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Balance Difference Equations

# Balance Difference Equations - Balance Difference Equations...

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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 time-invariant (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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2 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 Z-transform methods Solution as the sum of a zero-input solution and a zero-state solution Other approaches Starting with: A A B i B n n n - = - = + - - 1 ) 1 ( we will summarize briefly the classical approach.
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