### Uniform Series Present Worth Factor and Capital Recovery Factor ( P/A and A/P )

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 ). This results in Equation [2.7]. Now subtract the two equations, [2.6] from [2.7], and simplify to obtain the expression for P when i ≠ 0 (Equation [2.8])

The term in brackets in Equation [2.8] 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 extend-

ing for n periods. The cash flow diagram is Figure 2–4 a .

Figure 2–4 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 [2.8] 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.

The P/A and A/ P factors are derived with the present worth P and the first uniform annual amount A one year (period) apart. That is, the present worth P must always be located one period prior to the first A.

The factors and their use to find P and A are summarized in Table 2–2 and inside the front cover.

The standard notations for these two factors are ( P/A , i %, n ) and ( A /P , i %, n ). Tables at the end of

the text include the factor values. As an example, if i 15% and n 25 years, the P/A factor

value from Table 19 is ( P /A ,15%,25) 6.4641. This will find the equivalent present worth at 15% per year for any amount A that occurs uniformly from years 1 through 25.

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

Table 2–2 includes the PV and PMT functions.

**EXAMPLE 2.3**

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?

**Solution**

The cash flows follow the pattern of Figure 2–4 a , with A $600, i 16%, and n 9. The present worth is

P = 600( P/A ,16%,9) = 600(4.6065) = $2763.90

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

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