Modern finance measures risk by squaring. Across 68 futures markets and 42 years, here is what that square hides.
The Evidence Base:
Over 640,000 daily observations across 68 global futures markets, spanning more than four decades, from September 1984 to July 2026.
One consistent method throughout.
Start with a walk to the office
Anna lives 1 kilometre from the office. Ben lives 5 kilometres away.
Ben lives five times further. Obviously.
Now square both distances. Anna scores 1. Ben scores 25.
Ben is no longer five times further away. He is twenty-five times further. And someone living 10 kilometres out, ten times Anna’s distance, scores 100. They now count for a hundred times what Anna counts for.
Squaring does not merely notice the person who lives far away. It magnifies them, and the magnification grows the further out you go.
Finance does exactly this, every day, to measure risk.
The operation nobody questions
To work out how much an investment moves around, finance takes each day’s return, measures how far it landed from the average, and squares that distance. Add up the squares, take an average, take a square root. The answer is called standard deviation, or sigma, or volatility. In practice everyone just calls it risk.
That one squaring operation sits underneath almost everything the profession believes.
It is the risk term in Modern Portfolio Theory, which won a Nobel Prize and shapes how most pension money on earth is allocated. It is the bottom half of the Sharpe ratio, the score that decides which fund managers keep their jobs. It is the beta in the Capital Asset Pricing Model, which sets the cost of capital in most corporate valuations ever performed. It is the volatility input to Black-Scholes, which prices the world’s options. It is the engine inside Value at Risk, which regulators wrote into banking law.
Remove that one square, and much of modern finance has to be rebuilt.
Where this work begins, and where it doesn’t
Before arguing about theory, we wanted to know whether the real world actually behaves the way finance assumes it does.
So we went and looked. Sixty-eight global futures markets, 647,629 market-days, from September 1984 to July 2026. Currencies, government bonds, metals, grains, energy, equity indices. One method, applied to all of it.
The central claim, that risk and variance are different things, is not ours. Mandelbrot made it in 1963 looking at cotton prices. Taleb has spent thirty years making it. Ole Peters has done the hard mathematical work that we lean on in Episode Four.
What we are adding is the evidence. As far as we can tell nobody has run this particular audit across the whole futures complex, following one arithmetic operation from first principles all the way through to leverage traps, correlation failure, and the destruction of capital.
We will also tell you, in Episode Five, about five things we believed going in that the data destroyed. Some of them we have said in public. They were wrong, and we can now show you exactly how wrong.
The billionaire in the room
Put a hundred ordinary people in a room and work out the average income. You get something sensible, perhaps sixty thousand dollars.
Now Jeff Bezos walks in. The average income is suddenly over a billion dollars. The number is technically correct and completely useless. It describes nobody in the room, including Bezos.
Everyone understands this. It is why we report median house prices rather than average ones.
Here is what does not get said. The risk calculation has the same problem, and the square makes it worse. When Bezos enters, his distance from the average gets squared. He stops being one data point among a hundred and becomes, in effect, the entire calculation.
So the question is whether markets have a Bezos in the room.
They do, and he is in every room
Grey is what a well-behaved bell curve would produce. Red is what actually happened across 68 markets.
Look at the single most extreme day in each market’s forty-year history. In a bell-curve world that one day should carry about a fifth of one percent of the total risk number. In reality it carries almost a full percent, more than five times the weight it is supposed to.
Now widen it. Take the busiest 1% of days, roughly one afternoon per calendar year. A bell curve gives them 8.5% of the risk number. Reality gives them 20.0%.
One day in a hundred is doing a fifth of all the work in your risk measurement.
Sixty-eight markets, sixty-eight exceptions
There is a way to test for this that assumes nothing at all about the distribution you are dealing with. You compare two rulers.
The first is standard deviation, which squares the distances.
The second is mean absolute deviation, which is the plainer measure. It just averages how far things sit from the middle, without squaring anything. A day five times bigger than normal counts as five times bigger than normal.
If returns really followed a bell curve, the ratio between these two rulers would settle at a known constant: 1.2533. That is a mathematical fact about the bell curve, not an estimate.
Anything above 1.2533 means more extreme days than a bell curve permits.
Sixty-eight markets. Sixty-eight results above the line.
Currencies behave this way. So do government bonds, metals, grains, energy, equities. Four decades, asset classes with almost nothing in common, and not one exception anywhere.
We never had to claim that returns follow some exotic distribution. We held up two rulers and found they disagree, in the same direction, everywhere.
In defence of the square
The people who put the square there were not fools, and the mathematics they relied on is correct. In a bell-curve world it is not merely acceptable. It is provably the best thing you can do.
