The Vault

THE FRACTALS OF FINANCE | Research Series Phase 2 | Episode 4 of 9

Proving It Is Not Random

The variance ratio was supposed to kill the random walk. It does something better. It reveals the spectrum’s two forces in a completely different test.

Act I revealed the coupled spectrum. Episodes 1 through 3 showed that the near-zero autocorrelation conceals a structured oscillation between positive and negative feedback, that the oscillation runs in every asset class, that the sixty-eight markets are permanently coupled along a spectral axis, and that the coupling produces the crisis alpha that defines trend-following’s value to equity portfolios.

But revelation is not proof. The rolling windows, the regime classifications, the conditional correlations: these are descriptive tools. They show patterns. They do not formally rule out the possibility that a random process produced them.

Act II requires a different standard. The variance ratio test provides it.

The Test

The idea behind the variance ratio is simple. If a market is truly random, then a move that starts on Monday should have no tendency to continue or reverse by Friday. The size of weekly swings should be predictable from the size of daily swings, in the same way that flipping a coin fifty times produces a predictable range of outcomes from flipping it ten times.

If weekly moves are consistently larger than the daily moves predict, something is making moves persist. Prices are adding up. Positive feedback is at work. If weekly moves are consistently smaller, something is making moves cancel out. The market is oscillating back and forth rather than going anywhere. Negative feedback is at work.

The variance ratio puts a number on this. A value of 1.0 means the market behaves exactly like a coin flip: no persistence, no reversal, pure randomness. Above 1.0, moves are adding up: the trending end of the spectrum, positive feedback dominant. Below 1.0, moves are cancelling out: the oscillating end, negative feedback dominant. The further from 1.0, the stronger the departure from randomness.

This test has been applied to financial markets since Lo and MacKinlay published it in 1988. Its conclusion, that the random walk fails, has been known for decades.

We are not here to repeat that conclusion. We are here to do something nobody has done before.

We decomposed the variance ratio by market state, using a Donchian channel breakout to separate periods of genuine directional persistence from periods of range-bound oscillation. And the result is not just a rejection of the random walk. It is the spectrum itself, visible in a completely different test.

The Full-Sample Picture

We computed the variance ratio at eight horizons, from two days to one year, for all sixty-eight contracts in the universe.

To know whether the results are meaningful, we needed a baseline: what would the variance ratio look like if markets were purely random? We built one by taking the S&P 500’s actual daily returns and shuffling them two thousand times, like cutting a deck of cards repeatedly. Each shuffle keeps the same returns but scrambles the order. Any genuine persistence in the data disappears. What remains is the variance ratio you would expect from randomness alone, and the range across two thousand shuffles forms a confidence interval. If the real variance ratio falls outside that range, the sequence itself is doing the work, not just the size or shape of the returns.

Figure 4.1 Variance ratio curves for all sixty-eight futures markets, plotted against horizon from two days to one year. Grey line at 1.0 marks the random walk null. Grey band is the 95 percent bootstrap confidence interval. Gold line is the universe mean. Red line is the S&P 500.

The universe mean sits below 1.0 at every horizon. The S&P 500 falls below the confidence interval at seven of eight horizons, reaching a minimum of 0.677 at three months. At the two-day horizon, seventy-six percent of markets are outside the random envelope.

The random walk fails across the full universe, at every horizon. That has been known since Lo and MacKinlay. This is not news.

What follows is.

The Average Lies Again

Taken at face value, the full-sample results suggest the S&P 500 is mean reverting. Variance grows slower than a random walk predicts. Moves are being consumed. That is the obvious conclusion, and it is wrong.

The central thesis of this series is that averages conceal structure. Episode 1 showed it for autocorrelation. Episode 3 showed it for the trend-equity correlation. Both times, the full-sample statistic was approximately zero, and both times the zero hid two opposing forces.

We applied the same principle to the variance ratio. But instead of using the rolling autocorrelation to classify states, we used a tool from trend-following itself: the Donchian channel breakout. A twenty-day Donchian identifies moments when price makes a new twenty-day extreme, the classic signal that directional energy has arrived. When the system is in a breakout, positive feedback is active. When the system is flat, waiting for a breakout that has not come, negative feedback dominates and moves cancel within the channel.

We separated each contract’s return history into breakout periods, when a Donchian position was active, and flat periods, when no breakout signal was present. Then we computed the variance ratio separately within each state.

