Two observations are in order concerning this example of multiple IRRs. First off, the example dashes any hopes that you might have that perhaps there will always be one “plausible” IRR and then one or more absurdly small or large ones that could be ignored. Not so. The two values of 5.5% and 8.66% of the example are both perfectly plausible for the given account.
Secondly, the IRR chart on the previous page provides some intuition for how the IRR's treatment of negative account balances causes the multiple solutions issue. Imagine that you start with the 5.5% graph, then increase the interest rate so as to morph it into the 8.66% graph. The positive account values during the first three years will increase faster, while the negative account values during the subsequent five years will decrease faster. That kind of “opposite movement” makes it possible for the ending balance to be the same for the two rates. The FREQ, by contrast, treats the cost of borrowing as an external quantity that is independent of the measurement of the account's performance. Therefore, the “opposite movement” described above will not occur.