テクニカル分析

Fibonacci Theory

A bit of history of Fibonacci

Before we get in too much about what Fibonacci is, let’s first answer the question “who is Fibonacci?” Leonardo Pisano, or Leonardo Fibonacci as he is most widely known, was a European mathematician in the Middle Ages who wrote Liber Abaci (Book of Calculation) in 1202 AD. In this book he discussed a variety of topics including how to convert currencies and measurements for commerce, calculations of profit and interest, and a number of mathematical and geometric equations. However, there are two things that jump to the forefront of our discussion in today’s world. First, in the beginning portions of Liber Abaci he discussed the benefits of using the Arabic numeral system. At the time, the influence of the defunct Roman Empire was still strong, and the preference of most European citizens was to use Roman numerals. However, in Liber Abaci, Fibonacci provided a very powerful, influential, and easy-to-understand argument for using the Arabic numeral system. From that point on, the Arabic numeral system got a strong foothold in the European community and soon became the dominant method of mathematics in the region and eventually throughout the world. It was so strong that we still use the Arabic numeral system to this day.

The second important section of Liber Abaci that we use today is the Fibonacci sequence. The Fibonacci sequence is a series of numbers where each number in the series is the equivalent of the sum of the two numbers previous to it.

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and so on, out to infinity

As you can see from this sequence, we need to start out with two “seed” numbers, which are 0 and 1. We then add 0 and 1 to get the next number in the sequence, which is 1. You then take that value and add it to the number previous to it to get the next number in the sequence. If we continue to follow that pattern we get this:

1 + 1 = 2; 1 +2 = 3; 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13; 8 + 13 = 21; 13 + 21 = 34; 21 + 34 = 55; etc.

The Fibonacci sequence is so important to this discussion because we need those numbers to get our Fibonacci ratios. Without the Fibonacci sequence, the Fibonacci ratios wouldn’t exist.

What Makes a Fibonacci Ratio?

With the advent of the internet, there has been a lot of misinformation on which values make up Fibonacci Ratios. Proliferation of Fibonacci analysis, particularly in the realm of trading, has encouraged misinterpretations and misunderstandings of how and what makes a Fibonacci ratio. Let’s look at what a Fibonacci ratio is, how it is created, and some examples of those that are not really Fibonacci ratios at all.

Fibonacci Ratios

The math involved behind the Fibonacci ratios is rather simple. All we have to do is take certain numbers from the Fibonacci sequence and follow a pattern of division throughout it. As an example, let’s take a number in the sequence and divide it by the number that follows it.

0 ÷ 1 = 0
1 ÷ 1 = 1
1 ÷ 2 = 0.5
2 ÷ 3 = 0.67
3 ÷ 5 = 0.6
5 ÷ 8 = 0.625
8 ÷ 13 = 0.615
13 ÷ 21 = 0.619
21 ÷ 34 = 0.618
34 ÷ 55 = 0.618
55 ÷ 89 = 0.618

Notice a pattern developing here? Starting at 21 divided by 34 going out to infinity you will ALWAYS get 0.618!

We could do this with other numbers in the Fibonacci sequence as well. For instance by taking a number in the sequence and dividing it by the number that precedes it, we see another constant number that develops.

1 ÷ 0 = 0
1 ÷ 1 = 1
2 ÷ 1 = 2
3 ÷ 2 = 1.5
5 ÷ 3 = 1.67
8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
21 ÷ 13 = 1.615
34 ÷ 21 = 1.619
55 ÷ 34 = 1.618
89 ÷ 55 = 1.618
144 ÷ 89 = 1.618

Another pattern develops out of the numbers of the Fibonacci sequence. Now 1.618 actually holds even more significance because it is also called the Golden Ratio, the Golden Number, or the Divine Ratio, but I could go on for many more pages about that subject.

Here are some more examples of patterns that develop by taking numbers in the Fibonacci sequence and dividing them in a pattern with other numbers within the sequence.

