[F008] The Fibonacci Ratios Carry No Privilege

Abstract

This paper asks whether the Fibonacci ratios are more special than other ratios — the answer is that they are not. It claims neither a new mathematical theorem nor a new law.

First, for the five Fibonacci ratios (0.236, 0.382, 0.500, 0.618, 0.786) and five fake ratios placed in the same band (0.150, 0.310, 0.440, 0.690, 0.860), the median touch rates are 53.5887% and 52.7018%, a difference of 0.8870 points (Section 3).

Second, and this is the core of the paper, sorting all ten ratios by distance from the middle interleaves the real and the fake. The correlation between distance and touch rate is −0.9083. What decides is not the name of the ratio but where the level sits (Section 4).

Third, the median bounce rate (the share retracing by half an ATR after a touch) is 81.1463% for the real and 81.0171% for the fake, a difference of 0.1292 points (Section 3).

Fourth, the median directional information is 0.4755 for the real and −0.4974 pips for the fake, a difference of 0.9730 pips. The daily t reaches 2 in absolute value in 2 of 20 real cells and 1 of 20 fake ones (chance gives about 0.9) (Section 3).

Fifth, the bounce rate of 81% is not owed to the ratios. The fake ratios also give 81%. Retracing half an ATR after touching a level happens often as ordinary price movement (Section 3).

Sixth, the same measurement on synthetic random walks gives a touch rate for the real 0.618 of 54.5621% to 54.7001%. That all but overlaps the measured 53.8237% to 57.0917% (Section 6).

Scope of this paper (scope note): No new mathematical theorem and no new law is claimed — levels split by a ratio, the true range and random walks are all known. No particular product or proponent is discussed — no real course, author, or vendor is the subject here, and none is named. What is treated is the public drawing method itself. Trading is not discussed — what is measured is touch rates, bounce rates and directional information, not trading performance including commissions, fills and money management. The scope is intraday and a small number of instruments — four series, six years, one-hour bars. This paper does not contradict results documented at longer holding periods, such as monthly time-series momentum. One way of taking the high and the low is fixed — the highest high and lowest low of the last n bars, which does not reproduce the swing endpoints a person picks by eye. That is the chief limitation of this paper (Section 8). Extension ratios — those drawn beyond the move rather than back into it — are not treated. No forecast is claimed. Relation to earlier papers: this series always reports, alongside a rate, what that rate would be by chance. F006 read a touch rate beside the position — this paper does the same with a different baseline, the fake ratios. Paper 423 showed that a wrong conclusion can come from correct parts. What is added here is placing fake ratios in the same band, sorting all ten by position and reporting the correlation, and showing that the high bounce rate is not owed to the ratios.

1. Notation and the question

A common practice draws levels between a recent high and low at these ratios.

Observation 1.1: these ratios come from the ratios of successive terms in a certain sequence. That the sequence appears in a market, however, has not been shown.

Observation 1.2: something must be watched here. The observation "price stopped at this level" says nothing on its own. Price stops somewhere, so drawing many levels guarantees that it stops at one of them.

Observation 1.3: the question settles by placing fake ratios alongside. Take ratios far enough from the Fibonacci ones, in the same band (not too near an edge), and make the same measurement. If no difference appears, the ratios carry no privilege.

Question: are the Fibonacci ratios more special than other ratios?

2. Writing the quantity and the design in advance

Design (declared before running and unchanged at run time)

Observation 2.1: the directional signal is placed on the bar of the touch, not on the bar before it. Conditioning on a touch and then measuring from before it puts "price moved toward the level" into the condition, so betting against that move loses by construction. Measure from after the event used as the condition.

Observation 2.2: the fake ratios are placed far enough from the real ones and not too near an edge. Near an edge, a separate effect — being hard to reach at all — enters.

The bar counts are 36,534 for USDJPY, 36,554 for EURUSD, 36,535 for GBPUSD and 34,892 for XAUUSD.

3. Placing the real ratios beside the fake — this is the separator

Calculation 3.1 The five Fibonacci ratios (evaluation period)

Calculation 3.2 The five fake ratios (same period, same procedure)

Observation 3.3: the median touch rates are 53.5887% for the real and 52.7018% for the fake, a difference of 0.8870 points.

Observation 3.4: the median bounce rates are 81.1463% and 81.0171%, a difference of 0.1292 points.

Observation 3.5: note the bounce rate of 81%. It invites the reading "it bounces eight times in ten". But the fake ratios also give 81%. Retracing half an ATR after touching a level happens often as ordinary price movement. It is not owed to the ratios.

Observation 3.6: the median directional information is 0.4755 for the real and −0.4974 pips for the fake, a difference of 0.9730 pips. The daily t reaches 2 in absolute value in 2 of 20 real cells and 1 of 20 fake ones. Chance gives about 0.9, so neither can be told apart from chance.

Observation 3.7: XAUUSD's 0.618 looks large at −11.2310 pips, but its t is −1.59. The same instrument's fake 0.440 gives −7.3910 with t −1.67 — a larger t for the fake. The size comes from the size of gold's moves.

