Math 420
Loans(Amortization)
Problem Set 4a
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)
Example: 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 1) Prospective Method, and 2) Retrospective Method
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
1
Math 420
Loans(Amortization)
Problem Set 4a
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
TOTALS
2
PMT
Interest Paid
Principal Repaid
Outstanding Balance
Math 420
Loans(Amortization)
Problem Set 4a
Generalized Amortization Schedules
Loan amount:
Period
3
PMT
Eff. Rate per period:
Interest Paid
Pmts:
Principal Repaid
Outstanding Balance
Math 420
Loans(Amortization)
Problem Set 4a
1. Ted, Rob and Will each borrow 5000 for 5 years at a nominal interest rate of 8%, compounded
semiannually. Ted pays all the interest and principal as a lump sum payment at the end of 5 years.
Rob pays interest semiannually as it accrues and principal at the end of 5 years. Will repays his loan
with 10 level payments at the end of every 6-month period. Calculate the total amount of interest
paid on all three loans.
2. Tim borrows X for four years at an annual effective interest rate of 8%. He repays the loan with equal
installments at the end of each year. The outstanding balance at the end of 2nd year is 1075 and at the
end of the third year is 559. Calculate the principal repaid in the first payment.
4
Math 420
Loans(Amortization)
Problem Set 4a
3. A 6,000 loan is being repaid with regular payments of X at the end of each year as long as necessary
plus a smaller payment one year after the final regular payment. Immediately after the ninth
payment, the outstanding principal is three times the size of the regular payment. If the annual
interest rate is 10%, calculate X.
4. Paula and Chris have a 30-year 150,000 mortgage with an 8% interest rate convertible monthly.
Immediately after the 120th payment, they refinance the mortgage. The interest rate is reduced to
6.5%, convertible monthly, and the term is reduced to 20 years. They also make an additional
payment of 20000 at the time of refinancing. Calculate their new monthly payment.
1
Math 420
Loans(Amortization)
Problem Set 4a
5. A loan, at a nominal interest rate of 24% convertible monthly, is to be repaid wth level annual
payments at the end of each month for 2n months. The nth payment consists of equal payments of
interest and principal. Calculate n.
6. A 35-year loan is to be repaid in equal annual installments. The amount of interest paid in the 8th and
25th installments is 135 and 108, respectively. Calculate the amount of interest paid in the 29th
installment.
1
Math 420
Loans(Amortization)
Problem Set 4a
7. A loan of 1000 at a nominal rate of 12% convertible monthly is to be repaid by six monthly
payments with the first due at the end of 1 month. The 1st three payments are x each, and the final
three payments are 3x each. Determine the sum of the principal repaid in the third payment and the
interest repaid in the 5th payment.
8. Ted had just purchased a home with 100,000 mortgage. The nominal interest rate is 9%, convertible
monthly. He will pay monthly payments of X for 30 years.
If instead, he made payments of X/2 every two weeks, how many years will it take for him to pay off
the mortgage. (Assume that a year consists of 52 weeks and two weeks is NOT half a month).
2
Math 420
Loans(Amortization)
Problem Set 4a
9. Timmy borrows 55,000 to be repaid with monthly payments of 500.38 at the end of each month, for
n years, based on a nominal interest rate of i, compounded monthly. He skips the first payment but
makes all the other payments on time. Because he skipped the 1st payment, he still owes 3077.94 at
the end of n years. Calculate i.
10. Wayne takes out a 15-year, 100,000 loan with monthly payments beginning one month from today.
He makes regular payments until and including the 1st payment in which the principal portion
exceeds the interest portion of the payment. If the nominal interest rate is 4.8%, convertible monthly,
then determine the total interest Wayne paid on the loan after making the 1st payment in which the
principal portion exceeds the interest portion of the payment.
3