Amortization of Loans Amortization of Loans 3 3 McGraw-Hill Ryerson© 14 - 1 Chapter 14 of

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McGraw-Hill Ryerson©

Chapter 14 of

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Calculate

Learning ObjectivesLearning Objectives

…the principal balance after any payment using both the Prospective Method and the Retrospective Method

After completing this chapter, you will be able to:

… the principal and interest components of any payment

And…And…

… the final loan payment when it differs from the others

LO 1.LO 1.

LO 2.LO 2.

LO 3.LO 3.

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Learning ObjectivesLearning Objectives

Calculate

LO 4.LO 4.

LO 5.LO 5. … mortgage loan balances and amortization periods to reflect

prepayments of principal

… mortgage payments for the initial loan and its renewals

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A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its

20-year amortization period. (1) Calculate the monthly payment.

(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ from $20,000?

2409

0

20 000

PMT = -179.9512

n =12* 20 = 240PV = $20000 FV = 0

1.1.

2.2. && 3.3.

LO 1.LO 1.

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(2) Using the monthly payment from part (1), calculate the PV of all payments. (3) Why does the answer in (2) differ

from $20,000?

2.2.

3.3.

179.95

179.95

n =12*20 = 240PV = ? FV = 0 PMT = 179.95

PV = 20,000.5345

The difference of $0.5345 is due to rounding the monthly payment to the nearest cent!

The difference of $0.5345 is due to rounding the monthly payment to the nearest cent!

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Calculate the exact balance after 5 years assuming the final payment will be adjusted for

the effect of rounding the regular payment.




20-year amortization period.

Calculate the exact n for monthly payments of $179.95 to repay a $20,000 loan...

20 000N = 239.982

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20-year amortization period.

After 5 years, 239.982 – 60 = 179.982 payments remain. Therefore, balance (after 5 years)

= PV of 179.982 payments of $179.95

60

N = 239.982N = 179.9821P/V = 17,741.05

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An Original Loan =

Consider that…Consider that…

The PV of ALL of the Payments

(discounted at the contractual rate of interest on the loan)

Also, that…Also, that…

A Balance = The PV of the remaining Payments

(discounted at the contractual rate of interest on the loan)

Then…Then…

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…this can be expressed as …the Statement of Economic Equivalence…this can be expressed as …the Statement of Economic Equivalence

(Original Loan)

Focal Date…Focal Date…

PV of first x Payments

PV of the

Balance just

after the xth Payment

For a focal date of the original date of the loan,

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of the xth payment,

the Statement of Economic Equivalence becomes…

Retrospective Method for Loan Balances

RetrospectiveRetrospective

Suppose we locate the Focal Date…

Balance

This is now rearranged to isolate the “Balance”This is now rearranged to isolate the “Balance”

Balance

FV of the

Original Loan

FV of the Payments

already made

FV of the

Original Loan

FV of the Payments

already made

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RetrospectiveRetrospective

… is based on PAYMENTS ALREADY MADE!`

Prospective Method for Loan Balances

… is based on PAYMENTS YET to be MADE!`

ApplicationApplication

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Calculate the exact balance after 5 years. Calculate the exact balance after 5 years.

A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95

during its 20-year amortization period.

Solve using…Retrospective Method

Prospective Method

Then compare…

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Balance = FV of $20,000 – FV of first 60 payments

60179.95

12

9

20,000

12 * 5 Years12 * 5 Years

FV= 17,741.05

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12* 20 Years = 24012* 20 Years = 240Total payments =

180179.95

12

9

PV= 17,741.88

0

- 60 made = 180 remaining

Balance = PV of remaining 180 payments

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Difference ($0.83) is because the Prospective Method assumes that the final payment is the same as all the others.

The Retrospective Method is based on payments already made.

FV= 17,741.05 Retrospective Method

for Loan Balances


PV= 17,741.88 Prospective Method for Loan Balances


Comparison of MethodsComparison of Methods

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Final Payment = (1+i) * (Balance after 2nd to last payment)

Balance after 239 payments = FV of $20,000 after 239 months – FV of 239 payments

239

179.95

129 FV= - 175.42

20,000

Final Payment = (1+0.09/12) * 175.42

= $176.74= $176.74

Calculate the size of the final payment. Calculate the size of the final payment.


during its 20-year amortization period. LO 2.LO 2.

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Meditech Laboratories borrowed $28,000 at 10%, compounded quarterly, to purchase new testing equipment.

Payments of $1,500 are made every 3 months.A. Calculate the balance after the 10th payment.

B. Calculate the final payment.




Balance after 10 payments = FV of $28,000 after 10 quarters – FV of 10 payments

101500

4

10

FV= - 19,037.29

28,000

A.

B.2.2.1.1. 3.3. Needed

Balance after 10 payments

Balance after 10 payments

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…the number of payments1.1. Calculate

0

N = 25.457 FV = -673.79

25

…the balance after the 2nd to last payment2.2. Calculate

3.3.


