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Thursday 24 September 2009

Cyclical Growth, Form and Fibonacci

Ancient Ideas, Modern Setting

As an up-to-date in-your face sort of blog we like to make sure our readers are well informed about the financial world as we see it. So, starting back in Ancient India in 200BC and taking in medieval Italy and some early twentieth century anti-Darwinian evolutionary thinking let’s take a look at plant growth and snail shells, how twentieth century humanity’s inclination to see the Man in the Moon translates into modern financial theory and why physics may simply be wishful thinking.

At the root of this journey is a simple mathematical progression named after a man who never discovered it and was more concerned with accountancy than trading. Still, he’s still remembered a millennium after his death, which is more than most of us can ever aspire to.

The Golden Ratio

In 1202 the Italian mathematician Leonardo of Pisa, aka Fibonacci, wrote Liber Abaci, a book which has three claims to fame in financial circles. Firstly it was one of the first books to introduce the Arabic numbering system to the West. Secondly it laid out the foundations of modern bookkeeping. Thirdly it presented the number pattern known as the Fibonacci sequence, although this had been known long before by Indian mathematicians. Only the latter has little significance in the development of science and business but, naturally, it’s the one that’s received the most attention.

The Fibonacci sequence is that that which starts with 0 and 1 and proceeds by adding the previous two numbers in the sequence to create the next one. So you get: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc, etc. Divide any Fibonacci number by the one below it and, as you get higher up the sequence, you get closer and closer approximations to the so-called golden ratio of 1:1.618.

Fibonacci in Living Form

As a piece of pure mathematics the sequence is unremarkable until you start digging around in the real world and find it appears regularly in nature. It can be found in many forms – in the arrangements of leaves around a stem, in the structure of sunflowers and pine cones and the shape of snail shells. The appearance of the golden ratio in nature has led many people to suspect that it embodies some kind of mystical significance. These people are idiots, of course, but idiocy has never been a bar to success in this world.

The true explanation for the presence of the Fibonacci sequence in nature was provided by D’Arcy Thompson back at the beginning of the twentieth century. In an age where evolution was accepted but the genetic code underlying it was not widely known Thompson was not alone in trying to find alternative explanations for natural selection. In the unique and beautifully written On Growth and Form he showed how many of nature’s creatures were adaptations of simple geometry rather than difficult evolutionary changes.

Essentially, Thompson argued that there were a few geometric forms suitable to life on Earth and by changing the morphology of these you could explain the majority of physical animal shapes. As an example, he showed how changing a few physical parameters could generate the whole range of different crab shapes found in nature. On a similar note, he showed how the golden ratio is the best physical adaptation to the shape of snail shells.

Order Out of Chaos

The existence of Fibonacci sequences and the golden ratio in nature is therefore not the result of some divinely inspired meddling but the process of natural selection figuring out the best forms for survival. It so happens that Fibonacci numbers offer the most optimal form of growth or packing for certain creatures and that natural selection has, though its normal process of trial and error, figured this out. It’s not that the golden ratio is “out” there, it’s simply that nature finds an efficient way of managing its resources to best effect.

Indeed the golden ratio is a trend rather than a fixed rule. Many flowers, for instance, don’t use Fibonacci numbers and the oft-quoted “fact” that the golden ratio defines the proportions of a human is simply wrong. In fact, judging by the preference of artists down the ages, it’s not even the preferred ratio of human beauty. Statements alleging that the golden ratio is everywhere and unavoidable are, simply, nonsense propagated by mystics and financial theoreticians. Not that there’s much difference between the two, usually.

Elliot Waving Not Drowning

The appearance of this mathematical sequence in nature, however, convinced many people that there was something special about it. As we’ve seen in recent times there’s no industry more attracted to mathematical solutions to complex problems than the financial sector and this isn’t simply a recent trend. The search for the magic formula for making money with no risk and less brainpower has been around for about as long as money has.

Back in the 1930’s Ralph Nelson Elliot developed a technique for forecasting market price movements which he, at some point, decided was based on the Fibonacci sequence. The principle of the Elliot Wave is that collective human psychology drives moves from mass optimism to mass pessimism and then back again. In Elliot’s reconstruction the ebb and flow of the markets is done in an eight step process, five up and three down, ranging over timescales from minutes to a grand supercycle of multiple centuries. According to Elliot the Fibonacci numbers simply appeared out of his theory, although there’s no known underlying principle to explain this. It’s just the way the world works, presumably: shake your chakra, baby.

For reasons we don’t quite understand humanity seems to have a drive to see cyclical behaviour in nature. Going all the way back to Ancient Greece the circle was viewed as the symbol of perfection and a whole model of physics was generated out of this, complete with circular orbits around the Earth. When the universe failed to play ball with the theory, by making planets seen from the Earth go backwards the scientists naturally refused to change their theory and instead developed an elaborate and wondrous system of epicycles, circles within circles.