Two numbers say everything
A bell curve is completely described by two numbers: its average and its standard deviation. Not approximately. Completely. Give a statistician those two numbers and they can rebuild the entire distribution with nothing left over.
If returns were normal, sigma would not be an approximation of risk. It would be a complete description of it.
In a bell curve, the extremes really are negligible
The left panel shows where the risk number comes from in a bell curve. Roughly three quarters of it comes from ordinary days, within two standard deviations of the average. Beyond three standard deviations you are down to under 3%. Beyond four, about a tenth of one percent.
So in a bell-curve world the squaring distorts nothing. It magnifies the extremes, but the extremes are so rare that even multiplied by twenty-five they never amount to anything. The ordinary days dominate, and in a bell curve the ordinary days are the distribution.
The square is not just adequate. It is the best available.
In 1920 Ronald Fisher proved that when data is genuinely normal, standard deviation is the most efficient measure of spread that exists. It squeezes more information out of each observation than the plain ruler does.
Our own simulation confirms it. On clean data the square recovers the truth with an error of 2.26%, against the plain ruler’s 2.38%. The square wins.
And the practical advantages pile up from there. Squared quantities add neatly across assets, which is the only reason a portfolio calculation works at all. They can be optimised, because the mathematics is smooth. They have exact solutions everywhere.
Given a bell-curve world, anyone would choose the square.
The knife-edge
All of that holds at the bell curve. The question nobody asked for forty years is how far you can drift away before it stops holding.
John Tukey asked in 1960.
Take clean, normal data and contaminate it slightly. Swap out a tiny handful of observations for wilder ones, the statistical equivalent of the occasional strange day. Then see which ruler still recovers the truth.
The right-hand panel above shows what happens. At perfect normality the square wins, exactly as Fisher proved. A whisper of contamination and it still wins. A little more and the two rulers tie. A little more again and the square loses, permanently, by a widening margin.
The crossover arrives at roughly one contaminated observation in three hundred.
That is not a margin of safety. It is a knife-edge.
Where markets actually sit
In our 68 markets, 0.41% of all trading days lie beyond four standard deviations. One day in every 243.
Markets are already past Tukey’s crossover on those days alone, before you count anything else.
And look what those days do. In a bell curve they carry about a tenth of one percent of the risk number. In real futures markets they carry 12.1%.
One hundred and seven times the load the model assumes.
Compare the two columns of that left-hand panel. In the bell curve, the risk number comes from where the data actually is. In real markets, an eighth of the whole thing is generated by four tenths of one percent of the days, and the square is straining to measure an event the model treats as a curiosity.
The same property, in two different worlds
Here is what makes the square treacherous rather than simply wrong.
The property that makes it powerful and the property that makes it dangerous are the same property. Squaring extracts the maximum possible information from each observation, which is why Fisher could prove it optimal on clean data. Squaring also amplifies contamination by the square of its distance, which is why it collapses when the data is not clean.
You cannot keep one and discard the other. It is the same mathematics, evaluated in two different worlds.
Finance chose the square for a world where the extremes are a rounding error, then took it into a world where the extremes carry an eighth of the risk and all of the ruin.
Next, in Episode Two: what this costs a manager whose livelihood depends on rare, large, favourable events. We follow 3,696 trades across four decades and find that 4.6% of them produce 100% of the profit. Then we look at what the industry’s scoreboard pays for catching one.
Richard Brennan writes on systematic trading, complex adaptive markets, and the philosophical foundations of trend following at atstradingsolutions.com. His books include The Fractals of Finance, Complex Adaptive Markets, Carved by Impossibility and The Aussie Turtles Trend Following Guide.
Want to explore why structure exists at all?
Carved by Impossibility: What Remains When Everything Else Is Eliminated
The book explores the architecture of constraint, emergence, and reality itself, and what it means for how we understand markets, life, and the universe.
Available now on Amazon in paperback, hardcover, and Kindle.
Want the theoretical foundation for why trend following works?
The Fractals of Finance: Determinism, Adaptation and the Geometry of Markets bridges complexity science with practical trading implementation. With a foreword by Jerry Parker, original Turtle Trader.
Available now on Amazon in paperback, hardcover, and Kindle.
Want a practical field manual for trading trends and capturing outliers?
The Aussie Turtles Trend Following Guide: A Field Manual for Hunting Outliers adapts the timeless principles of the original Turtle traders into a systematic, rules-based approach for modern markets. Co-authored with Adam Havryliv.
Available now on Amazon in paperback, hardcover, and Kindle.