Leonardo Pisano, nicknamed Fibonacci, was an Italian mathematician born in Pisa in the year 1170. His father Guglielmo Bonaccio worked at a trading post in Bugia, now called Béjaïa, a Mediterranean port in northeastern Algeria. As a young man, Fibonacci studied mathematics in Bugia, and during his extensive travels, he learned about the advantages of the Hindu-Arabic numeral system.

The Golden Ratio

In 1202, after returning to Italy, Fibonacci documented what he had learned in the “Liber Abaci“ (“Book of Abacus“). In the “Liber Abaci,” Fibonacci described the numerical series that is now named after him. In the Fibonacci sequence of numbers, after 0 and 1, each number is the sum of the two prior numbers. Hence, the sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 and so on, extending to infinity. Each number is approximately 1.618 times greater than the preceding number.

This value:1.618 is called Phi or the “Golden Ratio“. The Golden Ratio mysteriously appears frequently in the natural world, architecture, fine art, and biology. For example, the ratio has been observed in the Parthenon, in Leonardo da Vinci’s painting the Mona Lisa, sunflowers, rose petals, mollusk shells, tree branches, human faces, ancient Greek vases, and even the spiral galaxies of outer space.

Fibonacci Levels Used in the Financial Markets

In the context of trading, the numbers used in Fibonacci retracements are not numbers in Fibonacci’s sequence; instead, they are derived from mathematical relationships between numbers in the sequence. The basis of the “golden” Fibonacci ratio of 61.8% comes from dividing a number in the Fibonacci series by the number that follows it.

For example, 89/144 = 0.6180. The 38.2% ratio is derived from dividing a number in the Fibonacci series by the number two places to the right. For example: 89/233 = 0.3819. The 23.6% ratio is derived from dividing a number in the Fibonacci series by the number three places to the right. For example: 89/377 = 0.2360. 

Fibonacci retracement levels are depicted by taking high and low points on a chart and marking the key Fibonacci ratios of 23.6%, 38.2%, and 61.8% horizontally to produce a grid. These horizontal lines are used to identify possible price reversal points.

The 50% retracement level is normally included in the grid of Fibonacci levels that can be drawn using charting software. While the 50% retracement level is not based on a Fibonacci number, it is widely viewed as an important potential reversal level, notably recognized in Dow Theory and also in the work of W.D. Gann.

Fibonacci Retracement Levels as Trading Strategy

Fibonacci retracements are often used as part of a trend-trading strategy. In this scenario, traders observe a retracement taking place within a trend and try to make low-risk entries in the direction of the initial trend using Fibonacci levels. Traders using this strategy anticipate that a price has a high probability of bouncing from the Fibonacci levels back in the direction of the initial trend.

For example, on the EUR/USD daily chart below, we can see that a major downtrend began in May 2014 (point A). The price then bottomed in June (point B) and retraced upward to approximately the 38.2% Fibonacci retracement level of the down move (point C). 

In this case, the 38.2% level would have been an excellent place to enter a short position in order to capitalize on the continuation of the downtrend that started in May. There is no doubt that many traders were also watching the 50% retracement level and the 61.8% retracement level, but in this case, the market was not bullish enough to reach those points. Instead, EUR/USD turned lower, resuming the downtrend movement and taking out the prior low in a fairly fluid movement.

The likelihood of a reversal increases if there is a confluence of technical signals when the price reaches a Fibonacci level. Other popular technical indicators that are used in conjunction with Fibonacci levels include candlestick patterns, trendlines, volume, momentum oscillators, and moving averages. A greater number of confirming indicators in play equates to a more robust reversal signal.

Fibonacci retracements are used on a variety of financial instruments, including stocks, commodities, and foreign currency exchanges. They are also used on multiple timeframes. However, as with other technical indicators, the predictive value is proportional to the time frame used, with greater weight given to longer timeframes. For example, a 38.2% retracement on a weekly chart is a far more important technical level than a 38.2% retracement on a five-minute chart.

Using Fibonacci Extensions

While Fibonacci retracement levels can be used to forecast potential areas of support or resistance where traders can enter the market in hopes of catching the resumption of an initial trend, Fibonacci extensions can complement this strategy by giving traders Fibonacci-based profit targets. Fibonacci extensions consist of levels drawn beyond the standard 100% level and can be used by traders to project areas that make good potential exits for their trades in the direction of the trend. The major Fibonacci extension levels are 161.8%, 261.8% and 423.6%.

Let’s take a look at an example here, using the same EUR/USD daily chart:

Looking at the Fibonacci extension level drawn on the EUR/USD chart above, we can see that a potential price target for a trader holding a short position from the 38% retracement described earlier lies below at the 161.8% level, at 1.3195.

The Bottom Line

Fibonacci retracement levels often indicate reversal points with uncanny accuracy. However, they are harder to trade than they look in retrospect. These levels are best used as a tool within a broader strategy. Ideally, this strategy is one that looks for the confluence of several indicators to identify potential reversal areas offering low-risk, high-potential-reward trade entries.

Fibonacci trading tools, however, tend to suffer from the same problems as other universal trading strategies, such as the Elliott Wave theory. That said, many traders find success using Fibonacci ratios and retracements to place transactions within long-term price trends.

