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The Math of the Fixed Rate Equivalent (FREQ)

In plain language, the definition above says,“The internal rate of return is the rate that makes the sum of the net present values of the cash flows equal to zero, where ‘present’ refers to the time of the first cash flow.

As we will explain in more detail below, one can rearrange Equation (1) in such a way that it reveals the following alternate interpretation of the internal rate of return: “The IRR is the interest rate of the hypothetical fixed rate investment which, when subjected to the same cash flows as the real investment, results in the same gain or loss as the real investment.” For ease of terminology, we will also use the term replication to describe this situation: the IRR of an investment is the fixed interest rate that replicates the investment. This interpretation has the added side benefit that it is intuitive for the investor who does not have a background in mathematics or economics: “Had I put my money in a savings account, this is the rate that would have gotten me to where I am now.”

Upon closer inspection, however, there is an issue with the internal rate of return under this point of view. The fixed rate investment that the IRR uses to replicate the actual investment has the property that it always applies the same rate, even if the balance becomes negative. That is not how real-life fixed rate investments behave: the rate that a financial institution charges on a negative balance is substantially different from the one that it pays on a positive balance. Moreover, the IRR may well be negative. In that case, the IRR’s model amounts to paying an investor for being in debt. This is all the more troubling because—as we will demonstrate later—the balance of the modeling fixed rate account can become negative even if the investor’s account did not.

In [1] and [2], Teichroew et al. have shown that charging the same interest rate on positive and negative balances is also the root cause of the well-known fact that Equation (1) may have more than one solution. 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 view of this situation, it seems natural to measure the performance of financial accounts by a rate that is modelled after the IRR, except that the replicating fixed rate account charges the actual, realistic cost of borrowing on negative balances.

Definition 1 Suppose that we are given the following information about an individual investor’s financial account:

• A begin date and balance.
• An end date and balance.
• The dates and amounts of all deposits and withdrawals in between.

Then we define the fixed rate equivalent, or FREQ for short, of the account for that time period as the annual interest rate of the fixed rate account which, when seeded with the same beginning balance and subjected to the same deposits and withdrawals as the real account, results in the same ending balance as the real account. Here, the fixed rate account is such that in case of negative balances, the actual cost of borrowing is charged.

As a measure of performance for an individual investor’s financial accounts, the FREQ has the following advantages: