Fibonacci and the Golden Ratio

How This Natural Ratio Is Used in Investing

Reviewed by Gordon Scott
Fact checked by Vikki Velasquez

From the spirals of seashells to the arrangement of sunflower seeds, nature consistently follows a remarkable mathematical pattern known as the golden ratio. While this ratio has fascinated mathematicians and artists, it’s also found a surprising home in financial markets, where traders use it to analyze price movements and potential market turning points.

The golden ratio emerges from the Fibonacci sequence, where each number (with the exception of the first two) is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, 13…). Financial analysts have developed several technical analysis tools based on this mathematical relationship to identify potential support and resistance levels in asset prices. While some critics question its reliability, the ratio’s influence on market psychology and trader behavior has made it a persistent feature of technical analysis.

Below, we review the tools developed to take advantage of this pattern.

Key Takeaways

• The golden ratio is an irrational number equal to (1+√5)/2, or 1.6180.
• Technical analysts use four main Fibonacci-based techniques: retracements, arcs, fans, and time zones to identify potential support and resistance levels.
• The most commonly used Fibonacci percentages in trading are 38.2%, 50%, and 61.8%, derived from the sequence’s mathematical relationships.
• While popular in technical analysis, Fibonacci tools should be used with other indicators rather than as stand-alone trading signals.

History of Fibonacci’s Golden Ratio

Mathematicians, scientists, and naturalists have known about the golden ratio for centuries. The Fibonacci sequence is named after the mathematician Leonardo of Pisa, who lived from around 1175 to 1250. (Only later would he be known by the name Fibonacci.)

Although Fibonacci introduced these numbers to the Western world, they were 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 (except the first two) is simply the sum of the two preceding numbers (1, 1, 2, 3, 5, 8, 13, etc.).

Note

The golden ratio appears in the arts because it’s said to be 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 said to be based on the golden ratio.

The essential part is that as the numbers get larger, the ratio 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.

Important

The exact value of the golden ratio can be calculated as follows: ϕ = (1+√5) / 2.

Trading With the Golden Ratio

The golden ratio isn’t just a mathematical curiosity—it’s become a practical tool in financial markets through various trading strategies and analysis techniques. Technical analysts use this ratio and its related Fibonacci numbers to help predict potential support and resistance levels as possible turning points.

Traders typically focus on the key percentages 38.2%, 50%, and 61.8%. These numbers aren’t arbitrary but derived from mathematical relationships within the Fibonacci sequence. While some traders also use extended ratios such as 23.6% or 161.8%, the three core percentages form the backbone of most Fibonacci-based trading strategies.

The ratio’s effectiveness in financial markets stems partly from mass psychology. Traders and investors tend to cluster their trading decisions around these widely recognized levels. As more market participants bet on these levels as support or resistance, prices will be more likely to react when reaching these points.

However, it’s important to note that Fibonacci tools shouldn’t be used in isolation. Most traders incorporate them as just a part of a broader technical analysis strategy, using them to confirm signals from other indicators or to identify potential entry and exit points.

Note

Fibonacci levels often act as self-fulfilling support and resistance points because so many traders watch them.

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%, etc. Meanwhile, there are four ways that the Fibonacci sequence can be applied to charts: retracements, arcs, fans, and time zones. However, not all of these might be available, depending on your charting application.

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 a trend on a 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 shift up or down, the new support and resistance levels are often at or near these lines.

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 and trading ranges.

3. Fibonacci Fans

Fibonacci fans are composed of diagonal lines. After the high and low points on the chart are 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 point. These lines indicate areas of support and resistance.

4. Fibonacci Time Zones

Unlike the other Fibonacci methods, time zones are a series of 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 at which major price movements can be expected.

Critiques of the Fibonacci Sequence

Critics argue that using Fibonacci ratios in trading is more numerology than sound analysis. They point out that with so many potential Fibonacci levels—23.6%, 38.2%, 50%, 61.8%, etc., and their extensions—price is bound to bounce near one of these numbers simply by chance. Mathematician Keith Devlin, a well-known NPR contributor and writer of a biography of Leonardo of Pisa, has been particularly vocal about the misuse of Fibonacci sequences, noting that people tend to notice patterns that confirm their beliefs while ignoring countless examples where the ratios fail to predict market movements.

In addition, since these levels are widely known and watched by traders, any perceived effectiveness might reflect collective market psychology rather than any mathematical basis to price movement. Of course, this self-fulfilling aspect would make these tools valuable regardless of their theoretical validity.

The most salient critique of the renown of the Fibonacci ratios is that it leaves out Leonardo of Pisa’s true mathematical legacy, which lies not in the sequence that bears his nickname but in his revolutionary work introducing practical arithmetic to medieval Europe. His 1202 book Liber Abaci (“Book of Calculation”) helped replace cumbersome Roman numerals with the Hindu-Arabic number system (zero through nine) we use today. Before this, the mathematics of basic commerce relied on area “experts” with rudimentary abacuses for anything but the most basic calculations—and much of the wider European population was essentially innumerate.

Leonardo’s contributions transformed commerce and banking by making calculations far more efficient; we can fairly compare the change to the difference between mathematics before and after computers revolutionized the world in the second half of the 20th century. The mathematician demonstrated practical applications for merchants, including converting currencies, calculating profit margins, and computing interest—tools that formed the foundation of modern commercial mathematics. It’s ironic that he’s best known today for the Fibonacci sequence—a supposed hidden feature of nature—when his most significant contribution was making everyday arithmetic accessible to European merchants and many everyday people.

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 about 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’s derived from the Fibonacci series of numbers, where the preceding entries recursively define each entry. The golden ratio is also used in technical analysis, likely at least in part because traders behave predictably near the psychologically important Fibonacci lines.

The Bottom Line

Fibonacci analysis offers traders and investors a structured approach to identifying potential price levels where markets might reverse or pause. While its mathematical origins are fascinating, the ratio’s real value in financial markets comes from its practical application through retracements, arcs, fans, and time zones—tools used for pinpointing possible support and resistance levels.

However, any success with Fibonacci techniques requires using them alongside other forms of technical analysis.