This document discusses two methods for calculating the outstanding principal on an amortizing loan: the prospective method and the retrospective method. It also provides examples of calculating outstanding principal using each method. Additionally, it reviews the formulas for calculating interest paid and unpaid balance on a loan over time. It concludes by explaining the steps for creating an amortization schedule, including calculating the interest and principal portions of each payment and updating the outstanding balance.

3. Two Methods
 Prospective method:
outstanding principal at any point in time is equal to
the present value at that date of all remaining
payments
 Retrospective method:
outstanding principal is equal to the original principal
accumulated to that point in time minus the
accumulated value of all payments previously made
Note: of course, this two methods are equivalent.
However, sometimes one is more convenient than the
other

4. Examples
• (prospective) A loan is being paid off with payments of 500 at the
end of each year for the next 10 years. If i = .14, find the
outstanding principal, P, immediately after the payment at the end
of year 6.
• (retrospective) A 7000 loan is being paid of with payments of 1000 at
the end of each year for as long as necessary, plus a smaller
payment one year after the last regular payment. If i = 0.11 and the
first payment is due one year after the loan is taken out, find the
outstanding principal, P, immediately after the 9th payment.

5. Formulas
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n
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P
nt
nt 11
1
Simple Interest Amortized Loan Formula
Interest Paid

6. Formulas
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n
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n
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pymt
n
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P
nT
nT 11
1balanceunpaid
Unpaid Balance Formula
T is the number of years
from the beginning of the
loan to the present

7. AMORTIZATION SCHEDULE
• GOAL: DIVIDE EACH PAYMENT (OF ANNUITY) INTO TWO PARTS –
INTEREST AND PRINCIPAL
• AMORTIZATION SCHEDULE – TABLE, CONTAINING THE FOLLOWING
COLUMNS:
• PAYMENTS
• INTEREST PART OF A PAYMENT
• PRINCIPAL PART OF A PAYMENT
• OUTSTANDING PRINCIPAL
Duration Payment Interest Principal
Repaid
Outstanding
Principal
0 5000.00
1 13875.05 600.00 787.05 4212.95
2 13875.05 505.55 881.50 3331.45
3 13875.05 399.77 987.28 2344.17
4 13875.05 281.30 1105.75 1238.42
5 13875.05 148.61 1238.44 0
Example:
5000
at 12 % per year
repaid by 5 annual
payments
Amortization schedule:

8. Example
t - 1 t
Payment
X
Outstanding
principal
P
Interest earned during
interval (t-1,t) is iP
Therefore interest portion
of payment X is iP
and principal portion is
X - iP
A 1000 loan is repaid by annual payments of 150, plus a
smaller final payment. If i = .11, and the first payment is made
one year after the time of the loan, find the amount of
principal and interest contained in the third payment
Recall: in practical problems, the outstanding principal P can
be found by prospective or retrospective methods

9. Amortization Schedule Steps:
• Find the interest on amount use – Use the simple
interest formula.
• Principal portion is payment minus interest portion.
• New balance is previous balance minus principal
portion.
For the last period
• Principal portion is previous balance.
• Total payment is sum of principal portion and interest
portion.

10. Amortization Schedule Steps:
Payment
Number
Principal
Portion
Interest
Portion
Total
Payment
Balance
0 --------- --------- --------- loan amount
first
through
next-to-last
total
payment
minus
interest
portion
simple
interest on
previous
balance
I = Prt
use simple
interest
amortized
loan formula
previous
balance
minus this
payment’s
principal
portion
last previous
balance
simple
interest on
previous
balance
I = Prt
principal
portion plus
interest
portion
$0.00

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