Chapter 7-6 Fibonacci

In the 13th century AD a young mathematician named Leonardo Pisano Bigollo, most commonly known simply as Fibonacci, introduced the western world to an ancient number system in an effort to predict population growth.  The number system is quite simplistic- each number in the sequence is the sum of the previous two numbers beginning with 0 then 1.  The first 15 numbers of the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and 377.  The numbers themselves are of little consequence, but the relationships between them can hold incredible values when applied properly.  The most common application of the Fibonacci sequence is the calculation of the golden ratio.  Equal to 1.6180339887, this number has fascinating aesthetic and mathematical qualities that have shown up over the course of history countless times.  The further along the Fibonacci sequence you travel the closer the ratio between any two numbers gets to the golden ratio.  For example:

8/5 = 1.6       34/21 = 1.6190       377/233 = 1.6180

For financial applications the inverse of the golden ratio is much more relevant as it falls between 0 and 1.  Consider the inverses of the previous relationships:

5/8 = .6         21/34 = .6176         233/377 = .6180

In addition to the golden mean, another very significant relationship occurs between any given Fibonacci number and the number two places before it.

5/13 = .3846       21/55 = .3818       233/610 = .3820

Notice the relationship between the two number sets, particularly the final set?  Curiously enough the two Fibonacci ratios total 1, representing equal distance (.3820) from the upper and lower constraints of 1 and 0.

At this point in time, you’re probably wondering what any of this has to do with trading stocks in the 21st century.  Well, that is exactly what we are leading up to- the use of the Fibonacci derived golden mean, more specifically the inverse of it, as a powerful tool for technical analysis.  The most common application of the golden ratio in investing is through the calculation and charting of Fibonacci retracements.  Fibonacci retracements operate on the assumption that stocks and other securities do not go up in a straight line indefinitely.  Even in the strongest (or weakest) of markets the price action of a security ebbs and flows as buyers and sellers interact with one another.  Examples of this include mutual funds reallocating assets, institutional investors establishing positions, earnings reports and other market moving events, and widespread profit taking.  Fibonacci retracements are based on the idea that support and resistance areas will formulate after a significant move in a securities price.  These areas, for whatever reason, often happen at two common intervals between the high and low point of a move in a stock’s price- 61.80% off of the top and 38.2% off of the bottom.  In addition, many traders also consider the third order Fibonacci number, 23.60%, although its application is often used only in conjunction with another indicator.


Fibonacci retracements occur on all time frames and can be charted on any type of stock chart.  They can occur on a minute by minute basis visible on a 1 minute chart after just a few bars or over the course of several week or months as seen on a daily chart covering the entire recorded history of a stock’s price action.

Consider the following numeric examples of intraday Fibonacci retracements in action:

  • US Steel, X, opens up the day at $50
  • X dells of to $49.50 and puts in a low for the day
  • X rallies to $50.50 and puts in a high for the day
  • At this point the measured move is $1
  • Day traders take profits and X begins to pull back a bit
  • X settles at $50.12 and consolidates for fifteen minutes

As an observant day trader, you notice this price as a near perfect retracement to the 61.80% Fibonacci level.  You measure this by taking the difference between the high and the low (the size of the move; $50.50 – $49.50) and subtracting 38 cents from the top (or adding 62 cents from the bottom).  So, where’s the trade here?

If the market is strong the best available trade is to buy X at the retracement level under the assumption that it will hold, and hopefully retest the highs at some point in time throughout the day.  This is a low risk trade because a tight stop can be placed slightly below the Fib level.  Furthermore, the highs of the day now represent a breakout opportunity.  Under the best case scenario X will move aggressively through the highs and you will realize significant gains aided by an early entry at a lower price.

This can be shown graphically on a one-minute chart:


If the market is weak the best available trade is to short X through the retracement level under the assumption that the Fibonacci retracement has failed and the current high of the day will hold for the remainder of the trading session.  Once the .6180 Fib level has failed it is not uncommon for a stock to test the .3820 and even the lows of the day.  This is also a fairly low risk trade as a stop can be placed slightly above the highs of the day and then lowered as the price action heads south.  Under the best case scenario X will move aggressively through the .3820 level and then the lows of the day and you will realize significant gains aided by an early entry at a higher price.

