What Is a Fibonacci Retracement? The Levels, and a Coin-Flip Test

Draw a Fibonacci retracement on almost any chart and you'll find a pullback that stopped right on one of the lines. It can feel like the market knows the numbers.

Look at the chart below. Price climbed from $91.42 to $103.22, pulled back, and stopped at $95.92. The 61.8% line was at $95.93. Then price went on to a new high. It's a textbook bounce.

We didn't find that chart on a real stock. We made it with a random number generator. Every candle is a string of coin flips, so no line on it can mean anything.

So here's how Fibonacci retracement works, where the numbers come from, and what a bounce like that is actually worth.

What is a Fibonacci retracement?

Prices don't move in straight lines. Price runs up, falls back some of the way, then often carries on. That fall back is called a pullback, or a retracement.

A Fibonacci retracement measures how far back a pullback has gone. You pick a swing on the chart, and the tool draws lines across it at five depths: 23.6%, 38.2%, 50%, 61.8% and 78.6%.

Traders watch those lines as places where a pullback might stop and the trend might start again. A shallow pullback to 38.2% is often read as a sign of a strong trend. One that goes past 78.6% is often read as a trend in trouble.

How to draw a Fibonacci retracement

Most charting apps have a Fibonacci tool. In an uptrend, you click the swing low, then the swing high. The tool draws the lines between them, below the high, where a pullback might stop.

In a downtrend it's the other way round. You click the swing high, then the swing low. The lines sit above the low, where a bounce might stall.

You can also work the lines out yourself. Take the size of the swing, multiply it by the percentage, and take that off the high. Our swing ran from $91.42 to $103.22, a move of $11.80. 61.8% of $11.80 is $7.29, so the 61.8% line is at $103.22 − $7.29 = $95.93.

The pullback stopped at $95.92. That's 61.86% of the way back, 1 cent under the 61.8% line.

Where the Fibonacci numbers come from

The lines are named after Leonardo of Pisa, known as Fibonacci, who wrote about this number sequence in 1202. Each number is the sum of the two before it:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

Divide a number by the next one and you get about 0.618: 89 ÷ 144 = 0.618. Divide by the number two places on and you get about 0.382: 55 ÷ 144 = 0.382. Three places on gives about 0.236: 34 ÷ 144 = 0.236.

0.618 is 1 divided by the golden ratio, 1.618, which is why the 61.8% line gets the most attention. 78.6% is the square root of 0.618.

50% isn't a Fibonacci number at all. Traders added it because halfway is an obvious place to look.

None of this explains why prices would care about these numbers, and there's no accepted reason they should. So the useful question is a practical one. When price stops at a line, how surprised should you be?

The lines cover most of the swing

Start with where the lines sit. They're spread across the middle of the swing, and the widest gap between two of them is 16.8%, from 61.8% to 78.6%.

So a pullback that stops anywhere between 23.6% and 78.6% is never more than 8.4% of the swing from a line. The furthest spot is 70.2%, halfway between 61.8% and 78.6%. On our $11.80 swing, that's 99 cents.

Now say "at the line" means within 2% of the swing, which is 24 cents on our chart. A band that wide round each of the five lines covers 20% of the whole swing. So if pullbacks stopped at random places, about 1 in 5 would land "at a Fibonacci level" anyway.

You also get more than one try. Draw the tool from a different low or high and every line moves. With a few swings to choose from, one of them usually lines up with something.

We tested it on a price made of coin flips

One chart that agrees with the lines doesn't tell you much. So we counted.

We built a price that moves at random, 4 million candles long. We marked every swing on it, the way a zigzag indicator does, and measured how far each pullback went back.

That gave 100,962 pullbacks. Half of them went right through the start of the swing, which is normal on a random price. We kept the 50,508 that held: a higher low in an uptrend, or a lower high in a downtrend.

22.6% of those pullbacks ended within 2% of the swing from a Fibonacci line. That's a little more than the 20% you'd get from random stopping places, because random pullbacks tend to run deep.

So we moved all five lines 3% of the swing down, then 3% up. Those made-up lines caught 21.4% and 23.8% of the pullbacks. The real ones landed in between.

