Fibonacci Retracement

Reviewed 12 August 2026 · Sourced from Leonardo of Pisa's Liber Abaci, a 2006 Cass Business School study and our own arithmetic

What those lines on the chart are

You are watching a video. The analyst pulls up a chart, clicks one tool, and five horizontal lines appear across the pullback with percentages printed on them. “Watch 61.8,” they say. Price is at $25.40. One of the lines sits at $25.42. It looks, for a moment, as though the chart told them something.

Here is everything the tool did. It took two prices the analyst chose — the high of a move and the low of it — measured the distance between them, and drew lines at fixed percentages of that distance. No volume went into it. No earnings, no rates, no order book, no other stock, no other day. Two numbers in, five lines out. If the analyst had picked a different high or a different low, every line would be somewhere else.

Fibonacci retracement is that convention: horizontal levels drawn across a pullback at set percentages of the prior move, conventionally 23.6%, 38.2%, 50%, 61.8% and 78.6%, on the expectation that price may pause or turn near one of them. A retracement is just the pullback itself — price giving back part of a move without giving back all of it.

The percentages are not arbitrary, and that is the interesting part. Four of the five are exact arithmetic on a number called the golden ratio, which comes out of a mathematical sequence described in Italy in 1202. The remaining one, 50%, is not. This page walks the arithmetic, shows which parts are solid, and is straight with you about the part where the arithmetic stops and the assumption starts.

The sequence, and the number it converges on

Leonardo of Pisa lived from roughly 1170 to roughly 1250. In 1202 he finished Liber Abaci, and its purpose was not what people assume: it was written to introduce the Hindu-Arabic place-value decimal system and Arabic numerals to Europe, which at the time still ran on Roman numerals. It is a book about how to do arithmetic.

In its third section there is a worked exercise about rabbits. The University of St Andrews history of mathematics archive gives its opening as: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs…” Work the months through and the pair counts run 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and onward, each number the sum of the two before it. That is the Fibonacci sequence, and it appears in the book as the answer to a breeding puzzle, not as a discovery about anything.

Two footnotes that most trading pages skip. He did not invent it. The same sequence was described in Indian mathematics centuries earlier — by Virahanka, Gopala and Hemachandra among others — while counting the possible arrangements of long and short syllables in Sanskrit meter. Parmanand Singh laid this out in “The so-called Fibonacci numbers in ancient and medieval India,” Historia Mathematica 12 (1985). And “Fibonacci” was not his name. It is a nickname, from filius Bonacci, applied long after his death.

Now the number the charts care about. Divide each term by the one before it and the answers stop wandering:

They converge on a single irrational number, the golden ratio, written phi = (1 + sqrt(5)) / 2 = 1.6180339887… It has a property that matters in a minute: 1 / phi = 0.6180339887… — the same digits after the decimal point. Phi is the only positive number whose reciprocal is itself minus one.

All of that is checkable and none of it is in dispute. It is also, so far, pure mathematics with no opinion about stocks.

Where each level comes from

Being able to derive the levels is the difference between using a tool and being handed one. Every number on the standard retracement tool except 50% is 1 / phi raised to a power, or a root of it. Here they are:

Read those again, because each is doing something different. 61.8% is the ratio itself, inverted. 38.2% is that ratio squared — and it also happens to be exactly what is left over from 100% after you take 61.8% away, because phi has the unusual property that 1/phi + 1/phi^2 = 1. Those two facts are the same fact wearing different clothes. 23.6% is the ratio cubed. 78.6% is the odd one of the four: it is not a power of the ratio, it is a square root of it, which is a different operation someone chose because it filled a gap in the middle of the chart.

You will also meet levels the derivation produces but not every platform ships:

76.4% and 78.6% sit two-tenths of a percent apart and come from completely different operations. Some tools draw one, some the other, a few draw both. That should tell you something about how settled the list is.

The last two rows are extensions. Same convention, run past 100%: instead of asking how far back a move comes, you project the same multiples beyond where the move started, and 127.2% and 161.8% are the usual two. On our $18-to-$30 example below, a 161.8% extension measured up from the low sits at 18.00 + (1.618034 × 12.00) = $37.42. The arithmetic is identical to the retracement arithmetic. So is everything this page says about it.

