Technical analysis teaches traders to use Fibonacci retracement price levels (61.8%, 38.2%) when measuring possible pullbacks. If you couldn’t find these numbers when looking at the Fibonacci sequence—you wouldn’t be alone.

These ratios are drawn from another mathematical constant, the Golden Ratio Φ (also golden mean, golden section or divine proportion). How are these numbers connected?

## Raising Rabbits

In the 13th Century, a young man named Leonardo Pisano turned his mathematical aptitude for computation and numbers to the matter of raising rabbits. It was a very practical study in the Middle Ages, since the citizens of Pisa needed food. Starting with 1 pair of rabbits, he counted how many pairs of rabbits he obtained at the start of each month n. Each newborn pair of bunnies could breed in as little as 2 months, so they soon contributed to the rapid growth witnessed in the local rabbit population. Leonardo found that the population of mating pairs fit this linear recurrence relation:

F(1) = F(2) = 1

F(n) = F(n-1) + F(n-2)

Month n |
F(n) |

1 | 1 |

2 | 1 |

3 | 2 |

4 | 3 |

5 | 5 |

6 | 8 |

7 | 13 |

8 | 21 |

9 | 34 |

10 | 55 |

11 | 89 |

12 | 144 |

13 | 233 |

## Fibonacci Sequence

Leonardo Pisano first published his findings in Liber Abaci (1202) which literally means The Book of Abacus (since this was before graphing calculators—much less graphing smartwatches). However, it wasn’t until the 19th Century that they were named after him by French mathematician Edouard Lucas. At least, they were named after Fibonacci (which literally means “Son of Bonacci”). He was his father’s son, after all.

As an aside, Lucas researched sequences and series. He’s best known for his Lucas numbers whose linear recurrence relation is equivalent to that of the Fibonacci sequence, but using different values for F(1) and F(2). The Fibonacci sequence is an example of his more generalized Lucas sequence, with U_{n}(1,-1).

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, …

Fibonacci series Σ F(n), n→∞

Lucas numbers 2, 1, 3, 4, 7, 11, 18, …

## Golden Ratio

The Golden Ratio has a long history reaching back into Ancient Greece, having been studied extensively by Euclid. Suppose we have two measurements (these can be lengths, areas, etc.) a and b. Since a one-to-one ratio would be uninteresting, assume a and b are different, and that a is larger than b. These values are related by the Golden Ratio if the ratio of a+b to a is the same as the ratio of a to b.

One sense in which this relationship is believed to be so aesthetically pleasing to the eye is evident when you look at the a in the middle. The Golden Ratio gracefully collapses or expands like a telescope from the sum total of the two quantities (a+b) to the larger quantity (a), in precisely the same proportion as that of the larger quantity (a) to the

smaller (b). No other ratio scales up and down like this, and when you see its equation hopefully you’ll see why. Art and architecture utilizing the Golden Ratio often make use of this “recurrence” by nesting successive generations of “Golden” rectangles or “Golden” spirals.

1/φ + 1 = φ

Solving for Φ can be easily achieved by multiplying both sides by φ, rearranging terms and using the quadratic formula. This gives two roots (1.61803… and -0.61803…) for which the positive root is typically taken as Φ, although the negative root is also of interest to technicians.

## Ratio of Last 2 Fibonacci Numbers is Golden

What can I say about the ratio of the last two numbers in the (un-ending) sequence of Fibonacci numbers? If you take the ratio of the last two numbers F(n) and F(n-1) for sufficiently large n then you can approximate Φ. You can see from the following table that n doesn’t even need to be that large (n=10) to give an approximation accurate to three decimal places (ε<0.001).

n |
F(n) |
F(n-1) |
F(n) / F(n-1) |
|ε| |

2 | 1 | 1 | 1.000 | 0.618 |

3 | 2 | 1 | 2.000 | 0.382 |

4 | 3 | 2 | 1.500 | 0.118 |

5 | 5 | 3 | 1.667 | 0.049 |

6 | 8 | 5 | 1.600 | 0.018 |

7 | 13 | 8 | 1.625 | 0.007 |

8 | 21 | 13 | 1.615 | 0.003 |

9 | 34 | 21 | 1.619 | 0.001 |

10 | 55 | 34 | 1.6176 | 0.0004 |

Scottish mathematician Robert Simson was the first to find this relationship between the Fibonacci sequence (which hadn’t yet been named by Lucas) in 1753.

As n approaches infinity, ε approaches 0, so there would be no difference between the ratio of the last two Fibonacci numbers (were it possible to take the last two numbers of an infinite sequence).

Interestingly, the situation Simson found wasn’t special about only the Fibonacci sequence (they weren’t called that yet!), but instead to any sequence generated by the linear recurrence relation,

F(n) = F(n-2) + F(n-1)

It doesn’t matter what starting values you choose for F(1) and F(2), which are what distinguished the Fibonacci’s from other sequences like the Lucas numbers. The essential point that leads to the Golden Ratio from these sequences is that each successive element of the sequence is the sum of the preceding two elements.

It could be argued therefore that the connection between the Fibonacci sequence and the Golden Ratio really isn’t specific to the Fibonacci numbers.

It exists for a much broader class of sequences. Perhaps the reason this belief is so widely held (and attributed to Fibonacci), is because Leonardo Pisano was the first mathematician who published his research using this linear recurrence relation.

## Fibonacci Retracement Levels

From the root φ you can see the two irrational “Fibonacci” retracement levels most frequently used in technical analysis:

- 61.8%
- 38.2% (1.0 – 0.61803…)

Market technicians have thrown in 100% (1), 50% (½) and 0% (0) for good measure, giving you:

- 100%
- 61.8%
- 50%
- 38.2%
- 0%

## Explanation for these other retracement levels

Clearly, you can have a whole retracement or no retracement. The half-retracement actually comes from Gann theory rather than the Golden Ratio. Depending upon who you ask, retracement levels can additionally involve the one-third (33%), three-eighths (37.5%), five-eighths (62.5%), or two-thirds (67%).

I’ve always believed it confounds any attempt to objectively determine which technique applies (a Gann or Fibonacci analysis) when there is such confluence of technical levels seen at 33-38.2% and 61.2-67%. I guess that when you’re a market technician making a living from forecasting the future from charts (or crystal balls, or sacrificial goat entrails for

that matter), it’s prudent to allow yourself some wiggle-room for when “market noise” proves a forecast wrong.

Similar to the connection between Fibonacci numbers and the Golden Ratio, the connection here between the Golden Ratio and what market technicians use as their retracement levels today when forecasting potential price action is likewise tenuous. Two numbers (61.8% and 38.2%) are clearly derived from the Golden Ratio, while others have a much more muddy history. Some have been proposed by numerous technicians in the past like W. D. Gann. In a noisy financial market, holding a rigid conviction in the meaningfulness of any particular retracement level may lead to more disappointment than success.

## Conclusion

Fibonacci Day is a good day to remind ourselves of the mathematics underpinning some of the phenomena we observe in our day-to-day experience. Technical analysis purporting to be based upon the Fibonacci sequence, and used by traders, are seldom subject to a rigorous (or even casually critical) scrutiny. We must apply math appropriately when constructing our abstract models of the real world. Knowing the math, and knowing how it does (and does not) relate to our models of the real world, is crucial for making intelligent decisions.