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Mathematical Models Are Poor Predictors in Finance

Mathematical models in finance are elegant — and terrible at actually predicting outcomes.

Math in economics — as well as its smaller cousin finance — is seductive for a number of reasons.

One is simply our human brain. Illusory pattern perception is a well-documented occurrence whereby our minds often seek to find patterns in otherwise random data. Pareidolia, or the tendency to see a meaningful image in a random pattern, is a familiar instance of this phenomenon.

While randomness in financial markets has long been debated (merely to what extent; that randomness exists is beyond debate), work by several researchers suggests a clear link between our need for a sense of control and our perceptions. When we feel as though we do not have control over our environment, we have an increased tendency to perceive nonexistent patterns, which in turn provides us a false sense of control.

Jennifer Whitson, then of the University of Texas, and Adam Galinsky, then of Northwestern University, constructed a clever experiment to measure this effect. The researchers separated subjects into two groups based on control. Whereas the in-control group was given a menial test with accurate feedback, the lack-of-control group experienced random feedback from the same task, not contingent on their performance.

Subsequently, the lack-of-control crowd scored much higher on a test designed to measure both groups’ need to structure the world. Even more importantly, the lack-of-control group demonstrated greater illusory pattern perception relative to the in-control group, reporting more images that didn’t exist in digital pictures and consistently selecting certain stocks based on what was actually random, uncorrelated data. Time and again, the group that felt less control claimed to see images where there were none and found stock patterns that didn’t exist.

In capital markets, humans deal with making decisions about an uncertain future given an imperfect set of information — a perfect condition for increased illusory pattern perception. And mathematics applied from physical sciences often provides a nice, theoretical framework for the pattern, along with that sense of control. While it is seductive to apply things like geometric Brownian motion to describe random continuous price paths, it is also dangerous because finance is a social science, not a hard science. There is a big difference between what is likely to happen and what must happen.

In physics, net force equals mass times acceleration. Fnet = m(a), always. In chemistry, hydrochloric acid plus sodium yields with certainty sodium chloride (salt) and hydrogen gas.

On the other hand, in finance, math can provide relationships that are only tendencies, not certainties. For example, geometric Brownian motion, a model used to simulate price movements in assets, doesn’t hold perfectly because stocks exhibit discontinuities, or price gaps.

Moreover, analysts often get lost in the elegance of the math — and in so doing they completely lose sight of reality. Theorems like Modigliani-Miller Capital Structure Irrelevance are supported by completely valid mathematical arguments but are based on completely unsound premises.

I’ve often been struck by how wildly inaccurate such models are at predicting real-world outcomes and how such lack of empirical validation must surely challenge the original model to any rational observer. But alas, that’s often not the case.

Perhaps a more prosaic analogy will help the reality strike home. Let’s assume we have two individuals facing a giant boulder: a theoretical physicist and a farmer. Both are asked to predict how far a hard shove will move the boulder.

The theoretical physicist takes a moment to look the boulder up and down and considers his words carefully before he speaks.

“Well, without being able to accurately measure certain things, I must make some assumptions. Since I cannot calculate a coefficient of friction between the boulder and surface of the road, I will for the sake of argument set that to zero. Likewise, I have no anemometer, so I will assume no wind drag coefficient. Without a scale or means to measure the mass, I cannot accurately quantify the effects of gravity on the boulder, and so, assuming no other forces acting in other vectors, any amount of force applied to it would push it in the opposite direction from the force forever.”The physicist smiles, pleased with the logical infallibility of his technically correct response.

The farmer, on the other hand, stares at the boulder and then looks at the physicist, his disbelief apparent.

“You ain’t gonna move that rock.”

And of course the farmer is right.

It’s important to recognize that financial theory is just that: theory. It’s not the real world. Every time I am introduced to a shiny new model that promises to indulge my need for control and define with precision how the financial world works, I make sure to find a farmer who can tell me how it actually does work.

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