Fibonacci Ratios

There is a great deal of misinformation online about what ratios are correct Fibonacci ratios. Some charting packages have ratios that are not correctly Fibonacci ratios in their Fibonacci tool kits.

So in the interests of clarity and mathematical accuracy, this article will explain what is a Fibonacci ratio and how it is derived.

From a purely mathematical perspective the Fibonacci sequence is “seeded” by 0 and 1 and the subsequent numbers in the sequence by adding the previous two numbers

f_(n+1) = f_n + f_(n-1)

so the sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

When talking about Fibonacci ratios we are strictly just talking about the numbers that are *converged* on when you divide a number by its predecessor (or its predecessor’s predecessor, i.e. its predecessor two back etc). Or, conversely, we can do the reverse (which results in the inverse number) and divide by a number’s successor (or its successor two forward etc). This results in the two ratio sequences:

Mathmaticians were quite interested in the fact that the 1st ratio 0.618’s inverse was 1+0.618 (also known as the Golden Ratio or Golden Mean). This can be proved by finding the limit of the ratio f_(n+1)/f_n for the Fibonacci sequence when n tends to infinity. i.e. it is no accident, it is due to the recurrance relation f_(n+1) = f_n + f_(n-1).

The important thing to note here is that correct Fibonacci ratios are numbers that are converged upon when one divides one Fibonacci number by another.

Therefore 0.5 is not a Fibonacci ratio although it can be obtained by dividing 2 (the third number in the sequence) by 1 (it’s predecessor). This is not what the number converges upon as one moves up the Fibonacci number sequence. Once you start down this track, you could just as easily divide the third by the fourth and get 0.66666, etc.

0.786 is therefore not a Fibonacci ratio either, although many charting packages include it in their Fibonacci tools. We can obtain 0.786 from the square root of 0.618 but this is not the correct method to find a Fibonacci ratio. 0.786 cannot be obtained by dividing one Fibonacci number in the sequence by another.

3.618 is also not a Fibonacci ratio as it cannot be obtained by dividing one Fibonacci number in the sequence by another. Likewise 1.382 and 1.236 are also not Fibonacci ratios.

Fibonacci ratios cannot be derived by manipulating the above list of correct Fibonacci ratios; by squaring them, finding the square root of, multiplying one by another or otherwise manipulating Fibonacci ratios to find other ratios. Such derived ratios are not correctly Fibonacci ratios. They may still have a use and a place in technical analysis, but they cannot be referred to as Fibonacci ratios.

A big thank you to Alicia Monteith (Phd) for providing mathematical clarity for this article.