Uniform Series Formulas (P/A, A/P, A/F, F/A)

The equivalent present worth P of a uniform series A of end-of-period cash flows (investments) is shown in Figure (2–2 a). An expression for the present worth can be determined by considering each A value as a future worth F, calculating its present worth with the P/F factor, Equation (2-3), and summing the results.

The terms in brackets are the P/F factors for years 1 through n, respectively. Factor out A.

To simplify Equation (2-6) and obtain the P/A factor, multiply the n -term geometric progression in brackets by the (P/F, i %,1) factor, which is 1/(1 +i). Simplify to obtain the expression for P when i ≠ 0 (Equation (2-7)).

The term in brackets in Equation (2-7) is the conversion factor referred to as the uniform series present worth factor (USPWF). It is the P/A factor used to calculate the equivalent P value in year 0 for a uniform end-of-period series of A values beginning at the end of period 1 and extending for n periods. The cash flow diagram is Figure (2–2 a).

Figure (2-2): Cash flow diagrams used to determine: (a) P, given a uniform series A, and (b) A, given a present worth P

To reverse the situation, the present worth P is known and the equivalent uniform series amount A is sought (Figure (2–4 b)). The first A value occurs at the end of period 1, that is, one period after P occurs. Solve Equation (2-7) for A to obtain

The term in brackets is called the capital recovery factor (CRF), or A/P factor. It calculates the equivalent uniform annual worth A over n years for a given P in year 0, when the interest rate is i.

Spreadsheet functions can determine both P and A values in lieu of applying the P/A and A/P factors. The PV function calculates the P value for a given A over n years and a separate F value in year n, if it is given. The format, is

Similarly, the A value is determined by using the PMT function for a given P value in year 0 and a separate F, if given. The format is

The simplest way to derive the A/ F factor is to substitute into factors already developed. If P from Equation (2-3) is substituted into Equation (2-8), the following formula results.

The expression in brackets in Equation (2-11) is the A/F or sinking fund factor. It determines the uniform annual series A that is equivalent to a given future amount F. This is shown graphically in Figure (2–3 a), where A is a uniform annual investment.

Equation (2-11) can be rearranged to find F for a stated A series in periods 1 through n (Figure 2–3 b).

For solution by spreadsheet, the FV function calculates F for a stated A series over n years. The format is

The P may be omitted when no separate present worth value is given. The PMT function determines the A value for n years, given F in year n and possibly a separate P value in year 0. The format is

If P is omitted, the comma must be entered so the function knows the last entry is an F value.

Figure (2-3): Cash flow diagrams to (a) find A, given F, and (b) find F, given A

Table (2–2) summarizes the standard notation and equations for the A/P, P/A, F/A and A/F factors.

Table (2-2): P/A, A/P, A/F and F/A Factors: Notation and Equations

How much money should you be willing to pay now for a guaranteed $600 per year for 9 years starting next year, at a rate of return of 16% per year?

The cash flows follow the pattern of Figure 2–2 a , with A = $600, i = 16%, and n = 9. The present worth is

The PV function = PV(16%,9,600) entered into a single spreadsheet cell will display the answer P = ($2763.93).

The president of Ford Motor Company wants to know the equivalent future worth of a $1 million capital investment each year for 8 years, starting 1 year from now. Ford capital earns at a rate of 14% per year.

The cash flow diagram (Figure (2–4)) shows the annual investments starting at the end of year 1 and ending in the year the future worth is desired. In $1000 units, the F value in year 8 is found by using the F/A factor.

F= 1000( F/A, 14%,8) = 1000(13.2328) = $13,232.80

Figure (2–4): Diagram to find F for a uniform series, Example (2-3).