This can be shown graphically on a one-minute chart:


This example demonstrates the second most common use for Fibonacci levels- the Fibonacci breakout.  When trading Fibonacci levels it is important to consider the overall sentiment of the market as a metric for which direction the trade is likely to work out in.  In the previous example we see a security that has tagged its Fib level and bounced, but failed to make a real push northbound.  In fact, as the bumps its head against the level the bounces get smaller and smaller.  This is bearish activity and indicates that the Fibonacci level is more likely to behave as a breakout level than a pivot point.  As we indicated earlier, the market as a whole was weak.  This is another heads up that playing a bounce is less likely to work than playing a breakout.  If the market isn’t giving any indication then the trade should be passed on- there’s nothing wrong with leaving a set up alone if there is no information available to give you an upper hand.  As Fibonacci level breakouts often occur violently, the levels make very good spots for stop loss orders.  This is beneficial if you trade a pivot and it turns out to be a breakout. Although the trade is not going to work you are able to control your losses and move on to the next opportunity.

Fibonacci retracements are very helpful when considering completely unexpected moves in the market.  Often times, after a news event for example, securities will exhibit extremely abnormal behavior.  Often this is based on some sort of unexpected event that causes immediate panic in the markets while institutions try and work out what has occurred.  This type of price action usually demonstrates the rubber band effect- a stock will move violently in one direction as though it were stretching out a rubber band, then snap back equally as violently after an exhaustion print.  Consider the following example that takes place a one-minute char to aid your understanding:


At 2:53 on this day Massey Energy, MEE, announced that it was considering a takeover offer from a rival mining company.  MEE skyrockets, printing up almost $4 (9%) in the same minute immediately following the news release.  As more and more traders get involved the price action levels out.  As a trader you are now looking for an entry, you’ve missed the initial pop but you still want to get involved.  This is a momentum play and the momentum is upwards, now the tough part is the timing.  Notice where the price is at 3:00?  The price has pulled back to the .6180 Fibonacci level to the penny, this is your low risk entry.  Agile traders would have realized a $.90 profit within the same minute of entry.  A breakeven at worst stop loss order turns this into a risk free trade with the logical exit at or above the high of the initial move.

As with all other market indicators, the greater the time frame a signal occurs on the stronger the signal is.  The previous examples demonstrated how to use Fibonacci retracements on an intraday basis.  These trades will present themselves several times a day, some will work, and some won’t.  As a result, investors who do not day trade the market choose to focus on a security’s daily chart to find Fibonacci retracements that take place over days, weeks, and even months.  These signals will be much rarer, but also much stronger.

Consider the following numeric examples of Fibonacci retracements in action on a higher time frame:

  • After some consolidation, Netflix, NFLX, tests support at $90, reaching $95
  • Investors turn bullish on the stock, and NFLX makes a new 52-week high of $141
  • Buying slows down a bit and some profits are taken off the table
  • NFLX settles around $123 and consolidates for a week
  • Buying begins again and NFLX tests its high, breaking out with strength

This can be shown graphically on a daily chart:


As you can see here the Fibonacci level held up well.  NFLX only closed below it one time before gearing up and taking off.  A stop at the 50% retracement level ($118) was never in jeopardy.  Notice also how the .6180 level also almost perfectly lines up with the short term resistance area that occurred in July.  This makes the buy signal even stronger as we’ve learned that resistance becomes support once violated.  Once the price action turned bullish NFLX behaved about as well as you could ask for.  New highs were made aggressively, tested and confirmed within two weeks, and then made again with three very strong days.


