Measure the Performance of Your Investments
The FREQ Vs. the IRR (Internal Rate of Return)

The IRR's concept of the modeling fixed rate account becomes particularly problematic when the interest rate is negative. When a negative account balance is encountered in that case, the IRR calculation will apply the negative interest rate as the “cost” of borrowing, thereby assuming that the modeling account pays the account holder for being in debt. By contrast, the FREQ treats the cost of borrowing as an external quantity that is supplied to the calculation.

We believe that the issue of negative account balances as described above is sufficient reason to prefer the FREQ over the IRR. But it gets worse for the IRR. It is a well-known fact that the IRR is not always unique: there can be more than one IRR for one and the same account and the same begin and end date. This is a common objection to the use of the IRR in the world of individual investing: how can an advisor discuss the performance of an account with a client when, according to the IRR, that performance is 5% as well as 8%? How does one compare the performance of two accounts when the performance of one is 5% and 8%, and the performance of the other is 4% and 9%? In a pair of 1965 research articles by Daniel Teichroew, Alexander A. Robichek and Michael Montalbano in the journal “Management Science” (available here and here), it is shown that the IRR's treatment of negative account balances is the root cause of the multiple solutions problem, and that the FREQ's way of charging interest on negative balances does not suffer from that problem anymore. In short, the FREQ is always unique, the IRR is not.

The abovementioned research articles by Teichroew and coauthors were written in the context of capital budgeting and financing decisions. Alternate proofs, geared more towards application to measuring the performance of individual investors' accounts, can be found here.

To summarize, the FREQ provides two improvements over the IRR, the second one being a consequence of the first: the FREQ uses a more realistic model when replicating the account's performance with a fixed interest rate, and it no longer suffers from the multiple solutions issue that plagues the IRR. One could say that this is a case of two birds with one stone: by making the fixed rate model of the IRR more realistic, which is a good thing in and by itself, one also gets rid of the IRR's multiple solutions problem.

On the downside, the FREQ has the disadvantage that it requires an additional input, namely, the cost of borrowing (possibly varying over time) for the reporting period. In practice, that turns out to be much less of a burden than it would seem. What one does is to start with some default value for the cost of borrowing. In fact, that default could be anything at all; it does not even have to be realistic. Next, one calculates the FREQ for that cost of borrowing. For real-life individual investor accounts, it is extremely likely that the balance of the replicating fixed rate account never goes negative, so that the cost of borrowing never applies. Modifying the supplied value of the cost of borrowing does not change that. The unique FREQ has been found, and one need not worry about the cost of borrowing at all. This is almost always the case when measuring an individual investor's performance.

In those rare instances where the FREQ calculation reveals that the replicating fixed rate investment did become negative, that is, the cost of borrowing was applied at some point, things get a bit tricky for performance measurement. Suppose we go back and supply the real-life cost of borrowing. This does not have to be a single number; it may vary over time as is the case in real life. Using that cost of borrowing, we may then calculate a realistic FREQ: the annual interest rate of the fixed rate account that replicates the investment's performance and charges the actual cost of borrowing on negative account balances. The problem is that this information, the FREQ and the cost of borrowing, still