Fortunately, there is a second option for dealing with the case where the replicating fixed rate account encounters negative balances. If you look at the example given in the appendix of this paper, you’ll see that the negative balance of the replicating fixed rate account is caused by the fact that a period of very good performance was followed by a large withdrawal, which was then followed by a period of poor performance. This kind of extreme behavior appears to be typical of investments where the replicating fixed rate account goes negative. Therefore, one may, if it is deemed acceptable, try to split the analysis period up into two or more subperiods of more homogeneous performance. There is a good chance that this will once again lead to FREQ calculations that do not use the cost of borrowing.
In this appendix, we give an example of a brokerage account where the replicating fixed rate account as used by the FREQ and the IRR encounters negative balances despite the fact that the original brokerage account did not. We then demonstrate how the FREQ and the IRR deal differently with this situation.
Consider a brokerage account with the following beginning and ending balance and deposits and withdrawals in between:
The scenario of Table 1 above is not entirely unrealistic. Someone could have opened a brokerage account in 1992 with $5,000, more than doubled the account balance during the bull market of the 1990’s, and then taken $10,000 of gains off the table in 1995. After another fairly large deposit in January of 2000, the investor ended up losing almost everything in the crash of the spring of 2000. Note that in this scenario, the investor's account balance remained positive at all times.
For the FREQ calculation, let us assume a cost of borrowing of 20% for the period 1/1/1995 through 12/31/1996, and 5% for the period 1/1/1997 through 12/31/1999. We chose these somewhat extreme values for the sake of better visualization. The FREQ, that is, the interest rate that replicates the account's performance under these assumptions, equals 10.41907%.
The blue line in the chart below shows the account value over time of a fixed rate account which