Chart tools · Levels
Fibonacci Retracements Examined
Two of the ratios come from the sequence, one does not, and the rest are square roots of the first two. Draw them all and the market is never far from a level, which is the problem.
Which ratios are actually Fibonacci ratios
The sequence (1, 1, 2, 3, 5, 8, 13 and so on, each term the sum of the two before it) has the property that the ratio of consecutive terms converges on about 0.618. Square that and you get 0.382; the next step down gives 0.236. Those three are the genuine ratios.
| Level | Origin | Fibonacci? |
|---|---|---|
| 23.6 % | 0.618 cubed, a further step down the sequence ratio | Yes |
| 38.2 % | 0.618 squared, equivalently one minus 0.618 | Yes |
| 50 % | The halfway point. A long-standing observation that predates Fibonacci analysis entirely | No, the sequence never produces one half |
| 61.8 % | The limit of the ratio of consecutive terms | Yes |
| 78.6 % | The square root of 0.618, a derived figure | Derived, not a sequence ratio |
The 50 per cent line is worth a sentence of its own, because it is the most used of the five and it is misattributed everywhere. A market retracing half of a move is an old and reasonable observation; it arrived long before anyone connected retracements to the sequence. Using it is fine. Calling it a Fibonacci level is not, and the calculator on this site labels it accordingly.
What five levels do to a chart
Here is the arithmetic that the literature rarely puts in one place. Take a swing from 100 to 148, a 48 per cent advance, an ordinary size for a multi-week move, and draw all five retracement levels on it.
The problem gets worse rather than better as more levels are added. Each additional derivation (the square root of a ratio, then the square root of that) is arithmetically valid and makes the tool harder to be wrong with. That direction of travel is the thing to be suspicious about, in this tool and in any other.
The mechanism that is real
None of the above means the levels are inert, and the reason is worth stating carefully because it is the strongest thing that can be said for them.
A great many participants draw the same lines from broadly the same swings, and orders accumulate where people are watching. That makes a popular level a genuine feature of the order book: supply and demand really do change there, not because 0.618 governs anything but because a crowd agreed in advance to stand at that price. The effect is self-fulfilling, it is not weak, and it is the honest argument for knowing where the widely used levels sit.
It also implies a limit. A self-fulfilling level works to the extent that the crowd is large and agrees on the swing, so the levels drawn from an obvious, widely discussed move are the ones with a mechanism behind them, and the ones drawn from a swing only you can see have none at all.
The choice that decides everything
Two analysts with the same chart routinely produce incompatible level sets, both computed correctly, because they chose different swings. There is no published rule for the choice in ordinary practice; the instruction is to use the "significant" or "obvious" move, which is a judgement made after seeing what happened.
That is the same problem the trendline pagedescribes for a choice of two points, and it has the same fix: state the rule in advance. "The highest high and lowest low of the last sixty sessions" is mechanical, reproducible, and testable. Once the rule is fixed, the tool has become an indicator and can be evaluated like one, which is exactly what most of the published evidence for Fibonacci levels avoids.
Using them without overclaiming
Keep the count small. The two genuine ratios and the halfway point are enough. Every extra line increases the chance that price is near one and decreases what that fact tells you.
Record the swing and the scale. A level is only checkable later if the inputs are written down, and a chart drawn on a log scale gives different lines from one drawn on a linear scale over a large move.
Pair a reaction with volume. A turn at a level on expanding volume is a different event from one on a quiet session: the first says a lot of stock changed hands at that price, and the second says almost nobody was there. This is the addition that makes a drawn level into something with a fact attached, and it costs nothing but the habit of looking at the second panel.
Frequently asked questions
Where do the ratios come from?
From the Fibonacci sequence, in which each number is the sum of the two before it. The ratio of consecutive terms converges on about 0.618, and 0.382 is that number squared, or equivalently one minus it. Those two, with 0.236 as the next step, are the genuine Fibonacci ratios used in retracement analysis. The sequence itself is an ordinary piece of mathematics with real applications elsewhere; whether it says anything about markets is a separate question with far less behind it.
Is 50 per cent a Fibonacci ratio?
No. It is the halfway point, it appears on every retracement tool, and it has nothing to do with the sequence, the ratio of consecutive Fibonacci terms never equals one half. It is included because a market retracing half of a move is a long-standing observation in its own right, predating Fibonacci analysis by decades. Nothing is wrong with using it; describing it as a Fibonacci level is simply inaccurate, and the calculator on this site labels it as what it is.
What about 78.6 per cent?
It is the square root of 0.618, which makes it a derived figure rather than a sequence ratio. Several other levels circulate on the same basis; 88.6 per cent is the square root of that again. Each derivation is arithmetically valid and each additional level makes the tool harder to be wrong with, which is the direction of travel worth being suspicious about.
Why is having many levels a problem?
Because it removes the possibility of failure. The figure on this page computes the spacing on a realistic swing: five levels cover a substantial part of it, and the average gap between adjacent lines is a few per cent of price. A market moving inside that zone is almost always near a level, so "price reacted at a Fibonacci level" is close to unfalsifiable, and a claim that cannot fail cannot be evidence.
So do the levels work?
There is a mechanism, and it is not the one usually offered. If a large number of participants place orders at widely published levels, those levels become places where supply and demand genuinely change, self-fulfilling rather than predictive. That effect is real and worth knowing about. What the evidence does not support is the stronger claim that a mathematical constant governs price movement, and the tests that have looked for it have generally not found more than the crowding effect would explain.
Which swing should the levels be drawn from?
This is the question that decides the answer, and it has no mechanical rule in ordinary practice. Different analysts choose different highs and lows and get incompatible level sets from one chart, all of them correctly computed. If you want a defensible version, fix the rule in advance — the highest high and lowest low of a stated lookback, for instance — which converts the tool into an indicator and makes it testable.
Do extensions work differently from retracements?
They are the same arithmetic projected beyond the swing rather than inside it, at ratios above one, 127.2 per cent, 161.8 per cent and so on. The self-fulfilling mechanism applies equally, and the falsifiability problem is worse, because an extension level has open space around it rather than a bounded range. Treat them as places other people are watching, which is the only claim the construction supports.
What is the honest way to use them?
As a record of where a crowd is likely to be watching, written down before the fact. Note the swing you used and the scale of the chart, keep the count of levels small, the two genuine ratios and the halfway point are enough, and pair any reaction with a volume reading, because a turn on expanding volume is a different event from one on a quiet session. That is a modest use and it survives scrutiny, which the stronger versions do not.