Math 325
Ch 5 – Amortization Schedules and Sinking Funds
This chapter introduces two main methods of loan repayment:
1) The Amortization Method
2) The Sinking Fund Method
Goal: Finding the outstanding balance at any time t.
We'll discuss the Amortization method first!
Amortization Method:
• The borrower repays the lender by means of installment payments at periodic intervals.
• These installment payments form an annuity whose PV is equal to the original amount of the
loan.
• Each payment is partially repayment of principal and partially payment of interest.
An amortization schedule is a table which shows the division of each payment into principal and
interest, together with the outstanding loan balance after each payment is made.
Note: “Outstanding loan balance,” “outstanding principal,” “unpaid balance,” and “remaining loan
indebtedness” all mean the same thing.
Methods for determining Outstanding Loan Balance(OB)
1) Prospective Method: The OB at any point in time is equal to PV at that date of the remaining
installment payments.
2) Retrospective Method: The OB at any point in time is equal to the original amount of loan
accumulated to that date minus the accumulated value at that date of all installment payments
previously made.
Notation: Bt = Outstanding loan balance at any time t
Bt p = Outstanding loan balance according to prospective method
Bt r = Outstanding loan balance according to retrospective method
These two methods are equivalent. So usage of superscripts p and r is optional.
Equivalence of prospective and retrospective method
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Example 1. A loan is being repaid with 10 payments of $2000 followed by 10 payments of $1000 at
the end of each half-year. If i(2) = 10%, find the outstanding loan immediately after 5 payments have
been made using
a) prospective method
b) retrospective method
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Example 2: A loan is being repaid with 20 annual payments of $1000 each. At the time of the 5 th
payment, the borrower wishes to pay an extra $2000 and then repay the balance over 12 years with a
revised annual payment. If the effective rate of interest is 9%, find the amount of revised annual
payment.
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Section 5.3 – Amortization Schedule
As mentioned earlier, An amortization schedule is a table which shows the division of each payment
into principal and interest, together with the outstanding loan balance after each payment is made.
Example 3. A loan of $5000 is being repaid with level annual payments at the end of each year for 4
years at 6% per year convertible semiannually. Make an amortization schedule for this loan.
Period
4
PMT
Interest Paid
Principal Repaid
Outstanding Balance
Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Generalized Amortization Schedules
Loan amount:
Period
5
PMT
Eff. Rate per period:
Interest Paid
Pmts:
Principal Repaid
Outstanding Balance
Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Example 4. A loan is being repaid with a series pf payments at the end of each quarter for 5 years. If
the amount of principal in the 3rd payment is $100, find the amount of principal in the last 5 payments.
The interest rate is i(4) = 10%.
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Example 5. A $20,000 loan is being repaid by payments of $400 at the end of each month for as long
as necessary, plus a smaller final payment. If i(12) = 12%,
a) Find the outstanding balance after the 17th payment
b) Find the amount of principal and interest owed in the 18th payment
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Section 5.4: Sinking Fund
The borrower pays the lender by means of one lump sum payment at the end of the term of the loan,
along with interest payments made in installments over this period. It is also assumed that borrower
makes periodic payments into a fund called a “sinking fund,” which will accumulate to the amount of
the loan to be repaid at the end of the term of the loan.
The interest that is paid in the sinking fund method is sometimes called service on the loan.
If the rate of interest paid on the loan equals the interest earned on the sinking fund, then the sinking
fund method is equivalent to the amortization method.
Example 6. A loan of $5000 is being repaid with level annual payments at the end of each year for 4
years at 6% per year convertible semiannually. Make a sinking fund schedule for this loan. Assume tge
sinking fund earns I = 0.0609.
Step 1: Find the amount of sinking fund required to accumulate $5000
Step 2:
Period Interest Paid Sinking Fund Int. earned on SF
Deposit
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Amount in SF
Net amount of Loan
Math 325
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Ch 5 – Amortization Schedules and Sinking Funds
Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Notes: In Example 6, since rates were equal,
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Sinking Fund Method With Different Interest Rates
Let i = rate of interest paid on the loan
j = rate of interest earned on the sinking fund.
Suppose a loan of 1 is to be repaid in n periods via the sinking fund method with above interest rates.
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Math 325
Ch 5 – Amortization Schedules and Sinking Funds
Example 7. A payment of $36,000 is made at the end of each year for 31 years to repay a loan of
$400,000. If the borrower replaces the capital by means of a sinking fund earning 3% effective, find the
effective rate paid to the lender of the loan.
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