## Refinancing Mortgage:

A mortgage is refinanced when a consumer takes a new loan to repay the outstanding balance of an existing loan, usually at a lower interest rate. In some cases, refinancing requires an upfront fee.

The monthly saving is \$37.99.

We first compute the monthly payment of the original loan. We can use the following formula to compute the monthly payment for a loan with principal {eq}P {/eq}, monthly interest rate {eq}r{/eq} and number of monthly payments {eq}T{/eq}:

• {eq}\displaystyle \frac{Pr}{1 - (1 + r)^{-T}} {/eq}

The original loan has a principal of 125,000, 15*12 = 180 monthly payments. Applying the formula, the monthly payment is:

• {eq}\displaystyle \frac{125,000*5.65\%/12}{1 - (1 + 5.65\%/12)^{-180}} = 1031.33 {/eq}

Two years into the mortgage, there are 180 - 12*2 = 156 monthly payments left. The outstanding balance is the present value of the remaining monthly payments, i.e.,

• {eq}\displaystyle \frac{1031.33*(1 - (1 + 5.65\%/12)^{-156})}{5.65\%/12} = 113,777.28 {/eq}

If the loan is refinanced with a principal equal to the outstanding balance, but a lower annual interest rate of 5% and a shorter term of 13 years, then the new monthly payment is:

• {eq}\displaystyle \frac{113,777.28*5\%/12}{1 - (1 + 5\%/12)^{-156}} = 993.34 {/eq}

Thus the monthly saving is:

• 1031.33 - 993.34 = 37.99