Internal Rate of Return (irr): Sometimes "Wrong"

    In this decisionmaking mechanism, the stream of returns (discounted at a rate to be solved for) are set equal to zero to determine (by solving a polynomial equation) what rate of return a project is returning--if ra > rb , this criteria says take A, the project yielding the higher internal rate of return.  This approach is commonly taught in business schools, despite some disadvantages, in addition to being more complicated to solve.

    IRR =  (PVb - PVc)/(1 + k) = 0, solve for k  (if k is greater than the opportunity cost, r, proceed with the project)

Example:                 Year 0              Year 1            irr
Project A               -$100              +$100               -0-
Project B               -$100              +$130               30%

Problems:
    First, there is a problem with multiple roots--consider a project with the following pattern of costs and benefits:

                           Year 0              Year 1               Year 2
Project              -$100            +$300/(1.0)1        -$200/(1.0)2      =   0    suggests that irr = 0
but,                   -$100            +$300/(2.0)1        -$200/(2.0)2      =   0    suggests that irr = 100%!
(In complex cases of many sign switches, it can be difficult to know "the" answer--and, of course, the calculations are tedious without a computer).
    Second, this approach shares the problem that B/C had for projects that are strict alternatives but of different size--the higher B/C ratio will have a higher internal rate of return, regardless of its NPV.  (See earlier example, or the simpler one here)
                                 Year 0          Year 1      Year 2      Year 3      Year 4    etc.
Project A                 -$1,000       +$300       +$300      +$300       +$300      ...
Project B                 -$5,000       +$1,000    +$1,000   +$1,000    +$1,000   ...

The internal return of Project A is 30% (ka = $300/$1,000) while the internal rate of return on Project B is only 20% (kb = $1,000/$5,000).  However, if we calculate the net present value of Project A using a 10% discount rate, we find that it is equal to $2,000 while the net present value of Project B is $5,000.  [This example presupposes, anticipating questions, that the difference in cost between Project A and Project B, $4,000, would either be invested in a project or the capital market and would yield a NPV of zero (as necessary if the discount is in fact chosen to reflect the opportunity cost of capital].  There are further problems with this approach (for strict alternatives with different lengths of life or for different timings of the same project, inherently strict alternatives--but, we don't need to go into those problems here!)