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1
Balance Difference Equations
Copyright 2006, David R. Brown
For use only in EE 366 or EE 366K
(revised 6/26/06)
It is informative and useful to consider economic equivalence relationships from a
difference equation perspective.
In essence, the equation governing loan balances,
investment plan balances, and project balances is a difference equation.
Any of the
techniques covered in EE 313 (Signals and Systems), EE 362K (Automatic Control), EE
351M (Digital Signal Processing) and other courses involving discrete time analysis and
design are relevant and applicable.
Loan Balances
The difference equation governing a loan balance is considered first:
period
time
the
of
end
at the
(credited)
loan
on the
made
payment
the
is
period
per time
interest
is
period
time
previous
the
of
end
at the
balance
loan
the
is
period
time
the
of
end
at the
balance
loan
the
is
:
where
1
1
1
th
n
n
th
n
n
n
n
n
n
A
i
B
n
B
A
iB
B
B




+
=
This equation is a first order, linear, and timeinvariant (LTI) difference equation.
Simple manipulation of the equation yields:
n
n
n
A
B
i
B

+
=

1
)
1
(
or
n
n
n
A
B
i
B

=
+


1
)
1
(
Specifying an initial condition,
B
0
, permits solution of the equation in a number of
ways.
For simple loan situations,
A
n
is often a constant as we might have in making equal
monthly payments on a loan.
Thus,
A
n
= A
.
(We do not have to impose this condition.
One might choose to make arbitrary payments on the loan if the lender permits it.)
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If the initial condition is based on an amount
P
borrowed at
n = 0
, the initial condition
is:
P
B
=
0
One solution strategy is to do an iterative solution (as we did on the first Excel exercise).
We might set up the iterative table as follows:
.
.
.
.
.
.
A
B
i
B
A
A
B
i
B
A
A
P
i
B
A
P
A
B
i
B
A
n
n
n
n
n

+
=

+
=

+
=

+
=

2
3
1
2
1
1
)
1
(
3
)
1
(
2
)
1
(
1
0
0
)
1
(
It is easy to include extra columns showing the amount of each payment going to
interest and the amount applied to reducing the loan balance (as we did in the first Excel
exercise).
The iterative solution is conceptually simple and easy to implement with spreadsheet
programs.
An analytical solution of the difference equation provides a “closed form” solution for
the balance
B
n
.
We might do this in a number of ways:
A classical approach:
solution as the sum of a forced plus natural
response.
Solution by Ztransform methods
Solution as the sum of a zeroinput solution and a zerostate solution
Other approaches
Starting with:
A
A
B
i
B
n
n
n

=

=
+


1
)
1
(
we will summarize briefly the classical approach.
The natural response will be of the form:
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This note was uploaded on 01/20/2010 for the course EE 366 taught by Professor Pore during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Pore

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