Various methods of repaying a loan are possible. With the amortization method the borrower repays the lender by means of installment payments at periodic intervals.

1
Sections 5.1, 5.2
Various methods of repaying a loan are possible.
With the amortization method the borrower repays
the lender by means of installment payments at
periodic intervals.
It is often desirable to determine the
outstanding loan balance (which can also be
called the outstanding principal, unpaid
balance, or remaining loan indebtedness) at a
particular point in time. There are two
approaches which can be seen by considering the
following
Present Value of All Payments Amount of Loan
Accumulate each side of the equation to the
desired current date
Current Value of Payments Accumulated Value of
Loan Amount
Accumulated Value of Past Payments Present
Value of Future Payments Accumulated
Value of Loan Amount
2
Present Value of Future Payments Accumulated
Value of Loan Amount Accumulated
Value of Past Payments
Prospective Method Retrospective Method
Each side of this equation represents one of two
approaches which are equivalent in general
With the prospective method, the outstanding loan
balance at any point in time is equal to the
present value at that date of the remaining
payments. With the retrospective method, the
outstanding loan balance at any point in time is
equal to the original amount of the loan
accumulated to that date less the accumulated
value at that date of all payments previously
made.
The outstanding loan balance at time t will be
denoted by Bt and when desirable, Btp and Btr
will be used to distinguish between the
prospective and retrospective methods
respectively.
3
Consider a loan of at interest rate i per
period being repaid with payments of 1 at the end
of each period for n periods. For 0 lt t lt n, the
outstanding loan balance after exactly t periods
is
a n
Btp from the prospective method,
a nt
Btr from the retrospective method.
(1 i)t
a n
s t
Prove algebraically that Btr Btp .
1 vn i
(1 i)t 1 i
(1 i)t
(1 i)t
a n
s t
(1 i)t vnt (1 i)t 1
i
1 vnt i
a nt
4
A loan is being repaid with 16 quarterly
payments, where the first 8 payments are each
200 and the last 8 payments are each 400. If
the nominal rate of interest convertible
quarterly is 10, use both the prospective method
and the retrospective method to find the
outstanding loan balance immediately after the
first six payments are made.
B6p
With the prospective method, we have
200(v v2)
400v2(v v2 v8)
400(v v2 v10) 200(v v2)
400 200
a 10
a 2
400(8.75206) 200(1.92742)
3115.34
that the original loan amount is
With the retrospective method, we have
200(v v2 v8)
400v8(v v2 v8)
400(v v2 v16) 200(v v2 v8)
400 200
400(13.05500) 200(7.17014)
3787.97
a 16
a 8
We now find
B6r
3787.97(1.025)6 200
s 6
4392.90 200(6.38774)
3115.35
5
A loan is being repaid with 15 annual payments of
500 each. At the time of the tenth payment, the
borrower wishes to pay an extra 500 and then
repay the balance over 6 years with a revised
annual payment. If the effective rate of
interest is 8, find the amount of the revised
annual payment.
The loan balance after ten years (prospectively)
is
B p 500
10
a 5
500(3.99271)
1996.355
With the extra 500 payment, the loan balance is
1496.35
If X represents the revised annual payment, then
X 1496.35
a 6
X 1496.35 / 4.62288 323.68