Theorem 2 Suppose we are given a series of cash flows as in Equation (1) , and the first cash flow is positive, while the last one is negative. Assume further that for each period of time between two cash flows, we know what the cost of borrowing is, i.e., the interest rate charged on an account with a negative balance. Then there is exactly one rate with the following property: the fixed rate investment whose interest rate on a positive balance equals and whose rate charged on a negative balance is the given cost of borrowing replicates the given investment. Moreover, there is an algorithm that calculates the rate from the given cash flows and the cost of borrowing.
Definition 3 We call the rate described above the fixed rate equivalent, or FREQ for short, for the given investment and cost of borrowing.
A proof of Theorem 2 can be found in the extended version of this paper. The assumptions about the first and last cash flow will be replaced with a more refined hypothesis. As with Theorem 1, it is not necessary to study our proof of Theorem 2 for mathematical verification. We are merely giving a differently structured presentation of the results of Teichroew et al. ( and ), geared toward the purpose of measuring the performance of investor accounts with deposits and withdrawals (money-weighted performance).
At first glance, it appears as if the calculation of the FREQ would be much harder than that of the IRR because of the need to provide the cost of borrowing. In practice, this turns out to be less of a burden than it would seem. What one should do is to start with some rough estimate of the cost of borrowing. In fact, that estimate could be anything at all; accuracy does not matter. Next, one calculates the FREQ for that cost of borrowing. For real-life investments, it is quite likely that the balance of the replicating fixed rate investment never goes negative, so that the cost of borrowing never applies. That means it would not have applied for any other estimate either: the cost of borrowing does not matter for this investment. The unique FREQ has been found, and by Theorem 2, the FREQ then also equals the unique IRR.
If, on the other hand, 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 subtle for performance measurement. Suppose we go back and supply the real-life cost of borrowing. (Recall that the cost of borrowing used may vary over time as is the case in real life.) With 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 does not capture the performance very well: much depends on how much time the replicating fixed rate account spent in positive and negative territory, respectively, and what the magnitude of the positive and negative balances were. One may well argue that the best one can do in this situation is to look at a chart of the account balance over time of the fixed rate account and thereby get a visual impression of the performance (see the chart below for an example). However, it is important to note that one may still use the FREQ for comparing the performance of investments with the same deposits and withdrawals: a higher FREQ is always better than a lower FREQ, assuming that the same cost of borrowing is used.
There is a second option for dealing with the case where the replicating fixed rate account encounters negative balances. In the example that we've been using throughout, 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 is 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.