Fibonacci Retracements and Extensions
Fibonacci tools use mathematical ratios derived from the Fibonacci sequence to identify potential support/resistance levels and price targets. The key ratios -- 23.6%, 38.2%, 50%, 61.8%, and 161.8% -- are among the most widely used technical levels in financial markets.
Last updated: 29 July 2026
Prerequisites
PrerequisitesThis sheet assumes you can already read a trend and judge its strength, together with support and resistance (ch. 2, Easy level). Fibonacci retracements analyse corrections *inside* a trend, so the trend itself has to be established before the tool means anything.
Definition
DefinitionFibonacci ratios come from the Fibonacci sequence, set out by Leonardo of Pisa — Fibonacci — in his Liber Abaci (1202). The rule is simple: each number is the sum of the two preceding it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...
The ratios used in technical analysis are derived from the arithmetic relationships between the terms of that sequence:
61.8% (the golden ratio, φ⁻¹): any term divided by the next converges to 0.618. For example, 144/233 = 0.6180. This is the single most important ratio, and practitioners treat it as the core Fibonacci level
38.2%: any term divided by the term two places further along converges to 0.382. It is the complement of 61.8% (1 − 0.618 = 0.382)
23.6%: any term divided by the term three places further along converges to 0.236
78.6%: the square root of 0.618 (√0.618 ≈ 0.786), used as a deep retracement
50%: not a Fibonacci ratio at all, but included as standard because Dow theory already identified the halfway retracement as a classic correction level
Murphy covers these ratios in chapter 13 of Technical Analysis of the Financial Markets (1999 edition, devoted to Elliott wave theory, for which Fibonacci numbers supply the mathematical foundation) as part of his discussion of trend retracements. He presents them as a mathematical filter that sharpens the support and resistance zones already identified by classical chart reading. Retracements are measured across the whole price swing — trough to peak for an advance — and the ratios mark where the correction is likely to end.
Why it matters
Murphy builds Fibonacci ratios into all of his retracement examples because they give a precise mathematical framework for anticipating where a correction should end. Beyond the tool itself, though, the real question is: why would these ratios work on financial markets at all?
Three complementary explanations:
1. Mathematical universality: the golden ratio (φ = 1.618) appears in unrelated natural phenomena — galactic spirals, human body proportions, plant phyllotaxis, the structure of DNA. This is not mysticism; it is an emergent property of systems that grow recursively. Financial markets, being complex adaptive systems with feedback loops, display comparable behaviour.
2. The self-fulfilling effect: Murphy stresses that millions of traders worldwide watch the same Fibonacci levels. As price approaches the 61.8% retracement, buy orders cluster and fire — which is what actually turns the price. The result is a convention effect that reinforces the levels' reliability. The academic literature, by contrast, is mixed to negative: the reference study by Batchelor and Ramyar (2006) on the Dow Jones finds no tendency for retracements to stop at Fibonacci levels more often than chance would predict. Their method is the one that matters here — comparing how often price actually turns at a Fibonacci level against how often it would do so at random — and once that comparison is made, the edge disappears. What practitioners observe is therefore best explained by that concentration of orders: the convention itself, not some hidden property of prices. The distinction is not academic hair-splitting, because it dictates how the tool should be used. If a level holds only because enough participants act on it, then the most-watched levels are the most dependable, and a level nobody is watching carries no particular weight. That is precisely why 38.2%, 50% and 61.8% behave differently from 23.6% or 78.6%: not because the mathematics ranks them, but because the audience does.
3. Crowd psychology: market corrections are not random — they reflect participants' state of mind. After a 100-point advance, a 38% pullback (a shallow retracement) signals strong buying conviction. A 62% pullback signals deep doubt, with the trend surviving only just. These proportions map intuitively onto the confidence levels of market participants.
Further reading: Robert Fischer, Fibonacci Applications and Strategies for Traders (1993), and Carolyn Boroden, Fibonacci Trading (2008), are the standard practitioner references on applying these ratios to trading.
