The idea

“Networks are everywhere. All you need is an eye for them.” Albert-László Barabási.

Albert-László Barabási is a Romanian-born Hungarian-American physicist, renowned for his pioneering discoveries in network science. In his seminal 1999 paper with Réka Albert, “Emergence of Scaling in Random Networks,” he reshaped how we understand connected systems. His 2002 book Linked carried the argument to a general audience, making the case that networks are everywhere once you develop an eye for them.

This week, we will implement the paper Network Momentum across Asset Classes by Pu, Roberts, Dong & Zohren (2023), from the Oxford-Man Institute.

Using graph learning, the authors build a dynamic network across 64 futures contracts in 4 asset classes, then trade the network momentum that propagates through it:

• 1.51 Sharpe and 22% annual return, 2000-2022 (out-of-sample),

• Only 65% correlated with classic momentum: a genuinely new signal, not a repackage,

• Cross-class links are the edge: connecting bonds → currencies → commodities beats trading each class in isolation,

• Networks stay remarkably stable, yet visibly reshuffle during the 2008 crisis and the 2020 crash.

No economic ties. No company data. Just prices, a graph, and a regression.

Here’s our plan:

1. First, we will quickly summarize the paper;

2. Next, we will implement the paper, step by step, showing our results throughout the journey;

3. Then, we will run some diagnostic analysis (turnover, costs, placebo, regime robustness), showing the signal survives;

4. We will then propose a signal-smoothing improvement to reduce turnover & costs;

5. Finally, we will wrap up with final thoughts and next steps.

Let’s get to it.

Paper summary

The big idea

• Momentum spillover is the phenomenon where a “winning” asset’s momentum propagates to connected assets, suggesting you should go long assets linked to past winners and short those linked to past losers.

• It was originally documented only in pairwise, fundamentally-linked pairs: a company’s stock and its bonds, or firms along a supply chain.

• This paper coins network momentum: a signal for a target asset built by averaging the momentum features of its network neighbors, weighted by connection strength. The key leap is extending it across four asset classes at once: commodities, equities, fixed income, and currencies.

How it works

• Universe: 64 liquid continuous futures contracts, spanning the four classes, 1990-2022.

• Features: 8 individual momentum features per asset: volatility-scaled returns over 1-day, 1-month, 3-month, 6-month, and 1-year horizons, plus 3 normalized MACD indicators at different time scales.

• Graph learning: A linear, interpretable convex-optimization model estimates a daily adjacency matrix where each asset is a node and edge weights reflect similarity in momentum features (non-negative, sparse, no self-links). They ensemble five graphs from different lookback windows (252 to 1260 days) for stability.

• Signal + portfolio: Network momentum features (neighbor-weighted averages) feed an OLS regression predicting each asset’s next-day volatility-scaled return. The long/short portfolio is called GMOM, with positions volatility-scaled to a 15% target.

Key results

• Sharpe ratio of 1.51 and ~22% annual return out-of-sample (2000–2022), after volatility scaling.

• Beats every benchmark tested: Long-Only (0.73), MACD (0.70), and, critically, individual momentum / LinReg (1.13), despite using the exact same input information.

• Also posts the smallest maximum drawdown (~20%) and shortest drawdown duration of the group.

• Only ~65% correlated with individual momentum (72% sign agreement), meaning the network layer generates a genuinely new signal rather than repackaging classic momentum.

What makes it work (robustness)

• Cross-class connections are the secret sauce. Restricting to a single class, or to intra-class edges only, degrades performance: the inter-class links bridging bonds, currencies, and commodities are what lift returns. Even single-asset trading benefits from them.

• Networks are stable but crisis-sensitive. Edge overlap between consecutive days is very high (Jaccard > 0.99), yet the structure visibly reshuffles during the 2008 crisis and 2020 crash.

• Cost resilience. Turnover is higher than model-free strategies, but GMOM keeps a positive Sharpe up to ~3 bps of costs and stays above 0.5 up to 2 bps. It’s more robust than plain individual momentum.

• Reversals differ from classic momentum. Recent 1-day returns load positively; 1-month returns load negatively (a reversal), and network momentum shows different reversal patterns than individual momentum at longer horizons.

• Combining individual + network features in one regression (RegCombo) edges out GMOM slightly (Sharpe 1.54), hinting nonlinear/ML models could squeeze out more.

Why it matters

• First demonstration that momentum spillover can be mapped across asset classes using nothing but prices, via an interpretable graph model.

• Delivers a practical multi-asset strategy that’s diversifying (low correlation to standard momentum) and reasonably cost-tolerant.

