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option. Thus, these comparisons may be considered as incremental investments. In order to use INPV, it is
                    necessary to know the internal discount rate and the time over which the comparison is to be made. This
                    method is illustrated in Example 10.13.


                    Example 10.13



                    Based on the information provided in Example 10.10 for an acceptable internal interest rate of 15% and
                    time n = 5 yrs, determine the most attractive alternative, using the INPV criterion to compare options.
                          For i = 0.15 and n = 5 the value for (P/A, i, n) = 3.352. (See Equation 9.14.)
                          Equation (10.4) becomes


                                                                  NPV = – PC + 3.352 YS
















                    From the results above, it is clear that Options 2, 3, and 4 are all potentially profitable as INPV > 0.
                    However, the best option is Option 2 because it has the highest INPV when compared with the do-nothing
                    case, Option 1. Note that other pairwise comparisons are unnecessary. We simply choose the option that
                    yields the highest INPV. The reason that the INPV gives the best option directly is because by knowing i
                    and n, each dollar of incremental investment is correctly accounted for in the calculation of INPV. Thus, if
                    the  incremental  investment  in  going  from  Option A  to  Option  B  is  profitable,  then  the INPV  will  be
                    greater for Option B and vice versa. It should also be pointed out that by using discounting techniques, the
                    best option has changed from Option 3 (in Example 10.11) to Option 2.


                    Operating  Cost  Methods.      In  the  previous  section,  yearly  savings  were  converted  to  an  equivalent
                    present value using the present value of an annuity, and this was measured against the capital cost. An
                    alternative method is to convert all the investments to annual costs using the capital recovery factor and
                    measure them against the yearly savings.


                    We develop the needed relationship from Equation (10.4), giving


                                                           INPV/(P/A,i,n) = – PC/(P/A,i,n) + YS


                    It can be seen that the capital recovery factor (A/P, i, n) is the reciprocal of the present worth factor (P/A,
                    i, n). Substituting this relationship and multiplying by –1 gives


                                                          – (INPV)(A/P,i,n) = (PC)(A/P,i,n) – YS


                    The term on the left is identified as the Equivalent Annual Operating Cost (EAOC). Thus, we may write

                    (10.5)
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