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

§  P_g = 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 P_g located at time 0. The relation to determine the total present worth P_g 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 P_g, and obtain

Solve for P_g 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 P_g.

The equation for P_g 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 P_g 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 P_g. Total P_T 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 P_g and total present worth P_T. To obtain P_T = $-17,999, three components are summed—first cost, present worth of estimated salvage in year 6, and P_g. 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).