Implementing the paper

A. Data, universe, and return methodology

The paper uses 64 ratio-adjusted continuous futures from Pinnacle’s CLC database across four asset classes. We mirror it with 60 Norgate continuous futures.

Norgate’s plain continuous symbol (e.g. &CL) is a spliced unadjusted series: at every roll it jumps from the expiring contract’s price to the next one’s without any P&L happening. Its pct_change is therefore not a return (for markets with steep carry the bias is enormous). The fix (standard in the futures literature, and equivalent to the paper’s ratio-adjusted data) combines both Norgate variants:

Features are computed on the synthetic ratio-adjusted index

The chart below shows what’s at stake for crude oil: the spliced series says -12% over 2005-2015; the true excess return was -76%.

B. Individual momentum features

Eight features per (date, asset), all from prices alone:

Vol-scaled returns over $\Delta \in \{1, 21, 63, 126, 252\}$ days:

Normalized MACD (Baz et al. 2015) for $(S,L) \in \{(8,24), (16,48), (32,96)\}$:

where $m(t,J)$ is a price EWMA with $\alpha = 1/J$. Every feature is winsorized at ±5 EWMA standard deviations (half-life 252).

Here’s a sample of what we get:

C. Graph learning

This is the core of the paper. For each trading day $t$ and lookback $\delta$, stack each asset’s 8 features over the past $\delta$ days into $V_t \in \mathbb{R}^{N_t \times 8\delta}$ — each row is one asset’s momentum trajectory (for $\delta=1260$ that’s 10,080 columns). Only assets with complete windows qualify, so $N_t$ grows from ~33 (1990) to ~59.

The adjacency matrix solves the convex problem (Kalofolias 2016):

with $Z_{ij} = \|v_i - v_j\|^2$ (normalised by its mean so $\alpha,\beta$ are comparable across lookbacks).

Five graphs ($\delta \in \{252, 504, 756, 1008, 1260\}$) are averaged and degree-normalised:

Here’s the visualization of the learned graph for a given date:

These are the top-3 edges of the ensemble graph on 2008-01-02 (α=1, β=0.3), learned purely from the similarity of momentum-feature trajectories, with no tickers, sectors, or asset-class labels supplied:

• Wheat finds wheat. &ZW (CBOT soft red winter wheat) connects to &MWE (Minneapolis spring wheat) and &KE (Kansas City hard red winter wheat). The two other US wheat varieties, near-perfect substitutes trading largely on the same fundamentals, with soybean meal (&ZM) third, an adjacent CBOT feed-grain. The graph rediscovered the wheat complex from prices alone.

• The US equity block assembles itself. &ES (S&P 500) links to &RTY, &YM, &NQ (Russell 2000, Dow, Nasdaq), i.e., all four US index futures cluster while European and Asian indices stay outside its top-3, capturing the region-level factor structure.

• The Treasury curve clusters, with near-uniform weights. &ZN (10yr) connects to &ZF (5yr), &ZT (2yr), &ZB (30yr) at 0.15/0.14/0.14: the whole US duration complex, weighted almost equally, exactly what you’d expect for instruments driven by one yield curve.

• Crude finds its physical and statistical relatives. &CL (WTI) links to &BRN (Brent) and &GAS (gasoil) (the same barrel and its refined product) plus &GD (S&P GSCI index), which is itself ~70% energy by construction, so the edge is essentially mechanical exposure the graph had no way of knowing about except through prices.

• FX clusters by currency bloc. &6E (euro) picks &6B (sterling) and &6A (Aussie), co-movement against the common USD leg, plus a genuine inter-class edge to soybean oil (&ZL), plausibly the weak-dollar/commodity-boom co-trend of 2005-07 showing up in the momentum histories.

• Precious metals + the dollar. &GC (gold) connects to &SI (silver) and &PL (platinum), then crosses asset classes to &6E: sensible, since gold and the euro share the anti-USD trade. These inter-class edges (gold↔euro, equity↔FX below) are precisely the connections the paper argues carry the incremental signal that individual momentum can’t see.

• Asia-Pacific equities bridge to Europe. &HSI (Hang Seng) finds &YAP (Australian SPI 200) first (its regional partner), then &FDAX/&FESX, consistent with a global equity factor with a regional tilt.

Every neighborhood is economically legible (substitutes, curves, currency blocs, index overlap), yet the model was shown nothing but each contract’s own momentum history.
The economically sensible structure is an output, not an input.

D. Network momentum propagation

Each asset’s 8 network features are the edge-weighted average of its neighbors’ individual features (the asset itself is excluded, zero diagonal).