Epicycles in Finance

Of course, the model was wrong, being based on a false premise, just as models of stockmarket behaviour based on Fibonacci sequences are wrong. The golden ratio exists in nature because it’s the optimal solution to specific problems, not because the universe is designed that way. Our projection of pure mathematics onto the universe’s haphazard geometric best fit engineering solution seems like as good a metaphor as any for the way that modern physics is trying to understand how our universe is put together. Instead, maybe it’s just the only way of making the damn thing work.

In Elliot’s work and the development of multiple other stockmarket cycles – Kitchin, Kuznets, Kondratieff – we see humanity trying to impose order onto chaos. It’s in our nature to try to find patterns, because we’re pattern matching creatures: it’s our way of structuring the world around us. Sadly this often plays us wrong, causing us to see a Man in the Moon, castles in the clouds and cycles in stockmarkets.

Spurious Stockmarket Structures

Many of the more recent market failures have been caused by attempts to model stockmarkets on the assumption that there’s structure underlying them. This spurious quest for structure and precision where none really exists is dangerous for investors. From time to time any theory will work – possibly because enough people believe it, possibly by chance. Inevitably, however, all theories will melt into oblivion in the crucible of market madness.

Those ancient Indian mathematicians knew a thing or two, but they didn’t know about stockmarkets. Remember – the truth isn’t out there, it’s inside us. Trusting in invisible sequences in random systems won’t always be successful, even if those sequences genuinely appear in nature. Systems including humanity will never obey nature’s simple physical laws.

Related Articles: Technical Analysis, Killed By Popularity, Correlation Is Not Causality (And Is Often Spurious), Gaming The System


  1. "Of course, the model was wrong, being based on a false premise, just as models of stockmarket behaviour based on Fibonacci sequences are wrong."

    Sure, now we know the universe doesn't revolve around the Earth, what false premise flaws these market models? We know market movement is fractal in nature and Fibonacci sequences appear to play a role in their reiterations.

    "The golden ratio exists in nature because it’s the optimal solution to specific problems, not because the universe is designed that way."

    Since nature seems to act as an ultimate optimizer, the fact that Fibonacci ratios are pervasive through various natural systems suggests that the relationships in the sequence could be central to the relationships seen in these systems. Especially when we know human nature is not random, as evidenced by the inevitable cyclical nature of markets.

  2. On false premises, to quote Benoit Mandelbrot: "Elliott’s work fails the requirements of objectivity and repeatability: in his own words, “considerable experience is required to interpret [it] correctly” and “no interpretation [is] valid unless made by [him]."

    Other questions arising would be - "is nature really the ultimate optimiser?", "is human nature really not random?" and "are markets inevitably cyclical"? I can't see that any of these points can be taken as a given.

  3. Sadly this often plays us wrong, causing us to see a Man in the Moon, castles in the clouds and cycles in stockmarkets.

    Have you read the book Stock Cycles?

    Michael Alexander presents a compelling case that market returns are anything but random. The extent to which they follow a pattern in which a secular bull is followed by a secular bear and then a secular bear is followed by a secular bull can be measured mathematically. I think it would be fair to say that the idea that these patterns would repeat over and over again by pure chance is a one in a million shot.

    I'm not saying that I believe in the Eliot Wave stuff. That's a much more precise theory that I have serious doubts about (I haven't studied it enough to make dogmatic statements one way or the other). But if you are saying that you don't believe that secular bulls cause secular bears, then I disagree. I would put the odds that secular bears occur randomly (that is, that they are not caused by secular bulls) at the same place where I put the odds that the moon is made of green cheese. The mountain of evidence going the other way is so strong that it is just undeniable (in my assessment, to be sure!).

    Alexander explains why astrology was once viewed as a "science." Astrologers were able by looking at the placement of the moon and stars to tell people when it was best to plant their crops (that weather changes took place in seasonal patterns was not widely appreciated at this time -- many asserted in oh-so-confident tones that it was impossible to "time" one's crop planting endeavors). The astrologers were of course right -- there IS a pattern by which weather changes take place. Any "scientist" of the day could have pointed out that the astrologers got something wrong. But that wouldn't have changed the fact that the astrologers were right about something of huge practical importance.

    So it is with "timers" of today. Stock returns are not random. Stock returns follow inevitable cycles (secular bull followed by secular bear) that have been repeating since the first market opened for business. Those who have discovered flaws in a few overly aggressive timing schemes can point to those flaws as "evidence" that no timing plan can work. But they are of course only fooling themselves. Paying attention to the cycles that have always applied is the truly "scientific" way to go (according to me!), given the mountain of evidence that today exists showing that these cycles are an inevitable feature of markets.