Is there a rationale for the use of Fibonacci numbers in Technical Analysis?

Monument of Leonardo da Pisa (Fibonacci), by G...

Monument of Leonardo da Pisa (Fibonacci), by Giovanni Paganucci, completed in 1863, in the Camposanto di Pisa. (Photo credit: Wikipedia)

Fibonacci and his numbers

The Italian mathematician who came to be known as Fibonacci (from the Latin word filius for son, and his father’s name Bonacci) is credited with drawing attention to the series of numbers which bears his name. His real name was Leonardo Pisano Bigollo, and his father was an Italian merchant who operated a profitable trading post at a port (Bejaioa) near Algiers in Algeria.

The young  Leonardo learned the use of Hindu-Arabic numerals from Islamic mathematicians of his day, and he incorporated their concepts into his book entitled “Liber Abaci” published in 1202.

The sequence arrived at by adding the two  previous numbers to calculate the next number. Thus the Fibonacci numbers are 1, 2 (1+1), 3 (2+1), 5 (3+2), 8 (5+3), 13 (8+5), 21 ( 13+8), 34 (21+13), 55 (34+21), 89 (55+34), 144 ( 89+55) etc.  The curious phenomenon about the sequence is that the ratio between any two consecutive numbers, after the first few, is constant at 61.8%.

Why Fibonacci numbers are relevant

This ratio occurs frequently in nature’s designs.  e.g. the distribution of bird feathers , the  increase in the width of teeth, the number of nodes in the spiral rotations of a pine cone, and  the proportions of the lengths of the bones in the human hand.

Not surprisingly people have embraced Fibonacci proportions in design principles since the  ratio between breadth and length is aesthetically more pleasing than a simple square or a rectangle would be. Thus the dimensions of credit cards, TV and computer screens, and in building the pyramids of Egypt are all based on Fibonacci ratios.

Fibonacci numbers have been adopted by Technical Analysts  there being two commonly used charting tools, Fibonacci retracements, and Fibonacci extensions (projections)

Although mostly used for share-price movements, they can also be applied on time scales, to suggest when significant change might be expected.

Fibonacci extensions

When price movements exceed the limits of the high and lows points  of trends or lesser impulsive moves, Fibonacci numbers can  still be set. By projecting an existing up-trend,  Fibonacci levels can be set at 161.8%, 261.8%, and 423.6% of the previous high; and conversely for downtrends.

Fibonacci retracements in an uptrend

Retracement of the share-price in an uptrend is encouraging for continuation of the trend. Most moderate profit taking pull-backs statistically are between one-third and two-thirds of the previous impulsive movement higher. In Fibonacci terms this would correspond to a range between 61.8% and 38.2%.  Such price retreats enable new buyers who missed out on stock acquisition before, to buy at more realistic prices. In this situation there may be no change in sentiment towards the company. Knowing what is statistically likely may enable traders to sell after a significant lift in share-price, and to perhaps re-buy again more cheaply within a relatively short interval.

Fibonacci retracements after a significant down-trend.

Fibonacci retracement levels can be valuable in setting price targets for stocks that for one-reason or another have had their share-price savaged, resulting in loss of market favour, stop/loss selling and  perhaps short selling by hedge funds and traders seeking to exploit the price weakness.

A rebound in the share-price in such a scenario, particularly when the price decline has been precipitate, is likely to be protracted and tentative. Shareholders who have stayed with the company may be too disillusioned to risk more of their capital, while new-shareholders attracted by the low price, may be reluctant to buy until sure that the trend has changed.

In these situations when the trading outlook may have been irreversibly compromised, the price may never return to its former heights. A rally in the stock-price particularly if it exceeds a previous peak, could well entice buyers to re-rate the stock higher, and to exit when they think there is sufficient profit.

The 50% retracement level seems to be etched into the memory and the psyche of traders, as a reasonable compromise between full recovery of the share-price, and a return to the doldrums.

It may be relevant that when a share-holder who has been unable or unwilling to sell his/her stake in the company, chooses to buy more stock at or near the bottom of the cycle, the 50% recovery point may be a position to exit with perhaps little or no loss of capital.

Clearly there is never a critical price at which shareholders will act but rather a range. Once again the range is commonly between 38.2% and 61.8%.

These are levels to be monitored more closely since they are often positions where trend may undergo a short or longer-term change, depending on market sentiment.

When one target is attained, the position should be reviewed, and new target(s) established.


 My experience


Over a twelve month period of writing this blog, I have often used the Fibonacci ratios 38.2% and 61.8%, together with the 50% level, as targets in price predictions for stocks that have been oversold, bottomed-out, and started to trend upwards. From personal experience these levels are often correct in suggesting points where the rise of the share-price may stall, or even reverse. Alternatively if there is enough momentum, the share price may continue to rise to the next level.

The leading Australian exponent of the use of Fibonacci numbers in Technical Analysis is Alan Oliver whose book “Trading With The Gods” provides readily comprehended details of the use of Fibonacci tools such as Fibonacci retracement, and Fibonacci extension levels.

A link to his website is:



Categories: Technical Analysis, Trading opinion

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3 replies

  1. Thanks for explaining in such an easy to understand way how the Fibonacci sequence is calculated. So much was ‘over my head’ but I managed to follow on. I feel there is a lot to put to good use extending the grey matter.

    Thanks for visiting my blog – much appreciated. Cheers Susan

    • Thank you Susan for taking time to look at one of my posts. I apologize for the dull nature of my blog, but it is something I find pleasure and challenge writing. Congratulations on your attractive site. I look forward to seeing more of your posts. Regards. Ken.


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