Question

A homeowner took out a 30-year, fixed-rate mortgage of $145,000. The mortgage was taken out 6 years ago at a rate of 8.4 percent. If the homeowner refinances, the charges will be $2,550. What is the highest interest rate at which it would be beneficial to refinance the mortgage? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Answer 1

Annual payment (PMT) on mortgage can be calculated with the help of PV of an Annuity formula

PV = PMT * [1-(1+i) ^-n)]/i

Where,

Present value (PV) =$145,000

PMT = Annual payment =?

n = N = number of payments = 30 years

i = I/Y = interest rate per year = 8.4 %

Therefore,

$145,000 = PMT* [1- (1+8.4%) ^-30]/8.4%

= $13,369.12

The mortgage was taken out 6 years ago therefore the balance amount is the present value of loan calculated for 30 -6 = 24 years

PV = PMT * [1-(1+i) ^-n)]/i

Where,

Present value (PV) =?

PMT = Annual payment =$13,369.12

n = N = number of payments = 30 -6 = 24 years

i = I/Y = interest rate per year = 8.4 %

Therefore,

PV of loan (balance amount) = $13,369.12 * [1- (1+8.4%) ^-24]/8.4%

= $136,188.28

The remaining amount of loan is $136,188.28 but if the homeowner refinances, the charges will be $2,550.

Therefore total amount of loan = $136,188.28 + $2,550 = $138,738.28

The highest interest rate at which it would be beneficial to refinance the mortgage can be calculated by assuming same annual payment for remaining 24 years

PV = PMT * [1-(1+i) ^-n)]/i

Where,

Present value (PV) = $138,738.28

PMT = Annual payment = $13,369.12

n = N = number of payments = 24 years

i = I/Y = interest rate per year =?

Therefore,

$138,738.28 = $13,369.12 * [1- (1+i) ^-24]/i

Or i = 8.17%

If interest rate is 8.17% per year, then it is no profit no loss situation; therefore any interest rate below 8.17% it is beneficial to refinance the mortgage