Fibonacci Retracements and Extensions
Fibonacci tools draw correction and extension levels from Fibonacci ratios (23.6%, 38.2%, 61.8%, 78.6%) and Dow theory's 50% level; they ship with charting software and appear in market commentary. Murphy covers them in chapter 13 of Technical Analysis of the Financial Markets. Tested on the Dow Jones from 1915 to 2003, these ratios (except 23.6%, which the test leaves out) occur no more often than chance (Batchelor and Ramyar, 2006).
Last updated: 20 October 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% (φ⁻¹, the inverse of the golden ratio): any term divided by the next converges to 0.618. For example, 144/233 = 0.6180. Murphy names it, with 50%, as one of the two key ratios for setting retracement targets (cited by Batchelor and Ramyar, 2006)
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. It comes from Dow theory: Hamilton (1922) writes that secondary movements retrace 33% to 66% of the primary move, with 50% the typical amount (cited by Batchelor and Ramyar, 2006)
Murphy covers these ratios in chapter 13 (1999 edition), which is devoted to Elliott wave theory; Elliott (1940) held that successive price moves tended to conform to Fibonacci ratios (Batchelor and Ramyar, 2006). A retracement is measured across the whole price swing — trough to peak for an advance — and each ratio gives a price level where the method places a possible end to the correction.
Why it matters
Fibonacci levels are among the most cited tools in technical analysis. In the Batchelor and Kwan (2000) survey of practising analysts they are mentioned more often than moving averages, and Batchelor and Ramyar (2006) collect, over three days in October 2004, commentaries published by Reuters and Dow Jones among others that rely on them for currencies, interest-rate contracts, coffee and crude oil. The question is whether prices actually respect these levels. Three studies shed light on it:
1. Equities: Batchelor and Ramyar (2006) identify the successive moves of the Dow Jones from 1915 to 2003 and count how many retracements and projections fall within 2.5 percentage points of the ratios 0.382, 0.5, 0.618, 0.786, 1, 1.382, 1.618, 2, 2.618 and 4.236, in price and in duration. They compare that count with the count from series drawn at random from the same index: a few ratios stand out, but no more than chance produces over so many tests. Their conclusion: the idea that these ratios occur in the Dow can be dismissed.
2. Currencies: Osler (2000) tests the support and resistance levels published daily from 1996 to 1998 by six foreign-exchange firms (commercial banks, investment banks, information providers, in the United States and abroad) for the mark, the yen and the pound against the dollar. These levels predict intraday trend interruptions better than 10,000 sets of arbitrarily chosen levels. More than 70% of them end in 0: they are mostly round numbers.
3. The mechanism: Osler (2003) examines almost 9,700 stop-loss and take-profit orders placed at a large bank in three currency pairs. Take-profit orders cluster at round numbers, which tends to make the rate bounce there; stop-loss orders sit just beyond them, which tends to extend the move once the round number is crossed.
This concentration of orders is therefore established for round numbers in currencies; none of the studies cited here establishes it for Fibonacci levels.
Key points
Levels are computed on a complete swing: trough to peak for an advance, peak to trough for a decline. The drawing is the same on every timeframe and in every market
38.2% and 61.8% sit either side of Dow theory's 50%, inside the one-third to two-thirds range that Hamilton already gave for corrections (cited by Batchelor and Ramyar, 2006)
Extensions are used to set objectives: after a correction, the method targets 100% first (a return to the prior peak), then 161.8%. Batchelor and Ramyar also test these projections on the Dow Jones: they fall on the ratios tested no more often than chance
Two results need to be kept apart: a level that many traders watch can concentrate orders, as Osler (2003) shows for round numbers in currencies; the Fibonacci ratios tested, on the other hand, occur no more often than chance over nearly a century of the Dow Jones (Batchelor and Ramyar, 2006)
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 February 2022 low (~4,115) and then the June 2022 low (~3,637) settled beneath the 23.6% and 38.2% levels respectively. The low of 13 October 2022, at 3,491, came within 0.5% of the 50% level without approaching 61.8% (3,195), and the index rose above 5,000 in 2024. One case that lands on a level does not show that the level stopped the decline: only a count of every retracement, compared with chance, can settle the question, which is what Batchelor and Ramyar do for the Dow Jones from 1915 to 2003.
Common mistakes
CautionTreating a Fibonacci level as a point where price will stop. On the Dow Jones from 1915 to 2003, retracements stop at the Fibonacci ratios tested no more often than chance (Batchelor and Ramyar, 2006)
Judging the tool by examples. One case that lands on a level says nothing about frequency: every case has to be counted, including those that fall between two levels, and the count compared with chance
Crediting Fibonacci levels with an effect established for round numbers. Osler (2003) shows that orders cluster at round numbers in currencies; none of the studies cited in this sheet shows the same clustering at Fibonacci levels
Sources
ReferencesBatchelor, R. and Ramyar, R. (2006), "Magic numbers in the Dow", Cass Business School, City University London. openaccess.city.ac.uk.
Osler, C. L. (2003), "Currency Orders and Exchange Rate Dynamics: An Explanation for the Predictive Success of Technical Analysis", The Journal of Finance, vol. 58, no. 5, pp. 1791-1819.
Osler, C. L. (2000), "Support for Resistance: Technical Analysis and Intraday Exchange Rates", Federal Reserve Bank of New York, Economic Policy Review, July 2000.
📊 Fibonacci — Retracements and the Golden Zone
- Impulse
- Retracement
- Extension
Market impact
MarketsFibonacci tools ship as standard with charting software, and market commentary quotes these levels: Batchelor and Ramyar (2006) collect examples on currencies, interest-rate contracts, coffee and crude oil, published by Reuters and Dow Jones among others over three days in October 2004. Widespread use does not prove an effect on prices: on the Dow Jones from 1915 to 2003, the ratios tested occur no more often than chance (Batchelor and Ramyar, 2006), and the concentration of orders documented by Osler (2003) concerns round numbers in currencies.