Fibonacci Numbers and Their Application in Trend Analysis
By Arthur W. Hafner, Ph.D.
September 1, 2000
Leonardo Fibonacci is a mathematician born around 1175 in Pisa, Italy. He died in 1250. In 1202 he investigated a problem about how fast rabbits could breed in ideal circumstances. Although this problem may not sound too interesting, it turns out that it opened a whole new avenue of ideas that have been the object of study over the past 800 years.
The purpose of this piece is to describe Fibonacci numbers and to show one of their applications in trend analysis.
The problem goes something like this: How many pairs of rabbits will there in one year if the following assumptions hold:
1. We begin with a newly-born pair of rabbits, one male and one female.
2. Rabbits can mate at the age of one month so that the female can produce another pair of rabbits at the end of its second month, producing a male and female rabbit.
3. Questions of genetics are ignored.
4. The rabbits never die, none escape, and all females reproduce.
With these assumptions, how many pairs of rabbits will there be in one year?
This is a fun and relatively easy problem so that the interested reader might want to take a pencil and piece of paper and spend a few minutes on it. An illustrated solution to Fibonacci's Rabbits is instructive. <http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits > The number of pairs of rabbits in the field at the start of each month is
<1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610. >, a series of numbers that has come to be called Fibonacci Numbers.
Interesting points to notice about Fibonacci Numbers
I. The series begins with 1 and each successive number is the sum of the two previous numbers.
II. The series is made up of even, odd, and prime numbers.
Except for the primes, all Fibonacci Numbers can be factored.
III. Except for the first few numbers, (a) any given number in the sequence is about 1.618 times the proceeding number, and (b) about 0.618 times the following number.
IV. Looking at the values that are next to each other and dividing by their sums, we see the following:
(1, 2, 3) 1/3 = 0.333 2/3 = 0.666
(2, 3, 5) 2/5 = 0.400 3/5 = 0.600
(3, 5, 8) 3/8 = 0.375 5/8 = 0.625
(5, 8, 13) 5/13 = 0.385 8/13 = 0.615
(8, 13, 21) 8/21 = 0.381 13/21 = 0.619
(13, 21, 34) 13/34 = 0.382 21/34 = 0.618
(21, 34, 55) 21/55 = 0.382 34/55 = 0.618
If more decimal places were used, one would see the interesting fact that the numbers begin to converge to 0.382 (38.2%) and 0.618 (61.8%).
V. Other Fibonacci percents can be determined by dividing numbers in the sequence above them. For example:
(a) Divide the first number by the second. This is 1/1, which is 100%.
(b) Divide the second number in the sequence by the third. This is 1/2 or 0.50, which is 50%.
(c) Taking numbers that are not at the beginning of the sequence, as illustrated in III above, divide the number by the one following it, and the numbers converge to 61.8%.
(d) Taking numbers that are not at the beginning of the sequence, as illustrated in III above, divide a number by two numbers above it, and the numbers converge to 38.2%.
(e) Taking numbers that are not at the beginning of the sequence and divide a number by three numbers above it, the numbers converge to 23.6%.
(f) If one calculates the value of 1 minus each of these Fibonacci calculations, one obtains the values 38.2%, 50%, 61.8%, and 76.4%.
are the Fibonacci percents:
This is all pretty interesting analysis. Since the beginning of the Thirteenth Century when Fibonacci did his work, fellow mathematicians and other scientists and scholars have greatly extended it. Fibonacci patters have been identified in nature (leaves, petal arrangements, pine cones, pineapples, seeds, and shells) and they pop up in many other mathematical relationships.
Fibonacci Retracement Levels and Regions
The Fibonacci Percentages <0%, 23.6%, 38.2%, 50%, 61.8%, 76.4%, 100%> are called Fibonacci Retracement Levels. In trend analysis, these levels serve as an heuristic to set and explain price levels and to forecast points of resistance. The idea of retracement is that a stock, for example, may rise to a certain point and then profit taking will force its price down. Eventually, the issue again begins to rise in value, i.e. it retraces its former path back to the old high.
Fibonacci Retracement Levels or Regions are viewed by some as precursors of a renewed up-trend. Early identification of these can identify issues that are "on the move." The way that Fibonacci Retracement Levels work is that they are calculated on the basis of the range of the price change, where Range = Highest Price - Lowest Price. This range, which must be retraced as the issue reclaims its old high, is divided into percentage levels. The assertion is that the stock will likely make a new level by having successfully achieved an earlier lower level.
The balance of this essay illustrates the application of Fibonacci Retracement Levels. Disclaimer: No statements in this paper should be construed or thought to represent investment advice of any type since the sole purpose of the explanation is to illustrate the technique.
Technique for Calculating Fibonacci Retracement Levels
Many stocks can serve as an example, such as Exodus Communications (EXDS) and Sycamore Networks (SCMR). This latter issue will be used to demonstrate how to calculate the retracement levels.
1. Go to <www.BigCharts.com > and look at the chart for SCMR that shows the period of March 2 2000 through April 17 2000, where the high on March 2 was $199.50 and the low was $47.25 on April 17. We are interested in what happens following this period as the issue begins to "retrace" the ground that it lost when it fell from $199.50 to $47.25.
2. To see the high and low more clearly, look at a six month chart. Print it out since you might want to make some horizontal lines on it.
3. Calculate the Range for the stock, i.e. calculate the difference between its high and low.
The range is R = $152.25 <$152.25 = $199.50 - $47.25>
4. The Fibonacci Retracement percent points are 0%, 23.6%, 38.2%, 50%, 66.8%, 76.4%, and 100%, where 0% means no retracement from the high and 100% means that the replacement will have to be the whole range.
This is how the levels are computed:
L(0) = $199.50 - (0.00)($199.50 - $47.25) or $199.50
L(1) = $199.50 - (.236)($199.50 - $47.25) or $163.57
L(2) = $199.50 - (.382)($199.50 - $47.25) or $141.34
L(3) = $199.50 - (.500)($199.50 - $47.25) or $123.38
L(4) = $199.50 - (.618)($199.50 - $47.25) or $105.41
L(5) = $199.50 - (.762)($199.50 - $47.25) or $ 83.49
L(6) = $199.50 - (1.00)($199.50 - $47.25) or $ 47.25
From the theory, SCMR fell from $199.50 to $47.25, which is from L(0) to L(6). Since the stock is at its bottom, L(6), in its retracement the first point of resistance is to achieve L(5). If the stock successfully completes breaking through to L(5), then there is an expectation that its value may extend to break L(4), and if it makes that level, then on to L(3), etc. Of course, the point in using the Fibonacci Retracement Levels is to identify when the issue breaks to L(5) or L(4) and to take profitable advantage of this observation.