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I have referred to Fibonacci retracements and phi-based rebound much in the technical analysis posts I have been writing of late. In so doing, I have been assuming that readers understand what the Fibonacci sequence is and how it applies to the price activity in negotiated markets. Given how esoteric it actually is, I thought I'd take the time to explain first what the Fibonacci sequence is and, tomorrow, how it applies to negotiated financial markets.

First, an explanation of the Fibonacci sequence itself. Leonardo Pisano Fibonacci, a medieval businessman and mathematician, came upon the "Fibonacci" sequence when thinking about the hypothetical question how fast rabbits could bread in "ideal" circumstances. He set up the following initial conditions: two pairs of rabbits would mate and produce another pair every month starting once the rabbits reach 1 month of age; no rabbits ever die, and one female and one male are always produced. Then he asked the question: how many rabbits would exist after one year?

The answer? 144 rabbits.

The Fibonacci sequence of numbers that "explains" the answer to this particular problem was born, and it is thus:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, infinity

As you can see, each Fibonacci number is the sum of the two numbers before it.

There are some important and interesting relationships between and among these numbers that are non-obvious. To wit: starting at 1, if we take the next Fibonacci number and divide by the previous Fibonacci number, we get the following:

1/0 = null
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666
8/5 = 1.60
13/8 = 1.625
21/13 = 1.61538
34/21 = 1.619
55/34 = 1.6176

If this process is taken to infinity, the series narrows and reaches what has been termed the golden ratio: 1.618034... Like Pi, it has no end as the number continues on into infinity. For our purposes however, the ratio 1.618 is a close enough approximation.

Now what happens when we take the Fibonacci sequence and instead of dividing one of the Fibonacci numbers by the number preceding it, we divide by the number after it?

0/1 = 0
1/1 = 1
1/2 = .5
2/3 = .666
3/5 = 0.60
5/8 = 0.625
8/13 = 0.61538
13/21 = 0.619
21/34 = 0.6176
34/55 = 0.6182

As you can see, if we take this process to infinity, the series narrows and reaches 0.618034... This is called phi: like Pi and the golden ration above, it has no end as the number continues on into infinity. For our purposes however, the ratio 0.618 is a close enough approximation.

So what? Why does a mathematical curiosity matter? Interestingly, phi, or .618 and its close cousin the golden ratio 1.618, can be found in nature. In fact, it can be found almost everywhere in nature. The pattern exhibited by the seeds in the head of a sunflower or by a nautilus shell, the arrangement of petals around flowers, patterns of pinecones and pineapples, quarks and other subatomic particles disintegrate in phi patterns, even the relationships between human body parts (size and spacing of eyes, ears, the nose) etc.: all these are governed by Phi.

There are literally limitless examples where natural patterns are governed by the Fibonacci sequence; the reason this is is that nature abhors waste. Patterns of seeds, flower petals, etc. need to arranged in a way such that no space is wasted. In other words, they need to be packed together in the most efficient way. Phi, or .618, does just that. Nature uses phi to govern the placement of objects in relation to each other and to govern the movement of objects too (as in our quark example above). The Greeks were aware of this and in fact used phi in their architecture and in their paintings: it is said that the Mona Lisa is considered a timeless beauty precisely because da Vinci, aware of the importance of Phi, made sure he used it in the spacing of her features on the canvas.

Nature, art, music, mathematics, physics: phi is seemingly everywhere. But finance?; tomorrow we'll explore how and why this important ratio can be applied to the prices of financial instruments in negotiated markets.
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