Question

Ten years ago you obtained a 30-year mortgage for $400,000 with a fixed interest rate of 3% APR compounded monthly. The mortgage is a standard fixed rate mortgage with equal monthly payments over the life of the loan.

1. What are the monthly fixed mortgage payments on this mortgage (i.e., the minimum required monthly payments to pay down the mortgage in 30 years)?

2. What is the remaining loan balance immediately after making the 120th monthly payment (i.e., 10 years after initially obtaining the mortgage)? Assume that you made all the mortgage payments on time and you only made the minimum required monthly payment over the last 10 years.

c. Today, a mortgage broker contacts you and says that you can reduce your monthly mortgage payments by refinancing the mortgage you obtained 10 years ago. The broker suggests that you refinance your current loan amount into a new 30-year mortgage with a fixed interest rate of 3.6% APR compounded monthly. The mortgage is a standard fixed rate mortgage with equal monthly payments. The mortgage will be fully paid off after 30 years. What are the monthly fixed mortgage payments on this refinanced mortgage?

d. To refinance the loan, you need to pay refinancing costs of $3,000 upfront. Assume you will own your home for an additional 30 years until the new mortgage is fully paid off and that you do not refinance the mortgage again in the future. Is it beneficial to refinance your current loan from part (a) with the new loan from part (c)? What is the NPV of refinancing your loan if your discount rate equals to the interest rate on the new mortgage (i.e., 3.6% APR with monthly compounding)?

Answer 1

Ten years ago you obtained a 30-year mortgage for $400,000 with a fixed interest rate of 3% APR compounded monthly. The mortgage is a standard fixed rate mortgage with equal monthly payments over the life of the loan.

Loan amount, PV = 400,000

Nper = number of months in the original term to maturity = 12 x 30 = 360

Rate = interest rate per month = APR / 12 = 3% / 12 = 0.25%

Part (a)

Monthly mortgage payment = PMT (Rate, Nper, -PV) = PMT (0.25%, 360, -400000) = $1,686.42

Part (b)

Nper = number of months in the original term to maturity = 12 x (30 - 10) = 240

the remaining loan balance immediately after making the 120th monthly payment = - PV (Rate, Nper, PMT, FV) = - PV (0.25%, 240, 1686.42, 0) = $304,079.24

Part (c)

Nper = number of months in the original term to maturity = 12 x 30 = 360

Rate = interest rate per month = APR / 12 = 3.6% / 12 = 0.3%

Loan amount = PV obtained in part (b) = 304,079.24

Monthly mortgage payment = PMT (Rate, Nper, -PV) = PMT (0.3%, 360, -304079.24) = $1,382.48

Part (d)

It's not beneficial to refinance your current loan from part (a) with the new loan from part (c), because you are refinancing a lower cost loan with a higher cost loan. Loan being refinanced is at 3% APR is lower than the interest rate of 3.6% APR applicable on the refinancing loan. Since, you are replacing a cheaper loan with an expensive one and that also by paying some refinance cost, hence, it's not beneficial to refinance the loan.

NPV = PV of the balance monthly payments at new interest rate - loan outstanding - upfront refinancing cost = -PV (Rate, Nper, PMT, FV) - 304,079.24 - 3,000 = - PV (0.3%, 240, 1686.42, 0) - 304,079.24 - 3,000 = $288,221.34 - 304,079.24 - 3,000 = - $18,857.89

(Please note that there is a negative sign before the final answer above)