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…the final payment3.3. Calculate







Final Payment = (1+0.10/4) * 673.79

= $690.63= $690.63

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A $9,500 personal loan at 10.5% compounded monthly is to be

repaid over a 4-year term by equal monthly payments.

A. Calculate the interest and principal components of the 29th

payment.

B. How much interest will be paid in the second year of the loan?

LO 3.LO 3.

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A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term

by equal monthly payments.A. Calculate the interest and principal components

of the 29th payment. B. How much interest will be paid in the second year of the loan?



of the 29th payment. B. How much interest will be paid in the second year of the loan?

First: … find the size of the monthly payment

PV = n = i =9500 12(4) = 48 .105/12

4812

10.5

PMT = - 243.23

9500 0

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of the 29th payment.



of the 29th payment.

A.

First: … find the balance after the 28 payments

28

PMT = - 243.23

243.23

FV = -4445.06

Interest Component of Payment 29 = 0.105/12* 4445.06 = $38.89

= i * Balance after 28th payment

Principal Component = PMT – Interest Component= $243.23 - $38.89

= $204.34= $204.34

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by equal monthly payments.B. How much interest will be paid in the

second year of the loan?


by equal monthly payments.B. How much interest will be paid in the

second year of the loan?

First:… find the balance after 1 Year, and the balance after 2 Years

12

FV = -7483.53

Total Principal paid in year 2 = $7,483.53 - $5,244.84

= $2,238.69= $2,238.69

24

FV = -5244.84

Total Interest paid in year 2 = 12($243.23) - $2,238.69

= $680.07= $680.07

Balance after 1 year

Balance after 1 year

Balance after 2 years

Balance after 2 years

B.

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… is a loan secured by some

physical property

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MORTGAGE APPLICATION


Mortgage Loans…Basic Concepts and Definitions

…the borrower is called

the mortgagor

…the lender is called

the mortgagee

Borrower Lender

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Mortgage Loans…Basic Concepts and Definitions

Face Value of mortgage = original principal amount

Term … From … date on which loan advancedTo … date on which the remaining Principal Balance is due and payable

…most common periods are 20 and 25 years. …most common periods are 20 and 25 years.

Interest Rate …usually a lender will commit to a fixed interest rate for only a shorter period

or term (6 months to 7 years)

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A Mortgage Loan at 8.5% compounded semiannually with a 25-year amortization period

A Mortgage Loan at 8.5% compounded semiannually with a 25-year amortization period

Graphic IllustrationsGraphic Illustrations

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Years

Inte

rest

%

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25

The Composition of Mortgage Payments during a 25-year

Amortization

The Composition of Mortgage Payments during a 25-year

Amortization

Principal Component

Approximately 40%

Approximately 40%

Approximately 60%

Approximately 60%

Year 14

Interest Component

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McGraw-Hill Ryerson© 0 5 10 15 20 25Years

Pri

nci

pal

Bal

ance

$

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

100,000

Mortgages Declining Balance during a 25-year

Amortization

Mortgages Declining Balance during a 25-year

Amortization

Principal declines slower in

earlier years

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MORTGAGE APPLICATION …need to satisfy all 3 of the following Ratios…

Loan-to-Value Ratio (LVR)Loan-to-Value Ratio (LVR)

Gross Debt Service Ratio (GDS)Gross Debt Service Ratio (GDS)

Total Debt Service Ratio (TDS)Total Debt Service Ratio (TDS)

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Principal Amount of LoanLending Value of Property

x 100% 75%75%

Total monthly payments for Mortgage, Property taxes, and Heat

Gross Monthly Incomex 100% 32%32%

Total monthly payments for Mortgage, Property taxes, Heat and Other Debts Gross Monthly Income

x 100% 40%40%

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You have saved $35,000 for the down payment on a home.

You want to know the maximum conventional mortgage loan for which you can qualify

in order to determine the highest price you can pay for a home.

What maximum monthly mortgage payment do the GDS and TDS ratios permit?

What maximum monthly mortgage payment do the GDS and TDS ratios permit?

… gross monthly income is $3,200… 18 payments of $300 per month remaining on

a car loan … property taxes of $150 per month and heating costs of $100 per month … the bank has upper limits of 32% for the

GDS Ratio and 40% for the TDS Ratio

… gross monthly income is $3,200… 18 payments of $300 per month remaining on

a car loan … property taxes of $150 per month and heating costs of $100 per month … the bank has upper limits of 32% for the

GDS Ratio and 40% for the TDS Ratio

Personal Data

Personal Data

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Total monthly payments for Mortgage, Property taxes, and Heat

Gross Monthly Incomex 100% 32%32%

Maximum Mortgage payment + 150 + 100

$3,200 = 32%32%

Maximum Mortgage payment = .32(3200) - 250

= $774

= $774

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Total monthly payments for Mortgage, Property taxes, Heat and Other Debts Gross Monthly Income

x 100% 40%40%

Maximum mortgage payment + 150 + 100+ 300

$3,200 = 40%40%

Maximum Mortgage payment = .40(3200) - 550

= $730 = $730

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What is the maximum mortgage for which you qualify?

