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The Math of the Fixed Rate Equivalent (FREQ)
• Grasping the intuitive meaning of the FREQ does not require any mathematical understanding at all. It can be explained to the investor by saying, “Had you put your money in a savings account, this is the rate that would have taken you to the same ending balance.”
• According to  and , the FREQ does not suffer from the multiple solutions problem that plagues the IRR. Moreover, a unique FREQ exists except in certain corner cases, such as the investor always being in debt.
• The FREQ is consistent with the ordinary annualized return as defined in modern portfolio theory: in the absence of any deposits and withdrawals, the FREQ equals the annualized return.

The purpose of this paper is to explain and illustrate the mathematical underpinnings of the FREQ. There are no new mathematical results here; the relevant work has already been done in the papers  and  by Teichroew et al.. However, those papers are written in the context of capital budgeting and financing decisions. Applying the results to the performance measurement of investor accounts with deposits and withdrawals requires some adapting. The purpose of this paper is to provide a self-contained presentation that saves the reader the trouble of having to perform this adaptation.

The present paper explains the mathematics of the FREQ without giving formal proofs. There is an extended version in PDF format available that provides the mathematical proofs as well.

## A Closer Look at the Internal Rate of Return

Since the FREQ is a modification of the IRR, we begin by taking a closer look at the IRR. In order to do so, we need to address two trivial but potentially confusing issues. Firstly, in the world of capital budgeting, cash flows into an investment are usually given as negative numbers (money out of the investor’s pocket), while cash flows out of the investment are given as positive numbers (money into the investor’s pocket). In the world of individual investing, it is the other way round. On a brokerage statement, deposits into an account are positive numbers and withdrawals are negative numbers. Throughout this paper, we will use the individual investors’ convention, counting inflows to the investment as positive numbers, outflows as negative numbers. Needless to say, switching between the two points of view amounts to no more than multiplying both sides of Equation (1) by a factor of −1, which has no bearing on the solvability and solutions of the equation.

The second issue that needs clarification is even more trivial than the first one. It is clear that only real numbers greater than or equal to −1 make sense as rates of return. Moreover, the classical definition of the IRR as shown in Equation (1) above implicitly excludes the number −1 as a solution by putting the expression in the denominator. We will therefore use the following terminology throughout this paper:

Definition 2 By a rate, we mean a real number greater than −1. Specifically, a solution to Equation (1) that is greater than −1 will be called a solving rate for the equation.

In Equation (1) above, we have already stated the most common definition of the IRR. The first step towards the FREQ is to transform Equation (1) in such a way that it reveals the alternate interpretation of the IRR as the interest rate of the fixed rate investment that replicates the actual investment. In order to keep track of what the individual pieces of the equation mean, we have to perform the transformation in several small steps.