On its own, the 61.8% line caught 5.9% of the pullbacks. A line at 60% caught 5.8%, and one at 65% caught 6.2%. There's no bump at 61.8%. Pullbacks just get a little more common the deeper you go, right past 78.6%.

The typical pullback that held went 67% of the way back, deeper than the 61.8% line. Random prices make deep pullbacks all the time.

Does the 61.8% level hold?

Traders also talk about a level holding. Price touches the line, turns, and goes back to the high. So we tested that too.

Each time price touched a line, we checked what came first: a return to the swing high, or a break of the swing low. We ran it for the five Fibonacci lines and four we made up: 30%, 45%, 55% and 70%.

Every line held about as often as 1 minus its depth. The 61.8% line held 37.6% of the time, and 1 minus 0.618 is 38.2%. The 50% line held 48.8%. The made-up 55% line held 44.0%, right where you'd expect between them.

That isn't luck. On a random price, the chance of reaching one price before another depends only on how far away each one is. From the 61.8% line, the high is 61.8% of the swing away and the low is only 38.2% away. The low is closer, so it usually comes first.

Buying the 61.8% line is a 1:1.62 trade

This links straight to the risk-reward ratio. Say you buy at the 61.8% line, put your stop under the swing low, and set your target at the old high.

You're risking 38.2% of the swing to make 61.8%. That's a risk-reward ratio of 1:1.62, the golden ratio itself. It breaks even if you win 38.2% of the time.

On the coin flip it won 37.6%, so it made about nothing: −0.02R a trade. Every other line did the same, from −0.04R to −0.01R. Deeper lines pay more and hold less, by just enough to cancel out.

Every line lost a little. Part of that is chance. Part is the fill: a buy order at a line fills as price falls through it, so you're often a little underwater straight away. Real orders work the same way.

It's the same reason every ratio broke even in our risk-reward test. On a random price, a bigger target is reached less often by exactly enough to cancel it out.

So do Fibonacci retracements work?

This test can't tell you they don't work on real charts. Real prices aren't coin flips. Trends, news and other traders' orders all matter.

The usual argument for the lines is that so many traders watch them that orders gather there. That could be true. But one bounce can't show it, because a random price bounces off them too.

What would show it is a test on real prices. The swings would be picked by a fixed rule before anyone looks at the lines, and the Fibonacci lines would have to beat made-up ones. We haven't run that here. A bounce on one chart is the start of a question, not an answer.

The tool is still useful as a plan. It turns "wait for a pullback" into prices you can set alerts and orders at, with an obvious place for a stop. Just don't treat a line as a reason to trade on its own.

How to use it without fooling yourself

• Pick the swing before you look at the lines. Use the most obvious swing low and high, and don't redraw it to make a line fit.
• Expect price to stop near some line. Five lines cover most of the swing, so a touch is likely by chance.
• Look for a second reason at the same price, like an old support level, a retest or a trendline.
• Plan the trade in R. Buying the 61.8% line with a stop under the swing low is a 1:1.62 trade, so it has to win more than 38.2% of the time.
• Keep score, misses included. If your lines don't hold more often than 1 minus their depth, they aren't helping.

Method, and what this cannot tell you

We built a price that moves at random. Each candle is 16 small random moves, up or down with equal odds, so it has a real open, high, low and close. We made 4 million candles from a fixed random seed, so anyone can run it again and get the same answer. A swing counted once price turned by about six times a typical candle's move, the way a zigzag indicator marks swings. We measured every pullback from wick to wick, in both directions.

• Real prices aren't a coin flip. Trends, news and other traders' orders all change the odds. This shows what you'd see with nothing behind the price at all.
• It can't tell you whether Fibonacci lines work on real charts. It tells you how often they'd seem to work by chance, which is the number any real test has to beat.
• Where pullbacks end depends on how you pick the swings. We used one fixed rule. A bigger or smaller swing size moves the depths around, but no rule makes a coin flip respect one line more than its neighbours.
• The hold test fills every order at the line and every stop at the swing's start, with no costs and no gaps. Real fills are worse.