Moving Averages Follow Price — the plain-English companion

the paper (DOI) · code & data repository

Companion to the research paper of the same title. This is education, not investment advice. Nothing here tells you what to buy, sell, or predict. It explains what the paper found, how the checking worked, and what the results do and do not mean.

The claim in one sentence

When a price chart “comes back to” its moving average, it is mostly the average doing the moving — not the price.

Start with the picture everyone has seen

Turn on any financial news channel and wait. Before long, someone will show a price chart with a smooth line running underneath it — the 200-day moving average, or the 50-day. The price wanders above the line. And the person on screen says something like: “price is stretched here, we expect it to revert to the mean.”

Then, a few weeks later, price and line meet. The prediction looks brilliant. The chart seems to prove that prices get pulled back toward their averages, like a ball rolling downhill. Millions of trading decisions, thousands of strategy books, and an entire vocabulary — “overextended,” “reversion,” “the pull of the mean” — are built on this picture.

The paper asks a question almost nobody asks about that picture: when the price and the line meet, who traveled? Did the price come down to the line? Or did the line climb up to the price?

That question turns out to have an answer. Not an opinion — an answer, one you can compute. And the answer, on 155 years of data across four continents, is that most of the traveling is done by the line.

What a moving average actually is

To see why, you have to look inside the line — because a moving average is not a mysterious market force. It is a bucket of old prices.

A 200-day simple moving average works like this: take the last 200 daily closing prices, add them up, divide by 200. That number is today’s value of the line. Tomorrow, the oldest price falls out of the bucket, the newest price drops in, and you divide by 200 again. Every day, one old day out, one new day in.

Notice what this means. The average is built entirely out of the past. It contains no forecast, no opinion, no force. It is arithmetic performed on history. And that arithmetic has an unavoidable consequence. Whenever the price moves somewhere and stays there, the bucket slowly fills up with prices from the new level — and the average is dragged there, mechanically, automatically, with the certainty of long division.

The paper calls this property adaptation. The average adapts to the price. It cannot do otherwise. It is made of the price.

There are fancier averages — the exponential moving average (which weights recent days more heavily), the weighted moving average, the Hull moving average (a fast construction popular with traders). The paper handles all of them. They differ in speed, but they share the same nature: they are recipes applied to past prices, and they all get dragged toward wherever the price goes.

The thought experiment that cracks the myth open

Here is the paper’s central move, and it needs no statistics at all — just a scenario simple enough to settle by hand.

Imagine a price that has sat at 100 forever. One day it jumps to 110 — and then never moves again. Not one tick. It sits at 110 for the rest of time.

At the moment of the jump, there’s a gap: price at 110, the 200-day average still down near 100, because its bucket is full of old 100s. Now watch what happens over the following days. Each day, one stale 100 falls out of the bucket and one fresh 110 drops in. The average creeps upward: 100.05, 100.10, 100.15… After 200 days, every old price has been flushed out, the bucket holds nothing but 110s, and the average sits at exactly 110.

The gap has fully closed. Price and line have met. On a chart, it looks exactly like “reversion to the mean.”

But the price never moved. One hundred percent of the gap was closed by the average traveling to the price. Zero percent by the price traveling to the average. If you had predicted “price is stretched, it will come back to the line,” you would have looked right — and been wrong about everything. Nothing came back. The line went and got it.

This is the content of the paper’s first four lemmas — one each for the simple, exponential, weighted, and Hull averages. (A lemma is a small proven mathematical statement, a building block.) For every one of these filter types, a one-time permanent jump produces complete gap closure with the average doing all of the work. That’s not an empirical finding subject to luck; it is algebra, as certain as 2 + 2. The exponential average closes the gap along a smooth curve; the simple average closes it along a straight ramp; the Hull average overshoots slightly and settles. All of them arrive. The price does nothing.

The lesson of the thought experiment is precise and narrow: the meeting of price and average is, by itself, zero evidence of any force pulling on the price. The meeting is guaranteed by the construction of the average. Convergence is what trailing averages do for a living.

But real prices wiggle

A fair objection: real prices don’t jump once and freeze. They jiggle every day. Maybe in real markets, the price genuinely does some of the traveling.

The paper meets this objection twice — once in theory, once in data.

The theory step asks: suppose prices moved like coin flips — each day up or down at random, with no memory and no force pulling anywhere. (Mathematicians call this a random walk, and it is the standard “no forces here” baseline.) If a random-walk price finds itself far from its average, what happens next, on average?

The paper proves (Theorem 2) that under a random walk, the gap still closes — and the average share of the closing, measured across many such episodes, equals exactly one hundred percent. The price, having no memory, wanders with no preference for the direction of the line; its expected contribution to closing the gap is zero. The average, being made of the price’s past, grinds toward it regardless. So even with realistic jiggling, the pure no-forces world produces exactly the pattern that chart-watchers call reversion: gaps that reliably close, closed entirely by the line. A further result (Theorem 3) puts guardrails on how this works for a much broader family of price processes, separating the closure into a mechanical part, a drift part, and unavoidable noise.