Divide by 2nd following Divide by 2nd preceding Divide by 3rd following Divide by 3rd preceding
0 ÷ 1 = 0 1 ÷ 0 = 0 0 ÷ 2 = 0 2 ÷ 0 = 0
1 ÷ 2 = 0.5 2 ÷ 1 = 2 1 ÷ 3 = 0.33 3 ÷ 1 = 3
1 ÷ 3 = 0.33 3 ÷ 1 = 3 1 ÷ 5 = 0.2 5 ÷ 1 = 5
2 ÷ 5 = 0.4 5 ÷ 2 = 2.5 2 ÷ 8 = 0.25 8 ÷ 2 = 4
3 ÷ 8 = 0.375 8 ÷ 3 = 2.67 3 ÷ 13 = 0.231 13 ÷ 3 = 4.33
5 ÷ 13 = 0.385 13 ÷ 5 = 2.6 5 ÷ 21 = 0.238 21 ÷ 5 = 4.2
8 ÷ 21 = 0.381 21 ÷ 8 = 2.625 8 ÷ 34 = 0.235 34 ÷ 8 = 4.25
13 ÷ 34 = 0.382 34 ÷ 13 = 2.615 13 ÷ 55 = 0.236 55 ÷ 13 = 4.231
21 ÷ 55 = 0.382 55 ÷ 21 = 2.619 21 ÷ 89 = 0.236 89 ÷ 21 = 4.231
34 ÷ 89 = 0.382 89 ÷ 34 = 2.618 34 ÷ 144 = 0.236 144 ÷ 34 = 4.235
55 ÷ 144 = 0.382 144 ÷ 55 = 2.618 55 ÷ 233 = 0.236 233 ÷ 55 = 4.236
89 ÷ 233 = 0.382 233 ÷ 89 = 2.618 89 ÷ 377 = 0.236 377 ÷ 89 = 4.236
144 ÷ 377 = 0.382 377 ÷ 144 = 2.618 144 ÷ 610 = 0.236 610 ÷ 144 = 4.236

As you can see, we could get many different numbers by just taking numbers within the Fibonacci sequence and developing a divisory pattern within the sequence. However, this is not the only way to come up with Fibonacci ratios. Once we have the numbers from dividing, we can then take the square roots of each of those numbers to get more numbers. See the chart below for some examples of those values.

Fibonacci Ratio Operation Result
0.236 Square root of 0.236 0.486
0.382 Square root of 0.382 0.618
0.618 Square root of 0.618 0.786
1.618 Square root of 1.618 1.272
2.618 Square root of 2.618 1.618
4.236 Square root of 4.236 2.058

The last part of making these numbers Fibonacci ratios is to simply turn them into percentages. Using that rationale 0.236 becomes 23.6%, 0.382 becomes 38.2%, etc. So looking at our analysis we can then see that 23.6%, 38.2%, 48.6%, 61.8%, 78.6%, 127.2%, 161.8%, 205.8%, 261.8%, and 423.6% are our bona fide Fibonacci ratios.

What about 50%?

While the 50% ratio is often used in Fibonacci analysis, it is not a Fibonacci ratio. Some say that the 50% level is a Gann ratio, created by W.D. Gann in the early 1900’s. Others call the 50% level an inverse of a “sacred ratio.” Just like the Fibonacci ratios, many people will either take the inverse or square root of the “sacred ratios” to form more values. Some examples can be found in the table below.

Sacred Ratio Operation Result Inverse of Sacred Ratio
1 Square root of 1 1 1
2 Square root of 2 1.414 0.5
3 Square root of 3 1.732 0.333
4 Square root of 4 2 2.236
5 Square root of 5 0.25 0.2

Whatever the source, the 50% ratio seems to be a rather important and relevant level when trading, so often times it is included in Fibonacci analysis as if it were a Fibonacci ratio. Some of the other numbers included in the table have been mistaken as Fibonacci ratios as well, but obviously are not.

Whatever the source, the 50% ratio seems to be a rather important and relevant level when trading, so often times it is included in Fibonacci analysis as if it were a Fibonacci ratio. Some of the other numbers included in the table have been mistaken as Fibonacci ratios as well, but obviously are not.

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