Fibonacci retracements occur on both up moves and down moves.  The identification and action is the same in both cases, only reversed.  Consider the following example of a down move Fibonacci retracement breakout:


  • After topping out at $124, Potash, POT, begins selling off
  • POT sells off $40 and eventually finds a base down at $84
  • After failing to go lower, buyers return and POT retraces to the .3820 Fib level
  • Every time POT drops buyers return, and the drops get smaller and smaller
  • Finally, sellers give up and POT blasts through $100
  • POT stalls again at the .6180
  • An outside bid comes in and POT prints up to the $150’s


This can be shown graphically on a daily chart:


In this case, the Fibonacci level acted as resistance, as it should, but ended up as a breakout level rather than a pivot point.  As you can see, POT bumped its head up against this level several times.  Each time it would hit sellers would return and it would be driven back down.  However, each time it dropped the decline became smaller and smaller.   This should have been interpreted as bullish activity and indicated that a Fibonacci breakout was the higher probability trade.  Eventually sellers gave up and POT blasted through the resistance, demonstrating significant strength and followed by 5 very bullish days.  On August 17th a bid for Potash came in and the stock skyrocketed.

Fibonacci Expansions

Thus far we have learned the value of Fibonacci levels as key indicators for entering a position.  However, their value does not end there.  This next section will detail how the Fibonacci sequence can be used as a valuable tool in identifying exits for positions that have already been established.  For many traders this represents one of the most challenging aspects of trading.  Every trader will tell you time and time again about a huge profit they had in a stock that evaporated virtually instantly because they didn’t know when to get out.  Fortunately there is a simple yet successful method for exiting positions based on the same principles of the Fibonacci retracement.

Fibonacci expansions are based on the exact same principles as the retracements, only using the proper form of the golden ratio rather than the inverse.  Consider what we have learned so far- when a security has made a measurable move, as seen through a marked high and low, the price is likely to settle between the two poles, most often at 38.2% and 61.8% of the range.  This is calculated through the inverse of the golden mean 1/1.6180 such that the retracement level falls within the two extremes.  The same principle holds true when price action breaks out of a consolidation area.  Once a breakout occurs, the measured move often reaches 1.6180 times the size of the consolidation making this an ideal location to take profits off the table.


Consider a stock that has been trading between $49 and $50 for the better part of the morning.  The consolidation area measures $1 and a big move in either direction is likely to occur when it is broken.  When the price action breaks out of the consolidation range it is likely to do so violently, quickly, and then test the level it has just violated.  The worst thing that can happen is that you get in on the breakout and then see a large paper profit disappear as you fail to take profits appropriately.  In this instance, the Fibonacci expansion level would be equal to $50.62.  This can be calculated two ways- as 1.6180 times the move off the bottom, equal to 1$, or an additional 61.80% of the move added on to the top.  In both cases the result is the same- $50.62.

The Fibonacci expansion level of 161.81% is a great place to take profits as it represents an achievable target as well as a likely pivot point.  As with the 61.8% and 38.2% internal levels, the 161.80% level often acts as both a pivot point and a potential breakout area.  Market sentiment will help dictate which direction the price action will go when it reaches the Fibonacci level, however, as this is a congestion area volatility will be high and the price action may be very unpredictable.  As a result, the Fibonacci expansion is a very logical target when taking profits.  Some traders choose to take off 1/3 or 1/2 of their position with the hopes that the remaining portion will break out and turn the trade into a home run.  This is perfectly acceptable as the locked in profits will turn the trade into a risk free venture.  Tight stop loss orders should also be used when the target is reached to avoid giving back significant amounts of unrealized gains.

Consider the following chart demonstrating a Fibonacci extension on an intraday basis:


  • This example above shows a stock that was bouncing between $52.15 and $53.45
  • The consolidation area in this case measures $1.30; equal to $53.45-$52.15
  • Eventually sellers outmuscled buyers and the stock breaks down
  • The trade here is to short the stock through the lows of the day
    • Governed by the Fibonacci expansion, the profit target in this case would be $51.35; equal to $53.45- ($1.30 * 1.6180)
    • The stock breaks down and the target of $51.35 is hit relatively quickly.
    • Price action exceeds the Fibonacci expansion level by less than ten cents before reversing.  Fortunately, this does not bother us because we were prepared and took profits aggressively.

