6

The Math of the Fixed Rate Equivalent (FREQ)
Theorem 1 Let be a rate that solves Equation (4) . Suppose that has the following property: the expressions

as defined in (5) are all non-negative, that is, the balance of the fixed rate account by which the IRR models the investment never becomes negative. Then is the only rate that solves the equation.

A proof of Theorem 1 can be found in the extended version of this paper. Let us emphasize again that it is not necessary to study our proof of Theorem 1 for mathematical verification. The core mathematics is already contained in the papers [1] and [2] by Teichroew et al.. Our presentation is merely structured differently, according to our purpose of adapting the results of Teichroew et al. to the problem of measuring the performance of investor accounts with deposits and withdrawals (money-weighted performance).

There is a subtle strength in Theorem 1 that is easy to overlook. A casual reader might misinterpret the statement of the theorem as follows: for a given investment, there can be at most one internal rate of return for which the balance of the replicating fixed rate account never becomes negative. However, the theorem is much stronger than that. It says that if there is an IRR for which the balance of the replicating fixed rate investment never dips into the red, then there can be no other IRR at all.

The converse of Theorem 1 is not true: if we have an investment for which the IRR is unique, then it is not necessarily true that the balance of the replicating fixed rate account is never negative. (See the extended version of this paper for an example.) Therefore, hunting for mathematical conditions that guarantee uniqueness of the IRR is not, in our opinion, the best way to resolve the multiple solutions issue. Instead, one should modify the model by which the fixed rate is calculated on positive and negative balances. (See Theorem 2 below.)

The mathematical proof of Theorem 1 is not trivial. However, the example of Table 1 can be used to gain an intuition for why and how a negative balance in the fixed rate account that replicates the investment creates the multiple solutions issue for the IRR. Recall that in this example, the IRR has the two values 5.5% and 8.66%. From Theorem 1, we already know that for neither one of the two rates, the balance of the replicating fixed rate account stays non-negative all the way. In both cases, there is compounding on a negative balance going on. Otherwise, there could not have been more than one solving rate.

Let us now look at the account balances over time in the fixed rate accounts for both rates. We’ve already calculated two of these balances in (7) and (8) above. Continuing in the same way, we obtain the following table of account balances. The balance on each date is the balance after the deposit or withdrawal on that date has been processed, with the exception of the last date, where we show the balance just before the $672.08 withdrawal that empties the account.

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

Account Balance at 5.5% | $5,000 | −$4,128.79 | $603.83 | $672.08 |

Account Balance at 8.66% | $5,000 | −$3,584.73 | $569.19 | $672.08 |