Fibonacci Analysis Page 3
The key observation to draw from the Nasdaq chart in Figure 1.4 is that each price range identified—A, B, C, and range D—is subdivided into three ratios, 38.2 percent, 50 percent, and 61.8 percent, relative to its starting and ending price levels. The calculations that result from subdividing each range are entirely dependent on the range first selected, and these ranges require thought as they become critical to our success. We will not be starting at the price high and then using the extreme price low as favored by our industry to apply Fibonacci analysis. Figure 1.4 is a good introduction but more is involved. The correct ranges to use will be addressed in detail within the next chapter when we focus on the concept of market expansion and contraction.
FIGURE 1.4 Nasdaq Composite Index Monthly Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software
Leonardo Fibonacci (circa 1170-1250) was not the first to document the Fibonacci number sequence. That honor falls to ancient Hindu mathematicians.13
FIGURE 1.5 Leonardo Fibonacci—(Leonardo da Pisa), by Giovanni Paganucci, in the Camposanto di Pisa Cemetary, Italy
Source: Private photograph collection of Connie Brown
Fibonacci might have lost his identity with the number sequence entirely had it not been for Édouard Anatole Lucas14 (1842-1891), a French mathematician who rediscovered the Fibonacci sequence in the late nineteenth century. It was Lucas who attributed the number series to Fibonacci’s book Liber Abaci (Book of the Abacus, 1202), thereby establishing the name of the numbers as the Fibonacci Number Sequence. Lucas is known for defining a formula to find the nth term of the Fibonacci sequence.
Liber Abaci demonstrates routine computations that merchants performed when converting currencies. Fibonacci showed the advantages of using the Hindu-Arabic number system compared with the Roman numeral system. In hindsight, Fibonacci made things much harder than they needed to be by frequently expressing fractions as unit fractions. As example: One quarter and one third of a tree lie below ground, a total of twenty-one palmi in length. What is the length of the tree?15 Fortunately, Fibonacci’s book made a huge impact, and as a result, we do not have to trade markets using Roman numerals. But sadly Fibonacci’s studies through Mesopotamia have been lost. The library and museum of Mesopotamian artifacts vanished in the twenty-first century, when the Iraq National Museum in Baghdad was looted and burned after the 2003 Gulf War invasion. In Chapter 4, you will see a valuable artifact within the British Museum of great interest, but an Egyptian Egyptologist was near tears when she told me during my 2007 visit to Cairo that all the manuscripts and papyrus they had in Iraq are now lost forever.
CHAPTER NOTES
Abbreviations are references to the bibliography contained at the back of this book. As example, Leh, 8, 17 refers to the alphabetized bibliography code, Leh, which further describes: Lehner, Mark. The Complete Pyramids: Solving the Ancient Mysteries. London: Thames and Hudson Ltd., 1997.
1 The first clear definition of what later became known as the Golden Ratio was given around 300 BC by the founder of geometry, Euclid of Alexandria. Euclid defined a proportion derived from a simple division of a line into what he called its “extreme and mean ratio.” In Euclid’s words:A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
—HEATH
2 Hall, 191-223; see also Taylor.
3 Taylor.
4 Hall; Taylor.
5 Aristotle, Metaphysica, 1-5.
6 Stroh, 82-85.
7 Robin.
8 El-Daly, 60.
9 Mad, 5. The story of phi begins in the fourth dynasty of Egypt, about 2500 BC. The last of the original great Seven Wonders of the Ancient World are the extraordinary pyramids of Egypt. The Great Pyramid of Khufu (called Cheops by the Greeks), the Pyramid of Khafre (Chephren), Khufu’s grandson, and the smallest of the three called the Pyramid of Menkaure, are located on the Giza plateau near Cairo.Everyone agrees on the meticulous precision with which the pyramids were built. Khufu is oriented to within three feet six inches of an arc from true North. In addition, it is well known that Khufu contains the phi proportion between its sides and base. With only rudimentary arithmetic and only ropes or sticks for measuring, this seems like an improbable result. But we will study this geometry further in Chapter 4.
Khufu has been measured repeatedly since the seventeenth century AD, each time with greater accuracy. A right angle could be constructed by folding a rope with thirteen equally spaced knots, such that three spaces between knots form one side of a triangle, four spaces form another side, and five spaces form the hypotenuse. Several researchers have found murals and papyruses that show knotted ropes. Several pyramids were built on the proportions of the three-four-five triangle. The height, half-base, and slant angle of Khafre have been measured at 143.5 meters, 107.5 meters, and fifty-three degrees ten feet. The ratio of these, 143.5/107.5 = 1.33488, is very close to 4/3 as verified by the angle. (Leh, 7) This ratio of 1.33 will take on greater significance later in the book.
10 Merriam-Webster’s Collegiate Dictionary: Tenth Edition shows that Phi (φ) is pronounced, “fi” like fire, and not pie. The reciprocal phi or 1/(φ) is pronounced, “fee” like bee.
