Benoit Mandelbrot saw fractals everywhere. Time wasn't one of them.
Though time does not run in a straight line, it changes according to price and stretches and shrinks like a balloon rubber. It speeds up and slows down. Mandelbrot, the 1993 Wolf Prize winner focused all his mathematics on price and not on time. The seriousness of his work suggests that either time was not important enough to be featured or he did not consider time to have a mathematical fluctuation or in other words an “organic life”.
Others ignored it too
Barring a few, majority of behavioral finance researchers might consider fractal nature of emotions as a ridiculous idea, leave aside fractal nature of time. Isn’t it strange, we have so many organic systems and entities built in and around time such as the society, a market, nature, humans and the only thing, which we assume could not have life was time itself. The few, who thought, researched and wrote about time as a pulse, a cycle never counted for anything.
How did Mandelbrot miss it?
Mathematicians claim themselves to be purists working without assumptions. Mandelbrot cracks a joke on economists as the ones who assume in his book, ‘The (Mis)behavior of markets’, missing a key and rather significant thought. The slip-up was monumental.
Equality, mean, price independence have been the corner stones of modern finance. Mandelbrot challenges the modern finance techniques and calls them as “foolhardy underestimation of risk of ruin”. The ‘Bell Curve’ has an equal portion of standard deviations on each side. Tests proved that in real situations the ‘Bell Curve’ fails miserably. Prices cluster and do not take a bell shape form. Mandelbrot also says that prices are not predictable or controllable, therefore seeing nature through probability theory is tough.
Further, he also said that fractal geometry’s multi-fractals begins with the realisation that mean in not golden. As long as his analysis was limited to modern finance, it was fine, but it seems Mandelbrot assumed a similar equality for time, which was static and perfect rather than proportional or fractalled as he saw it in price.
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What about proportion?
The power law was more about proportion than mean or equality. The law applies to negative and positive price movements, leaves room for big price changes far away from mean. If everything in nature was proportional (language, income, markets) based on the Pareto’s 80-20 statistical distribution and the unchallenged ZIPF law, then why was time harshly judged on equality?
Cyclical periodicity did not exhibit perfect regularity but an average periodicity. Business cycles observations, according to Mandelbrot, occurred and disappeared. And strangely all the above cycles had a mathematical proportion X, X/3, X/9, X/27. Despite such research, Mandelbrot did not mention time. For him time was not alive, not proportional, and had nothing to do with fluctuations that were just meant for price and nature.
The partial pattern
There were further partial omissions. Mandelbrot gives a large weightage to pictures in identifying fluctuations. He believed that fractal geometry was about spotting repeating patterns and his idea of “price dependence” also gives credit to history. However, in this past-to-future linkage, cyclicality was just limited to identifying the fluctuation patterns.
This was partial acceptance of pattern recognition on one side, and on the other, he said there was no predictable pattern. Price had the memory of 1929, time had no memory. On one side he said, “The wigglies (fluctuations) looked the same in different times” and on the other he trashes all ‘head and shoulder’ watching.
“My mathematical models can generate charts that, purely by the operation of random process, appear to trend and cycle”. A similar exercise of computer generated charts were put in Hersh Shefrin’s book ‘Greed and Fear’. The aim was to highlight that conventional chartists can’t differentiate original from fake.
The question of when?
Though pattern watchers are pulled down by both psychologists and mathematicians, it’s only the pattern watchers who are left to predict and claim that forecasting works. Despite all the mathematics and psychology understanding, both behavioral finance and fractal geometry schools say that predictability is an illusion. Mandelbrot even goes ahead and says that ‘fluctuations’ are predictable not what is between them.
With all this behavioral and mathematics understanding, the only lesson fractal geometry teaches us is how to avoid losing money. But you can only avoid losing money if you invest first. Whom should the investor turn to? Behavioral finance says advisors don’t know, just diversify, buy the losers, sell the winners and save. The majority will herd and bubbles will come again as fluctuation and cyclical pattern unfold. Both pattern bashers never address the ‘when’.
Time fractals vs price fractals
In his 1999 article, “A multi-fractal walk down Wall Street”, Mandelbrot used a three wave pattern, the first and last being in the direction of the general trend and the middle against the general trend to illustrate how market fractalled by subdividing in similar forms. He focused on the price and not the time.
This is the reason why price fractals and time fractals seem disconnected. We redrew Mandelbrot’s multi-fractals from a time cycle (up leg and down leg) rather than from a price trend (up leg-down leg- up leg) perspective (right hand side). Time fractals break down into X, X/3, X/9, X/27 and further on just like the documented economic cycles. A triangle breaking down into three smaller triangles and so on. Multi-fractals have a symmetry that relates a whole to its part, so why can’t time similarly be a fractal. Why did we not see them? Because we assumed time to be equal and not proportional. We were looking for more regularity or equality than what was witnessed.
“Fractals are visual and hence intuitive, it’s a straight forward idea.” What you can see on time (RHS) is visual and intuitive. Call it coincidence but Mandelbrot used a three leg generator to make the multifractals cases just like, Waclaw Sierpinski and Helge von Koch used a triangle. The idea of nature and price fractal is challenged as we introduce the time fractals for the first time. One triangle sub dividing into 3, 9, 27, 81 triangles and so on.
X- and Y axis
So now either time is fractalled or price and nature is. It’s a simple mathematical idea that both X and Y axis cannot have a fractalled pattern at the same time. The Y axis contains all of Mandelbrot’s nature symmetry and beauty while the X axis just has time. One of the ideas is wrong, either time is not fractalled or only time is fractalled giving everything a beauty and a form. Just to prove our case we starting looking for time triads (time fractals) on a ratio line. We were not surprised. The triads were there too. Time was indeed a three, working independent of the Y axis. Draw a rate of change on a price or a price ratio or on any time series, you will see them. Tony Plummer in his book Forecasting financial markets does give reference to time cycles as a triad of patterns, but does not extend the work to fractals.
Conclusion and chaos
Time fractals change the way we look at nature. It brings order in the disorder of the Chaos Theory. The butterfly effect suddenly starts to lose its mystery. The sensitivity dependence Chaos Theory talks about is due to the genetic code of time, which is always different as there is an infinite time above and below us. We may have some time till we find the next theory of everything, but till then start watching the fluctuation.
The author is CEO, Orpheus CAPITALS, a global alternative research firm