## Cash Flow Series with a Special Pattern

Whenever one can identify patterns in cash flow transactions, one may use them in developing concise expressions for computing either the present or future worth of the series. For this purpose, we will classify cash flow transactions into three categories: (1) equal cash flow series, (2) linear gradient series, and (3) geometric gradient series. To simplify the description of various interest formulas, we will use the following notation:

1. Uniform Series: Probably the most familiar category includes transactions arranged as a series of equal cash flows at regular intervals, known as an equal-payment series (or uniform series) (Figure 17.2.3a). This describes the cash flows, for example, of the common installment loan contract, which arranges for the repayment of a loan in equal periodic installments. The equal cash flow formulas deal with the equivalence relations of P, F, and A, the constant amount of the cash flows in the series.

2. Linear Gradient Series: While many transactions involve series of cash flows, the amounts are not always uniform: yet they may vary in some regular way. One common pattern of variation occurs when each cash flow in a series increases (or decreases) by a fixed amount (Figure 17.2.3b). A 5-year loan repayment plan might specify, for example, a series of annual payments that

FIGURE 17.2.2 Decomposition of uneven cash flow series into three single-payment transactions. This decomposition allows us to use the single-payment present worth factor.

increased by \$50 each year. We call such a cash flow pattern a linear gradient series because its cash flow diagram produces an ascending (or descending) straight line. In addition to P, F, and A, the formulas used in such problems involve the constant amount, G, of the change in each cash flow.

3. Geometric Gradient Series: Another kind of gradient series is formed when the series in cash flow is determined, not by some fixed amount like \$50, but by some fixed rate, expressed as a percentage. For example, in a 5-year financial plan for a project, the cost of a particular raw material might be budgeted to increase at a rate of 4% per year. The curving gradient in the diagram of such a series suggests its name: a geometric gradient series (Figure 17.2.3c). In the formulas dealing with such series, the rate of change is represented by a lowercase g.

Table 17.2.1 summarizes the interest formulas and the cash flow situations in which they should be used. For example, the factor notation (F/A, i, N) represents the situation where you want to calculate the equivalent lump-sum future worth (F) for a given uniform payment series (A) over N period at interest rate i. Note that these interest formulas are applicable only when the interest (compounding) period is the same as the payment period. Also in this table we present some useful interest factor relationships. The next two examples illustrate how one might use these interest factors to determine the equivalent cash flow.

(a) Equal (uniform) payment series at regular Intervals

(b) Linear gradient series where each cash flow In a series increases or decreases by a fixed amount, G.

(c) Geometric gradient series where each cash flow in a series increases or decreases by a fixed rate (percentage), g.

FIGURE 17.2.3 Five types of cash flows: (a) equal (uniform) payment series; (b) linear gradient series; and (c) geometric gradient series.