Stock Market Analysis, Phi and the Fibonacci Sequence

Human expectations occur in a ratio that approaches Phi.

Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative.

Phi and Fibonacci numbers are used to predict stocks

Phi (1.618), the Golden Mean and the numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, …) have been used with great success to analyze and predict stock market moves, known as retracements. Forbes ASAP featured a story on the work of scientist Stephen Wolfram in cellular automata (underlying rules that determine seemingly random phenomenon) stating “This seashell may hold the secret of stock market behavior, computers that think and the future of science.”

Markets may be as geometrically perfect as a spider’s web

Ermanometry Research shows the markets to be perfectly patterned, explaining that humans, being part of nature, create perfect geometric relationships in their behaviors, not unlike a spider spinning a geometrically perfect web with no conscious awareness of its amazing feat. Ermanometry applies the logarithmic spirals found in sea shells with dynamic ratios in 3D to relate one market move to others.

Phi, or Golden Ratio, patterns often define the timing of highs and lows and price resistance points

The golden ratio, or phi, appears frequently enough in the timing of highs and lows and price resistance points that adding this tool to technical analysis of the markets may help to identify fibonacci retracements, the key turning points in price movements. The photos below illustrate how the Golden Mean Gauge and Phi-based analysis software (PhiMatrix) can be used to identify these turns in the market. The middle arm of the gauge keeps the phi point of the outer arms as the gauge is opened and closed. The lines of the phi-based software are all in phi relationship to one another. The ratios of Fibonacci numbers, commonly used in technical market analysis, converge on phi as explained on the Fibonacci Series page.  Click on each photo to enlarge.

DJIA Daily Chart
from 1/2004
through 11/04
using PhiMatrix
DJIA Monthly Chart
from 1/2000
through 6/2003
using a Golden
Mean Gauge

Phi and Fibonacci numbers define the price movements of stocks in Elliott Wave Theory

Fibonacci numbers were used by W.D Gann and R.N. Elliott, pioneers in technical analysis of the stock market.  In Elliott Wave Theory, all major market moves are described by a five-wave series, adding to the potential to identify the turns described above. The classic Elliott Waveseries consists of an initial wave up, a second wave down (often retracing 61.8% of the initial move up), then the third wave (usually the largest) up again, then another retracement, and finally the fifth wave, which would exhaust the movement. In addition, each of the major waves (1, 3, and 5) could themselves be separated into subwaves, and so on, and exhibit other Fibonacci relationships.A sample stock price wave analysis could look something like this:

Elliott wave in stock prices and the Fibonacci series and retracements based on phi, the golden ratio

Major, minor and sub waves are shown in red, yellow and green, and the total number of increases and decreases (2, 5 or 8) is a Fibonacci number. Note too that the predicted end result is based in the Fibonacci series as well as the end price is 61.8% of the high and 0.618  is equal to 1/Φ and 0.382 is 1/Φ2.

For additional information on Elliott Wave Theory, its application and related concepts, please consult the resources below.

Free E-book



Beautiful Pictures – A gallery of simple, clear graphs that show how Elliott waves are in Fibonacci proportion to each other in time and price back to 1932.  Click HERE to order.


Elliott Wave Principle – Described as the “the definitive textbook on the Wave Principle,” this classic is the most useful and comprehensive guide to understanding and applying the Elliott Wave Principle.  Click HERE to order.


Socionomics: The Science of History and Social Prediction – Illustrates the historical correlation between patterned shifts in social mood and their most sensitive register, the stock market. It also includes essays, based on over 20 years of research, that correlates social mood trends to music, sports, corporate culture, peace, war and macroeconomic trends.  Click HERE to order.



How to Forecast Gold & Silver Using the Wave Principle – Robert Prechter’s work in publishing specific gold and silver forecasts for 22 years during one of the metals’ most historically baffling periods and his correct calling of nearly every major turn and trend during that time.  The years in question ran from 1979-2001, a period book-ended by gold’s $850 all-time high in 1980, and its low near $250 in 2001. “How to Forecast Gold and Silver” will shows what matters and what doesn’t when you want to invest in precious metals, looking in one place to predict where precious metals would go: to those markets themselves, and how to do it right.

Other books click HERE.

Tutorials / extracts




Note:  The above information is presented for educational purposes.  No express or implied warranty is given as to the results that will be achieved by its application and no responsibility is accepted for financial decisions based on this information.
U.S. Government Required Disclaimer – Trading of any security in any market potential rewards, but also potential risks. You must be aware of the risks and be willing to accept them in order to invest in the markets. Don’t trade with money you can’t afford to lose. This is neither a solicitation nor an offer to buy/sell futures, stocks or options. No representation is being made that any account will or is likely to achieve profits or losses similar to those discussed on this web site. The past performance of any trading system or methodology is not necessarily indicative of future results. No guarantee is made that you will be able to replicate the same results.




  1. Leong says

    I work in a telecommunications company and while i was reading this paper i was surprised to see that between the top three operators in the 40 countries studied over the past decade, in order to achieve an equilibrium point in the mobile market (where prices are competitive and the revenue allows companies to still invest in this market) the average ratio of the market share between the first and second operator and also between the second and third operator is 1.6!!! Coincidence?

  2. corey92 says

    Very interesting read, I have recently joined the stock market and look forward to testing this strategy for trading!

  3. Gytana says

    I would like more info on how to use the matrix with the markets. Do you just place this grid on top of the market interactive chart? Seems that the matrix can be interpreted to what you want to see. Anyone who can explain how to use would be appreciated.

    • Chartsky says

      Gytana . . .

      I’ve found that using Fibonacci Levels in conjunction with other technical analysis (i.e., support and resistance levels, trend lines, price patterns, etc.) works very well. The more confirmation price is expected to hold at a particular level the more trustworthy that level should be.

      Good Luck!

  4. says

    This site has certain ethical considerations concerning disseminating information about PHI and its’ relationship to financial markets, for obvious reasons. It’s probably because of this, that the above information places the majority (>61.8%) of the movements of the market primarily on human behavior, as it mirrors other natural systems. Those who post comments on this site, have none of these ethical considerations. For example, the difference today, is that high speed proximity traders, dark pools, and sovereign wealth funds use complex mathematical algorithms based on PHI to exert a much greater influence on market behavior than humans. Back in the day when investors would own a stock for weeks, months, and years, this may have held true.
    The average has changed dramatically, however, to just days, hours, and minutes. The reality of individual stockholders in large numbers determining the movements of the market has changed dramatically, as well. Otherwise, how else could the Shanghai Composite Index, for example, on June 4, 2012, the 23rd Anniversary of the Tiananmen Square Massacre of June 4, 1989, open at 2346.98, and be down at the close, exactly 64.89? The Friday, 3-6-6 S&P generational bottom, hit 666.666…The market reversed on 3-9-6 and has been going up ever since. How much you ask? Five years (1826.18 days) to the day later, 3-7-14, the S&P was at 1884, up 1218 points. Per day, 1218/1826=.666… Now, the funny PHI part, at least one of them, is that 366 days X .18=the 66th day, or March 6, (3-6)! Just recently, on 3-2-15, near the 6th Anniversary, the market closed at
    2117.39, or 3.1618 X 666.69 (the 3-6-6 closing value)!
    Behavioral Finance is real, but so is PHI. In this context, PHI is the Z-axis, the TIME-axis, the only axis algos care about, no matter what investors are caring about.


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