AMORTISING LOAN

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

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.

Formulas

PntpymtI paid *

nr

nr

pymtnrP

nt

nt 111

Simple Interest Amortized Loan Formula

Interest Paid

Formulas

nr

nr

pymtnrP

nT

nT 111balanceunpaid

Unpaid Balance Formula

T is the number of years from the beginning of the loan to the present

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 PrincipalRepaid

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 yearrepaid by 5 annual

payments

Amortization schedule:

Example

t - 1 t

PaymentX

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

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.

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 balanceI = Prt

use simple interest

amortized loan

formula

previous balance

minus this payment’s principal portion

last previous balance

simple interest on previous balanceI = Prt

principal portion plus

interest portion

$0.00

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