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9
The Math of the Fixed Rate Equivalent (FREQ)

Because of the fact that the replicating fixed rate account used by the FREQ rarely becomes negative, there is not much visible difference between the FREQ and the IRR: for almost all real-life investments, the IRR is unique and equals the FREQ. However, this does not mean that the FREQ provides little or no gain in practice. The IRR is typically calculated using numerical methods such as Newton’s method. The investor is then told that this is “the” IRR. However, there is no information as to whether this IRR value is unique, or, more importantly, whether the underlying replicating fixed rate investment did or did not encounter negative balances. The FREQ, on the other hand, always gives the guarantee of uniqueness, and, for that matter, of having used a realistic and intuitive model for the fixed rate replication. Loosely speaking, one could make the case for the FREQ like this: the FREQ is unlikely to change your IRR, but it is the only way of making sure that your IRR is good.

To illustrate the FREQ and give an intuition for why it does not suffer from the multiple solutions issue that plagues the IRR, let us return to the example of Table 1. Recall that in Chart 1, we have shown how and why there are two IRRs in that case. Now 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 investment under these assumptions, equals 10.41907%. In the chart below, we also show the account balances over time for accounts with a 12% interest rate and a 9% interest rate, respectively. This demonstrates that the “opposite movement” that lead to the multiple IRR values as shown in the Chart 1 cannot happen anymore when the FREQ is used: a higher interest rate will lead to a higher ending balance, and a lower interest rate will lead to a lower ending balance.

Chart 2

The table below shows the exact balances for the accounts of Chart 2 above.