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2.2 Amortization
In this application of difference equations, we examine simple problems of
finance that are of major importance to every engineer, on both the personal
and professional levels. When the purchase of any capital equipment or real
estate is made on credit, the assumed debt is normally paid for by means of
a process known as amortization. Under this plan, a debt is repaid in a
sequence of periodic payments where a portion of each payment reduces the
outstanding principal, while the remaining portion is for interest on the loan.
Suppose that the original debt to be paid is C and that interest charges are
compounded at the rate r per payment period. Let y(k) be the outstanding
th
th
principal after the k payment, and u(k) the amount of the k payment.
After the k payment period, the outstanding debt increased by the inter-
th
est due on the previous principal y(k – 1), and decreased by the amount of
payment u(k), this relation can be written in the following difference equa-
tion form:
y(k) = (1 + r) y(k –1) – u(k) (2.2)

We can simplify the problem and assume here that the bank wants its
money back in equal amounts over N periods (this can be in days, weeks,
months, or years; note, however, that whatever unit is used here should be
the same as used for the assignment of the value of the interest rate r). There-
fore, let

u(k) = p for k = 1, 2, 3, …, N (2.3)
Now, using Eq. (2.2), let us iterate the first few terms of the difference
equation:

y(1) = (1 + r)y(0) – p = (1 + r)C – p (2.4)

Since C is the original capital borrowed;

At k = 2, using Eq. (2.2) and Eq. (2.4), we obtain:

y(2) = (1 + r)y(1) – p = (1 + r) C – p(1 + r) – p (2.5)
2
At k = 3, using Eq. (2.2), (2.4), and (2.5), we obtain:

3
2
y(3) = (1 + r)y(2) – p = (1 + r) C – p(1 + r) – p(1 + r) – p (2.6)
etc. …
and for an arbitrary k, we can write, by induction, the general expression:

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