Dave takes out a 23-year mortgage of 200000 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 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?
how to do this math
To find out the size of Dave's final loan payment, we will follow these steps:
1. Calculate the original monthly payment:
Dave has a 23-year mortgage, which is 23 x 12 = 276 months.
The interest rate of 13.2% compounded monthly is equivalent to a monthly interest rate of 13.2 / 12 / 100 = 0.011.
We can use the formula for the monthly payment of a mortgage:
Monthly Payment = P * r * (1 + r)^n / ((1 + r)^n - 1),
where P is the principal (loan amount), r is the monthly interest rate, and n is the total number of payments.
Plugging in the values, we get:
Monthly Payment = 200000 * 0.011 * (1 + 0.011)^276 / ((1 + 0.011)^276 - 1).
Calculate this expression to find the original monthly payment amount.
2. Determine the remaining loan balance after making the 65th payment:
Dave has made 65 payments, so there are 276 - 65 = 211 payments remaining.
We can use the formula for the remaining loan balance of a mortgage after making a specified number of payments:
Remaining Balance = P * ((1 + r)^n - (1 + r)^p) / ((1 + r)^n - 1),
where P is the original principal, r is the monthly interest rate, n is the total number of payments, and p is the number of payments made.
Plugging in the values, we get:
Remaining Balance = 200000 * ((1 + 0.011)^276 - (1 + 0.011)^65) / ((1 + 0.011)^276 - 1).
Calculate this expression to find the remaining loan balance.
3. Calculate the new monthly payment:
Dave wants to reduce his monthly payment by $500.
Subtract $500 from the original monthly payment to find the new monthly payment amount.
4. Determine the number of payments required to pay off the remaining balance at the new interest rate:
We need to find the number of payments that will fully pay off the remaining balance.
Use the formula for the number of payments required to fully pay off a loan:
Number of Payments = -log(1 - (r * Remaining Balance) / Monthly Payment) / log(1 + r),
where P is the remaining balance, r is the new monthly interest rate, and Monthly Payment is the new monthly payment amount.
Plugging in the values, we get:
Number of Payments = -log(1 - (0.0072 * Remaining Balance) / New Monthly Payment) / log(1 + 0.0072).
Calculate this expression to find the number of payments required.
5. Calculate the final loan payment:
The final loan payment will be the remaining balance after the calculated number of payments.
Subtract the remaining balance from the product of the number of payments and the new monthly payment to find the final loan payment amount.
Perform these calculations to find out how large Dave's final loan payment will be.