11 Fibonacci ratios can be added, subtracted, multiplied, or divided and the result will always be another Fibonacci ratio. As examples:
0.618 × 0.618 = 0.382 0.618 - 0.382 = 0.236
0.382 × 0.618 = 0.236 0.382 - 0.236 = 0.146
0.236 × 0.618 = 0.146 etc. 0.236 - 0.146 = 0.090
12 Ibid.
13 Gies, 57, 110, and Dunlap, 35. The Italian mathematician born 1170 AD in Pisa, Italy, was commonly known as Fibonacci, which was a shortened form of Filius Bonaccio (son of Bonaccio). Fibonacci’s book Liber Abaci (Book of the Abacus) introduced in 1202 the Hindu-Arabic system of numbers to Europe. Fibonacci states in his introduction that he accompanied his father Guilielmo, on an extended commercial mission in Algeria with a group of Pisan merchants. There, he says, his father had him instructed in the Hindu-Arabic numerals and computations. He continued his studies in Egypt. (Cooke, 289) (Fibonacci was recognized as a great mathematician, but his solution to the rabbit problem was overlooked when he determined the number of immortal pairs of adult and baby rabbits each month over a one year interval revealing the Fibonacci number sequence 0, 1, 1, 2, 3, 5, 8, 13, . . . and so on.) See also Parm and Parm, 85, and Pin, 45. In the classical period of Indian mathematics (400 AD-1200 AD), important scholars like Aryabhata, Brahmagupta, and Bhaskara II made early contributions to the study of the decimal number system, zero, negative numbers, arithmetic, and algebra. Trigonometry was introduced into ancient India through the translations of Greek works, further showing that Fibonacci applied the number sequence to solve the rabbit problem, but he did not discover it.
14 Hark, 276-288.
15 Cooke, 289.
CHAPTER 2
The Concept of Market Expansion and Contraction
FIGURE 2.1 Iceland Low
Source: Jacques Descloitres, MDOS Rapid Response Team, NASA/GSFC
TO UNDERSTAND MARKET EXPANSION and contraction, you need to understand the differences between ratios, means, and proportion. In this chapter, the concepts of market expansion and contraction will be evaluated within bar charts, and then you will learn how to develop the unique support and resistance price grid a specific market uses to create its next market swing. The voids between the levels of support and resistance help you work smarter with oscillators and with other technical tools as well. So the best place to start is with a quick review of the terms ratio, mean, and proportion.
Ratio: A ratio between two numbers a and b is a : b or a/b.
The inverse ratio between the two numbers a and b is b : a or b/a.
The Fibonacci number series is created by adding two numbers together to form the next in the series, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . and so on. When you want
to know the relative relationship between two different numbers, you divide one by the other. If you divide 144 by 89, the result is a ratio of 1.6179. You can round this result to 1.618. If you divide 89 by 55, you obtain the ratio 1.61818181818. . . . In fact, if you divide the higher numbers after 233 by their lower number in the number sequence, you find a near exact 1.61805233 result, as an example, 233 divided by 144. Each pair of numbers will have the same corresponding relationship between them. So you know 1.618 is important but not why just yet.
If you divide 55 by 89, you produce a reciprocal relationship between the two values of 0.61797, or rounded 0.618. If you divide 89 by 144, you find a similar result of 0.61805. So 1.618 (Phi or φ) and the reciprocal 0.618 (phi, 1/φ) are clearly ratios of significance when you compare two nearby numbers in the Fibonacci sequence.
If you consider alternate relationships between two points, such as 21 divided by 55, you obtain the ratio 0.3818181818 or 0.382. This will be another ratio of great interest to you when you start analyzing markets.
Mean: If you want to find the mean b between a and c, you may consider three common approaches. There are others but you will not be encountering them.
The arithmetic mean b of a and c is b = (a + c)/2.
The geometric mean b of a and c is b = √ac.
The harmonic mean b of a and c will equal the following:
b = (2ac)/(a + c).
Statisticians commonly use the arithmetic mean, and it is usually the one implied when we just say, “mean.” The geometric mean is used to find average rates of growth. The growth of sales for a business or the growth of price swings in a rally uses the geometric mean.
When would one use the geometric mean as opposed to arithmetic mean? What is the use of the geometric mean in general?
The arithmetic mean is relevant any time several quantities add together to produce a total. The arithmetic mean answers the question, if all the quantities had the same value, what would that value need to be in order to achieve the same total?
In the same way, the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean answers the question, if all the quantities had the same value, what would that value need to be in order to achieve the same product?
For example, suppose you have an investment that earns 10 percent the first year, 60 percent the second year, and 20 percent the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers indicate is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant quantity is the geometric mean of these three numbers.
The question about finding the average rate of return can be rephrased as, by what constant factor would your investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and 1.20 the third? The answer is the geometric mean (1.10 × 1.60 × 1.20). If you calculate this geometric mean, you get approximately 1.283, so the average rate of return is about 28 percent (not 30 percent, which is what the arithmetic mean of 10 percent, 60 percent, and 20 percent would give you).
Any time you have a number of factors contributing to a product and you want to find the “average” factor, the answer is the geometric mean. The example of interest rates is probably the application most used in everyday life.
