Question

Precisely 15 years ago (you have just made the 180th monthly payment) you took out a
traditional, fixed rate 30-year mortgage for $500,000 at 6.80% APR. You are considering
whether to refinance this loan and replace it with a fixed rate 15-year mortgage (assume an
identical timing of the remaining payments). The closing costs are $3,000, which you intend to
borrow. Disregarding the psychic cost of the process, please find the breakeven interest rate (i.e.
the rate at which you would be indifferent between refinancing vs. keeping the existing
mortgage). Express your answer as an APR with 3 digits after the decimal point.

Answer 1

For old Mortgage, N = 30 years, APR = 6.80%, Principal Amount = $500,000

As initially no fees are mentioned, Any fees = 0,

Thus, monthly installment can be found using financial calculator or MS excel

Input data, N = 360, I/Y(interest rate for one period) = 6.80%/12 = 0.5667%, PV = -5,00,000, FV = 0, Calculate PMT = ?

Hence, PMT = $3259.63

In excel, the amount of principal paid in 180 installments CUMPRINC function can be used and the answer is   $132794.30

Excel Calculations
N	360
I/Y	0.005666667
PV	₹ 5,00,000.00
FV	0
PMT	$ -3,259.63
CUMIPMT	-132794.2996

Thus, the amount of principal to be paid $500000 - $132794.30   $367205.70

Now, to be indifferent between refinancing and keeping existing mortgage my annual outflow should be the same for both cases. Thus, PMT = $3259.63

For refinancing, I take a loan of $367205.70 in addition to closing costs of $3000. Thus total principal amount  ​​ $370205.70

For PMT (monthly payments) = $3259.63, Principal Amount = $370205.70, Periods = 180 months (left), New monthly interest rate can be calculated using Rate function of excel or I/Y function of financial calculator.

Annual interest rate = monthly interest rate *12 = 6.6702%

Loan amount Left	$ 3,67,205.70
Closing costs	$ 3000
New Principal	$ 3,70,205.70
New monthly interest rate	0.5558%
Annual interest rate	6.6702%
Interest Amount	$ 24,693.42
APR	6.725%

This interest rate times the principal amount of $370205.70 give annual interest amount = $24693.42.

This interest amount divided by the original loan amount left i.e. $367205.70 gives APR = $ 6.725%

Hence, after refinancing loan, the closing costs are to be added to the loan amount to get the principal amount.

But this principal amount is used to only calculate the interest amount paid (by finding interest rate first and interest amount form there.)

However to find APR, this interest amount should be divided by the original loan amount left.