In view of our upcoming discussion of the FREQ vs. the IRR, we need to point out an aspect of the FREQ that is so subtle that you may not believe it exists until you have seen an example. Suppose we're looking at a brokerage account for a certain period of time, and assume that there are deposits and/or withdrawals along the way. Here's a very simple example, where there is just one transaction along the way, a $5,000 deposit in December of 2002:
Next, we calcuate the FREQ and find it to be 10.07%. This means that a fixed rate account that pays 10.07% interest annually, starts out with the same beginning balance as the brokerage account, and undergoes the same deposits and withdrawals will have the same ending balance as well. We can now superimpose the account value of the replicating fixed rate account on the chart above:
We see that the blue and green lines end at the same point. That's the defining property of the fixed rate equivalent: the ending balances are the same.
Now here's the thing that's a bit odd and that you may not believe until you've seen an example. It is possible to have a brokerage account, just like the one in the example above, just with a more complex combination of deposits and withdrawals, with the following property: the brokerage account itself has a positive account balance throughout, and yet, the replicating fixed rate account encounters negative balances. In other words, the green line in the chart above would go below zero despite the fact that the blue one does not. Examples of this behavior are very rare in the world of real-life individual investing, but they have been known to occur. If you'd like to see one, go to the appendix of this paper.