Measure the Performance of Your Investments

The Math of the Fixed Rate Equivalent (FREQ)

A GreaterThanZero White Paper

What Does This Article Do for Me?

Explains the mathematical foundations of the FREQ (fixed rate equivalent), and how it relates to the IRR (internal rate of return). The reader should be mathematically inclined and comfortable with mathematical formalism. However, for this version of the paper, no knowledge of higher mathematics is required. An extended version is available in PDF format that provides the mathematical proofs as well.


A common way of measuring the performance of a financial account in a way that takes into account deposits and withdrawals (money-weighted performance) is to calculate an internal rate of return (IRR). This amounts to modeling the account’s performance with that of a fixed rate account: the IRR can be interpreted as the fixed interest rate that replicates the performance of the account. Generally speaking, this is the appropriate way of measuring money-weighted performance. However, in rare cases, it is possible for the modeling fixed rate account to encounter negative balances despite the fact that the original financial account did not. In this case the IRR calculation assumes that the same rate that is paid on positive balances is charged on negative balances. That is certainly not a realistic model of a fixed rate account. Moreover, in [1] and [2], Teichroew et al. have shown that this way of charging interest is also the root cause of the IRR’s mulitple solution problem, and that a more realistic way of charging interest on negative balances results in unique solutions.

The fixed rate equivalent, or FREQ for short, follows the insights of Teichroew et al. by charging an external cost of borrowing in the rare case of negative balances, while preserving the IRR's logic in all other cases. The FREQ thus offers two advantages over the IRR: it is more realistic and intuitve in its treatment of negative balances, and it is a unique measure of money-weighted performance without any mulitplicity issues. Providing an external cost of borrowing can be difficult in practice. However, the FREQ calculation allows us to flag the rare situation of negative balances, and the insights behind the FREQ offer advice on alternate ways of dealing with that case.

The articles by Teichroew et al. are written in the context and for the purpose of capital budgeting and financing decisions. The purpose of this paper is to explain all this in a manner that is geared towards measuring the performance of an individual investor’s accounts.


The internal rate of return (IRR), also known as the discounted cash flow rate of return (DCFROR), or the effective rate of return, is a measure of profitability that is used in capital budgeting as well as in the world of individual investing, where it sometimes appears under the moniker personalized rate of return. The IRR deals with situations where a sequence of cash flows to and from an investment occur at certain points in time. In the world of individual investing, this corresponds to accounts that undergo deposits and withdrawals.The IRR is most commonly defined as a real number greater than −1 that solves the equation

= number of cash flows
= amount of the th cash flow
= number of periods between the first and the th cash flow (hence 0)