Credibility of Fibonacci

For market spectators, it is an instantly recognizable constant: 0.618034...  Named after a 13^th 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 portion 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 modeling risk, which he or she may not appreciate.

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

In other natural applications we enjoy the fact that the Fibonacci ratio is one less than the Golden ratio φ.  Now looking at the approximation series, we can see that the following successive ratios approach, can only limit to the FR.  And along the way, the estimated proportion oscillates both above and below, it's 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…

In the color fuchsia, we have charted this damping process below.  But before we get to it, let's step back and appreciate the broader context of how price levels and volatility changes as time evolves.  This can 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 had a year-to-date peak.  Of those 14 days, 10 have been down days (71%).  Now eight or nine days, instead, would have been closer to the FR.  This is also a dismal science anyway, since 14 days isn't represented on the Fibonacci series count above.  How do we become more comfortable with a sampled ratio, from a time continuum, starting on either side of August 2?

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

We show in this research the mathematical credibility analysis that provides a helpful guide in understanding the confidence about the 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 other related time units (e.g., weeks, or minutes) assuming an equally rich data set exists for that time-frame.

In the illustration below we combine the two important numerical discussions we have had thusfar.  We have the dampening Fibonacci ratio.  And one can see that the horizontal axis is the number of trading days.  And using the variance formula just discussed, we also show in the same chart the standard error of this estimate.  See the lime color line.

It is clear from the illustration the troubling fact 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%.  But the standard error is even higher, at 27.2%.  Now both errors converge to zero over time, but we see the FR estimate actually converges more quickly.  This is troublesome from a probabilistic perspective when using this for market data estimation, so we now address the new approach we must 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.

As an example, to be within 10% of the true Fibonacci ratio, implies being between 51.8%, and 71.8%.  This is the top row of the table below.  In this table we calculate the minimal number of trading days, using a 2-tailed test (since the estimate oscillates either above or below the true FR).  Seven popular testing combinations are shown to facilitate our discussion.

	80% confidence	90% confidence	95% confidence
10.0%	0.10/1.28 ~7.8% standard error, or >34 days (as can be seen at the right of the graphic)	0.10/1.64 ~6.1% standard error, or >55 days	N.A.
5.0%	0.05/1.28 ~3.9% standard error, or >144 days	0.05/1.64 ~3.0% standard error, or >233 days	0.05/1.96 ~2.6% standard error, or <377 days
2.5%	N.A	0.025/1.64 =1.5% standard error, or 987 days	0.025/1.96 ~1.3% standard error, or <1597 days

It is clear from the table that the minimal sample size is typically in the hundreds or thousands of trading days, except for the loosest of confidence levels.  Even at an 80% confidence of being within 10% of the Fibonacci ratio, it doesn’t make sense to look at performance streaks of less than one month.  The corollary to this is that for strategies with less than a one-month horizon, in addition to any roll-over risks, one would have a wide level of model parameter risk or slack in estimation, which an individual may not appreciate as they evaluate technically trading strategies.