5

The Math of the Fixed Rate Equivalent (FREQ)
Let us consider an example. Suppose the following activities occur in an investor’s account:

Date | 1/1/1992 | 1/1/1995 | 1/1/2000 | 1/1/2002 |

Cash Flow | $5,000 | −$10,000 | $6,000 | −$672.08 |

Recall that the $5,000 on 1/1/1992 stand for the initial deposit, and the final withdrawal of $672.08 empties the account, meaning that $672.08 was the ending balance on 1/1/2002. The scenario of Table 1 is a rather realistic one for an individual investor. 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.

If you enter the numbers of Table 1 above into your favorite software for calculating the IRR, then, depending on which software you use and what initial guess you enter (if any), the answer could be 5.5% or 8.66%. We are looking at a case where there is more than one IRR.

As we have mentioned before, the root cause of the IRR’s multiple solutions issue is the fact that the IRR uses a model where the fixed rate account charges the same rate regardless of whether the balance is positive or negative. The point of the present example is to illustrate that phenomenon. Here’s the equivalent of Equation (4) for our present example:

First of all, one can use this to verify the two IRRs of 5.5% and 8.66%. (In fact, one must use a little more precision to get the end value right to the penny; use 8.663 instead of 8.66.) Recall that the left hand side of Equation (6) describes how the account balance of a savings account with a fixed annual interest rate of develops over time if we make the deposits and withdrawals of Table 1 above. Now let us use the equivalent of expressions (5) to follow the balance of that fixed rate account over time. We’ll do this for the 5.5% interest rate. For the first three years, we have $5,000 compounding at 5.5%, followed by a $10,000 withdrawal. The hypothetical account balance on Jan 1, 1995 is thus

For the next five years, we are looking at −$4128.79 compounding at 5.5%, followed by a $6,000 deposit into the account. Therefore, the calculation of the account balance for Jan 1, 2000 looks like this:

We see that as the IRR tries to model the actual investment as an account with a fixed rate, the balance of that fixed rate account drops into the negative. The model used by the IRR then applies ordinary compounding to that negative balance. That amounts to charging the same interest rate on a debt that is paid on a positive balance. Note that in the case of a negative IRR, this actually amounts to paying an investor for being in debt. The following theorem is a mathematically precise statement of the fact that the use of this particular fixed rate model is the root cause of the IRR's multiple solutions issue.