Risks Of Fibonacci Ratios In Short-Term Investing

Risks Of Fibonacci Ratios In Short-Term Investing

 

For market spectators, it is an instantly recognizable constant: 0.618034...  Named after a 13th century Italian mathematician, we generally see the Fibonacci ratio (FR) to help explain a variety of inner mechanics related to price levels within the market.  If a large enough set of the investment public subscribes to FRs, then these mathematical anchors reinforce real buy and sell signals.  Employing technical strategies based on the FR though exposes an individual to some additional degrees of model parameter risk, which he or she may not appreciate.

 

Despite the term “ratio”, the Fibonacci Ratio is actually an irrational function.  There are no two integers that can be divided in order to produce the FR.  And the simplest of all cute geometric formulas would be the solution to x2+x=1, which can not have an integer in the numerator since it involves 5.  The polynomial equation shown can resemble the variance formula for a Poisson distribution to explain queuing processes in a manufacturing plant.  And there are some linkages between the Poisson and binomial distribution approximations, which is helpful to explain how the FR comes about.

 

In other natural applications we enjoy the fact that the Fibonacci ratio is one less than the Golden ratio .  Returning to the approximation series, we can see that the following successive ratios approach, only in limit, the FR.  But along the way, the estimated proportion oscillates above and below, the true value.

 

Series:

1, 2(1+1), 3(1+2), 5(2+3), 8(3+5), 13(5+8), 21(8+13)…

 

Ratios estimates:

1:2<FR, 2:3>FR, 3:5<FR, 5:8>FR, 8:13<FR, 13:21>FR…

 

We chart these dampening ratios in the illustration below, but for now the broader context of how price levels and volatility changes over time occurs, can help bridge the gap between stochastic analysis and Fibonacci levels.  For example, through August 22, there have been 14 trading days since the S&P 500 peaked for the year thusfar.  Of those 14 days, 10 have been down days (71%).  While eight or nine would have been closer to the FR.  And all of this is a dismal science anyway, since we are at 14 days, and 14 is not in the Fibonacci series above.  How do we become more comfortable with a sampled ratio from a time continuum, before and after August 2?

 

And when we see current market declines that are now in a streak, how can we tell the level of confidence we have, such that the true Fibonacci ratio is within the confidence range of our estimate?  In most cases, this is a doubly interesting question as we noted no integer day combinations can possibly produce a FR and also there are a great many day counts for a investment idea that does not fit within a FR.  Another example is that the typical calendar month has just over 22 trading days, even though 22 is not in the Fibonacci series shown above.

 

We show in this research the mathematical credibility analysis that provides a helpful guide in understanding Fibonacci ratio approximations for limited time-frame strategies.  The statistical variance of a sampled proportional distribution, from which the FR is estimated, is p*(1-p)/n.  Where p are the fractional ratios that are observed in the market, and n is the integer number of days for the analysis.  If necessary, this computation can theoretically be scaled for time units in either direction (e.g., weeks, or minutes) assuming rich data for that time-frame exists.

 

In the illustration below we combine two important numerical information that we have discussed thusfar.  We have the dampening Fibonacci ratio, as the time series grows.  One can see the horizontal axis for number of trading days.  And using the variance formula just discussed, we also show in the same chart the standard error of this estimate.

 

 

It’s clear from the illustration that the standard error of the Fibonacci estimate is perpetually higher than the error itself.  As an example, at n=3, the FR estimate of 2/3 is off by 4.9%.  Yet the standard error is even higher, at 27.2%.  Now both errors converge to zero over time, but we see the FR estimate converges more quickly.  This is troublesome some a probability perspective, so we now address the approach we must therefore instead use.

 

With such large standard errors, we move away from the critical level of the appropriate confidence interval to use, and think instead about the credibility of looking at a Fibonacci ratio from performance data. This is a more rigorous test that will provide us with the minimal sample size needed, to be within a certain amount of the true FR.