5. Suppose that five years ago you borrowed $500,000 using a 30-year fixed-rate
ID: 2796490 • Letter: 5
Question
5. Suppose that five years ago you borrowed $500,000 using a 30-year fixed-rate mortgage with an annual interest rate of 7.00% with monthly payments and compounding. The interest rate on 30-year fixed- rate mortgages has fallen to 6.25% and you are wondering whether you should refinance the loan. Refinancing costs are expected to be 4% of the new loan amount.
What is the net present value of refinancing if you make all of the scheduled payments on the new loan?
What is the net present value of refinancing if you pay off the new loan at the end of the 6th year?
How many payments do you need to make on the new loan in order for refinancing to have a positive net present value?
Explanation / Answer
Amount Borrowed =$500,000
Interest Rate = 7% per annum or 0.5833% per month
Time = 30 years or 360 monthly periods
Level Monthly Payment = PMT(0.005833, 360, 500000) =$3,326.51
Loan outstanding balance after 5 years =$500,000x(1+0.005833)60 -3326.51x{((1+0.005833)60-1)/0.005833}
=$500,000x1.4176 - 3326.51x71.5929
=$708,812.63 - 238,154.68
=$470,657.95
a.) If refinanced, Refinancing Cost =0.04x470,657.95 =$18,826.32
New Interest Rate =6.25% per annum or 0.5208% per month
Balance Tenure =300 monthly periods
New Level Monthly Payment =PMT(0.005208, 300, 470657.95) =$3,104.79
Net Present Value of Refinancing = -18,826.32 + (3326.51 - 3104.79)x{(1-(1+0.005208)-300)/0.005208}
= -18,826.32 + 221.72x150.38
= -18,826.32 + 33,341.58
= 14,515.26
b.) If the loan is paid at the end of 6th year,
NPV of refinancing =-18,826.32 + (3326.51 - 3104.79)x{(1-(1+0.005208)-12)/0.005208}
= -18,826.32 + 221.72x11.5981
= -18,826.32 + 2,571.53
= -16,254.79
c.) Let n payments to be made for making NPV positive after refinancing,
0 = -18,826.32 + (3326.51 - 3104.79)x{(1-(1+0.005208)-n)/0.005208}
18,826.32 = 221.72x{(1-(1+0.005208)-n)/0.005208}
84.91 = {(1-(1+0.005208)-n)/0.005208}
0.4486 = 1-(1+0.005208)-n
(1+0.005208)-n = 0.5513
n = 113
Hence, total payments made after refinancing =113 or for 9.4 years
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