Dave takes out a 27-year mortgage of 200000 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 70th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 600 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 600 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?

Question

Dave takes out a 27-year mortgage of 200000 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 70th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 600 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 600 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?

Answer 1

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Answer 2

To find the final loan payment, we need to calculate the remaining balance on Dave's mortgage after making the 70th payment. Then, we can determine the monthly payment for the refinanced mortgage and calculate the final loan payment.

First, let's calculate the remaining balance on Dave's mortgage after the 70th payment. We can do this using the formula for the future value of an ordinary annuity:

Future Value = P * ((1 + r)^n - 1) / r

Where:
P = monthly payment
r = monthly interest rate
n = number of payments

For the original mortgage, P = unknown, r = 14.4%/12 = 0.012, and n = 27 * 12 = 324. We can rearrange the formula to solve for P:

200000 = P * ((1 + 0.012)^324 - 1) / 0.012

Now, solving for P:

P = 200000 * (0.012 / ((1 + 0.012)^324 - 1)

Using a calculator, we find that P ≈ 2069.78 dollars (rounded to the nearest cent). Therefore, Dave's original monthly payment is approximately 2069.78 dollars.

Next, let's calculate the refinanced monthly payment. The remaining balance after the 70th payment will become the principal of the refinanced mortgage. The new interest rate is 8.4%/12 = 0.007, and Dave wants to reduce his monthly payment by 600 dollars. Therefore, the new monthly payment is:

Refinanced P = 2069.78 - 600 = 1469.78

Now, we can calculate the remaining balance after the 70th payment using the same future value formula:

Remaining Balance = Refinanced P * ((1 + 0.007)^((27 * 12) - 70) - 1) / 0.007

Solving for the remaining balance:

Remaining Balance ≈ 1469.78 * ((1 + 0.007)^214 - 1) / 0.007

Using a calculator, we find that the remaining balance is approximately 118790.67 dollars (rounded to the nearest cent).

Lastly, to find the final loan payment, we divide the remaining balance into equal installments with the refinanced monthly payment of 1469.78 dollars until the balance is zero. Then, the final payment will be the remaining balance.

Number of equal payments to fully pay off the remaining balance = Remaining Balance / Refinanced P

Number of equal payments ≈ 118790.67 / 1469.78

Using a calculator, we find that the number of equal payments is approximately 80.93 (rounded to two decimal places).

Since Dave agrees to make equal monthly payments for as long as necessary, followed by a single smaller payment, the final payment will be the remaining balance after the 80th payment:

Final loan payment ≈ 118790.67 dollars

Therefore, Dave's final loan payment will be approximately $118,790.67.