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Transcript of Amortization of Loans Amortization of Loans 3 3 McGraw-Hill Ryerson© 14 - 1 Chapter 14 of.
14 - 1Amortization
of Loans
Amortization
of Loans 33 33
McGraw-Hill Ryerson©
Chapter 14 of
14 - 2Amortization
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Amortization
of Loans 33 33
McGraw-Hill Ryerson©
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.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for
the effect of rounding the regular payment.
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its
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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Calculate the exact balance after 5 years assuming the final payment will be adjusted for
the effect of rounding the regular payment.
Calculate the exact balance after 5 years assuming the final payment will be adjusted for
the effect of rounding the regular payment.
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments during its
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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Retrospective Method for Loan Balances
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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Retrospective Method for Loan Balances
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.
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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Prospective Method for Loan Balances
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.
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
Retrospective Method for Loan Balances
PV= 17,741.88 Prospective Method for Loan Balances
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.
A $20,000 mortgage loan at 9% compounded monthly requires monthly payments of $179.95
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.
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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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.
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.
…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.
B. Calculate the final payment.
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…the final payment3.3. Calculate
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.
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.
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?
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?
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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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.
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.
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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A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term
by equal monthly payments.B. How much interest will be paid in the
second year of the loan?
A $9,500 personal loan at 10.5% compounded monthly is to be repaid over a 4-year term
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 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 APPLICATION
MORTGAGE APPLICATION
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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MORTGAGE APPLICATION
MORTGAGE APPLICATION
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
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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MORTGAGE APPLICATION
MORTGAGE APPLICATION
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)
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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Gross Debt Service Ratio (GDS)Gross Debt Service Ratio (GDS)
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 Debt Service Ratio (TDS)Total Debt Service Ratio (TDS)
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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Loan-to-Value Ratio (LVR)Loan-to-Value Ratio (LVR)
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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MORTGAGE APPLICATION
MORTGAGE APPLICATION
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