THE FIBONACCI SEQUENCE AND ITS APPLICATION

Known for millennia by scientists, naturalists and mathematicians, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on to infinity is known today as the Fibonacci sequence. The sum of any two adjacent numbers in this sequence forms the next higher number in the sequence, viz., 1 plus 1 equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and so on to infinity. The ratio of any two consecutive numbers in the sequence approximates 1.618, or its inverse, .618, after the first several numbers. Refer to Figure 24 for a complete ratio table interlocking all Fibonacci numbers from 1 to 144.

1.618 (or .618) is known as the Golden Ratio or Golden Mean. Nature uses the Golden Ratio in its most intimate building blocks and in its most advanced patterns, in forms as minuscule as atomic structure and DNA molecules to those as large as planetary orbits and galaxies. It is involved in such diverse phenomena as quasi crystal arrangements, planetary distances and periods, reflections of light beams on glass, the brain and nervous system, musical arrangement, and the structures of plants and animals. Science is rapidly discovering that there is indeed a basic proportional principle of nature. The stock market has the very same mathematical base as do these natural phenomena.

At every degree of stock market activity, a bull market subdivides into five waves and a bear market subdivides into three waves, giving us the 5-3 relationship that is the mathematical basis of the Elliott Wave Principle. We can generate the complete Fibonacci sequence by using Elliott's concept of the progression of the market. If we start with the simplest expression of the concept of a bear swing, we get one straight line decline. A bull swing, in its simplest form, is one straight line advance. A complete cycle is two lines. In the next degree of complexity, the corresponding numbers are 3, 5 and 8. As illustrated in Figure 25, this sequence can be taken to infinity.

Figure 24

Figure 25

In its broadest sense, then, the Elliott Wave Principle proposes that the same law that shapes living creatures and galaxies is inherent in the spirit and attitudes of men en masse. The Elliott Wave Principle shows up clearly in the market because the stock market is the finest reflector of mass psychology in the world. It is a nearly perfect recording of man's social psychological states and trends, reflecting the fluctuating valuation of his own productive enterprise, and making manifest its very real patterns of progress and regress. Whether our readers accept or reject this proposition makes no great difference, as the empirical evidence is available for study and observation. Order in life? Yes. Order in the stock market? Apparently.

RATIO ANALYSIS

Ratio analysis has revealed a number of precise price relationships that occur often among waves. There are two categories of relationships: retracements and multiples.

Retracements

Fairly often, a correction retraces a Fibonacci percentage of the preceding wave. As illustrated in Figure 26, sharp corrections tend more often to retrace 61.8% or 50% of the previous wave, particularly when they occur as wave 2 of an impulse wave, wave B of a larger zigzag, or wave X in a multiple zigzag. Sideways corrections tend more often to retrace 38.2% of the previous impulse wave, particularly when they occur as wave 4, as shown in Figure 27.

Retracements are where most analysts place their focus. Far more reliable, however, are relationships between alternate waves, or lengths unfolding in the same direction, as explained in the next section.

Motive Wave Multiples

When wave 3 is extended, waves 1 and 5 tend towards equality or a .618 relationship, as illustrated in Figure 28. Actually, all three impulsive waves tend to be related by Fibonacci mathematics, whether by equality, 1.618 or 2.618 (whose inverses are .618 and .382). These impulse wave relationships usually occur in percentage terms. For instance, wave I from 1932 to 1937 gained 371.6%, while wave III from 1942 to 1966 gained 971.7%, or 2.618 times as much.

Wave 5's length is sometimes related by the Fibonacci ratio to the length of wave 1 through wave 3, as illustrated in Figure 29. In those rare cases when wave 1 is extended, it is wave 2 that often subdivides the entire impulse wave into the Golden Section, as shown in Figure 30.

Figure 28            Figure 29            Figure 30

In a related observation, unless wave 1 is extended, wave 4 often divides the price range of an impulse wave into the Golden Section. In such cases, the latter portion is .382 of the total distance when wave 5 is not extended, as shown in Figure 31, and .618 when it is, as shown in Figure 32. This guideline explains why a retracement following a fifth wave often has double resistance at the same level: the end of the preceding fourth wave and the .382 retracement point.

Figure 31

Figure 32

Corrective Wave Multiples

In a zigzag, the length of wave C is usually equal to that of wave A, as shown in Figure 33, although it is not uncommonly 1.618 or .618 times the length of wave A. This same relationship applies to a second zigzag (labeled Y) relative to the first (labeled W) in a double zigzag pattern, as shown in Figure 34.

Figure 33          Figure 34

In a regular flat correction, waves A, B and C are, of course, approximately equal. In an expanded flat correction, wave C is usually 1.618 times the length of wave A. Often wave C will terminate beyond the end of wave A by .618 times the length of wave A. Each of these tendencies are illustrated in Figure 35. In rare cases, wave C is 2.618 times the length of wave A. Wave B in an expanded flat is sometimes 1.236 or 1.382 times the length of wave A.

Figure 35

In a triangle, we have found that at least two of the alternate waves are typically related to each other by .618. I.e., in a contracting, ascending or descending triangle, wave e = .618c, wave c = .618a, or wave d = .618b. In an expanding triangle, the multiple is 1.618.

In double and triple corrections, the net travel of one simple pattern is sometimes related to another by equality or, particularly if one of the threes is a triangle, by .618. Finally, wave 4 quite commonly spans a gross or net price range that has an equality or Fibonacci relationship to its corresponding wave 2. As with impulse waves, these relationships usually occur in percentage terms.

These guidelines increase dramatically in utility when used together, as several are simultaneously applicable in almost every situation at the various degrees of trend.

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