Fibonacci and the Golden Ratio

How nature's ratio can be used for investing

There is a unique ratio that can be used to describe the proportions of everything from nature's smallest building blocks, such as atoms, to the most advanced patterns in the universe, like the unimaginably large celestial bodies. Nature relies on this innate proportion to maintain balance, but the financial markets also seem to conform to this "golden ratio." 

The golden ratio is derived from the Fibonacci numbers, a series of numbers where each entry is the sum of the two preceding entries. Although this sequence is associated with Leonardo of Pisa, the Fibonacci numbers were actually formulated for the first time by the Indian mathematician, Virahanka, 600 years prior to their introduction to the Western world.

Here, we take a look at some technical analysis tools that have been developed to take advantage of the pattern.

Key Takeaways

  • The golden ratio is an irrational number that is equal to (1+√5)/2, or approximately 1.618...
  • The ratio is derived from an ancient Indian mathematical formula which Western society named for Leonardo Fibonacci, who introduced the concept to Europe.
  • Nature uses this ratio to maintain balance, and the financial markets seem to as well.
  • The Fibonacci sequence can be applied to finance by using four main techniques: retracements, arcs, fans, and time zones.
  • Fibonacci numbers have become famous in popular culture, although some experts say their importance is exaggerated.

History of the Mathematics

Mathematicians, scientists, and naturalists have known about the golden ratio for centuries. It's derived from the Fibonacci sequence, named after the Pisan mathematician Leonardo Fibonacci, who lived from around 1175 A.D. until around 1250 A.D.

Although Fibonacci introduced these numbers to the Western world, they were actually discovered by Indian mathematicians hundreds of years earlier. The poet Pingala used them to count the syllables of Sanskrit poetry around 200 B.C., and the method for calculating them was formulated by the Indian mathematician Virahanka around 800 years later.

In this sequence, each number is simply the sum of the two preceding numbers (1, 1, 2, 3, 5, 8, 13, etc.).

Fibonacci borrowed heavily from Indian and Arabic sources. In his book Liber Abaci, he described the Hindu-Arabic numeral system represented by the numbers 0 through 9. He called this the "Modus Indorum," or the method of the Indians.

But this sequence is not all that important. The essential part is that as the numbers get larger, the quotient between each successive pair of Fibonacci numbers approximates 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, ϕ, and the divine proportion, among others.

So, why is this number so important? Well, many things in nature have dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function for the building blocks of nature.

The exact value of the golden ratio can be calculated by:


ϕ = (1+√5) / 2

Examples of the Golden Ratio

Don't believe it? Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get a number around 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different components throughout nature.

The golden ratio also appears in the arts, because it is more aesthetically pleasing than other proportions. The Parthenon in Athens, the Great Pyramid in Giza, and Da Vinci's Mona Lisa all incorporate rectangles whose dimensions are based on the golden ratio. It seems to be unavoidable.

But does that mean it works in finance? Actually, financial markets have the very same mathematical base as these natural phenomena. Below we will examine some ways in which the golden ratio can be applied to finance, and we'll show some charts as proof.

Trading and Investing With the Golden Ratio

The golden ratio is frequently used by traders and technical analysts, who use it to forecast market-driven price movements. This is because the Fibonacci numbers and the golden ratio have a strong psychological importance in herd behavior. Traders are more likely to take profits or cover losses at certain price points, which happen to be marked by the golden ratio.

Curiously, the widespread use of the golden ratio in trading analysis forms something of a self-fulfilling prophecy: the more traders rely on Fibonacci-based trading strategies, the more effective those strategies will tend to be.

Thanks to books like Dan Brown's The Da Vinci Code, the golden ratio has been elevated to almost mystical levels in popular culture. However, some mathematicians have stated that the importance of this ratio is wildly exaggerated.

The Golden Ratio and Technical Analysis

When used in technical analysis, the golden ratio is typically translated into three percentages: 38.2%, 50%, and 61.8%. However, more multiples can be used when needed, such as 23.6%, 161.8%, 423%, and so on. Meanwhile, there are four ways that the Fibonacci sequence can be applied to charts: retracements, arcs, fans, and time zones. However, not all might be available, depending on the charting application being used.

1. Fibonacci Retracements

Fibonacci retracements use horizontal lines to indicate areas of support or resistance. Levels are calculated using the high and low points of the chart. Then five lines are drawn: the first at 100% (the high on the chart), the second at 61.8%, the third at 50%, the fourth at 38.2%, and the last one at 0% (the low on the chart). After a significant price movement up or down, the new support and resistance levels are often at or near these lines.

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Image by Sabrina Jiang © Investopedia 2020

2. Fibonacci Arcs

Finding the high and low of a chart is the first step to composing Fibonacci arcs. Then, with a compass-like movement, three curved lines are drawn at 38.2%, 50%, and 61.8% from the desired point. These lines anticipate the support and resistance levels, as well as trading ranges.

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Image by Sabrina Jiang © Investopedia 2020

3. Fibonacci Fans

Fibonacci fans are composed of diagonal lines. After the high and low of the chart is located, an invisible horizontal line is drawn through the rightmost point. This invisible line is then divided into 38.2%, 50%, and 61.8%, and lines are drawn from the leftmost point through each of these points. These lines indicate areas of support and resistance.

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Image by Sabrina Jiang © Investopedia 2020

4. Fibonacci Time Zones

Unlike the other Fibonacci methods, time zones are a series of vertical lines. They are composed by dividing a chart into segments with vertical lines spaced apart in increments that conform to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). Each line indicates a time in which major price movement can be expected.

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Image by Sabrina Jiang © Investopedia 2020

What Is the Relationship Between the Fibonacci Series and the Golden Ratio?

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.

Why Is the Fibonacci Sequence So Important?

The Fibonacci sequence is a recursive series of numbers where each value is determined by the two values immediately before it. For this reason, the Fibonacci numbers frequently appear in problems relating to population growth. When used in visual arts, they are also aesthetically pleasing, although their significance tends to be highly exaggerated in popular culture.

Why Is 1.618 So Important?

The number 1.61803... is better known as the golden ratio, and frequently appears in art, architecture, and natural sciences. It is derived from the Fibonacci series of numbers, where each entry is recursively defined by the entries preceding it. The golden ratio is also used in technical analysis because traders tend to behave in a predictable way near the psychologically-important Fibonacci lines.

The Bottom Line

Fibonacci studies are not intended to provide the primary indications for timing the entry and exit of a position; however, the numbers are useful for estimating areas of support and resistance. Many people use combinations of Fibonacci studies to obtain a more accurate forecast. For example, a trader may observe the intersecting points in a combination of the Fibonacci arcs and resistances.

Fibonacci studies are often used in conjunction with other forms of technical analysis. For example, Fibonacci studies, in combination with Elliott Waves, can be used to forecast the extent of the retracements after different waves. Hopefully, you can find your own niche use for the Fibonacci studies and add it to your set of investment tools.

Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
  1. Historia Mathematica. "The So-Called Fibonacci Numbers in Ancient and Medieval India," Pages 229-232.

  2. Encyclopaedia Britannica. "Fibonacci."

  3. Archive of Physics. "The Golden Mean and the Physics of Aestetics."

  4. Wild Maths. "Fibonacci and Bees."

  5. Popular Mechanics. "The Mysteries of the Golden Ratio Explained by Math and Sunflowers."

  6. University of Georgia. "The Golden Ratio In Art and Architecture."

  7. Live Science. "Phi: The Golden Ratio."

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