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 first year of the project. A definition 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.
§ g = constant rate of change, in decimal form, by which cash flow values increase or decrease from one period to the next. The gradient g can be + or -.
§ A1 = initial cash flow in year 1 of the geometric series
§ Pg = present worth of the entire geometric gradient series, including the initial amount A1
Note that the initial cash flow A1 is not considered separately when working with geometric gradients. Figure (2–12) 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 (2–12 a) by the P/F factor 1/ (1 + i) n .
Multiply both sides by (1 + g)/ (1 + i), subtract Equation (2-27) from the result, factor out Pg, and obtain
Solve for Pg and simplify.
The term in brackets in Equation (2-27) 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 (2-28) and observe that the term 1/ (1 + i) appears n times.
Figure (2–12): 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.
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.
A coal-fi red power plant has upgraded an emission control valve. The modification costs only $8000 and is expected to last 6 years with a $200 salvage value. The maintenance cost is expected to be high at $1700 the first year, increasing by 11% per year thereafter. Determine the equivalent present worth of the modification and maintenance cost by hand and by spreadsheet at 8% per year.
1) Solution by Hand
The cash flow diagram Figure (2–13) shows the salvage value as a positive cash flow and all costs as negative. Use Equation (2-35) for g ≠ i to calculate Pg. Total PT is the sum of three present worth components.
Figure (2–13): Cash flow diagram of a geometric gradient, Example (2-6).
2) Solution by Spreadsheet
Figure (2–14) details the spreadsheet operations to find the geometric gradient present worth Pg and total present worth PT. To obtain PT = $ - 17,999, three components are summed—first cost, present worth of estimated salvage in year 6, and Pg. Cell tags detail the relations for the second and third components; the first cost occurs at time 0 .
The relation that calculates the (P / A, g, i%, n) factor is rather complex, as shown in the cell tag and formula bar for C9. If this factor is used repeatedly, it is worthwhile using cell reference formatting so that A1, i, g, and n values can be changed and the correct value is always obtained. Try to write the relation for cell C9 in this format.
Figure (2–14): Geometric gradient and total present worth calculated via spreadsheet, Example (2-6).