Geometric Gradient Series Factors

It is common for annual revenues and annual costs such as maintenance, operations, and labor to go up or down by a constant percentage, for example, +5% or -3% per year. This change occurs every year on top of a starting amount in the fi rst year of the project. A defi nition and description of new terms follow.

A   geometric  gradient  series is a cash flow series that either increases or decreases by a  constant  percentage  each period. The uniform change is called the  rate of change

Note that the initial cash flow   A1   is not considered separately when working with geometric  gradients.

Figure 2–21 shows increasing and decreasing geometric gradients starting at an amount   A1   in  time period 1 with present worth   Pg    located at time 0. The relation to determine the total present  worth   Pg  for the entire cash flow series may be derived by multiplying each cash flow in Figure 2–21  a  by the

The term in brackets in Equation [2.32] is the  (  P/A , g , i , n )  or geometric gradient series present  worth factor for values of   g  not equal to the interest rate   i.  When    g =  i , substitute   i  for   g  in Equation [2.31] and observe that the term 1/(1 + i) appears n times.

Figure 2–21
Cash flow diagram of (a) increasing and (b) decreasing geometric gradient series and present worth Pg.

 

The (P A,g,i,n) factor calculates Pg in period t = 0 for a geometric gradient series starting in period 1 in the amount A1  and increasing by a constant rate of g each period.

 The equation for   Pg and the (  P/A ,  g ,  i ,  n ) factor formula are

 

 It is possible to derive factors for the equivalent   A   and   F  values; however, it is easier to determine
the   Pg    amount and then multiply by the   A/P  or   F/P   factor. 

As with the arithmetic gradient series, there are no direct spreadsheet functions for geometric gradient series. Once the cash flows are entered,   P  and   A  are determined using the NPV and PMT  functions, respectively.