E. The regression

One pooled OLS across all assets and days: target $y_{i,t} = r_{i,t\to t+1} / \sigma_{i,t}$ . Models are refit every 5 years on expanding windows, features z-scored with training-window statistics only:

The standardized coefficients are a key replication checkpoint against the paper's Table 4: 1-day return positive, 1-month negative (short-term reversal), MACD(16,48) negative (mid-term reversal), with the reversal effects stronger in network momentum than individual momentum.

Both tables reproduce the paper’s pattern line for line: ret_1d positive everywhere, ret_1m negative (and roughly 2× stronger in GMOM than LinReg, the paper’s exact finding), the MACD +/−/+ signature across the three timescales, and all magnitudes decaying toward zero in later blocks as the anomalies weaken.

Now, let’s check the backtest.

Portfolio construction and backtest

Four strategies, all through the identical portfolio code (for the reference strategies we pass their signal as the “prediction”, so MACD uses the sign of the Baz $\phi$-rule rather than its continuous sizing):

• Long Only: always +1 (the market benchmark)

• MACD: model-free: $x = \tfrac13\sum_k \phi(y(S_k, L_k))$, $\phi(y) = y e^{-y^2/4}/0.89$

• LinReg: the regression on the asset’s own features (no graph)

• GMOM: the regression on network features

For comparability with the paper’s Panel B, each daily return series is rescaled to exactly 15% realized vol.

In summary:

• The paper’s ordering replicates exactly: GMOM > LinReg > Long Only ≈ MACD, with clear separation at every step. The graph adds ~+1.0 of raw Sharpe over the same regression run on the asset’s own features (2.11 vs 1.13): the network is doing the work, not the plumbing around it.

• The three reference strategies land on top of the paper’s numbers (Long Only 0.63 vs 0.60, MACD 0.62 vs 0.52, LinReg 1.13 vs 0.94; returns and drawdowns at 15% vol within a few points). Since these share every pipeline component with GMOM except the graph, that closeness validates the data, features, and backtest end to end.

• GMOM is also the best-behaved, not just the most profitable: smallest max drawdown of anything tested (-18% vs -28% LinReg, -42% Long Only at the same 15% vol) and the highest hit rate (56.7%).

• The result is not fading out of sample of the paper: extending 2022 → 2026 leaves GMOM essentially unchanged (2.115 → 2.106) while the model-free MACD baseline decays (0.62 → 0.51).

• Our GMOM exceeds the paper’s (2.10 vs 1.36), a gap large enough to warrant suspicion; this is why we ran a battery of diagnostics tests (in the next section: costs, hyperparameter luck, placebo graphs, regime concentration).

• Why sharpe_raw: all Sharpes here are computed on raw, unscaled signals, gross of costs (the same basis as the paper’s Table 1 Panel A) so the comparison against their numbers is apples-to-apples. The return and drawdown columns are stated at a 15% vol target (their Panel B convention). We will analyze costs in the next section.

Let’s also look at daily return correlations:

• GMOM-LinReg correlation is 0.68, matching the paper’s ~0.65, despite identical input features. Only about half the variance is shared (0.68² ≈ 0.46): propagating momentum through the graph produces a genuinely distinct return stream, not a reweighted copy of individual momentum. Position signs agree ~73% of the time, again on top of the paper’s ~72%.

• GMOM is nearly market-neutral (0.10 to Long Only). Its returns are essentially orthogonal to just holding the asset universe, so the Sharpe is coming from long/short timing, not embedded beta. LinReg carries more market exposure (0.26).

• The graph pulls the signal away from classic trend-following. LinReg still co-moves with the model-free MACD baseline (0.54), while GMOM’s correlation to MACD drops to 0.23. Network momentum is the furthest of the three model strategies from a vanilla trend signal.

• Diversification implication: with 0.68 correlation and both legs profitable, GMOM and LinReg combine usefully. This is consistent with the paper’s finding that RegCombo (both feature sets in one regression) edges out GMOM alone.

Diagnostics

A. Turnover and transaction costs

Our GMOM exceeds the paper’s. Let’s run diagnostic tests to thoroughly verify the results. We will run 4 independent ways to double-check them, starting with turnover and transaction costs:

Highlights:

• Costs are charged per unit of turnover: each strategy pays c bps on daily position changes, sweeping c = 0-5bps. The headline numbers are the c = 0 column; the full table is in the code notebook.

• At realistic costs for futures this liquid (0.5–1bp), GMOM nets Sharpe 1.74-1.37, still above the paper’s gross 1.36, while LinReg drops to 0.83-0.53.

• Cost-resilience ordering replicates the paper: GMOM stays positive to ~3bps, LinReg only to ~2bps. The network signal buys roughly an extra basis point of cost headroom at every level.

