Tom got a 30 year fully amortizing FRM for $1,500,000 at 6%, with constant monthly payments. After 3 years of payments, rates fall and he can get a 27 year FRM at 5%, but he must pay 2 points and $1000 in closing costs to get the new loan. Think of the refinancing decision as an investment for Tom, he pays a fee now but saves money in the future in the form of lower payments. What is the annualized IRR of refinancing for Tom assuming he pays through maturity?

Answer is explained in the explanation section below.

Explanation:

Solution:

Note: In this question, Cash Flow table is used as well, which I have attached below. Please refer to that table as well.

Monthly payment for the 30 year FRM loan:

PV = 1,500,000; r (monthly interest rate) = 6%/12 = 0.5%; n (number of monthly payments) = 30*12 = 360

PMT (monthly payment) =

PMT (monthly payment) = = (0.5%*1,500,000)/(1 -(1+0.5%)^-360) = 8,993.26

Balance remaining after 8 years (if refinancing is not done):

PMT = 8,993.26; r = 0.5%; n = 360 - (8*12) = 264

PV = PMT*(1 - (1+r)^-n)/r = 8,993.26*(1 - (1+0.5%)^-264)/0.5%  

PV = 1,316,585.31

Balance remaining after 3 years (if refinancing is done):

PMT = 8,993.26; r = 0.5%; n = 360 - (3*12) = 324

PV = PMT*(1 - (1+r)^-n)/r = 8,993.26*(1 - (1+0.5%)^-324)/0.5%  

PV =  1,441,261.05

Cost of refinancing = 2%*remaining balance + 1,000 = (2%*1,441,261.05) + 1,000

Cost of refinancing = 29,825.22

Monthly payment (if refinancing is done):

PV = 1,441,261.05; r = 5%/12 = 0.4167%; n = 27*12 = 324

PMT = (1,441,261.05*0.4167%)/(1 -(1+0.4167%)^-324)

PMT = 8,114.86

Balance remaining after 5 years (after refinancing):

PMT = 8,114.86; r = 0.4167%; n = 22*12 = 264

PV = 8,114.86*(1-(1+0.4167%)^-264)/0.4167%

PV = 1,297,794.91

Note: Cash flow table is attached below.

Using financial calculator:

CF0 = -29,825.22;

CF1 = 878.40; N0 = 59 (5 years less one month);

CF2 = 19,668.80; N2 = 1, solve for IRR.

IRR = 2.69%

Hence,  

Annual IRR = 2.69%*12 = 32.29%

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