© 2000 John Petroff

3)- Mutually exclusive projects with different lives

 

A peculiarity of equipment replacement projects analysis is that it is common to have several pieces of equipment available from different vendors. The lives of service of the various machines are not likely to be the same. A problem of unequal lives results from the fact that some machines would serve fewer years and require replacement sooner than others. A brief mention of how to deal with this situation was offered in Chapter 3, but a complete explanation was postponed until a full set of numbers could be used as an example, as we now have.

 

To make the alternatives comparable, several methods are possible:

a)- bringing the life to a common number of years with an assumption of as many replacements as needed,

 

b)- annualizing NPV obtained by dividing the NPV of each alternative by the number of years the machine will be in use. The annualized NPV is calculated as

 

ANVP = NPV / n

 

where n = length of project

 

c)- annuity equivalent NPV obtained by dividing the NPV of each alternative by the present value factor of an annuity, thus converting NPV into an annuity equivalent; its formula is

 

ANNUITY NPV = NPV / PVAF

 

where PVAF = present value annuity factor

 

d)- infinite life equivalent NPV obtained by dividing the annuity equivalent NPV by the discount rate (see annuity in perpetuity presented in Chapter 2); it is calculated with

 

INFINITE NPV = ANNUITY NPV / k

 

where k = discount rate assigned to project

 

e)- calculating infinite life equivalent NPV on a common number of years.

The last method is the most robust (i.e. least likely to leading to an error), but it is also the most involved and complex. It also requires the assumption of replacement several times in the future with exactly the same equipment, which is unlikely to be applicable in reality, but is arithmetically logical. The second method (dividing NPV by the number of years) is the simplest and is generally recommended, but it is not as robust.

The conflict between the choice of trucks 1, 2 and 3, gives a good opportunity to investigate how to deal with this type of decision. In Table T-10.6 we note that truck 3 has the highest NPV, but the lowest IRR and PI. Using NPV rather than any other statistic is usually recommended and will be advocated in subsequent paragraphs. Choosing truck 3 on the basis of the data in this case would not be correct, as it will shown below.

To conduct a full comparison of the different methods Table T-10.7 below presents several useful statistics in addition to IRR and NPV; they are

- ANVP: annualized NPV (as stated above, it is calculated by dividing NPV by the number of years of the project)

- ANNUITY NPV: annuity equivalent of NPV (see above)

- INFINITE NPV: infinite-life equivalent NPV (see above)

On the basis of Table T-10.7, it is clear that the right choice is truck 2, because it has the highest annuity equivalent NPV of $860, compared to $670 for truck 1 and $660 for truck 3. Most importantly, truck 2 has the highest infinite-life NPV of $8,600, compared to $5,580 for truck 1 and $6,600 for truck 3 . One may note that the same conclusion is reached using the annualized NVP. But, the higher NPV of truck 3 is still disturbing, so is the high IRR for truck 1.

Next we calculate the same statistics using a common number of years rather than the different lives. The common number of years is 48; this means that truck 3 will be replaced four times, truck 1 six times and truck 2 eight times. The results are presented in Table T-10.8 below.

On the basis of Table T-10.8 the selection of truck 2 is confirmed. This time all the statistics support the same choice, except IRR. Discussion of why IRR may not be the proper selection criterion is taken up in paragraph Subsection E-5-Comparison below. The lesson from Table T-10.8 is that truck 2 has the highest NPV when a common number of years is used, no matter how that NPV is restated: total, annualized, annuity equivalent or infinite life. Note that annuity equivalent NPV of $860 for truck 2 and infinite-life equivalent of $8,600 remain the same for 48 year as they were for six years; and so is it true for the other alternatives. This means that these statistics are the most robust. Also observe that the annualized NPV if different in this calculation compared to Table T-10.7 but it still gives the correct solution.

See review questions Q-10E3.1 through Q-10E3.10.

See research assignment R-10E3.1.

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