Key points
Murphy sets out a probability hierarchy for retracements: the minimum correction in a strong trend is roughly one third (33-38%), the moderate retracement is 50%, and the 62-66% band is the trend's last chance — his reversal threshold is two thirds, beyond which a reversal becomes more likely than a mere correction. The 78.6% level used today as a deep retracement is a later practitioner addition (harmonic approaches, Gartley patterns), not part of Murphy's framework
A Fibonacci retracement gains considerable weight when it coincides with other technical levels: a horizontal support, a moving average, or a trendline at the same price. This confluence principle sits at the heart of Murphy's method, and it follows directly from the self-fulfilling mechanism — where several independent methods point at the same price, several independent groups of traders place orders there
Fibonacci extensions turn a completed retracement into realistic price targets. If price corrects to 61.8% and then resumes the trend, the minimum objective is the 100% extension (a return to the prior peak), and the extended objective is the 161.8% extension
Fibonacci retracements work across every timeframe (intraday to monthly) and every market (equities, FX, commodities, crypto). As elsewhere in technical analysis, Murphy favours the longer timeframes — daily and weekly — whose signals are more reliable; practitioners attribute this partly to the larger volume that trades around those levels, and the same reasoning applies to instruments: a level on a heavily traded index has an audience that a thinly traded stock does not
The golden zone (50%-61.8%) is modern practitioner vocabulary, absent from Murphy: many traders regard it as the optimal correction area in a strong trend and concentrate their entries there, which reinforces the self-fulfilling effect further still
Concrete example
ExamplesAfter the March 2020 low at 2,191 and the January 2022 peak at 4,818, the S&P 500 entered a bear market. Applying Fibonacci levels to that advance gave: 23.6% = 4,199, 38.2% = 3,814, 50% = 3,505, 61.8% = 3,195. The arithmetic is worth following once, because everything else rests on it: the swing measures 4,818 − 2,191 = 2,627 points, so the 50% retracement sits at 4,818 − (0.50 × 2,627) = 3,505, and the 61.8% at 4,818 − (0.618 × 2,627) = 3,195. Note that the levels are subtracted from the peak, not added to the low — a retracement is measured against the move that produced it. The 23.6% and 38.2% levels were both broken in turn — the February 2022 low (~4,115) and then the June 2022 low (~3,637) settled clearly beneath them — exactly the cascade described in the key points above. It was the 50% retracement that halted the decline: the market bottomed at 3,491 on 13 October 2022, less than 0.5% from the level, without ever approaching 61.8% (3,195) — a sign that the primary uptrend remained intact. The rebound from there carried the index above 5,000 in 2024.
Common mistakes
CautionTreating Fibonacci levels as exact lines, accurate to the point. Murphy is clear that these are probability zones (±1-2%), not surgical levels. A retracement at 60.5% is every bit as meaningful as one at 61.8%
Applying Fibonacci to price swings that are too short or too insignificant. Retracements are more reliable on swings that follow a clear trend and cover enough ground (at least 20-30% on an index, or a clean swing in intraday trading)
Ignoring confluence. A 61.8% retracement standing alone carries far less weight than a 61.8% that coincides with former horizontal support and the 200-day moving average. Murphy insists on the point: Fibonacci levels are filters to be combined, not standalone signals
Treating every Fibonacci level as equal. In practice 38.2%, 50% and 61.8% are the three primary levels; 23.6% and 78.6% are secondary. Cluttering the chart with every level at once costs more in readability than it adds in information
Practical note
MurphyMurphy uses Fibonacci retracements as an anticipation tool once a significant price move has run its course. His method: as soon as a directional move ends and a correction begins, he immediately draws the Fibonacci levels between trough and peak (or the reverse). He then places his orders in advance at the confluence levels — Fibonacci plus horizontal support plus moving average. Preparing the ground this way keeps him from reacting emotionally as the correction unfolds, and lets him act on a plan settled beforehand — the classic distinction between planned trading and impulsive trading.
📊 Fibonacci — Retracements et Zone d'Or
Market impact
MarketsFibonacci levels are among the most widely used tools among institutional and retail traders alike — they ship as standard on professional charting platforms, and the technical research notes of the major banks refer to them routinely for the main indices. The 161.8% extension is frequently quoted as a price objective in research reports. Fibonacci clusters on the major indices (S&P 500, Nasdaq, DAX) concentrate orders that amplify price reactions at those levels. In October 2022, the 50% retracement of the 2020-2022 advance featured prominently in technical commentary on the market's direction.