Sinking Fund Factor and Uniform Series Compound Amount Factor ( A/F and F/A )

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 [2.9], the following formula results.

The expression in brackets in Equation [2.12] 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 2–6  a , where. A  is a uniform annual investment.

The uniform series   A  begins at the end of  year (period) 1 and continues through the  year of  the given     F.  The last. A value and F occur at the same time.

Equation [2.12] can be rearranged to fi  nd   F  for a stated   A  series in periods 1 through   n   (Figure 2–6  b ).

The term in brackets is called the   uniform series compound amount factor  (USCAF), or   F/A   factor.
When multiplied by the given uniform annual amount   A , it yields the   future worth of the uniform
series.  It is important to remember that the future amount   F  occurs in the same period as the last   A .

Standard notation follows the same form as that of other factors. They are (  F /A ,  i ,  n )  and
(  A/F ,  i ,  n ). Table 2–3 summarizes the notations and equations, as does the inside front cover.

As a matter of interest, the uniform series factors can be symbolically determined by using an  abbreviated factor form. For example,   F/A  = (  F/P )(  P/A ), where cancellation of the  P  is  correct.
Using the factor formulas, we have

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 deter-
mines 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 2–6 Cash  flow diagrams to (  a )  find   A,  given   F , and (  b )  find   F,  given   A .

EXAMPLE 2.5

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. 

Solution

The cash flow diagram (Figure 2–7) 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.

Figure 2–7 Diagram to find   F  for a uniform series, Example 2.5.