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.

Answer and Explanation:

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

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

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


Learn more about this topic:

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Buying a House: Mortgage Types & Loan Length

from Finance 102: Personal Finance

Chapter 7 / Lesson 4
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