Figure 4.2 Regime-conditional variance ratio profiles. Green: breakout periods (trending). Red: flat periods (oscillating). Grey dashed: full-sample average. The gap at three months is 0.847.

The green line is the variance ratio during breakout periods. It crosses above 1.0 at the one-month horizon and continues rising, reaching 1.13 at three months and 1.17 at one year. Variance is growing faster than linear. Trends are persisting. Moves are following through.

The red line is the variance ratio during flat periods. It collapses below 1.0 immediately and falls to 0.28 at three months. Variance is being consumed. Moves are reversing within the channel. Overshoots are correcting. The market is oscillating, not trending.

The grey dashed line is the full-sample average, sitting at 0.910 at three months. It tells you that variance grows slower than linear. It does not tell you why. The spectral decomposition answers that question: the market is not simply mean reverting. It oscillates between two states that produce opposite variance signatures. The full-sample value below 1.0 is the weighted average, because the market spends a substantial fraction of its time in the range-bound state where moves cancel.

The spectrum is not hidden in the average. It is the reason the average exists. Breakout periods produce variance ratio above 1.0. Flat periods produce variance ratio far below 1.0. The two forces, visible in a completely different test from the rolling autocorrelation, separated by the very mechanism that trend-following strategies use to distinguish signal from noise.

Two Forces, One Market

The spectral decomposition works not just on average but for individual contracts.

Figure 4.3 Scatter of per-contract VR at three months. Horizontal: breakout-period VR. Vertical: flat-period VR. The lower-right quadrant, where the spectrum predicts markets should cluster, contains the largest group. Sixty-five percent of markets show breakout VR above 1.0. One hundred percent show flat VR below 1.0.

Each dot is a market. The horizontal axis shows its variance ratio during breakout periods. The vertical axis shows its variance ratio during flat periods. The lower-right quadrant is where the spectral framework predicts markets should sit: breakout VR above 1.0 (trends persist) and flat VR below 1.0 (moves oscillate). The largest cluster sits exactly where the framework predicts.

This is not a statistical artefact of averaging across contracts. The two-force pattern appears in individual markets, across every asset class, at the level of single contracts. The spectrum is not an aggregate phenomenon. It is a market-by-market reality.

The S&P 500

Figure 4.4 S&P 500 variance ratio profile. Minimum VR of 0.677 at three months means three-month S&P 500 variance is only 67.7 percent of what a random walk would produce.

The S&P 500 is the most extreme case in the equity universe. Its variance ratio drops to 0.677 at three months, meaning that three-month S&P 500 variance is only 67.7 percent of what a random walk would produce. The remaining 32.3 percent of variance is being consumed by negative feedback: oscillation partially reversing the directional moves that positive feedback created.

This is exactly the mechanism that produces the crisis alpha from Episode 3. Positive feedback drives equities in one direction. Then the system shifts toward the negative-feedback end of the spectrum and oscillation sets in. Trend strategies, positioned for the directional move, get the full run. The variance ratio quantifies how much of that move is subsequently consumed by the reversal.

The Spectrum Across Four Decades

Figure 4.5 Universe mean variance ratio at three months by decade. The most recent decade shows the largest departure from the random walk across the full universe.

The decade view seeds what comes next. The universe mean VR at three months has varied across four decades, but the most recent decade, 2016 to 2026, shows the strongest departure from the random walk: 0.894, the lowest value in the series.

This is a preview, not a proof. The question of whether the feedback structure is accelerating requires the full amplitude analysis that Episode 5 delivers. But the direction is suggestive: the most recent decade, the one shaped by the largest policy interventions in history, shows the largest systematic departure from randomness.

State 1 Is Dead

The variance ratio results eliminate the first of our three states with finality. The random walk fails across the full universe, at every horizon, in every asset class.

But what this episode adds is not the rejection. That has been known since 1988. What this episode adds is the mechanism. The spectral decomposition shows that the variance ratio is not just a number. It is the weighted average of two opposing forces. Breakout periods create variance. Flat periods consume it. The full-sample VR sits between them.

This is the same finding as Episodes 1 and 3, arrived at through a completely independent test. The autocorrelation averages to near-zero because positive and negative feedback cancel. The trend-equity correlation averages to modestly negative because bull and bear markets produce opposing signs. The variance ratio averages below 1.0 because the two forces produce opposing variance signatures. Three different tests. Three different statistics. One spectrum.