That sets up the real question. The myth says price gets pulled toward the average. The mathematics says the pattern everyone sees would appear even if nothing pulled at all. The only way to find out what real markets do is to measure who travels — and that’s the empirical heart of the paper.

The scoreboard: measuring who closed the gap

The paper builds a scoreboard. Every time a price gets unusually far from its moving average — an “event” — the clock starts. Sixty-three trading days later — about three months — the paper measures how much of the original gap got closed. Then it splits the closure into two contributions: how far the price moved toward where the average had been, and how far the average moved toward where the price had been. The average’s slice of the total is called the MA share. If the myth were right, the price would do most of the traveling and the MA share would sit well below 50%. If adaptation dominates, the MA share sits above 50%. Events are spaced at least 63 days apart so no stretch of history is counted twice.

Before trusting this ruler on real data, the paper checks the ruler itself. It runs the same measurement on synthetic data where the true answer is known — pure random walks, plus simulations built to churn the way real markets do. The measured shares come back approximately unbiased. The ruler isn’t bent.

Then the measurement runs on reality: eleven instruments, chosen to span the world’s major markets. The S&P 500 with daily data back to 1871 — 155 years, more than 25,000 trading days. The NASDAQ-100. Japan’s Nikkei 225 back to 1949. Germany’s DAX. Britain’s FTSE 100. Hong Kong’s Hang Seng futures. Crude oil futures. Gold futures. Ten-year U.S. Treasury futures. Euro and yen currency futures. Six asset classes, four continents. Each instrument is tested against four moving averages spanning the speed spectrum — the fast Hull-50, the medium SMA-50 and EMA-50, and the slow SMA-200 — giving 44 instrument-filter combinations in all.

The result: the average’s share of gap closure exceeds 50% on all 44 combinations. All of them. The lowest score in the entire table is 63.5% (Hang Seng, Hull-50) — meaning even in the single friendliest case for the myth, the line did nearly two-thirds of the traveling. The highest is 166.8% (NASDAQ-100, SMA-200).

Wait — how can a share be more than 100%?

The number that buries the myth

A share above 100% means something wonderfully strange: the price didn’t just fail to travel toward the average — it moved further away, and the average had to cover more than the whole original gap to catch it. Imagine chasing someone who is walking backward, away from you. When you finally reach them, you have covered more than the distance that originally separated you. That’s an MA share above 100%: total closure, delivered entirely by the chaser, against a target in retreat.

Thirty-three of the 44 combinations — 75% — show an MA share above 100%. Read that again with the myth in mind. In three-quarters of the tested cases, at the very moments chart-watchers describe as “price reverting to its mean,” the price was on net moving away from the average while the average hunted it down. The folklore doesn’t just overstate the pull on price. In the typical case, the pull points the other way.

The second instrument: does tomorrow lean toward the line?

The gap-closure scoreboard is an attribution over three-month windows. The paper adds an independent, sharper test at the shortest possible horizon: when price sits far from its average today, does tomorrow’s move lean toward the average more often than a coin flip? If a genuine attractive force existed, this “toward rate” should sit meaningfully above 50%.

Under the fast Hull-50, eight of the eleven instruments are statistically indistinguishable from 50% — coin flips. Three show mild attraction, with toward rates around 53% — Hang Seng at 53.2%, gold at 52.9%, the euro at 53.9%. Those figures are reported without the statistical correction for having run many tests at once, so some of them may be luck. Under the slow SMA-200, the surprise runs the other direction: three instruments — the S&P 500 (47.8%), the Nikkei (45.8%), and the NASDAQ-100 (47.0%) — show mild repulsion: tomorrow leans slightly away from the line. After adjusting for each market’s long-run upward drift, two of those three fade to coin flips; only the Nikkei’s repulsion survives, and strongly.

The headline from this test is the absence of a headline: across eleven instruments and two very different filters, there is no broad, systematic tendency for price to move toward its moving average. A few small effects flicker at the edges, in both directions, mostly fragile. The mass of the evidence is a coin flip.

The self-fulfilling prophecy check

There’s a sophisticated version of the myth worth testing separately. It goes: fine, the average is just arithmetic — but millions of traders watch the same lines. If everyone buys at the 200-day average, their buying creates the bounce. The line becomes a self-fulfilling meeting point — what game theorists call a Schelling point, a spot people coordinate on simply because everyone expects everyone else to.

The paper tests this directly: if popular, round-number windows (like 50 and 200) act as coordination magnets, then price behavior at those averages should differ measurably from behavior at arbitrary unpopular windows (like 47 or 183) that no one watches. Across 137 instrument-window-timeframe combinations, the measured difference is minus 0.005 percentage points — indistinguishable from zero (a formal test agrees, p = 0.78; in plain terms, pure noise). At the one-day horizon this test can see, the watched lines behave exactly like the unwatched ones. If trader coordination at famous moving averages moves prices, it left no detectable fingerprint here.