One more thing worth naming: the 0% and 100% lines are not levels at all. They are the two prices the analyst picked. They cannot fail to be there.

The 50% level is not a Fibonacci ratio

State it plainly, because almost nothing else on the internet will. 50% is not a Fibonacci ratio. It is not a power of phi, it is not a root of phi, and it does not come out of the sequence the way the other four levels do. Put 0.5 next to the derivations in the section above and there is no expression that produces it. It sits on the tool, between 38.2% and 61.8%, wearing the same clothes as its neighbors and carrying none of their arithmetic.

One near-miss is worth disposing of, because someone will raise it. Both 1 and 2 are Fibonacci numbers, so 1 ÷ 2 = 0.5. True, and beside the point. Every early ratio in the sequence is a waypoint in something still converging, not the destination: 1/1 = 100%, 1/2 = 50%, 2/3 = 66.7%, 3/5 = 60%, 5/8 = 62.5%. Nobody draws 66.7% or 60% and calls it a Fibonacci level. Picking 50% out of that list and keeping it is a choice made for other reasons.

So why is it there? Two reasons, and both are honest ones. First, half of a move is the obvious place to look, and it was the obvious place to look centuries before anybody put ratios on a stock chart. Second, the idea that a pullback runs somewhere around a third to two-thirds of the prior move is older than Fibonacci retracement's popularity and comes from a different tradition entirely: the Dow-era literature. Robert Rhea's The Dow Theory (1932) is commonly credited with describing a secondary reaction in roughly those terms. We could not reach Rhea's original text to check the exact wording, so treat that attribution as unverified — but notice how neatly 38.2% and 61.8% bracket a third and two-thirds, and that the older observation came first. Our Dow Theory page covers that framework properly.

There is no mechanism

This is the sentence the page owes you, and it gets said once, cleanly, without a sneer.

There is no mechanism by which a ratio derived from a rabbit-breeding exercise written in 1202 governs the price of a share. Not a weak mechanism, not a disputed one. There is no proposed chain of cause and effect that survives being written down. A share price is the output of orders: people and programs deciding to buy or sell at a price, matched on an exchange. Somewhere in that chain something would have to consult the golden ratio, and nothing does.

The usual bridge offered is that phi turns up in nature — in shells, in the arrangement of seeds, in branching. Some of those examples are real and some are folklore repeated past the point of embarrassment, but it does not matter which, because the argument does not connect. Sunflower seed spirals and the Nasdaq do not share a generating process. Establishing that a number appears in phyllotaxis says nothing about whether it appears in a pullback.

Nor is there an authority here. No regulator, exchange or standards body defines Fibonacci retracement, publishes the level set, or sets a threshold for anything about it. That is not a knock — the same is true of support and resistance and of most chart conventions. It just means every number you are quoted comes from a practitioner, not a rulebook, and should be treated accordingly.

Here is the turn, though, and it is the part that keeps this page from being a debunking. The absence of a mechanism in the ratio is not the absence of a mechanism on the chart. Those are different claims, and the second one is genuinely interesting.

Why the levels sometimes appear to work anyway

Three reasons, in increasing order of how much they should change your mind.

1. Attention is a real mechanism

If enough traders draw the same line, some of them place orders near it. Orders that cluster at a price are what makes price hesitate there — the whole story behind support and resistance, and it does not require the ratio to mean anything. The level works because it is watched, not because it is correct. That is real, and it comes with a hard limit: crowding is an argument about the next few days, not about market structure, and it holds only while everyone is watching the same swing. As the next section shows, they usually are not.

2. Any dense set of levels catches some turns

This is the one that should do the most work on you, and you can check it.

That is the point. Five lines across a pullback will look prescient no matter what the five numbers are, because five lines leave nowhere for price to go that is far from a line. A chart with more levels on it is not a chart with more information on it.

3. They land on levels that matter for other reasons

61.8% of a move frequently lands near a round number, a prior high or low, the top of an old range, or a moving average many people watch. When a Fibonacci line and a prior high sit at the same price, the prior high has an independent reason to matter — real orders were filled there. The Fibonacci line is along for the ride and takes the credit. That is also why a level looks strongest exactly when it is least attributable, and why a breakout through a “Fibonacci” level is usually a breakout through something older.