Consider the following demonstration of a Fibonacci extension on a daily chart:


  • This example above shows a stock that made a strong move from $95 to $141
  • The consolidation area in this case measures $46; equal to $141-$95
  • The trade here is to buy the stock through the highs of the year
    • Governed by the Fibonacci expansion, the profit target in this case would be $169.43; equal to $95 + ($46 * 1.6180)
    • The stock breaks out and the target of $169.43 is reached
    • Price action exceeds the Fibonacci expansion level and then reverses the next day.  Fortunately, this does not bother us because we were prepared and took profits aggressively.

The previous chart should look very familiar, as it is the exact same NFLX chart we looked at earlier when learning about Fibonacci retracements.  This chart does a great job of demonstrating how price action behaves when approaching Fibonacci levels both inside and outside of a measured move.   Although this is not always the case, every Fibonacci retracement is not necessarily followed by an extension, it is important to understand how securities come to behave in the ways that they do.  Seemingly unpredictable events should start to appear clearer when applying these analytical techniques.  By knowing where the Fibonacci levels are both within a measured move and on either side of it you can greatly increase your chances of entering, exiting, and managing your positions profitably.

To better see the interaction between the two Fibonacci levels consider the NFLX chart again, but this time with both sets of Fibonacci numbers indicated:


This phenomenon is also demonstrated on the POT chart we used earlier:


Fibonacci Clusters

When deciding whether or not to enter or exit a position the more signals you have on your side the better.  When it comes to Fibonacci levels, this is demonstrated through a Fibonacci cluster.  A Fibonacci cluster is simply a price level that represents a Fibonacci level on multiple Fibonacci cycles.  Fibonacci cycles most often occur when a stock is making swing highs and lows within a larger trend.  In this instance there are multiple areas in which to draw a Fibonacci retracement between the various highs and lows.  When Fibonacci levels overlap amongst multiple retracement scales they become stronger symbols.  This is simply due to the fact that they represent a Fibonacci level on more than one timeframe and scale.

Consider the following intraday charts that demonstrate the construction of a Fibonacci cluster representing the intersection of the .3820 level of one measured area with the .6180 of another:


This example shows the 61.80% Fibonacci retracement of a local move intersecting with the 38.2% Fibonacci retracement of the overall down trend of the stock, marked by the orange line.  As you can see this level held up very well creating a 3 point move the first time the stock reversed and an 8 point move the second time.  Additionally, this is an excellent example because on top of the presence of a Fibonacci cluster, the secondary move occurs at the 61.80% retracement of the overall trend.  This obviously is not always the case, but certainly noteworthy and another demonstration of how powerful the Fibonacci levels are.

Fibonacci clusters also often occur in choppy, trendless markets.  These can be very helpful in identifying pivot points for use as entries.  Identifying potential areas for this to occur is relatively simple- look for a swing high followed by a swing low followed by a swing high.  That might sound wordy, but essentially you are looking for a V shape (or inverted V shape).  Fibonacci clusters, if present, can easily be identified by drawing Fibonacci retracements on both legs of the chart formation.  These clusters can be identified days, even weeks in advance as it will take some time for the price action work itself out before settling into a neutral territory.  For this reason this type of Fibonacci cluster is generally applied to daily charts.

Consider the following example of constructing a Fibonacci cluster on a Goldman Sachs (GS) daily chart:


In this case the Fibonacci cluster occurs at the intersection of the midpoint (50%) of the Leg 1 (up) and the 61.80% retracement of Leg 2 (down).  As you can see the cluster is fully formed and identified by the swing low that occurs on August 31.  However, as we’ve pointed out, it takes some time for this level to be tested.  In this case the stock begins another leg up, makes a lower high, and begins another leg down.  This time around it hits the cluster on September 23, bounces around for a week, and begins a very strong move upwards taking out all resistance along the way.