times 12 = 252 $

The formula for the monthly mortgage payment $ M $ is:

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

Plugging in the values, we can compute the original monthly payment.

### Step 2: Calculate the Remaining Balance After 65 Payments

After making 65 payments, Dave wants to refinance. To do so, we need to find the remaining balance of the loan after 65 payments.

The formula for the remaining balance $ B $ after $ k $ payments is:

\[
B_k = P \times \frac{(1 + i)^n - (1 + i)^k}{(1 + i)^n - 1}
\]

Here, $ k = 65 $ payments have been made, so we'll compute the remaining balance at that point.

### Step 3: Refinancing the Loan

Now, Dave refinances with a new interest rate of 8.4% compounded monthly, and he wants to reduce his monthly payment by $400. The new monthly interest rate is:

\[
i' = \frac{8.4\%}{12} = 0.007
\]

Let the new monthly payment be $ M' = M - 400 $, where $ M $ is the original monthly payment calculated in Step 1.

The remaining balance from Step 2 becomes the new loan principal, and Dave will make equal monthly payments of $ M' $ for a certain period, followed by a single smaller final payment.

To find out the size of the final payment, we'll calculate how long the reduced monthly payments would last and the remaining balance at that point, which will be Dave's final payment.

I'll compute each of these steps to get the exact value of the final payment. Let me run the calculations.### Step 1: Original Monthly Payment

Dave's original monthly payment is approximately:

\[
M = 2936.44 \, \text{dollars}
\]

### Step 2: Remaining Balance After 65 Payments

After making 65 payments, the remaining balance on Dave's mortgage is approximately:

\[
B_{65} = 232,437.38 \, \text{dollars}
\]

### Step 3: Refinanced Monthly Payment

After refinancing, Dave reduces his monthly payment by \$400. Therefore, his new monthly payment becomes:

\[
M' = 2536.44 \, \text{dollars}
\]

### Next Step: Calculate the Final Payment After Refinancing

Now we need to calculate how long Dave will make these reduced payments and what his final payment will be. I'll proceed with this calculation.### Final Step: Dave's Final Loan Payment

After refinancing, Dave will make reduced monthly payments of \$2536.44 for 148 months. His final payment will be a smaller amount of approximately:

\[
\text{Final Payment} = 124.37 \, \text{dollars}
\]

Would you like more details or further clarifications? Here are some follow-up questions you might explore:

1. How does refinancing affect the total interest paid over the loan term?
2. What if Dave had made a lump sum payment before refinancing?
3. How does the duration of the loan change with a different reduced payment amount?
4. What happens to the loan if the interest rate changes again in the future?
5. Can Dave pay off the mortgage earlier, and how would that impact his payments?

**Tip:** When considering refinancing, it's crucial to look at both the new interest rate and the total amount paid over time, as a lower rate might still result in higher costs if the loan is extended too long.

Learn how to calculate the final loan payment for Dave's 21-year mortgage after refinancing. This problem involves mortgage amortization, refinancing, compound interest, and loan payment calculations. Explore step-by-step solutions using formulas like the monthly payment formula and remaining balance formula. Suitable for Grades 11-12 or college students.

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