# Assume a borrower made a mortgage loan five years ago for $180,000 at 8% interest for 30 years...

## Question:

Assume a borrower made a mortgage loan five years ago for $180,000 at 8% interest for 30 years (monthly payment). After five years, interest rates fall, and a new mortgage loan is available at 7.5 percent for 25 years. The loan balance on the existing loan is $171,125.74. Suppose the closing cost for the new loan will be $5,500 plus $150 for recording fees. Should the borrower refinance?

## Interest Rates and Mortgages:

A mortgage is a secured loan that is granted to a borrower so that he can purchase a property. A lender will charge an interest rate on the mortgage loan that will be paid throughout the term of the loan. Each mortgage payment made by the mortgagor will be composed of both interest payment and principal repayment. Interest is calculated based on the outstanding value of the loan. This means that when the loan is the largest - at the beginning of the repayment period - the interest expense will be the highest. Each mortgage payment will decrease the outstanding principal amount, so the interest expense goes down incrementally with each payment.

A change in interest rate can represent a significant difference in a borrower's total interest expense over the course of a mortgage. When a borrower is given the option to refinance his mortgage, he must determine whether the cost savings he would experience on interest savings would exceed the expense of refinancing.

## Answer and Explanation:

To answer this question we must determine whether the cost savings of acquiring a lower rate of financing will offset the cost of attaining said financing. To do this we will determine the interest rate differential between the two rates and compare the cost savings on interest to the closing cost and recording fee expenses associated with the new loan.

We can calculate a mortgage payment amount using the following formula:

{eq}M = P ((r(1+r)^n) / ((1+r)^n - 1)) {/eq}

Where: M = Mortgage payment amount

P = Principal amount

r = Interest rate (expressed as a monthly rate)

n = Total number of payments

a). $171,125.74 for 25 years at 8%

We must first determine our monthly interest rate (r) and our total number of payments (n).

To determine our monthly interest rate, we can simply divide the annual interest rate by 12.

r | = 0.08/12 |

r | = 0.0.00666667 |

To determine our total number of payments, we multiply the number of years the loan will be in place by the number of payments that will be made each year.

n | = 25 x 12 |

n | = 300 |

We can now enter this data into our payment formula.

M = 171,125.74 ((0.00666667(1+0.00666667)^300) / ((1+0.00666667)^300 - 1))

= 171,125.74 ((0.00666667(7.340176) / (7.340176 - 1))

= 171,125.74 (0.0489345 / 6.340176)

= 171,125.74 x 0.00771816

= 1320.776217

= $1,320.78

The mortgage payment per month for maintaining an 8% interest rate will be $1,320.78. Note that this is the same mortgage payment that would be calculated if we had calculated the rate for a $180,000 mortgage at 8% over a 30 year time line.

b). $171,125.74 for 25 years at 7.5%

All of our data stays the same except for the interest rate.

r = 0.075 / 12

= 0.00625

M = 171,125.74 ((0.00625(1+0.00625)^300) / ((1+0.00625)^300 - 1))

= 171,125.74 ((0.00625(6.48288) / 6.48288 - 1))

= 171,125.74 (0.040518 / 5.48288)

= 171,125.74 x 0.0073899

= 1264.604122

= $1,264.60

If the borrower refinances, his new monthly mortgage payment will be $1,264.60

c). In order to determine which loan to choose, it is necessary to see whether the amount paid for refinancing is greater than or less than the amount of interest you save over the course of the loan.

The difference in the monthly payment is:

$1,320.78 - $1,264.60 = $56.18.

The total cost of refinancing the mortgage is $5,650 ($5,500 + $150). |

The break even point (the payment at which the interest rate savings equals the cost of the point) is:

$5,650 / $56.18 = 101 (rounded up to the next full payment).

This means that over the course of 300 payment (12 payments per year over 25 years) the savings from the lower interest rate will surpass the cost of refinancing at the 101 payment.

Because the break even point falls within the term of the loan, the borrower should refinance.

Total cost savings from the lower interest rate are:

$56.18 x 300 = $16,854.00

After paying $5,650 to refinance, total amount saved would be:

$16,854.00 - $5,650.00 = $11,204.00.

#### Learn more about this topic:

from Finance 102: Personal Finance

Chapter 7 / Lesson 4