Loans and money issues

The link between Fibonacci and financial markets comes through another school of thought for technical analysis, Elliott Wave Theory, named after Ralph Nelson Elliott (1871–1948). Elliott first made the connection between his Wave Theory and the Fibonacci sequence of numbers in his blog Nature’s Law — The Secret of the Universe (1946). Elliott Wave Theory suggests financial markets move in five waves of progression followed by three waves of regression. As such a 5–3 wave move completes a wave cycle. The five “up” waves are labelled 1–5, while the three “down” waves are labelled a–c. Of necessity, waves 1, 3 and 5 are seen as impulsive waves while waves 2 and 4 are seen as corrective.
Remembering the Fibonacci sequence, it should be immediately obvious that 1, 3 and 5 are Fibonacci numbers. Furthermore, if we break each wave down into sub-waves, we notice two things, firstly that each sub-wave conforms to the 5–3 wave pattern and secondly that when we add up these sub-waves we come to 21 impulsive and 13 corrective waves, making 34 in total. Once again, 13, 21 and 34 are all Fibonacci sequence numbers.
Fibonacci sequence numbers are also used in other technical indicators, such as in moving averages — e.g. 5, 13 and 21 moving averages, 21, 34 and 55 or 31, 55 and 144. Within the financial markets, the most widely used application of the Golden ratio is through the Fibonacci retracement, which relates to the fact that corrective waves have retraced the previous wave by 38.2%, 50% or 61.8%. Fibonacci fan lines provide key support or resistance corresponding to the Fibonacci retracement levels. Once such a Fibonacci fan line support or resistance has been broken, this tends to suggest the extension of a correction and thus a potential wave reversal. In sum, Fibonacci levels can provide crucial tops and bottoms in the market and are widely watched by both short- and medium-term currency market participants.
A final school of thought is Gann Theory, created by W.D. Gann (1878–1955), which seeks to predict future prices using specific geometric angles. Gann angles or Gann lines can be created by graphing price against time. The basic Gann angle or line is created by assuming an increase in one unit for both price and time, resulting in a line which is at a 45◦ angle to both axes. Because of the price and time increases involved, this is called a 1 × 1 angle. Gann lines are drawn off major price tops and bottoms. If the price is above the 1 × 1 line, this signals a bullish trend and conversely if it breaks below the line this signals a bearish reversal. Including the
1 × 1 angle, Gann identified nine significant angles or lines relating to price and time:
1 × 8 — 82.5 degrees
1 × 4 — 75 degrees
1 × 3 — 71.25 degrees
1 × 2 — 63.75 degrees
1 × 1 — 45 degrees
2 × 1 — 26.25 degrees
3 × 1 — 18.75 degrees
4 × 1 — 15 degrees
8 × 1 — 7.5 degrees
Each of the angles or lines can provide a support or resistance depending on the trend. Generally speaking, the 1 × 1 angle as reflected by a trend-line is not sustainable given the steepness of the angle involved. Prices cannot continue appreciating at a 45◦ angle forever. The 3 × 1 angle is generally viewed as more sustainable in terms of price trends over the long term.

Having gone through the basic building blocks of technical analysis and the technical indicators that are used, we will now look at the major technical schools of thought that have dominated the way technical analysts and traders look at price patterns. The first one to focus on is the Fibonacci school of thought, named after Leonardo Fibonacci, an Italian mathematician born in 1170. Fibonacci discovered a series of numbers such that each number is the sum of the two previous numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and so on . . . To some, these numbers may seem more or less random. In fact, they are actually far from random, containing important interrelationships, and they are found in a surprising number of real-life examples. Indeed, it is not too much of an exaggeration to suggest that these numbers represent the mathematical building blocks of life. For a start, note that any given number is roughly 1.618 times the previous one. Equally, any number is 0.618 times the following number. As it stands, this does not answer the question of how Fibonacci happened to found, albeit inadvertently, a type of technical analysis. For this, we have to look first at Fibonacci’s so-called “rabbit problem”, which relates to his attempt to demonstrate the application of Hindu–Arabic numerals through the example of rabbits. The mathematical problem that Fibonacci posed is that if two rabbits were put in an isolated place, how many pairs of rabbits could be produced from that pair in a year if every month each pair produces a new pair, which itself from the second month also becomes reproductive? At the start of the first month, there would only be the first pair. By the start of the second month, there would be the original pair plus one new pair, resulting in two pairs of rabbits. However, during that second month, the original pair will again produce another pair while the second pair is maturing. Thus, at the start of the third month, there should be three pairs, which brings us back to the Fibonacci number series. In terms of a mathematical formula, this can be expressed as: X n +1 = X n + X n−1 where X n is the number of pairs of rabbits after n months.
This became known as the Fibonacci sequence, as coined by the French mathematician Edouard Lucas (1842–1891). As the Fibonacci sequence progresses, a clear relationship between the numbers becomes apparent, as reflected by the 0.618 and 1.618 ratios mentioned above. The very fact that there can be a consistent ratio between numbers is itself “statistically significant”, confirming that there is more in this than just a random series of numbers. Note also that if you take any number and divide it by the number two higher in the sequence the ratio comes to 0.382. Not coincidentally, 38.2% and 61.8% are major Fibonacci retracement levels within the Fibonacci school of technical analysis.
While we look to Fibonacci and Lucas as the founders of modern-day Fibonacci analysis, it appears that long before them the importance of this sequence of numbers and ratios was well known and appreciated. Indeed, these ratios appear to have been used in the construction of both the Great Pyramid of Giza in Egypt and the Parthenon in Greece. The 0.618 or 1.618 ratio, also known as the Golden ratio, is commonly viewed in mathematics as one of the building blocks of natural growth patterns — in geometry as in life. Even the human body can be shown to contain elements of the Golden ratio, measuring the distance from the feet to the navel and in turn from the navel to the top of the head as a ratio. The basic building blocks of human beings, the DNA double helix, also contains the Golden ratio.