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# The FREQ Vs. the IRR (Internal Rate of Return)

A GreaterThanZero White Paper

Explains the differences between the FREQ (fixed rate equivalent) and the IRR (internal rate of return), and argues that individual investors should prefer the FREQ over the IRR. No math or other special expertise is required. A more rigorous mathematical presentation of the issues discussed in this article can be found here.

## Introduction

If you have a 401k account or some other kind of brokerage account at a financial institution, you may have seen a "personalized rate of return" on your monthly or quarterly statement, or online after logging into your account. This term refers to a rate of return that takes into account deposits and withdrawals, also known as money-weighted performance. How is this rate of return calculated? Somewhere in the small print of your account statement or on your brokerage's website, it will almost certainly say that the personalized rate of return, or money-weighted rate of return, is calculated as the internal rate of return, or IRR for short.

By and large, the IRR is indeed the appropriate measure of money-weighted performance. However, there is a rare corner case in which the IRR calculation is counterintuitive and inappropriate for measuring money-weighted performance. This white paper explains how the FREQ fixes this rare case, based on well-known existing mathematics, while preserving the IRR’s logic in all other cases. An important side effect of the fix is that the FREQ no longer suffers from the IRR’s multiple solutions problem. The fact that the FREQ is thus guaranteed to be a unique measure of money-weighted performance is perhaps the most important reason to prefer it over the IRR.

The presentation given here assumes no mathematical education or inclination on the part of the reader. To make the paper self-contained, we start by defining and explaining the FREQ and the IRR from scratch.

## Definition of the FREQ (Fixed Rate Equivalent)

Suppose we are looking at an individual investor's financial account, and we are given the following information:

• a beginning date and balance,
• an ending date and balance,
• the dates and amounts of the deposits and withdrawals in between.

Now imagine an account that pays a fixed annual interest rate, like a (somewhat idealized) savings account. Assume further that we give our imaginary account the same beginning balance as the investor's real account and subject it to the same deposits and withdrawals. Then we can ask, “What annual interest rate does the imaginary account have to pay to end up with same ending balance as the investor's real account?” That rate is the account's fixed rate equivalent, or FREQ for short. In other words, the FREQ is the annual fixed interest rate that replicates the real account's performance.

If you have a bit of a mathematical instinct, you are probably feeling uncomfortable at this point. The issue that we just glossed over is, does such a replicating fixed interest rate always exist, and if so, is it unique? The answer is, the FREQ exists except in some uninteresting corner cases. The FREQ is also unique. However, there is a lot more to say about uniqueness. It will feature prominently in our discussion of the FREQ vs. the IRR.