Extending time fractals to explain the bell curve transformation into a Pareto distribution reconciles the efficient, inefficient market debate.
Robert Brown, a Scottish botanist observed the random Brownian motion nearly 250 years back. Nearly the same time Carl Gauss, also known as the Princeps mathematicorum (prince of mathematicians) created the bell curve. After 100 years Louis Bachelier connected these two great works by pioneering the study of financial mathematics and stochastic processes. Bachelier also referred to as the father of modern finance inspired the efficient school.
In a similar time frame Vilfred Pareto with his Pareto distribution started the work on fractional Brownian motion. One can easily generate a random sample from Pareto distribution. Bachelier and Pareto were on two sides of the Brownian randomness. What started from Robert Brown split into the efficient and inefficient school of thought.
Two extremes
Efficient school of thought was the bell curve believers who were more concentrated just like Mandelbrot's clusters. Harry Markowitz, William Sharpe, Fischer Black, Myron Sholes and Eugene Fama are a few of the efficient market theorists. The moment you start watching behavioural patterns, market patterns or fractals, start making S curve forecasts, talk about proportionality and the law of large numbers, you are no more an efficient theorist focusing on mean, average, linearity and continuity. The Pareto rule drove the inefficient school of thought. It was widely distributed, de-clustered, characteristically proportional like the underlying law.
Mandelbrot might feel uneasy in the way I respectfully group him with the other luminaries like Thomas Malthus, Pierre François Verhulst, Charles Dow, Kingsley Zipf, Ralph N. Elliott, Robert Prechter, Daniel Kahneman and Robert Shiller, but the truth is that the father of fractals is a part of the pattern, fractal, anomaly, inefficiency watchers.
Did the inefficient school go too far in challenging the efficient model? A flapping butterfly causing a tornado in Texas seems like an exciting story, but how solid is the mathematics? Just because we see anomalies from normal, more volatility and extreme fluctuations, does the subject of value and calculus suddenly get trashed? Just because efficient theorists have failed to challenge the academic attacks, do all their work with CAPM model, beta, option pricing become worthless? Well, if you look at it from Mandelbrot's view, “pattern watchers are naive, bell curve is nonsense and conventional finance is based on unrealistic assumptions.”
Efficient theorists had the field to themselves. The inefficiency school can be given the benefit of doubt as they were not really noticed and were marginalised (compared to the efficient school) for a century. Now that the tide has turned and theories stand challenged, I would still think twice before calling conventional finance junk and Euclidean geometry as a redundant linear mathematics. Efficiency and inefficiency are thought to be two extremes that there could not be anything more sacrilegious in the academic world than trying to bridge the two disparate schools of thought.
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Time triads
Last time we talked about fluctuations, pulse and life in time rather than price or nature. We tried to make a case for a dynamic and proportional time that has a pattern. The movement of time we are talking about is called translation. This is different from the popular Google translation and should be one of the few references of its kind on the world wide web, like some other ideas like time triads, time fractals and Econohistory.
The time triads we illustrated last time are made of hierarchy of Euclidean equilateral triangles. If time was static, all triangles (triads) in the set would be equilateral. Mathematically the time between two bases of a triangle is rarely equal. This is because the larger degree time (the higher amplitude triangle) pushes the underlying smaller time (triangle) causing it to translate, move or shift. This self feeding change creates an imbalance pushing time ahead. Real markets are full of such cases, when one period between two lows of a time triangle is not the same as the period in the other base (A<B>C). This is one reason why an average 3.3 business cycles vary from 3 to 5 years. Though the average periodicity holds, it is irregular.
This irregularity kept mathematicians away from time and this might be the reason Mandelbrot talks about contracting, expanding, flexible, speeding, slowing time. He might be simply talking about the moving and translating triads. Fractals are caused by time translation. There are other reasons linked with the human tendency of looking for average returns, looking for order, etc. why time did not get the attention.
Mandelbrot’s asks a very similar question like Edward R Dewey asked 70 years back. Why do cycles repeat? Why do price and natural fractal exist everywhere? There are similar questions. Why do history, time, price, human mind and water claim to have memory? Is it not the memory associated with time?
The triads on one side demonstrate a clear power law behaviour 9X, 3X, X, X/3, X/9, larger size lower frequency (if there is one cycle of 9X period), lower size-larger frequency (there are 81, X/9 cycles) and on the other side triads suggest why a bell curve might be the idealized form and the power law curve a translated form. Moving and translating caused more distortions from the idealized bell curve and created the power law curve. This means that all the inefficiency, anomalies, fluctuations were never a separate set but part of the universal time fractal set.
Time translation explains why nature is similar not equal (same)? Why there are more curves in nature than straight lines? Why we see order in randomness and randomness in order? Why chaos is not about disorder but about different time frequencies interacting through nature? Why moving time is change? Why risk perception is not about price but time polarity? Why time arbitrage will take over statistical arbitrage as the next investment strategy? Why dumping pattern watchers might be the biggest mistake in research? They just might be holding the truth.
In conclusion, it was never about rationality or irrationality. Time translation and time fractals may take time to be understood and be accepted, not because it is too complex, but because the idea is too simple. While working on some random operations Mandelbrot and James R Wallis produced some records which all appeared to display a long, slow up down cycle of three. Upon these long waves, smaller and more numerous cycles seemed to interpolate themselves. When they looked at small section of the record, they again saw three waves, each a third shorter than the section. Now these are some time triads Mandelbrot would find it tough to call static. The time was alive and translating.
The author is CEO, Orpheus CAPITALS, a global alternative research firm