Finance and the Fibonacci Sequence By Benjamin R. Hull

History

• • Leonardo of Pisa (1170-1240) – Financial engineer; analyzed business problems – Brought Arabic Number system to the West Liber Abaci (1202) – Tools to calculate present value, compound interest, geometric series… – Rabbit problem • First western appearance of Fibonacci sequence

Liber Abaci

Rabbits!

The Golden Ratio

• • Biology Architecture • 𝜑 – = 1+√(5) 2 = 1.6180339… Approximate: Divide n+1 Fibonacci number by nth

The Fibonacci Sequence

• • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … 𝐹 𝑛 = 𝐹 𝑛−1 + 𝐹 𝑛−2 • 𝐹 𝑛 = 1 √5 ∙ ( 1+ 5 2 ) 𝑛 1 − √5 ∙ ( 1−√5 2 ) 𝑛

Technical Trading

• • Pioneered by Charles Dow A Random Walk Down Wall Street – Burton G. Malkiel – Cannot predict future trends based on past ones – Spread of information – Crowd instinct and mass psychology • Individuals want to break even ($40 to $50 drop example) • Price rise causes bandwagon effect

Strategy

• • Identify trends and act based on possible future movements Support and Resistance – Lines represent potential turnaround points – Support: Price passes and continues rising – Resistance: Price does not pass and drops

• • • • • • • • • • •

Fibonacci Ratios

Probabilities based on Fibonacci percentage Ratios – 100%, 61,8%, 38.2%, 23.6%, 0.0% – 50%, 78.6%, 76.4% ( ( ( Fibonacci Ratios: 1+√(5) ) 0 = 1 = 100% 2 1+√(5) ) −1 = 0.61803… ≈ 61.8% 2 1+√(5) ) −2 = 0.386196…≈ 38.6% 2 ( 1+√(5) 2 ) −3 = 0.23606

… ≈ 23.6% ( 1+√(5) 2 ) −∞ = 0 = 0.0% Other Ratios: 1 - ( 1+√(5) ) −3 = 0.76394… ≈ 76.4% 2 1 - ( 1+√(5) ) − 1 2 = 0.78615… … ≈ 78.6% 2 1 = 0.5 = 50.0% 2

Pictures

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Sources

• • • • • Books: “Fibonacci and the Financial Revolution”, Goetzmann, William N., The Origins of Value (pages 123-144). Oxford University Press Inc., New York, 2005. “Fibonacci Numbers”, N. N. Vorobev, Addison Wesley, Massachusetts, 2nd Edition, 1994. “A Random Walk Down Wall Street”, Malkiel, Burton G., W. W. Norton and Company, New York, 6nd Edition, 1996. “Technical Analysis from A to Z”, Achelis, Steven B., McGraw Hill, New York, Second Edition, 2001.

“Fibonacci and Lucas Numbers, and the Golden Section”, S. Vajda, Halsted Press: a division of John Wiley and Sons, New York, 2nd Edition, 1994. • • • • • • • • • • Pictures and Other Sources: http://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci.jpg/220px-Fibonacci.jpg

http://en.wikipedia.org/wiki/File:The_Parthenon_in_Athens.jpg

http://www.mathacademy.com/pr/prime/articles/fibonac/fibonac_8.gif

http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2F0%2F04%2FLiber_abbaci_magliab_f124r.jpg&h=z AQFTteev&s=1 http://upload.wikimedia.org/wikipedia/commons/1/1c/Fibretracement.png

http://stockcharts.com/school/doku.php?id=chart_school:chart_analysis:fibonacci_fan http://www.landlearn.net.au/newsletter/2008term3/images/rabbit-family-tree.png

http://theinsanium.blogspot.com/2011/01/fibonaccis-rabbits.html

http://en.wikipedia.org/wiki/Fibonacci_retracement http://blog.afraidtotrade.com/wp-content/uploads/112208-2335-fibonacci13.png