How they get drawn, and who chooses

The tool needs two clicks: a swing high and a swing low. Nothing in the tool, and nothing in the mathematics, tells you which two.

On any real chart there are several defensible candidates. The intraday spike high, or the highest close? The low of last month, or the low of the year? Where the move “really” began, or where the chart shows it beginning? Those are judgment calls, and they are made by a person with a view.

Now put the two problems together, because this is the strongest criticism of the whole method and it belongs on the page in plain words. Five levels per drawing, more than one defensible drawing, and price is essentially always near somebody's level. A claim that cannot fail is not a claim. It cannot be tested, it cannot be wrong, and therefore it cannot be evidence.

The shape of the argument you will be shown is worth learning to spot. After the turn, the analyst posts the drawing that worked. You never see the drawings that did not, because those were never posted. That is not dishonesty on anyone's part, necessarily — it is what happens when a method has enough freedom in it that a flattering version is always available.

This is a large part of why Markets · Technical shows live charts and breadth readings with evidence ratings attached, rather than lines somebody drew for you. Stage 5 of the Technical Analysis course covers the methods in this family and what they can and cannot be asked to do.

What is confirmed and what is convention

Somebody has tested this properly, and the result deserves to be quoted rather than paraphrased. Roy Batchelor and Richard Ramyar of Cass Business School published “Magic numbers in the Dow” in September 2006. They took 22,194 daily observations of the Dow Jones Industrial Average from January 1915 to June 2003 and asked whether the ratios of the length and duration of successive price trends cluster around round fractions (0.5, 1, 1.5) or Fibonacci ratios (0.382, 0.618, 0.786, 1.382, 1.618, 2.618, 4.236). Their abstract: “A few significant ratios appear, but no more than would be expected by chance given the large number of tests we conduct.” Their conclusion is blunter — “the idea that round fractions and Fibonacci ratios occur in the Dow can be dismissed.”

Three honest qualifications. It is one study of one index. It defined a trend algorithmically rather than by an analyst's eye, which a practitioner would call the whole problem — though notice that is the same discretion problem from the section above, pointing the other way. And a rebuttal exists: Robert R. Prechter Jr. wrote a paper titled “Elliott Waves, Fibonacci and Statistics,” which is indexed but which we could not reach in full text, so this page does not summarize an argument it has not read. What we can say is that the academic literature here is thin, and that the most substantial direct test we could locate found nothing.

Here is the ledger, and it is the most lopsided one on this site.

Read the two halves together. The mathematics is completely confirmed and the trading application is almost entirely convention. Both are true at the same time, and mistaking the first for evidence about the second is the specific error this whole page exists to prevent.

What trips people up

• Believing the 50% line is Fibonacci. It is the most-watched level on the tool and the only one with no derivation. Anyone who tells you otherwise has not checked.
• Thinking the tool measured something. It divided the distance between two prices you picked. That is the entire calculation. Nothing about the market entered it.
• Drawing the swing after the pullback. If you chose the high and low because you already knew where price turned, you have not predicted a turn, you have described one.
• Adding levels until one fits. Turn on 76.4%, 88.6% and a second drawing from a different swing and price is never more than pennies from a line. Each level you add makes the picture look better and the claim mean less.
• Accepting a quoted hit rate. “The 61.8% level holds about 70% of the time” and its cousins are everywhere and sourced nowhere. Ask who published it. Nobody has.
• Treating phi in nature as evidence about prices. Two unrelated systems, no shared process. The seashell is not an argument.
• Confusing “the level is watched” with “the ratio matters.” The first is a claim about crowd behavior that expires when attention moves on. The second has never been established. Only one of them is a reason to expect anything.
• Sizing a trade on the strength of a line. How much you commit is a decision with real arithmetic behind it — see position sizing and what a stop-loss order actually does when it triggers. A drawing on a chart is not an input to either.

The figures on this page are checked against the source that publishes them, and dated. Published rates move after the release named above — the linked source always carries the current number. This page explains a term; it does not recommend a product.