Question

When you buy a house, you also "buy" an interest rate. That is, depending on the conditions of the housing market and the current federal interest rate, you get locked into a rate for the duration of your loan agreement by the bank that sells you the loan. Though there are many home loan durations (often terminating at the time called the "pay-off" date), the most typical for home-buyers is 30 years.

Suppose you buy a new home for $200,000 at 4.5% APR. After having owned the home for 5 years, you are offered a refinance for 4.25% APR for the remainder of your loan duration. Of course, this reduces your payment... but, there are closing costs which amount to 3% of what you owe on the loan at the time of refinance. After the refinance, how long would you have to own the home for in order to "break even" with the closing cost fees you paid to refinance? (Assume you pay the refi costs out-of-pocket at the time of refinance.)

NOTE: The APR is calculated in a more complicated way than we will let-on here. Assume APR is the same rate concept developed in our modules on interest rates. Also, we can assume the interest is compounded monthly on a mortgage.

Answer 1

In the given scenario, first of all remaining balance of the loan at the end of 5 years needs to be calculated. For this, monthly payment towards loan must be calculated.

Formula to calculate monthly payments: Pmt = Ar / (1-(1+r)^-n)

The principal loan amount is A, the interest rate per period is r, the number of periods is t, and PMT is the payment per period.

Loan Amount = $200,000
Interest rate per month = 4.50% / 12 = 0.3750% or 0.00375
Number of periods = 30 Years x 12 Months = 360

Pmt => [($200,000 x 0.00375) / (1-(1+0.00375)^-360)]
=> $750 / 0.740104 = $1,013.37

Formula to calculate Remaining Balance: B = A*(1 + r)ⁿ – Pmt* {[(1+r)ⁿ – 1]/r}

The remaining balance is B, the principal loan amount is A, the interest rate per period is r, the number of periods is t, and PMT is the payment per period.

Loan Amount = $200,000
Interest rate per month = 4.50% / 12 = 0.3750% or 0.00375
Number of periods = 5 Years x 12 Months = 60
Pmt = $1,013.37

B = $200,000*(1 + 0.00375)⁶⁰ – $1,013.37* {[(1+0.00375)⁶⁰ – 1]/0.00375}
= $250,359.16 – ($1,013.37* 67.14555214)
= $182,315.83

Refinancing of the loan:

Closing Cost = $182,315.83*3% = $5,469.48

Monthly payment under new terms:

Loan Amount = $185,315.83
Interest rate per month = 4.25% / 12 = 0.354167% or 0.00354167
Number of periods = 25 Years x 12 Months = 300

Pmt => [($185,315.83 x 0.00354167) / (1-(1+0.00354167)^-300)]
=> $645.70 / 0.6537599 = $987.67

Amount saved every month = $1,013.37 - $987.67 = $25.70

Number of months required to “Break-even” = $5,469.48 / $25.70 = 212.85 Months or 17.73 Years