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

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. |

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