Our review of the paperwe backtested it

Halperin's matrix signals detect crises but forecast only 2008

A fixed-size view of S&P 500 correlation geometry, read across three crisis decades and one honest boundary.

2026-08-01 · 5 min read

Read the paper on arxiv

Our backtest of this idea

Our automated quick test, not the paper's

Observable Matrix Spectral Stress Sector Rotation for US Large-Cap Stocks

Backtest period 2015-01-01 to 2024-12-31 · hypothetical, net of modelled costs

Why these figures are not the paper's (1)

Our own audit found this run does not follow the paper faithfully (9)

Paper fixed S&P 500 membership over each ten-year period including two-year pre-roll. (invalidates: Paper full-universe sizes N=244, N=265 and N=309 no longer apply; exact Table 1 values and S&P 500 membership-specific sector attributions no longer apply.)
Wide panel of daily total returns. (invalidates: Exact paper correlation, market-share, PR, beta and Perron numerical levels no longer apply.)
Sector attribution by summing squared residual leading-eigenvector loadings within sector and dividing by sector share of names. (invalidates: Paper sector identity statements such as utilities, technology, energy, financials and REIT placement may not map exactly.)
Single-firm trajectory entropy production Σ_i = sum_t F, with thermodynamic force F_ab = log(μ_ab/μ_ba). (invalidates: Paper single-firm trajectory entropy-production leader results no longer apply.)

5 further finding(s) are described in the note.

These are our findings about our own implementation, not criticisms of the paper. Read the figures below as a description of what we ran.

StartTotal 120.4%End

Sharpe: 0.69
Total Return: 120.4%
Max Drawdown: -33.3%
CAGR: 8.2%
Volatility: 17.4%
Trades: 8,003

The one thing to take from this paper is the boundary it draws between detecting a crisis and predicting one, and it draws it with a number you can argue about. The correlation-fragility signals give a per-period AUC of about 0.72 at the 63-day horizon for 2008. For 2020 the same signals score 0.49, for 2001 about 0.46. So the machine reads an endogenous buildup and is blind to an exogenous shock or a dispersed unwind. If you already run an absorption-ratio style fragility gauge, that is the delta worth weighing: an honest map of when a concentration signal has anything to say.

A state descriptor for the cross section

Halperin tracks the S&P 500 cross section through three fixed-size matrices on a constant-membership universe (N=244, 265, 309 at the 504-day lookback). One is the arccos distance matrix of rolling correlations; the other two are Markov transition matrices of daily return and volatility ranks. The framing is that the object stays the same size while its spectrum carries the time dimension. Prior random matrix work reads a single snapshot or a slowly drifting one. The move here is to treat the whole trajectory as the observable and ask for quantities intrinsic to it: projector drift, a commutator norm against a pre-crisis anchor, entropy production of the rank chains.

The crisis signature in the correlation spectrum is familiar. Covid drove mean pairwise correlation from 0.29 in 2019 to 0.47 in March-April 2020, the market-factor share from 0.31 to 0.48, and the effective factor count from about nine to about four. The 2008 buildup ran the participation ratio from 8.3 to 5.3 and the market share from 0.34 to 0.43. Participation ratio and mean correlation sit at roughly -0.95 across all three periods, so the two are one reading. None of that is new. What is new is the lookback contrast and the market-factor removal.

The short window is where the information lives

At 126 days the Covid market-factor share spikes to about 0.73 against 0.52 at two years, and the effective factor count drops to about two against four. The short window also localizes the mid-2007 subprime tremor that the two-year window averages out. That is a concrete reason to run the fast probe: the standard two-year correlation window is a low-pass filter that smears onsets.

The 2001 case is the one that pays for the whole apparatus. At 504 days the effective factor count rose from 22.9 to 25.3 and the market share fell from 0.19 to 0.17. A dispersed unwind, opposite in signature to 2008 and 2020. A single rising average correlation would miss it. The spectral view separates a correlated crash from a decorrelated one, and that distinction is what defeats the forecast: there was no concentration to detect ahead of time.

Sector rotation after the market factor is stripped

Deflating the leading eigenpair exposes a sector geometry with its own clock. The raw market eigenvector holds anchor overlap between 0.93 and 0.99 through every crisis, so the market direction amplifies without turning. The market-removed sector eigenvector rotates hard, overlap falling to about 0.5 for Covid and below 0.2 for 2008, and the commutator norm steps from zero at the anchor to about 43 (Covid) and 30 (2008). Utilities is the most persistently concentrated residual cluster in every crisis, a rate-driven standing block on top of which each crisis lights its own sector.

The ranking chains are the part I would actually keep on a screen. The return chain forgets an ordering in about seven days (second eigenvalue near 0.86); the volatility chain holds one for thirty to forty days (second eigenvalue near 0.97). Volatility clustering as a mixing time. The volatility arrow of time flares to z about 8 at the 2002 bottom and z about 4 in 2007-08, and among single-session events only the 2021 Archegos liquidation left a clear arrow (z about 5). It stayed flat through Omicron, SVB and the hawkish-Fed repricing even as the VIX jumped. That selectivity, firing on sustained directional deleveraging rather than symmetric shocks, is a genuinely useful discriminator.

The paper reports no P&L, no costs, and no tradable strategy; it is a descriptive study, and the early-warning AUCs are in-sample and per-period, which the author states plainly.

What we built, and why the window fought us

We turned the spectral stress signal into a long-only US large-cap rotation: risk-on into the top three sectors by 63-day return with an inverse-volatility book, defensive into Utilities, Health Care and Staples at 55% gross when the market-factor share and participation-ratio z-score confirm stress. Top 300 names by annual cap, 10% position cap, 25% sector cap, daily close execution, four tenths of a cent a share commission and zero modeled slippage. The Sharpe, drawdown, and turnover for our 2015-2024 run are shown with this review. Those are ours, not the paper's, and they carry two problems worth stating in the same breath as the equity curve.

First, the window. Our backtest covers exactly the Covid decade, which is the one crisis the paper says its fragility signal cannot forecast (AUC 0.49). We built a stress-timing overlay and then tested it over the regime where the paper openly claims the timing has no lead. A defensive switch that fires coincidentally with a sharp exogenous crash cannot add much, and the 2008 buildup, the case the signal actually forecasts, sits outside our sample. Second, zero slippage flatters a daily-rebalanced book with a 60-name target and regime-driven turnover; real fills would eat into whatever the equity curve shows. This is one automated pass built from the paper's description, so read a weak result as evidence about our window and our sizing before reading it as a verdict on the method.

The idea I would carry forward is narrower than a strategy. Run the 126-day spectral concentration and the volatility-chain arrow as a two-part regime dashboard, use them to size gross exposure rather than to call turns, and expect a lead only when correlations are already tightening. The paper earns that much. It does not earn a claim to see 2020 coming, and it does not pretend to.