The Mathematics of “Anything Can Happen”

People like to say that life is random.  We accept that, even embrace it, because we like some variation in our lives; a chance meeting at the supermarket, running into an old friend at a party and so on.  Those little collisions make life more interesting.

Yet it is that same variability that drives us crazy in our professional lives.  When we launch a product, we want to know how it will sell.  When we invest our savings, we want to have some idea what the return will be.

A lot of smart, practical people have sought to solve the problem.  Clever math models and statistical techniques are employed to calculate the likelihood of positive ROI with the aim of creating more certainty.   The problem is that fate has a way of thwarting our best laid plans.  In truth, it is the whimsical mathematics of chaos that rules our destiny.

A Pregnant Postcard

In 1954, the prominent statistician Jimmie Savage sent a postcard to his friend, the eminent economist Paul Samuelson, that asked “ever hear of this guy?”  That terse, simple note would have consequences of historical proportions.

The ”guy” in question was Louis Bachelier, an obscure mathematician who wrote a dissertation in 1900 that described the fluctuation of market prices as a random walk. It was an impressive piece of work anticipating not only mathematical finance, but Einstein’s famous paper of Brownian motion published four years later.

Nevertheless, the paper merely got a passing grade and was soon forgotten.  That is, until our friend Savage came across it by chance and found it interesting, which led to the postcard he sent Samuelson.  Bachelier summarized the main idea in a single sentence:

The mathematical expectation of the speculator is zero.

Samuelson, who was pioneering the field of mathematical finance at the time, thought the paper was brilliant and began to actively promote it.  Soon after, Eugene Fama built Bachelier’s initial work into a full blown Efficient Market Hypothesis.

The idea of rational markets led to a whole slew of new theories and models including efficient portfolios, the capital asset pricing model (CAPM) and the Black-Scholes model which, in turn, gave birth to an entire industry of financial engineering and risk management. Nobel prizes were awarded, fortunes were made and all seemed well.

Until recently, of course, when a bunch of traders blew up the global financial system using those very same models.

The Difference between Randomness and Chaos

Randomness is actually fairly predictable, because it averages out (i.e. the mathematical expectation of the speculator is zero).  In other words, we expect things to move around a bit, but to be centered on a certain norm.  It’s a comforting concept that looks like this when we graph it:

Chaos is more like passing people on the street.  We expect that most people will live or work in the area and can calculate the average distance the vast majority have travelled, simply by surveying a random sample.  However, that average distance will tell us nothing about the possibility that a guy just flew in from Japan, although that’s surely possible.

Chaos takes extreme values into account and we describe them with power laws that look like this:

You can see the problem.  Randomness assumes that we can account for error and even when it happens, something else will balance it out.  Chaos makes no such assumption.  At some point, all bets are off and you’re on your own.  Averages have no meaning.

Noah Effects and Joseph Effects

Nobody thought more sensibly nor probed so deeply into chaos than Benoit Mandelbrot. He argued that the mathematical establishment had it backwards.  They dismissed those pesky little data points that didn’t fit their models as “outliers.”  He, on the other hand, felt that the data that seemed out of place was the stuff that was really important.

As a matter of fact, he thought that they were so consequential that he came up with a framework to describe them.  He called the forces that governed chaos “Noah Effects” and “Joseph effects:”

Joseph Effects: These are persistent.  Just like in the biblical story, where Joseph predicted seven fat years and seven lean years, events in a time series are highly dependent on what precedes them.

Noah Effects: These create discontinuity.   A storm comes and blows everything away; creating a new fact pattern that will be propagated through further Joseph effects.

For example, a weather forecast might tell us that there is a 70% chance of rain.  In actuality, it either will rain or it won’t.  If it does, the streets get wet and that makes them more dangerous.  If we get into a serious car accident, we can end up hospitalized for a few months, altering the course of our life.

An outlier?  Maybe, but that doesn’t make it any less important.

The Importance of Feedback

What Mandelbrot realized was that these types of effects would occur in any system that feeds back on itself.  He found them in things ranging from “noise” that cause errors in communication lines to the flooding of the Nile river to, most famously, financial markets.

Statistics can tell us a lot about the normal course of Joseph effects, but absolutely nothing about the Noah effects that can alter the tide of history.  A stock goes up because lots of people think it will go up.  A video goes viral and more people see it, so it gets passed on more often.

We can’t predict these events, but we need to account for them.  We’re all one terrible accident or one winning lottery ticket away from a completely different life.  Statisticians and economists fail to account for Justin Bieber, but every teenage girl knows that he’s real and that he matters (well, at least he does to teenage girls and to record companies).

Everyday is “Anything Can Happen Day”

Serious business people aren’t dreamers, they’re realists.  They know the odds and bet with the smart money.  The problem is that the smart money isn’t always so smart.

Mark Rubenstein, the financial genius who created portfolio insurance, estimated that the probability of market crash like the one we had in 1987 was so low that it should happen less than once in 20 billion years.  Yet we had already had one in 1929 and would have others in 2001 and in 2008.

I don’t know the odds of another crash happening in my lifetime, but I can safely assume that they are far from negligible.  Low probability events happen all the time for good or ill and they matter.  They change lives.  Dreams come true and so do nightmares.  One single company for Mark Zuckerberg, one big oil spill for BP and everything changes.

Celebrate today.  Anything can happen.

– Greg