It is known that the geometric mean is always less than or equal to the arithmetic mean (equality holding only when A = B). The proof of this is quite short and follows from the fact that (√A - (√B)2) is always a non-negative number. In the latter pages of this book, it will be important to know the difference, as you will begin to look at harmonic proportion within the markets.
The harmonic mean is used to calculate average rates such as distance per time, or speed. An analyst might consider the use of a harmonic mean when a third axis incorporates harmonic frequencies derived from ratios the market swings produce. This will be a new area of geometry for our industry discussed near the end of this book, but it is essential the concept of proportion is accurate in your mind.
Proportion: If we consider Plato’s1 request to make an uneven cut, we might realize an even cut would result in a whole: to a segment ratio of 2 : 1. The two equal segments would be 1 : 1. These ratios are not equal 1 : 1 :: 2 : 1, and so they cannot present a proportional relationship. There is only one way to form a proportion from a simple ratio, and that is through the golden section. Plato wants us to discover that only one special ratio exists such that the whole to the longer equals the longer to the shorter. (See Figure 2.2.) He knew this would result in the continuous geometric proportion found in nature without violating his secret oath.
Why did Plato use the division of a line rather than give us numbers to work from? The reason is the Fibonacci numbers give an irrational number that cannot be expressed as a simple fraction. Only by solving this problem geometrically can we discover that the longer segment will always have a value of 1.6180339, Phi (φ) relative to the whole. How this applies to chart analysis is this will be true regardless of what pivot points we use to define the range. The lesser segment of the line will always be 0.6180339 or phi (1/φ). The mean value is 1. The Pythagoreans attached great meaning to this mean value of 1,2 as it represents the unity that binds all living forms. When you see books on the Golden Ratio subdividing fish, horses, flowers, and shells, the math ratios and proportions are surprisingly similar. But what is critical in market analysis is to know what internal structures are significant to create the price range to be subdivided. There are clues everywhere to help us select the right ranges to subdivide, but they are rarely the most obvious price highs and lows. It is only through divine proportion can we see the unity of all in nature. Soon you will see markets abide by the same unity.
FIGURE 2.2 The Golden Ratio Proportions
There are two kinds of proportions to consider. The first is discontinuous (four-termed) such as 4 : 8 :: 5 : 10 or a : b :: c : d. This example has an invariant ratio of 1 : 2. The second kind of proportion is continuous (three-termed) or a : b :: b : c, which equals a : b : c, where b is the geometric mean of a and c. In market charts, you always select two market pivot price points, one being a price low and the other being a high or vice versa to subdivide the price range. The 61.8 percent ratio will be the farthest ratio from the starting point of the range. In the weekly chart for Caterpillar Inc. (CAT) stock in Figure 2.3, the distance from the price swing high for CAT to the 61.8 percent retracement level and the distance from the price low at B to the 38.2 percent retracement have equal lengths. It can be said of Figure 2.3 that the price high is to 61.8 percent as the price low is to 38.2 percent.
FIGURE 2.3 Caterpillar—Weekly Chart
Connie Brown, www.aeroinvest.com. Source: Copyright © 2008 Market Analyst Software
Plato went one step further by referencing an extended continuous geometric proportion such as 1 : 3 :: 3 : 9 :: 9 : 27. This proportion has an invariant ratio of 1 : 3 or 1/3. This ratio will have specific implications for how you can project future prices when the market is breaking into new highs. It also has an influence on where you should make that projection and often it is not from a price low that ends an old trend. As these concepts are used in the analysis ahead, you will find much of what you will do is visual. It will be applied geometry, and the actual math behind the ratios and proportions will be less difficult than you may think now.
The chart in Figure 2.3 shows how most traders create Fibonacci ratios within a bar chart. Actually, this chart begins to deviate from what most traders do slightly. Most traders will define the range by starting at the extreme price low of a major swing and then select the price high. You will soon see why you must never do this. After teaching numerous seminars this next point seems to catch everyone in application. May I suggest you underline or, even better, write this down on a separate note pad and attach it to your com
puter:When you want to find a price retracement level for support (under the current market price), start with a price high and move your cursor down toward a price low to define the range to be subdivided. (Also, the 38.2 percent, 50 percent, and 61.8 percent subdivision results must fall below the current price.)
Most quote vendors have taught you to start at the bottom because they draw their lines on your screen incorrectly by extending the lines away from the y-axis and the new price data. If they are wrong, by drawing the subdivided results behind your cursor rather than forward towards the most recent price bar,3 the problem can be easily fixed. Change your default settings so that the line ratios are drawn across the entire computer screen. If your vendor’s default setting includes more ratios than 38.2 percent, 50.0 percent, and 61.8 percent, remove the extra ratios. As an example, you will not use 75 percent.
When you want to find the resistance price level (over the current market price), you must start with a price low and move your cursor toward a price high. (The resulting subdivisions of the range must be above the current market price.)
Why? If you select your price range in the opposite manner, the method will force you to remain a beginner, as you will never be able to consider the internal proportions forming within the range of any price swing. This will become clearer as this chapter develops.