Fibonacci Number and Ratio Analysis
overlay indicators such as Bollinger Bands, Japanese Candlesticks, Ichimoku
Clouds, conventional trendlines, and price channels. To further help increase the
probability of identifying these potential reversals, traders and analysts often also
seek supporting evidence of overbought and oversold conditions in conventional
window oscillators such as Stochastics, commodity channel index (CCI), moving
average convergence‐divergence (MACD), rate of change (ROC), and the relative
strength index (RSI) during such reversals.10.1.3 philosophical significance of the Fibonacci
iterative series
We can clearly see that the next number in the Fibonacci series, Fn+1 (i.e., new
information), is the direct result of Fn (i.e., current information) interacting with
Fn−1 (past information).
F F F
Future Present Pastnnn+ 1 =+ − 1
( ) (=+) ( )In a sense, this mathematical expression uncannily reflects the commonly ac-
cepted philosophical premise that eventuality and causation are the direct result
of current action based on past conditions or experience. This approach seems
to fits very well with the study of technical analysis, as analysts and traders rely
wholly on past and current information in order to determine and predict future
market action.10.2 Fibonacci Ratios
One unique and rather mysterious feature of the Fibonacci series is that of the
ratio of the current Fibonacci number to its immediate previous number, that
is, the ratio (Fn+1/Fn) or (Fn /Fn−1). This ratio approaches 1.618 (to three decimal
places) as we progress further out along the series of Fibonacci numbers. In fact,
as it turns out, it really does not matter which two numbers were initially chosen
to start the sequence. As we progress further along the sequence, it will always
approach 1.618! In similar fashion, the Lucas numbers also approach this ratio
of 1.618 in the limit. This unique ratio is referred to as the Golden Ratio, or
Phi (Φ). This ratio is also sometimes referred to as the Divine Proportion. Please
refer to Figure 10.1.
As can be seen from Figure 10.1, both the Fibonacci and Lucas series ap-
proach the ratio of 1.618 as we progress further out along the sequence, that is,
dividing any Fibonacci number by the number that immediately precedes it. In
fact this ratio oscillates slightly as it asymptotically approaches 1.618, akin to
the mechanical dampening action as seen in vibrating springs. See Figures 10.2
and 10.3.
As an exercise, try selecting any two random numbers and generate a Fibo-
nacci sequence. You will inevitably discover that the ratio of any number to its
immediate previous number will always start to approach 1.618 as you progress
further along that particular sequence.