Figure 2: bounce rate by ratio (median of four series). The real and fake points sit at the same height.

4. Sorting all ten ratios by position

Calculation 4.1 Sorted by distance from the middle (0.5); touch rate is the median of four series

Observation 4.2: the real and the fake alternate almost exactly. They do not separate by kind; they order by distance.

Observation 4.3: the correlation between distance from the middle and touch rate is −0.9083. What sets the touch rate is not the name of the ratio but where the level is.

Observation 4.4: this is unremarkable in itself. A level nearer the middle is more easily reached. What this paper shows is that the Fibonacci ratios do not depart from that unremarkable fact.

Observation 4.5: anything that did depart would stand off this ordering. None of the ten stands off.

Figure 1: on the left, ratio against touch rate; on the right, distance from the middle against touch rate. On the right the real (dark) and fake (light) points fall on the same curve.

5. Changing the window

Calculation 5.1 Changing how many bars the high and low come from (evaluation period, 0.618 against 0.690)

Observation 5.2: widening the window from 24 to 96 lowers the touch rate from 65.5325% to 41.2052% for USDJPY. A wider window puts the level further away and makes it harder to reach.

Observation 5.3: in no window does the ordering of real against fake hold. USDJPY has the fake higher at all three windows; EURUSD and GBPUSD have the real higher at all three. The sign is set by the instrument, not by the kind of ratio.

6. The baseline — a random walk has no ratios and no levels

Calculation 6.1 Synthetic series whose within-bar movement is also random (10 realisations, evaluation period)

Observation 6.2: the random-walk touch rates fall between 54.1172% and 54.7001%. The measured ones run from 53.6062% to 57.0917%. They all but overlap.

Observation 6.3: a synthetic random walk has no Fibonacci sequence, no pullbacks and no support. The same rates appear anyway. The touch rate therefore says nothing about the market.

Observation 6.4: even on the random walk the real 0.618 comes out slightly above the fake 0.690 (54.7001% against 54.1172%, and so in all four series). That is because 0.618 is nearer the middle (distance 0.118 against 0.190). It agrees with the conclusion of Section 4.

7. Splitting by session

Design: five non-overlapping bins in Coordinated Universal Time. The signal is the bounce at the real 0.618.

Calculation 7.1 Directional information in each of the five bins (evaluation period)

Observation 7.2: the daily t reaches 2 in absolute value in 0 of the 20 cells. By chance about 0.9 would be expected, so the measured figure is below it.

Observation 7.3: XAUUSD is negative in all five bins, yet no t exceeds −1.79. Signs agreeing and being distinguishable from noise are two different things.

8. Limitations

1. ★ One way of taking the high and the low is fixed. Levels are drawn from the highest high and lowest low of the last n bars, which does not reproduce the "swing endpoints" a person picks by eye. In practice a person decides which swing to take. What this paper denies is the claim that the ratios themselves carry a privilege, not a practice that includes choosing the swing. This is the chief limitation.
2. Extension ratios — those drawn beyond the move rather than back into it — are not treated. Only retracements.
3. The fake ratios are placed one way. Another choice would give slightly different figures. The ordering of Section 4, however, is set by distance rather than by the choice.
4. The bounce is judged at half an ATR. Another width changes the bounce rate. The same width is used for real and fake, so the comparison stands.
5. Only one lattice is used. Bars are fixed at one hour.
6. The synthetic baseline uses independent normal increments. Volatility clustering is not included.
7. Four instruments over six years. Nothing is said about other markets or other periods.

Conclusion

Drawing levels at the five Fibonacci ratios, price touches them 53.5887% of the time and, having touched, retraces 81.1463% of the time. Numbers that read as "it bounces eight times in ten".

But the five fake ratios placed in the same band gave a touch rate of 52.7018% and a bounce rate of 81.0171%. The differences are 0.8870 points and 0.1292 points.

And sorting all ten by distance from the middle interleaves the real and the fake, with a correlation of −0.9083. A synthetic random walk gives the same rates.

One thing separates the cases — where the level sits. Not what it is called.

And what this paper could measure is the case where the recent high and low are taken mechanically. About a practice in which a person chooses which swing to use, nothing is answered here.

Data and reproduction

References

• Yamagishi, Y. (2026). The Table Is Right and the Trade Still Loses [Paper 423]. Zenodo. 10.5281/zenodo.22700114
• Yamagishi, Y. (2026). The %K of Stochastics Is the Position in the Range Itself [F006]. Zenodo. 10.5281/zenodo.22881505

Reference

Yamagishi, Y. (2026). The Fibonacci Ratios Carry No Privilege [F008]. Zenodo.
DOI: 10.5281/zenodo.22881668
YOI: https://www.yamagishiyuuki.com/yoi/F008
ORCID: 0009-0006-8587-3450

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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI F008. This paper is a preprint and has not been peer reviewed.