The short answer is that Net Present Value (NPV) is more reliable than Internal Rate of Return (IRR) and Payback for ranking the financial attractiveness of choices.
IRR and Payback may be very popular and point to the best solution MOST of the time, but in some circumstances mislead us. I'll give just two weaknesses of each:
One of IRR's errors is favoring a high return project that has a small impact on the company. As an extreme example, a company creates more value by pursuing a 20% return on a $10 million investment over a 40% return on a $1 thousand investment. But using pure IRR, the 40% option seems like a much better choice. It's obvious when the dollar amounts are so different, but less obvious when the values are closer together. NPV avoids this problem by providing the magnitude of value created for the company.
A second error of IRR occurs when one option has a large, early cash inflow. The IRR calculation assumes this is all reinvested at the same high rate of return in other projects. For example, are looking at various parcels of land to build a warehouse you plan to use for 10 years. One of your options shows a 25% return where you purchase land for $5 million and quickly sell off a portion you don't need for $2 million. A quirk of IRR assumes that you are able to reinvest that $2 million over the remaining 10 years of your warehouse model analysis -- a wonderful thought but rarely the reality. NPV avoids this because it makes no assumption of how that $2 million is reinvested.
Payback's downfalls are that it does not provide the magnitude of value each option creates (like the first IRR error), nor does it take into account the life beyond the payback period. For example, one IT solution may have a payback of 1 years, but has to be replaced at the end of the second year because the vendor will not support it any longer. Another option may have a 2 year payback but you can expect benefits to continue for 5 years.
In Payback's defense, when capital is extremely scarce you may need to sacrifice the longer-term value of the ideal solution to get the cash back quickly.
NPV isn't perfect (and trickier than that old Finance 101 course led us to believe) but it is more reliable than IRR and Payback for ranking options. If your folks are entrenched in these other two methods and need additional evidence I'd be happy to discuss further.
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