What's important is that when this happens, when the replicating fixed rate account encounters a negative balance, we must specify what the cost of borrowing is, that is, we must specify the rate that should be charged on negative balances. The virtue of the FREQ lies in the fact that it describes a concrete, albeit somewhat idealized, equivalent alternative to the actual investment. Therefore, the ideal cost of borrowing to be used is the actual cost of borrowing that was in effect in the real world at each point in time when our balance is negative. We'll discuss the choice of the cost of borrowing some more when we compare the FREQ to the IRR. For now, all that matters is that in rare cases, negative balances on the replicating fixed rate account can occur, and that a cost of borrowing must be provided for the calculation of the FREQ in that case.
The IRR is usually defined in the context of capital budgeting. In that context, it is the discount rate that makes the net present value of all cash flows from a project equal to zero (see e.g. the definitions on Investopedia and Wikipedia). Now "discount rate," "net present value," and "project" are not exactly terms that individual investors use to reason about the performance of their accounts. That begs the question if the above definition of the IRR can be restated in an equivalent form that is more intuitive for investors who wish to measure the performance of their financial accounts. Such an equivalent definition does indeed exist. It turns out that as a performance measure for accounts with deposits and withdrawals, the IRR is very similar to the FREQ. As a matter of fact, the two definitions look identical on the surface. The difference is in the small print.
Starting with the formal mathematical version of the above definition of the IRR, a rather simple rearrangement of the defining equation (see here for details) reveals that the IRR, just like the FREQ, is the fixed annual interest rate that replicates the real account's performance. In other words, it is the annual interest rate of the fixed rate account that, when subjected to the same deposits and withdrawals as the real account, has the same ending balance as the real account. To see where the difference between the FREQ and the IRR lies, recall from the previous section that it is possible for the replicating fixed rate account to encounter negative balances despite the fact that the real account does not. In that case, the FREQ calculation charges the cost of borrowing as specified by the calculation inputs on negative balances. The IRR calculation, on the other hand, treats negative balances the same as positive balances: it charges the same rate on negative balances that it pays on positive ones.
Recall that both the FREQ and the IRR use the model of a fixed rate account to measure the performance of any financial account. This reflects the rather obvious observation that a fixed rate account with a higher annual interest rate is preferable to one with a lower interest rate. Also, based on experience with savings accounts, certificates of deposit, mortgages, and the like, most people have at least a modicum of intuition for annual interest rates. Knowing that an account's performance was equivalent to that of a savings account with a 1% or 3% or 5% annual interest rate gives investors an idea of what happened to their money. In view of all this, it is plausible to request that the modeling fixed rate account should resemble a real-life fixed rate investment as much as possible. In that respect, we believe that the FREQ is superior to the IRR, for the following reason. Recall that the only difference between the two is the way that they treat negative balances of the modeling fixed rate account. The FREQ charges an externally supplied rate on negative balances, while the IRR does not distinguish between positive and negative balances. The latter is not the way fixed rate accounts work in real life: the rate that a financial institution charges on a negative balance is substantially different from the one that it pays on a positive balance.