# The mortgage on your house is 5 years old. It required monthly payments of 1,422, had an original...

## Question:

The mortgage on your house is 5 years old. It required monthly payments of 1,422, had an original term of 30 years, and had an interest rate of 10% (APR-monthly). In the intervening 5 years, interest rates have fallen and so you have decided to refinance - that is, you will roll over the outstanding balance into a new mortgage. The new mortgage has a 30-year term, requires monthly payments, and has an interest rate of 6.625% (APR-monthly).

a. What monthly repayments will be required with the new loan?

b. If you still want to pay off the mortgage in 25 years, what monthly payment should you make after you refinance?

c. Suppose you are willing to continue making payments of 1,422. How long will it take you to pay off the mortgage after refinancing?

d. Suppose you are willing to continue making payments of 1,422, and want to pay off the mortgage in 25 years. How much additional cash can you borrow today as part of the refinancing?

## EMI

EMI or equated monthly installment is defined as is one part of the equally divided monthly payments to clear off outstanding loan (principal and interest) within a predefined tenure.

EMI has two parts: (1) the principal and (2) the interest. The principal is paid against the loan amount and interest is paid as a cost of the loan outstanding. The interest can be charged on full amount or reducing balance amount.

Initially the APR = 10%

So monthly interest rate = 10%/12 =0.00833 per month

We know

We Know, {eq}EMI = [P * R * (1+R)^ {N} ]/[(1+R)^ {N} -1] .................(1) {/eq}

Where:

P = principal

R = interest rate per month = 0.00833

N = number of months = 12*30 = 360 Months

Putting the value we get,

{eq}$1,422= [P *0.00833*(1.00833)^ {360} ]/[(1+.00833)^ {360} -1 ]...................(2) {/eq} Solving the equation (2) for P we get, P$ 162,093

We make the loan amortization schedule for first 5 years as below:

 Month Beginning Balance EMI Principal Interest Ending Balance 1 $162,093$ 1,422 $72$ 1,351 $162,021 2$ 162,021 $1,422$ 72 $1,350$ 161,949 3 $161,949$ 1,422 $73$ 1,350 $161,876 4$ 161,876 $1,422$ 74 $1,349$ 161,803 5 $161,803$ 1,422 $74$ 1,348 $161,728 6$ 161,728 $1,422$ 75 $1,348$ 161,654 7 $161,654$ 1,422 $75$ 1,347 $161,578 8$ 161,578 $1,422$ 76 $1,346$ 161,502 9 $161,502$ 1,422 $77$ 1,346 $161,426 10$ 161,426 $1,422$ 77 $1,345$ 161,348 11 $161,348$ 1,422 $78$ 1,345 $161,271 12$ 161,271 $1,422$ 79 $1,344$ 161,192 13 $161,192$ 1,422 $79$ 1,343 $161,113 14$ 161,113 $1,422$ 80 $1,343$ 161,033 15 $161,033$ 1,422 $81$ 1,342 $160,952 16$ 160,952 $1,422$ 81 $1,341$ 160,871 17 $160,871$ 1,422 $82$ 1,341 $160,789 18$ 160,789 $1,422$ 83 $1,340$ 160,707 19 $160,707$ 1,422 $83$ 1,339 $160,623 20$ 160,623 $1,422$ 84 $1,339$ 160,539 21 $160,539$ 1,422 $85$ 1,338 $160,455 22$ 160,455 $1,422$ 85 $1,337$ 160,369 23 $160,369$ 1,422 $86$ 1,336 $160,283 24$ 160,283 $1,422$ 87 $1,336$ 160,197 25 $160,197$ 1,422 $88$ 1,335 $160,109 26$ 160,109 $1,422$ 88 $1,334$ 160,021 27 $160,021$ 1,422 $89$ 1,334 $159,932 28$ 159,932 $1,422$ 90 $1,333$ 159,842 29 $159,842$ 1,422 $90$ 1,332 $159,752 30$ 159,752 $1,422$ 91 $1,331$ 159,660 31 $159,660$ 1,422 $92$ 1,331 $159,568 32$ 159,568 $1,422$ 93 $1,330$ 159,476 33 $159,476$ 1,422 $94$ 1,329 $159,382 34$ 159,382 $1,422$ 94 $1,328$ 159,288 35 $159,288$ 1,422 $95$ 1,327 $159,193 36$ 159,193 $1,422$ 96 $1,327$ 159,097 37 $159,097$ 1,422 $97$ 1,326 $159,000 38$ 159,000 $1,422$ 97 $1,325$ 158,903 39 $158,903$ 1,422 $98$ 1,324 $158,805 40$ 158,805 $1,422$ 99 $1,323$ 158,705 41 $158,705$ 1,422 $100$ 1,323 $158,605 42$ 158,605 $1,422$ 101 $1,322$ 158,505 43 $158,505$ 1,422 $102$ 1,321 $158,403 44$ 158,403 $1,422$ 102 $1,320$ 158,301 45 $158,301$ 1,422 $103$ 1,319 $158,197 46$ 158,197 $1,422$ 104 $1,318$ 158,093 47 $158,093$ 1,422 $105$ 1,317 $157,988 48$ 157,988 $1,422$ 106 $1,317$ 157,882 49 $157,882$ 1,422 $107$ 1,316 $157,775 50$ 157,775 $1,422$ 108 $1,315$ 157,668 51 $157,668$ 1,422 $109$ 1,314 $157,559 52$ 157,559 $1,422$ 109 $1,313$ 157,450 53 $157,450$ 1,422 $110$ 1,312 $157,339 54$ 157,339 $1,422$ 111 $1,311$ 157,228 55 $157,228$ 1,422 $112$ 1,310 $157,116 56$ 157,116 $1,422$ 113 $1,309$ 157,002 57 $157,002$ 1,422 $114$ 1,308 $156,888 58$ 156,888 $1,422$ 115 $1,307$ 156,773 59 $156,773$ 1,422 $116$ 1,306 $156,657 60$ 156,657 $1,422$ 117 $1,305$ 156,540

Net outstanding after 5 years = $156,540 New monthly rate = 6.625%/12 = 0.00552 New {eq}EMI = [$ 156,540 *0.00552*(1.00552)^ {360} ]/[(1.00552)^ {360} -1 ] = $1,002.24 {/eq} #### b. If you still want to pay off the mortgage in 25 years, what monthly payment should you make after you refinance? 25 Years = 12*25 = 300 months So, New EMI {eq}= [$ 156,540 *0.00552*(1.00552)^ {300} ]/[(1.00552)^ {300} -1 ] = $1,069.13 {/eq} #### c. Suppose you are willing to continue making payments of 1,422. How long will it take you to pay off the mortgage after refinancing? We consider it will take N months So putting the value in equation (1) we get {eq}$1,422 = [$156,540 *0.00552*(1.00552)^ {N} ]/[(1.00552)^ {N} -1 ].................(3) {/eq} Solving the equation (3) by trial and error method or by using Microsoft Excel Goal Seek, we get, N = 170 Months So It will take 170 months more. d. Suppose you are willing to continue making payments of 1,422, and want to pay off the mortgage in 25 years. How much additional cash can you borrow today as part of the refinancing? We consider the principal = P So reconstructing the equation (3) {eq}$1,422 = P *0.00552*(1.00552)^ {300} ]/[(1.00552)^ {300} -1 ]............(4) {/eq}

By solving the equation (4) for P we get P = $208,206 So additional cash can be borrowed =$ 208,206 - $156,540 =$ 51,666 