Arithmetic Gradient Factors (P/ G and A/ G)

Assume a manufacturing engineer predicts that the cost of maintaining a robot will increase by  $5000 per year until the machine is retired. The cash flow series of maintenance costs involves a  constant gradient, which is $5000 per year.

An arithmetic gradient series is a cash flow series that either increases or decreases by a constant amount each period. The amount of change is called the gradient.

Formulas previously developed for an   A  series have year-end amounts of equal value. In the case of a gradient, each year-end cash flow is different, so new formulas must be derived. First,  assume that the cash flow at the end of year 1 is the   base amount  of the cash flow series and, therefore, not part of the gradient series. This is convenient because in actual applications, the base amount is usually significantly different in size compared to the gradient. For example, if you purchase a used car with a 1-year warranty, you might expect to pay the gasoline and insurance costs during the first year of operation. Assume these cost $2500; that is, $2500 is the base amount. After the first year, you absorb the cost of repairs, which can be expected to increase each year. If you estimate that total costs will increase by $200 each year, the amount the second  year is $2700, the third $2900, and so on to year   n , when the total cost is 2500 (  n -  1)200.  The  cash flow diagram is shown in Figure 2–11. Note that the gradient ($200) is first observed between year 1 and year 2, and the base amount ($2500 in year 1) is not equal to the gradient. Define the symbols  G  for gradient and  CFn  for cash flow in year  n  as follows.

Figure 2–11 Cash flow diagram of an
arithmetic gradient series.

G constant arithmetic change in cash flows from one time period to the next; G may be positive or negative.

It is important to realize that the base amount defines a uniform cash flow series of the size A that occurs eash time period. We will use this fact when calculating equivalent amounts that involve arithmetic gradients.   If the base amount is ignored, a generalized arithmetic (increasing) gradient cash flow diagram is as shown in Figure 2–12. Note that the gradient begins between years 1 and 2. This is called a   conventional gradient .

Figure 2–12 Conventional arithmetic
gradient series without
the base amount.

The    total present worth PT    for a series that includes a base amount   A  and conventional arithmetic gradient must consider the present worth of both the uniform series defi ned by   A  and the rithmetic gradient series. The addition of the two results in   PT.

where    PA is the present worth of the uniform series only,   PG   is the present worth of the gradient  series only, and the + or - sign is used for an increasing (   +G ) or decreasing (   -G )  gradient,  respectively.
 
The corresponding equivalent annual worth   AT    is the sum of the base amount series annual worth   AA    and gradient series annual worth   AG   , that is,

Three factors are derived for arithmetic gradients: the   P/G   factor for present worth, the   A/G
factor for annual series, and the   F/G  factor for future worth. There are several ways to derive
them. We use the single-payment present worth factor (  P/F ,  i ,  n ), but the same result can be ob-
tained by using the   F/P ,    F/A , or   P/A   factor.

In Figure 2–12, the present worth at year 0 of only the gradient is equal to the sum of the present worths of the individual cash fl ows, where each value is considered a future amount.

The left bracketed expression is the same as that contained in Equation [2.6], where the P/A  factor was derived. Substitute the closed-end form of the   P/A  factor from  Equation [2.8]

Figure 2–14
Conversion diagram from an arithmetic gradient to a present worth.

into Equation [2.23] and simplify to solve for P/G,  the present worth of the gradient series only.

Equation [2.24] is the general relation to   convert an arithmetic gradient     G   (not including the base amount) for    n   years into a present worth at year 0 . Figure 2–14  a  is converted into the equivalent cash flow in Figure 2–14  b .  The    arithmetic gradient present worth factor,  or   P/G  factor,  may be expressed in two forms:

 Equation [2.24] expressed as an engineering economy relation is

which is the rightmost term in Equation [2.19] to calculate total present worth. The G carries a
minus sign for decreasing gradients.

The  equivalent  uniform  annual  series    A/G    for an arithmetic gradient   G  is found by multiplying  the present worth in Equation [2.26] by the (  A/P ,  i ,  n ) formula. In standard notation form, the  equivalent of algebraic cancellation of   P  can be used.

 In equation form,

which is the rightmost term in Equation [2.20]. The expression in brackets in Equation [2.27] is  called the   arithmetic gradient uniform series factor   and is identifi  ed by ( A/G , i , n  ). This factor  converts Figure 2–15  a  into Figure 2–15  b .

The   P/G  and   A/G  factors and relations are summarized inside the front cover. Factor values
are tabulated in the two rightmost columns of factor values at the rear of this text.

Figure 2–15
Conversion diagram of an arithmetic gradient series to an equivalent uniform annual series.

There is no direct, single-cell spreadsheet function to calculate   PG    or   AG    for an arithmetic  gradient. Use the NPV function to display   PG    and the PMT function to display   AG    after entering  all cash fl ows (base and gradient amounts) into contiguous cells. General formats for these functions are

 The word entries in italic are cell references, not the actual numerical values. (See Appendix A,
Section A.2, for a description of cell reference formatting.)

An   F/ G   factor (  arithmetic gradient future worth factor ) to calculate the future worth   F/G    of a  gradient series can be derived by multiplying the   P/G  and   F/P  factors. The resulting factor,  (  F/G ,  i ,  n ), in brackets, and engineering economy relation is