Fibonacci retracement can become even more powerful when used in conjunction with other indicators or technical signals. Investopedia Academy’s Technical Analysis course covers these indicators as well as how to transform patterns into actionable trading plans.

The rectangle is a classical technical analysis pattern described by horizontal lines showing significant support and resistance. It can be successfully traded by buying at support and selling at resistance or by waiting for a breakout from the formation and using the measuring principle.

The Rectangle in Classical Technical Analysis

The rectangle formation is an example of a “price pattern” in technical analysis. Price patterns derive from the work of Richard Schabaker, considered the father of technical analysis, and Edwards and Magee, who wrote what many consider the bible on the subject.

This period of technical analysis derives from a time when charts were kept by hand on graph paper and even simple moving averages (SMA) had to be maintained by hand or with the use of a large, clunky adding machine.

Rather than modern technical analysis, which relies on indicators, such as moving average convergence divergence (MACD), technical analysts assumed that price patterns repeat themselves over and over throughout time. Pattern recognition meant pattern prediction and thus trading profit.

Many of the price patterns are based on geometrical figures. There are ascending, descending and symmetrical triangles, pennants and wedges. Occasionally, more fancifulshapes are seen, such as the head-and-shoulders formation.

The Rectangle: Supply and Demand in Balance

A price chart or graph may be thought of as an X-ray of supply and demand. Figure 1 describes a rectangle pattern where supply and demand are in approximate balance for an extended period of time. The shares move in a narrow range, hitting resistance at the rectangle’s top and finding support at its bottom. The rectangle can occur over a protracted period of time or form quickly amid a relatively wide-ranging series of bounded fluctuations. Schabaker notes that it can approach a square in its proportions.

In any case, it is a pattern which shows trader indecision, one in which the bulls and bears are approximately equally powerful.

Most technicians agree, the rectangle can serve as either a reversal or continuation formation. As a reversal pattern, it ends a trend either up or down. As a continuation pattern, it signifies a pause in the prevailing trend, with the expectation that the prior trend will eventually resume. In either case, the rectangle shows a tug of war between buyers and sellers. Ultimately, either accumulation or distribution prevails, and the shares breakout or breakdown.

“Significant” Support and Resistance

The concepts of support and resistance are critical to understanding the rectangle formation.

• Support is defined as any price point below the current market price where buying should emerge to create, at least temporarily, a pause in a downtrend.
• Resistance, on the other hand, is any price above the current market price where selling should emerge to create, at least temporarily, a pause in an uptrend.

In a rectangle, what may be referred to as “significant” support or resistance emerges – that is, a price level returned to again and again. Whereas trendlines in technical analysis are typically drawn on a diagonal, the diagramming of support and resistance requires horizontal trendlines.

ImClone Systems: an Example of a Rectangle Formation

Figure 2 of ImClone Systems (IMCL) employs open-high-low-close bars (rather than candlesticks) and is absent of any indicators, such as MACD. The only addition is a 30-week moving average (MA), which could have been calculated in the classical era.

Several observations are worth making on this chart. First, note that an intermediate uptrend line, in force for approximately one year, is broken. The break shows the uptrend has ended. Thus, the prolonged rectangle can either be a reversal or consolidation formation. Until there is a breakdown or breakout from the confines of the rectangle – roughly $37.50 to $47.50 – the pattern’s interpretation is uncertain.

Second, horizontal lines drawn on the chart denote significant support and resistance. Significant support was first established in September, tested twice in the early part of the year and retested in June. At each test of support, there was sufficient buying interest to drive the stock higher.

Significant resistance at $47.50 was first touched in August, then probed in October, April and July. At each juncture, the sellers overwhelmed buyers, and the stock receded. This vacillation between significant support and resistance creates the rectangle shape.

One final observation is the slope of the 30-week MA. Of all moving averages, this may best describe the trend. It relates to the rectangle by showing the sideways nature of the formation. In an uptrend or downtrend, the 30-week MA will slope up or down, not sideways. Note how in the early stages of the chart it sloped higher, mimicking the uptrend. Later it flattened and began to slope sideways, showing the prolonged consolidation.

Trading the Rectangle

The following are two basic strategies for trading a rectangle:

• The first is to buy at support and sell at resistance (one can also sell short at resistance and cover the short sale at support). To mitigate risk, in case the stock breaks down from support, a very tight stop can be employed of perhaps 3%. For example, if one bought ImClone at $37.50, the stop-loss would be 3% lower than $37.50 or $1.12. The trader would exit the position if the stock hit $36.38 ($37.50-$1.12).
• Another method to trade the rectangle is to wait for the breakout. As with all technical patterns, this breakout should ideally occur on above-normal volume. To know when to consider exiting the trade, the trader could use the measuring principle described below.

The Measuring Principle

The measuring principle allows you to set a specific minimum price target. Such a target should give you the objectivity to hold during periods of minor countertrend movement.

The measuring principle works with any well-defined technical analysis pattern, such as a rectangle or triangle. To calculate the minimum target, first establish the height of the pattern. In the case of ImClone Systems Figure 3 shows the calculation as follows: