Fibonacci Retracements: The Levels and the Evidence

Take the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. Divide each term by the one after it. The answer settles fast: 34 / 55 = 0.6182, then 55 / 89 = 0.6180. That limit is the whole mathematical content of the tool. Two more numbers come out of it by squaring and by taking a root, and everything else on the chart is a decision you made.

Four of the five levels are powers of one number

0.618 is the reciprocal of 1.618, the value those consecutive ratios converge toward. Square it and you get 0.382. Cube it and you get 0.236. Written out: 0.618 x 0.618 = 0.3819 gives 38.2% and 0.618 x 0.618 x 0.618 = 0.2360 gives 23.6%. The 78.6% line on most default menus is the square root, sqrt(0.618) = 0.7861. One constant and its powers account for the whole set.

The justification usually offered runs through nautilus shells, sunflower heads and the proportions of Greek buildings, and it arrives at the claim that the same ratio governs how a crowd of traders prices a company. The arithmetic in the first half of that argument is exact. The connection in the second half has never been supplied by anyone. Both things are true at once, and holding them together is the only honest starting position: the ratio is real, and its authority over a share price is assumed.

Drawing the swing is where the subjectivity lives

Pick the low that started the move and the high that ended it. Those two prices are the whole input. Everything after that is level = high - (high - low) x ratio, applied once per ratio.

Work it on a clean example. A stock based at 40.00, ran to 70.00 and is now pulling back. The range is 70.00 - 40.00 = 30.00.

Ratio	Arithmetic	Level
23.6%	70.00 - 30.00 x 0.236	62.92
38.2%	70.00 - 30.00 x 0.382	58.54
50%	70.00 - 30.00 x 0.500	55.00
61.8%	70.00 - 30.00 x 0.618	51.46
78.6%	70.00 - 30.00 x 0.786	46.42

Now shift one endpoint. Suppose you anchor the low at 42.50, where a secondary base formed on the way up, which makes the range 27.50 and sends the 61.8% level to 70.00 - 27.50 x 0.618 = 53.01. That is 1.55 above the first answer. On a stock trading near 51.46 the gap between the two versions is around 3% of its price, which is wider than the stop most people would set on the trade, so two charts carrying the same five labels describe entirely different trades.

50% is on the tool by habit

50 is absent from the sequence. The early pairs give 1 / 2 = 0.5 and 2 / 3 = 0.667 before the terms grow large enough to settle at 0.618, so 0.5 shows up once, at the front, as an artefact of small integers doing what small integers do.

It survives on the menu for two plainer reasons. A halfway pullback is obvious to watch. The idea long predates charting software. And most packages ship 50% inside the default set, which puts the level on more screens than any of the genuine ratios, including the 61.8% line the sequence actually produces.

The second reason is the stronger one. It also cuts against the mystique. If a level matters because thousands of people have it drawn in the same place on the same swing and leave orders near it, then the mechanism is crowding, and the sequence is decoration on top. This also explains why customising the ratios is a poor idea. A 0.55 line is yours alone, and nobody else is trading against it.

Depth describes what the pullback has already done

A retracement that stops around 38.2% has given back a small share of the advance, which is the shallow, orderly pullback people describe as a trend holding. A move through 61.8% has handed back most of the swing. At 78.6% almost nothing remains. Below 40.00 the swing is finished, since nothing is left to retrace, and the tool has run out of chart to divide.

The 40.00 price is worth more than the five lines above it, because it is the only one of the six that ends the idea, and an invalidation price is the thing a position is actually built from. Read the levels as description. They tell you how much of a completed move has been undone. Other people are looking at the same scale.

Timeframe decides which people. A swing measured on the weekly chart and a swing measured on a five minute chart generate two grids that share no prices at all, and the grid that matters is the one visible to the participants who trade the timeframe you are trading. The support and resistance guide covers what puts orders at a price in the first place: memory of fills, breakeven selling and round numbers.

Extensions point at targets above the high

Extensions use three points. Call them A at 18.00, B at 24.00 and C at 21.70. C is the low of the pullback. The measured leg is 24.00 - 18.00 = 6.00.

Ratio	Arithmetic	Target
1.000	21.70 + 6.00 x 1.000	27.70
1.272	21.70 + 6.00 x 1.272	29.33
1.618	21.70 + 6.00 x 1.618	31.41

1.272 is the square root of 1.618, which keeps the family consistent. The trouble is the anchor. Some platforms project from C and some project from B, so the same menu item that returns 29.33 on one chart returns 24.00 + 6.00 x 1.272 = 31.63 on another, a difference of 2.30 per share on a stock in the twenties. Check which convention yours uses.

This is the measured move from the chart patterns guide, with a multiplier bolted on. The reasoning is the same and it is just as loose: a move of a certain size tends to be followed by a move of comparable size. Treat the number as a scale for deciding whether the trade is worth taking.

Confluence is the one defensible use

A Fibonacci level on its own is a line drawn from two prices you picked. It becomes worth trading when it lands on evidence that exists independently of the tool: a zone where price previously turned, a moving average that other people watch, a round number where limit orders collect. The ratio drew your attention. The zone is what you are acting on.

Item	Value	Arithmetic
61.8% level	51.46	70.00 - 30.00 x 0.618
Prior resistance zone	50.90 to 51.60	The base built before the run
50-day average	51.20	Rising into the level
Entry	51.80	Taken once the zone is holding
Invalidation	50.40	Below all three pieces of evidence
Risk per share	1.40	51.80 - 50.40
Risk budget	$300	1% of a $30,000 account
Position size	214 shares	300 / 1.40 = 214

That table is the output worth having. The ratio contributed one row. Everything else came from the price series, from the size of the account, and from a decision about where the idea would be wrong. The moving averages guide has the calculation behind that 50-day line.

What can be tested, and what the circulating numbers are missing

The arithmetic is reproducible to the cent. Hand two people the same swing high and swing low and they will return identical levels every time, which is the last reproducible step in the process. Testing whether price respects 61.8% would require a written rule for selecting the swing, a written rule for what counts as respecting a level, a tolerance band, a date range and a defined universe of stocks, and the percentages that circulate on this subject supply none of those five things, which leaves an outsider with nothing to verify and no way to repeat the work.

The tolerance band by itself decides the answer. Allow 1% either side of 51.46 and you are testing a zone 1.03 wide. Tighten it to 0.25% and the zone is 0.26 wide. The loose version will look impressive around almost any level, including one drawn at random, and choosing the band after seeing the data lets you produce whatever figure you wanted. So this guide quotes no hit rate for anything. What it hands you is a level, the two prices that generated it, and the price at which the idea is finished, which is enough to size a position and nowhere near enough to justify one.

One effect does survive scrutiny, and it is the unglamorous one. A large number of traders open the same default tool, apply it to the same obvious swing on the same daily chart, and leave orders near the same three or four prices. That is enough to produce reactions at those prices. No property of the sequence is involved.

Where the method loses money

It needs a swing. In a stock that has chopped sideways for two months there is no completed move to measure, and any endpoints you pick are arbitrary in a way the tool will happily disguise with precise decimals.

Gaps ignore the levels. A stock that opens 12% lower on news has passed through 38.2%, 50% and 61.8% at prices that never traded, and no order resting at any of them was filled.

The deep levels flatter themselves. A pullback to 78.6% at 46.42 looks like a discount, and a stop that respects the swing low has to sit below 40.00, so an entry at 46.80 risks 46.80 - 39.60 = 7.20 per share. The same $300 budget then buys 300 / 7.20 = 41 shares against the 214 the shallower entry allowed. Deeper levels come with weaker evidence and a wider stop at the same time, which is the opposite of the bargain they appear to offer.

The expensive failure is the one the grid invites. Buy 100 shares at 58.54 because the shallow retracement held. Add 100 at 55.00 when that level gives way. The next line is right there. Add 100 more at 51.46. Average cost is (58.54 + 55.00 + 51.46) / 3 = 55.00 on 300 shares, and the position reached full size at the exact moment the case for it was weakest, which is the structural problem with any ladder of prices drawn in advance.

Fibonacci retracements earn their place as a fast way of generating candidate prices on a chart you already had reasons to open, and they earn nothing at all as a forecast, since the confident percentages attached to them cannot be checked by you or by anybody else. Read the technical analysis pillar for how trend, levels and momentum are meant to sit together, use the stock screener to find stocks pulling back inside an established trend before you start drawing anything, and test the measured move arithmetic against the chart patterns and indicators quiz.