The Stock Market Is Not Normal, and It Shows Traces of History
There is a picture of the stock market that everyone learns first, because it is simple and almost true. In this picture, each day’s return is like a coin flip: sometimes up, sometimes down, with no memory of yesterday and no knowledge of tomorrow. The picture has a precise consequence. If returns really were memoryless, then holding an investment five times longer would give you exactly five times the variance — risk would accumulate like the steps of a drunkard’s walk, no faster and no slower. Statisticians call this the random-walk benchmark. I will call it the coin-flip market. “Normal” describes it twice over: the returns follow the normal distribution, and nothing unusual ever carries over from one period to the next.
In 1988, Andrew Lo and Craig MacKinlay built a famous test around that consequence — the paper is bluntly titled “Stock Market Prices Do Not Follow Random Walks”. Take the variance of long-horizon returns, divide by the horizon times the variance of short-horizon returns, and check whether the ratio is one. For the U.S. market, it wasn’t — one of the first cracks in the random-walk picture. But their test, and most tests since, looks at a single series: an index, one line drawn through thousands of stocks. A market is not one line. It is a high-dimensional object with internal structure — a market-wide mode that moves everything together, industry and style groupings beneath it, and dozens of deeper, more idiosyncratic directions below those.
A new paper of mine generalises the variance-ratio test to that whole structure at once. Instead of one ratio for one index, it computes a ratio for every principal direction of the market’s covariance — every eigenmode, from the market mode down to the deepest — at every horizon from a day to five years, and in two separate channels: one for returns (which way prices move) and one for volatility (how violently they move). And because a single half-century sample can fool you, every number is computed a thousand times over on bootstrap re-samples of the data, so that everything comes with error bars. The data is the standard academic workhorse: daily returns on 49 U.S. industry portfolios from 1969 to 2026, cross-checked against 100 portfolios sorted by size and value, against each half of the sample separately, and against 25 European portfolios.
Under the coin-flip market, every one of those thousands of numbers should equal one.
Not normal
They do not equal one, and the sizes of the violations are worth savouring.
In the returns channel, the deviations are real but restrained. The market mode’s ratio rises above one at horizons of months — the echo of momentum, the tendency of recent winners to keep winning — and then sinks below one at horizons of years, the long reversal documented in the 1980s by De Bondt and Thaler. The deepest modes drift steadily below one: mean reversion, the slow snap-back of relative prices. These are the classic anomalies, and the framework recovers them all — but as modest bends in the curve, tens of percent, not orders of magnitude. The direction of the market, it turns out, is almost memoryless. That part of the textbook survives contact with the data surprisingly well.
The volatility channel is another world. At a five-year horizon, the market mode’s volatility ratio — the same statistic, computed for the sizes of moves rather than their signs — comes out at about 48. Where the benchmark says 1.0, the market delivers fifty. That is not a crack in the random-walk picture; it is the picture torn in half. Turbulence is not sprinkled evenly through time the way coin flips would sprinkle it. It arrives in waves, and the waves stack into longer waves, and the stacking goes on for years.
Traces of history
The mathematician Benoit Mandelbrot, who spent a career cataloguing the ways markets are wild rather than mild, gave the two great departures from normality biblical names. The Noah effect is the fat tails — the flood-sized moves that the bell curve says should never happen. The Joseph effect is the memory — seven fat years followed by seven lean years, prosperity and famine arriving in runs. His inspiration for the second came from outside finance entirely: the hydrologist Harold Hurst had spent decades studying eight centuries of Nile flood records and found that wet years cluster with wet years, dry with dry, over spans far longer than any weather system could explain. Hurst’s memory exponent — 0.5 for a memoryless process, higher for persistence, lower for anti-persistence — became the standard ruler for measuring how much of the past a process carries with it.
Held against that ruler, the market’s returns measure just above memoryless: the paper’s persistent factor has a Hurst exponent of about 0.52 to 0.57, and its anti-persistent factor — the mean-reverting undertow — sits far below, around 0.17 to 0.27. But the market’s turbulence is Joseph through and through. The volatility process behaves like a cascade: fast components with memory of days, slower ones with memory of months, and beneath them a component whose memory runs to years. Calm years genuinely follow calm years; turbulent years follow turbulent. Last decade’s storms are written into this year’s weather. That is the sense in which the market shows traces of history: its covariance today is partly a fossil record.
Five factors — one persistent, one anti-persistent, a central long-memory factor, and two volatility-cascade companions — turn out to reproduce seven classic statistical signatures of equity markets simultaneously: momentum, long-horizon reversal, factor momentum, deep-mode mean reversion, volatility clustering, multi-year volatility memory, and the concentration of turbulence onto the market mode. And the same five-factor structure, with nearly the same exponents, fits the industry portfolios, the size-and-value portfolios, each half of the sample, and the European panel. Whatever this structure is, it is not an American quirk or an artefact of one way of slicing the market.
The memory has a history of its own
Here is the finding I find most striking. The market’s memory is not a constant of nature — it changed, and the change can be dated.
Slide a 28-year window across the full sample, two years at a time, and refit the model inside each window, a thousand bootstrap replicates per window. Watch the slowest component of the volatility cascade. In windows centred in the early 1980s, its memory runs a little over two years. In windows centred from the mid-1990s onward, it runs to four. The transition concentrates in the late 1980s, and the uncertainty bands from the early windows and the late windows do not overlap — this is not noise wandering about, but a regime change, localised as sharply as a 28-year window permits. Notably, the change does not sit at the obvious calendar landmark (the sample’s 1998 midpoint, where a naive split-half comparison would put it). The usual suspects assemble themselves — the 1987 crash, the spread of index products and program trading, the changing microstructure of the market — but dating a shift is not the same as explaining it, and the paper is careful to do only the first.
One more twist. The model initially assumed that a stock’s exposure to the long-memory factor is one number — that the industries carrying the return-side memory are the same industries carrying the volatility-side memory. The data rejects this, decisively (the formal test clears a Bonferroni-corrected p-value of 0.0004). The two memories are carried by different parts of the market — anti-aligned, if anything. The market keeps two sets of books, written in different hands: one recording where returns have been trending, another recording where turbulence lives, and the entries do not match.
Why it matters
If risk compounded like coin flips, a five-year risk estimate would just be a one-year estimate times five. It is not — at the market mode, in the volatility channel, it is off by a factor of fifty. Anyone whose horizon is measured in years rather than days — a pension fund, a sovereign fund, a household saving for retirement — lives inside the long-horizon covariance, and the long-horizon covariance is shaped by memory that the standard scaling rules simply erase. The practical moral is not that markets are unknowable; it is the opposite. The departures from normality are not formless noise. They are structured — a small number of memory components, stable across markets and decades, measurable with honest error bars, and legible once you take the covariance matrix apart direction by direction.
The market is not normal. It remembers. And with the right instrument, you can read what it remembers — and even see the moment, somewhere in the late 1980s, when it began remembering differently.
The paper — “A Spectral Generalisation of the Variance Ratio: Eigenstructure of Long-Horizon Portfolio Covariance and a Multi-Memory Factor Model of U.S. Equity Returns” — is available on the spectral portfolio project page, where it joins the theoretical companion on spectral portfolio theory.