Fibonacci Numbers
Introduction
What can you say about the following series of numbers?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,...
Is it a random set? Or is it governed by a rational principle?
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If you examine the Fibonacci series from left to right, at the top of the page, you'll see that, after the first two numbers, the values increase in a very systematic way. Each number, in turn, is determined by the sum of the previous two numbers.
Taking the numbers in order, 2 is the sum of 1 and 1. This is followed by 3, which equals 2 + 1. Then we have 5 which is 3 + 2. Then comes 8 = 5 + 3. And so on.
Known as the Fibonacci summation series, this number sequence was discovered centuries ago by the mathematician, Fibonacci, and has several fascinating properties. We focus, here, on just one property: the Fibonacci ratio. And we see how it applies to the market.
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The numbers in the Fibonacci series grow indefinitely -- they have no upper limit. Even so, the ratio of each successive number in the series to its predecessor is not without bound. While the numbers themselves grow infinitely large, the ratio does not. In fact, the ratio approaches the number defined as half the sum of 1 and the square root of 5, an irrational number. It can't be written out completely, but can be approximated (at this point, to three decimal places) by the rational number 1.618. (A rational number is one that can be written as a fraction, whereas an irrational can't.)
Computing the successive ratios from the beginning, we get the series:
1/1 = 1 (<1.618)
2/1 = 2 (>1.618)
3/2 = 1.5 (<1.618)
5/3 = 1.666 (>1.618)
8/5 = 1.6 (<1.618)
13/8 = 1.662 (>1.618)
etc.
The ratio oscillates about 1.618 with each step. But the quotient gets closer to 1.618 (i.e., to the irrational number 1/2[1 + Ö 5] represented by 1.618 ). The values approach this limit asymptotically, or closer and closer.
The same way, the inverse ratio, namely the ratio of successive numbers to its following (larger) number, approaches the limit .618.
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OK! Now we know about the Fibonacci ratio and its inverse! Big deal! What does this have to do with stocks and bonds, or anything else, for that matter?
That, indeed, is the question. The point is, it's all about models -- about models that help predict what stocks will do, where prices will be tomorrow or the next day.
It is the Fibonacci ratio that can be used to project the prices! How effectively, though, is a judgment call, though no more a judgment call than any other model for playing the market. The ratio purportedly represents nature's law and human behavior. According to Robert Fischer, the Fibonacci ratio is best used in price correction targets. But it's also used to analyze price targets on wave extensions. Corrections and extensions are special characteristics of wave patterns of prices. See Elliott Wave Theory.
First, though, let me state Fischer's main forecasting rule, the one he draws from a statement made by R N Elliott, as follows:
A cyclical pattern or measurement of mass psychology is 5 waves upward and 3 waves downward, total 8 waves. These patterns have forecasting value -- when 5 waves upward have been completed, 3 waves down will follow, and vice versa.
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Like others before him, Elliott adopted the Dow theory and identified bull and bear market cycles by distinguishing among major, intermediate and minor waves. Expanding on this, Elliott characterized bull markets as having five major waves and bear markets as having three major waves, as in the following figure.
There are many possible trend levels, hence many 5-3 combinations, so you must be careful to stay at the same level when counting these waves, and that ain't easy!
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The Fibonacci forecasting procedure for the first rule is to wait to the end of a 5-wave move, because three waves in the opposite direction can be expected. So, if P1 and P5 are the prices for the stock at the beginning and end of waves 1 through 5, the expected price PC after the end of corrective waves A through C is:
PC = P5 - .618(P5 - P1)
In other words, this is roughly a 62% correction..
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A wave is a continuous surge in one direction, up or down, so it could be as small as the difference between the prices of two successive trades made by the floor broker at different prices. Dealing with "waves" of this size, though, would have no trading value for you. You need waves of sufficient amplitude to let you pay the trading costs and still make a profit. For the size to be meaningful, you need to relax the meaning of a surge to allow the prices to backslide for brief moments. The reactions have to be small enough to be regarded as noise.
By accepting this bit of noise, you "smooth out" the price movement and establish a minimum size requirement for a wave. The smoothing can be accomplished by using a moving average. The average heightens the wave structure and lets you determine more easily when a wave has occurred. With sufficient clarification of the waves, you can more easily decide whether five of them have occurred -- presuming you can decide whether or not wave 5 has an extension!
One way to get a more precise wave count is to base it on the number of consecutive moves of the price bar, either to the upside or the downside. (The price bar is the vertical line on the bar chart that gives the range of prices occurring in a defined interval of time -- a trading time period.) The Gann format, for example, identifies minor, intermediate, and main trend indicator charts, respectively, in terms of one-, two, and three-bar movements, for a presumed trading time interval.
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