Suppose that five years ago you borrowed $300,000 using a 30-year fixed-rate mortgage with an annual interest rate of 10% with monthly payments and compounding. the interest rate on 30-year fixed-rate mortgages has fallen to 8.5% and you are wondering whether you should refinance the loan. refinancing costs are expected to be 5% of the new loan amount. a. what is the net present value of refinancing if you make all of the scheduled payments on the new loan? b. what is the net present value of refinancing if you pay off the new loan at the end of the 3rd year? c. how many payments do you need to make on the new loan in order for refinancing to have a positive net present value?

Please check the explanation below.

Explanation:

Rate of Interest =10% or 0.83% monthly

Monthly Payment under this plan=PMT(0.0083, 360, 300000) =$2,632.71

Loan outstanding after 5 years of payments =$289,723

New Interest Rate =8.5% or 0.7083% monthly

Balance Tenure= 25 years

New Monthly Installment =PMT(0.007083,300,289723) =$2,332.93

Monthly savings in installment reduction =$2,632.71 - 2,332.93 =$299.78

a. Net present value of refinancing = -0.05x289,723 + 299.78x{(1-(1+0.007083)-300)/0.007083}

                                                  = -14,486.15 + 299.78x124.1886

                                                  = -14,486.15 + 37,229.25

                                                  = 22,743.10

b. With new monthly installment, balance outstanding at the end of 8th year =$278,258

Net Present Value of Refinance = -0.05x289,723 + 299.78x{(1-(1+0.007083)-36)/0.007083}

                                                  = -14,486.15 + 299.78x31.68

                                                  = -14,486.15 + 9,446.46

                                                  = -4,989.68

c. For refinance loan to have net present value positive, let n payments are required,

NPV = -0.05x289,723 + 299.78x{(1-(1+0.007083)-n)/0.007083}

14,486.15 = 299.78x{(1-(1+0.007083)-n)/0.007083}

14,486.15x0.00783/299.78 =(1-(1.007083)-n)

0.3423 = 1-(1.007083)-n

(1.007083)-n = 0.6577

(1.007083)n = 1.5204

Taking Log both sides,

n = log(1.5204)/log(1.007083)

n = 59.36

Hence, he would need to make 60 payments for making NPV of refinance as zero.