State 1 is dead. The spectrum is real, confirmed by three independent methods. The only question remaining is whether forty years of adaptive pressure have worn it down.

Episode 4 killed the random walk and revealed the spectrum’s signature in the variance ratio. Episode 5 answers the question that determines the future of trend-following: has four decades of exponential growth in systematic capital eroded the feedback structure? Has the adaptive markets hypothesis done its work? Or has the structure survived the onslaught intact? The answer is in the data.

The spectrum is real. Three tests confirm it. Episode 5 asks: has forty years of adaptive pressure worn it down?

Endnotes

Methodology

Variance ratio VR(q) = Var(q-period return) / (q × Var(1-period return)), using overlapping multi-period returns for maximum statistical power. Under the IID null, VR(q) = 1.0 for all q. VR < 1 indicates negative feedback dominance (oscillation at that timescale). VR > 1 indicates positive feedback dominance (trending). The test was introduced by Lo and MacKinlay (1988). Eight horizons tested: 2, 5, 10, 21, 42, 63, 126, 252 trading days.
Bootstrap CI: S&P 500 daily returns reshuffled with replacement 2,000 times, destroying temporal dependencies while preserving the marginal distribution. VR computed at all horizons on each shuffle. The 2.5th and 97.5th percentiles form the 95% CI. S&P 500 VR profile: VR(2d)=0.929, VR(5d)=0.849, VR(10d)=0.779, VR(1m)=0.761, VR(2m)=0.716, VR(3m)=0.677, VR(6m)=0.706, VR(1y)=0.768. Below CI at 7 of 8 horizons.
Regime-conditional VR: returns classified into breakout periods (Donchian channel position active) or flat periods (no position) using a 20-day Donchian channel with mid-channel exit. Long when price exceeds the 20-day high; short when price falls below the 20-day low; exit to flat when price returns to the channel midpoint. VR computed separately within each state for each contract. Universe means at 3-month horizon: breakout VR = 1.130, flat VR = 0.282, gap = 0.847. Breakout VR > 1.0 at 65% of contracts (universe mean above 1.0 at all horizons from 1 month onwards). Flat VR < 1.0 at 100% of contracts. Sixty-eight contracts had sufficient data in both states for this analysis.
S&P 500 VR(3m) = 0.677 implies that the 3-month variance is 67.7% of the IID prediction. The ‘missing’ 32.3% represents variance consumed by negative feedback partially offsetting positive-feedback-driven moves. This is the same mechanism that produces crisis alpha: the system drives prices in one direction via positive feedback, then partially reverses via negative feedback, creating the directional persistence that divergent strategies harvest and the reversal that equity portfolios absorb.
Universe mean VR(3m) by decade: 1986–1995: 1.050 (n=59 contracts), 1996–2005: 0.911 (n=68), 2006–2015: 0.998 (n=68), 2016–2026: 0.894 (n=68). The most recent decade shows the strongest departure. However, decade-level analysis is limited by sample size and is presented as suggestive rather than definitive. Episode 5 provides the formal amplitude analysis using rolling windows with much finer temporal resolution.

Data

Same dataset as Episodes 1 through 3: sixty-eight CSI ratio-adjusted continuous futures, September 1984 to January 2026. All returns are daily log returns. The Donchian breakout classification uses closing prices and a 20-day lookback window with mid-channel exit, applied independently to each contract.

Figures

Figure 4.1: 68 VR curves overlaid, coloured by asset class. Grey band = 95% bootstrap CI. Gold = universe mean. Red = S&P 500.
Figure 4.2: Regime-conditional VR profiles. Green = breakout-period mean (68 contracts). Red = flat-period mean. Grey dashed = full-sample mean. Gap annotated at 3-month horizon.
Figure 4.3: Scatter of per-contract VR(3m) during breakout periods (x-axis) vs flat periods (y-axis). Lower-right quadrant = spectral prediction confirmed.
Figure 4.4: S&P 500 VR at 8 horizons with bootstrap CI. Gold circles = below lower CI bound.
Figure 4.5: Universe mean VR(3m) by decade. Four bars. Most recent decade shows strongest departure.

This research series is drawn from The Fractals of Finance: Determinism, Adaptation and the Geometry of Markets

The book explores the full architecture of feedback, fat tails, and fractal structure in financial markets, and what it means for how we trade, invest, and understand risk.

Available now on Amazon in paperback, hardcover, and Kindle.

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