Factor Values for Untabulated i or n Values

  Often it is necessary to know the correct numerical value of a factor with an   i or n  value that is
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.