Use a 25-year amortization and an interest

rate of 8% compounded semiannually for a five-year term.

12

28 0

730

300

P/Y = 12 C/Y = 2 0P/V= 95,648.21 Maximum Mortgage

Maximum Mortgage

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xPrincipal Amount of LoanLending Value of Property

100% 75%75%

$95, 600Minimum house value = 75%75%

Minimum house value = $95, 648

75%75%= $127,530.67

Based on a $35,00 down payment and the maximum loan possible, what is the highest price you can pay for a home?

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At this price, the minimum down payment is:

$127,531– $95,648= $ 31,883

… the maximum price you can afford to pay for a home is…

$35,000 – 31,882 (Minimum down

payment )= over DP

$35,000 – 31,882 (Minimum down

payment )= over DP

Based on a $35,00 down payment and the maximum loan possible, what is the highest price you can pay for a home?

+ $127, 531 = $130,649$3,118

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Common Prepayment Common Prepayment

Privileges

Penalties&

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Partially OpenFully Open

Common Prepayment Common Prepayment PrivilegesPrivileges

PenaltiesPenalties&&

No restrictions or penalties

on extra payments by the

borrower!

Limited penalty-free prepayment

Lump or Balloon Payments10% or 15% of

the original amountIncreasing the Regular

Payment…permanentlyOnce a year by 10% or 15%

“Double-Up”Pay twice the amount for

any monthly payment

Closed

No prepayment

without a penalty

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Financial PenaltiesFinancial Penalties

Common Prepayment Common Prepayment PrivilegesPrivileges

PenaltiesPenalties&&

Contract provides

for a financial

penalty on any

prepayment not

specifically

permitted

The most common prepayment penalty is the greater of:

Three months’ interest on the amount prepaid,

or

Three months’ interest on the amount prepaid,

or

The lender’s reduction in interest revenue from the prepaid amount

(over the remainder of the

mortgage’s term)

The lender’s reduction in interest revenue from the prepaid amount

(over the remainder of the

mortgage’s term)ExampleExample

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The interest rate for the first 5-year term of a $100,000 mortgage loan is 7.5% compounded semiannually.

The mortgage requires monthly payments over a 25 year amortization period.

The mortgage contract gives the borrower the right to prepay up to 10% of the original mortgage loan,

once a year, without interest penalty.

Suppose that, at the end of the second year of the mortgage, the borrower makes a

prepayment of $10,000.

a) How much will the amortization period be shortened?

b) What will be the principal balance at the end of the five-year term?

a) How much will the amortization period be shortened?

b) What will be the principal balance at the end of the five-year term?

Solving StepsSolving Steps

LO 4.LO 4.

LO 5.LO 5.

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…the payments based on a 25-year amortization

…the balance after 24 payments

- Reduce this balance by $10,000

…the number of monthly payments needed to pay off this new balance

…the reduction in the original 25-year amortization period

2.2.

3.3.

4.4.

5.5.

1.1.

Calculate

Calculate

Calculate

Calculate

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300

100,000

12

7.5

PMT= -731.55

0

PV = n = i = c =100,000 25*12 = 300 .075/2 2/12

2

…the payments based on a 25-year amortization1.1. Calculate

monthly payment monthly payment

2.2. 3.3.

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731.55

FV= -97,007.25

24

10,000

87,007.25

…the balance after 24 payments2.2. Calculate

- Reduce this balance by $10,0003.3.

4.4. 5.5.

balance after 24 payments

balance after 24 payments

balance after the prepayment

balance after the prepayment

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87,007.250

…with the prepayment: 24 + 215 = 239 Total payments

Therefore, 300-239 = 61 months saved... i.e. 5 yrs 1 month

…the number of monthly payments needed to pay off this new balance

4.4. Calculate

…the reduction in the original 25-year amortization period

5.5. Calculate

N= 214.60 215 more payments to pay off the mortgage 215 more payments to pay off the mortgage

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Click On:Click On: Student Centre

Select:Select:

Interactive Mortgage Payoff Chart…online www.mcgrawhill.ca/college/jerome/

Interactive Mortgage Payoff Chart…online www.mcgrawhill.ca/college/jerome/

Click On:Click On:

-or--or-

Select:Select:4th Edition

Click On:

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This completes Chapter 14This completes Chapter 14

Documents

Amortization of Loans Amortization of Loans 3 3 McGraw-Hill Ryerson© 14 - 1 Chapter 14 of