# Guilty as charged – the misconception of mean-reversion

Have you ever heard people talk about “keeping up with changing times?” As an agency, it is our (one of our many) job to stay ahead of trends, psychographics, demographics, technology, regulatory changes, etc. to make sure we understand the needs of our clients and the best way to build strategy that takes these variables into account. It seems like it’s getting harder and harder to do these days. I mean, it’s hard enough to keep up with advances in technology (related, my iPhone 5 is due any day) and the “we’re recovering / not recovering” economy, but at least when it comes to our financial services clients, we never had to worry about edits to the formulas we had drilled into us a B-school. Until now.

Iddo I Eliazar of the Holon Institute of Technology (Israel), and Morrel H Cohen, Rutgers University, have published “The misconception of mean-reversion,” and it’s going to blow the doors off the financial industry. The abstract of their paper outlines the impact of their efforts:

The notion of random motion in a potential well is elemental in the physical sciences and beyond. Quantitatively, this notion is described by reverting diffusions—asymptotically stationary diffusion processes which are simultaneously (i) driven toward a reversion level by a deterministic force, and (ii) perturbed off the reversion level by a random white noise. The archetypal example of reverting diffusions is the Ornstein–Uhlenbeck process, which is mean-reverting. In this paper we analyze reverting diffusions and establish that: (i) if the magnitude of the perturbing noise is constant then the diffusion’s stationary density is unimodal and the diffusion is mode-reverting; (ii) if the magnitude of the perturbing noise is non-constant then, in general, neither is the diffusion’s stationary density unimodal, nor is the diffusion mode-reverting. In the latter case we further establish a result asserting when unimodality and mode-reversion do hold. In particular, we demonstrate that the notion of mean-reversion, which is fundamental in economics and finance, is a misconception—as mean-reversion is an exception rather than the norm.

I’ll skip the math, but mean-reversion is a fundamental concept in economics and finance proposing that, over time, prices and indices revert to their historical means and is modeled by the Ornstein-Uhlenbeck process. The process has a wide spectrum of applications ranging from physics to biology, and engineering to economics. From a physical-sciences perspective, the process models the motion of a particle trapped in a harmonic potential well and perturbed by random thermal fluctuations. One of the most popular applications of the process in finance is the Vasicek model for spot interest rates. Mean-reversion is a rather particular behavior displayed by the Ornstein-Uhlenbeck process – which is a very specific reverting diffusion characterized by a linear restoring force and by a constant noise magnitude. The goal of the paper is to demonstrate that the concept of mean-reversion is, in effect, a misconception – as mean-reversion is an exception rather than the norm.

The authors analyze general reverting diffusions and establish that if the magnitude of the diffusion’s perturbing white noise is constant then the diffusion’s stationary density is always unimodal, and the diffusion is always mode-reverting. Additionally, if the magnitude of the diffusion’s perturbing white noise is non-constant then, in general, the diffusion’s stationary density is not necessarily unimodal, and the diffusion is not necessarily mode-reverting. I simply cannot do justice to their paper, and strongly suggest that, if you deal in finance (or other sciences in which mean-reversion is factored) you download their paper.

The results and examples provided in their paper demonstrate that the concept of mean-reversion is a misconception – as mean-reversion is an exception rather than the norm. The conclusion of the paper is that the notion of mean-reversion should be replaced by the notion of mode-reversion, and the latter notion should be considered only within its admissible realm – whose boundaries are precisely prescribed by equation.

The adjustment to mode-reversion will have dramatic impact in the development of strategy for our financial clients – not only in their own businesses (and the impact that will have on consumers), but also in the way that data is reported / marketed to consumers in the first place. Not every client operates exactly the same, so it’s not possible to make a blanket statement of how strategy will change, but the impact is that marketing and measurement efforts, that use mean-reversion as a baseline for data and success, are no longer valid. That does affect everyone – especially us.

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