### Factor Values for Untabulated i or n Values

not listed in the compound interest tables in the rear of the book. Given specific values of i and n , there are several ways to obtain any factor value.

• Use the formula listed in this chapter or the front cover of the book,

• Use an Excel function with the corresponding P , F , or A value set to 1.

• Use linear interpolation in the interest tables.

When the formula is applied, the factor value is accurate since the specific i and n values are input. However, it is possible to make mistakes since the formulas are similar to each other, es-

pecially when uniform series are involved. Additionally, the formulas become more complex

when gradients are introduced, as you will see in the following sections.

A spreadsheet function determines the factor value if the corresponding P , A , or F argument in the function is set to 1 and the other parameters are omitted or set to zero. For example, the P F factor is determined using the PV function with A omitted (or set to 0) and F = 1, that is, PV( i %, n ,0,1) or PV( i %, n ,0,1). A minus sign preceding the function identifier causes the factor to have a positive value. Functions to determine the six common factors are as follows.

Figure 2–9 shows a spreadsheet developed explicitly to determine these factor values. When it is made live in Excel, entering any combination of i and n displays the exact value for all six factors. The values for i = 3.25% and n = 25 years are shown here. As we already know, these same functions will determine a ﬁ nal P , A , or F value when actual or estimated cash ﬂ ow amounts are entered.

**Linear interpolation**for an untabulated interest rate i or number of years n takes more time to complete than using the formula or spreadsheet function. Also interpolation introduces some level of inaccuracy, depending upon the distance between the two boundary values selected for i or n , as the formulas themselves are nonlinear functions. Interpolation is included here for indi- viduals who wish to utilize it in solving problems. Refer to Figure 2–10 for a graphical description of the following explanation. First, select two tabulated values ( x1 and x2 ) of the parameter for which the factor is requested, that is, i or n , ensuring that the two values surround and are not too distant from the required value x . Second, ﬁnd the corresponding tabulated factor values ( f1 and f2 ). Third, solve for the unknown, linearly interpolated value f using the formulas below, where the differences in parentheses are indicated in

**Figure 2–10**as a through c.

The value of d will be positive or negative if the factor is increasing or decreasing, respectively, in value between x1 and x2.

Determine the P /A factor value for i = 7.75% and n 10 years, using the three methods described previously.

**Solution**

Factor formula: Apply the formula from inside the front cover of the book for the P A factor. Showing 5-decimal accuracy,

**Comment**

Note that since the P/A factor value decreases as i increases, the linear adjustment is negative at 0.2351. As is apparent, linear interpolation provides an approximation to the correct factor value for 7.75% and 10 years, plus it takes more calculations than using the formula or spread- sheet function. It is possible to perform two-way linear interpolation for untabulated i and n values; however, the use of a spreadsheet or factor formula is recommended.

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