What the data says is actually real

A paper that only demolished something would be suspect — reality usually contains something. The analysis surfaced two genuine, forward-looking regularities, and the paper reports both, along with every result that cuts against its own thesis.

First: distance from the average predicts turbulence, not direction. When price is unusually far from its moving average, the following weeks tend to be unusually volatile — bigger swings, both ways. This relationship (a rank correlation of +0.54 on average) is positive on all eleven instruments. And it survives the paper’s toughest grading: of 38 tests scored under rules that correct for market data’s long memory, 33 remain significant — including every full-sample and every out-of-sample test. A stretched chart, it turns out, is telling you to expect drama, not to expect a particular direction.

Second: real mean-reversion does exist — somewhere else. The S&P 500 shows genuine reversion at the quarterly scale: a return over one 126-day stretch is negatively related to the return over the next (correlation −0.28). This is a different animal at a different timescale, driven by a different mechanism than the day-by-day mechanical adaptation that dominates the charts. The paper’s point was never “no force ever pulls on prices.” It is that the convergence you see between a price and its trailing average is not evidence of such a force. Where a real force does show up, it shows up in tests built to detect motion of the price itself — not in the meeting of a price with its own past.

And the fine print is stated in the paper’s own abstract rather than buried. Sliced into rolling ten-year windows, the S&P’s aggregate MA share dips below 50% in 5 of 18 decades — and in one decade, 1991–2001, the scoreboard’s split actually goes negative. In the modern era (1990–2026), the S&P shows a mild but statistically significant attraction — a toward rate of 53.1% — reported unadjusted for multiple testing. The dominant pattern across 155 years is adaptation; the paper prints its own exceptions at full size.

Why this matters beyond trading charts

“Reversion to the mean” is one of the most repeated phrases in finance, and this paper’s evidence says the chart pattern behind it is mostly an optical illusion — a mechanical echo mistaken for a force. That has teeth. Trading strategies premised on the average “pulling” price are premised on a chase they have backwards. Backtests that celebrate convergence are often celebrating arithmetic. And the deeper habit — treating the meeting of a measurement with reality as evidence that reality obeyed the measurement — extends far beyond markets. Any trailing average of anything — sales figures, temperatures, test scores — will converge to its series by construction, and every such convergence is available to be misread as a story about forces. This paper is the series’ foundation stone because it establishes, in the cleanest possible setting, the general principle: trailing measurements follow reality; reality does not follow them.

What the checking caught in this paper

The general machinery every paper in this series runs through — the hash-pinned inputs, the machine-checked ledger of numbers, the verification program, the adversarial review — is described once in the series’ shared verification note, which follows every companion on LaggingTruth.com. What belongs here is what the process caught and changed in this paper specifically.

The paper’s statistical claims were re-run under a corrected inference regime that respects the memory in market data — the kind of correction that routinely kills fragile findings. Five of 38 volatility-prediction tests fell; the paper reports the survivors (33, including every full-sample and out-of-sample test) alongside the casualties. A late strengthening pass replaced three of the paper’s own hedged caveats with completed analyses, registering three new ledger claims in the process — the final ledger binds 24 claims under 73 machine checks, all green at the publication freeze. And every one of the paper’s 31 citations was verified against its published source before release. The findings that cut against the thesis — the sub-50% decades, the negative 1991–2001 window, the modern-era attraction, the Nikkei’s stubborn repulsion — are printed in the abstract, not appendixed.

What this cannot do

The result is an attribution of the past, not a crystal ball. It tells you who did the traveling in historical gap closures; it does not forecast the next one. The 63-day measurement horizon is a choice — robustness checks at other horizons agree in direction, but the specific share numbers depend on it. The aggregate share weights big events more heavily; the median event tells a more modest version of the story, and the paper reports both. The self-fulfilling prophecy test sees only next-day effects; slower coordination dynamics would need a different instrument. And none of this says moving averages are useless — they remain fine summaries of where price has been. The claim is about what their convergence with price means, which is: by itself, nothing.

The takeaway

A moving average is a bucket of the past, and it follows the present wherever the present goes — that is what it is for. On a century and a half of data across six asset classes, when price and average met, the average did the walking — more than half the distance in every one of 44 tests. In three-quarters of them it walked all of the distance and then some, chasing a price that was moving away. Tomorrow’s price does not lean toward the line. The famous lines behave no differently from the forgotten ones. What a stretched chart actually foretells is turbulence, with no preferred direction.

So the next time someone points at a chart and says the price came back to its average — check who traveled. The mountain did not come to the average. The average went to the mountain. It always does. It’s made of it.

This companion is licensed CC BY-NC 4.0. The research paper it accompanies is licensed CC BY-NC-ND 4.0, and the analysis and verification code is MIT-licensed. Education, not advice: nothing in this document is financial advice, an investment recommendation, or a forecast.