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

The equivalent present worth   P  of a uniform series. A of end-of-period cash flows (investments)  is shown in Figure 2–4  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 ). 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).