• The honest caveat is turnover: GMOM trades 1.46 σ-scaled units/day (vs 1.00 for LinReg, and ~0.5 in the paper). Per unit of gross alpha we churn harder than the paper: the signal is fast, flipping signs frequently.

• That churn is addressable: signal smoothing is the paper’s own top future-work suggestion, and with 2.1 gross Sharpe there is ample room to trade alpha for cost room. We implemented that, it’s our last section (a 2-day EWM cuts turnover ~30% for only ~0.13 of gross Sharpe).

B. Hyperparameter sensitivity: Is 2.1 a lucky spike?

The graph depends on the choices for alpha and beta (their product). Sweeping 5 products, here’s what we found:

• One sweep covers the paper’s entire 11×11 grid: the degree-normalized graph depends on α and β only through their product, so the 1-D sweep over a 100× range of αβ spans everything the paper tuned over, from sparse graphs (~6 neighbors per asset) to nearly complete ones (density ~0.93).

• The result is a plateau, not a spike: raw Sharpe moves only 1.94 → 2.12 → 2.01 across that whole range, and every single untuned point clears the paper’s tuned GMOM (1.36). The excess over the paper cannot be hyperparameter luck: there is essentially nothing here to overfit.

C. Placebo graphs: Is the learned structure doing anything?

If GMOM’s edge survived information-free graphs, it would just be a clever way of smoothing cross-sectional momentum. Let’s check 2 placebos: a uniform graph (network feature = cross-sectional mean excluding self) and random Erdős-Rényi graphs at the ensemble’s density, both degree-normalized identically and run through the identical regression pipeline.

Here are the results:

The highlights are striking:

• Information-free graphs lose ~80% of the network edge. Replacing the learned graphs with a uniform graph (cross-sectional mean of everyone else) or random Erdős-Rényi graphs at the same density (run through the otherwise identical pipeline) collapses Sharpe from 2.12 to 1.24-1.32. Graph mechanics (any averaging) is worth ~+0.2 over LinReg’s 1.13; the learned similarity structure adds the other ~+0.8.

• The placebos aren’t just weaker, they’re riskier: worst drawdowns of anything tested (-32/-36% at 15% vol), because averaging over the wrong neighbors smears bad signals across the whole book. GMOM’s edge is genuinely the learned structure, not a smoothing artifact.

D. Regime robustness: Is the edge concentrated in one era?

Here are the annual results:

• The edge is broad, not episodic: 26 of 27 years positive at 15% vol (the sole exception is the partial 2026, roughly flat), versus five losing years for LinReg. No single era carries the result.

• Crisis convexity, amplified: the biggest years are exactly the trend-following crises: the dot-com unwind (2001–03), +65% in 2008, 2011, +85% in 2020, 2022. The weakest stretch is the 2013-2018 “momentum winter”, where GMOM grinds out small positive years while LinReg loses money.

• But there’s a decay: first-half Sharpe 2.46 vs second-half 1.73: the softening is real, but even the recent-decade figure still exceeds the paper’s tuned full-sample number.

Let’s see now how can we improve the high-turnover.

Signal smoothing

As we saw previously, the strategy churns 1.46 units/day. This is the flip side of a fast signal. The paper’s own top future-work item is turnover regularization.

The simplest version is smoothing the prediction with a per-symbol EWM before signing it:

Positions then flip only when the smoothed prediction changes sign, not on every day-to-day wobble. The sweep below compares spans 1 (no smoothing) through 21 on gross Sharpe, turnover, and net Sharpe at each cost level:

Highlights from the sweep:

• Span 2 is close to a free lunch: turnover falls 30% (1.46 → 1.03) for only 0.13 of gross Sharpe (2.10 → 1.97), and it dominates the unsmoothed strategy at every cost level ≥ 1bp: at 2bp it holds Sharpe 0.91 vs 0.60, at 3bp it stays positive (+0.38) where unsmoothed goes negative. Breakeven cost moves from ~2.9bp to ~3.5bp, and MDD even improves slightly. Recommended default: span 2 (span 3 if trading nearer 2-3bp).

• The cost regime picks the span: at ≤0.5bp smoothing is a wash; only above 3bp do long spans (10-21) win, and at those costs this daily strategy shouldn’t be traded at all.

• The alpha is fast. GMOM’s gross Sharpe halves (2.10 → 1.08) as the span stretches to a month, while LinReg only drops 1.12 → 0.83. The network signal decays faster than the individual one. Propagation is capturing short-lived cross-asset lead-lag effects (consistent with the regression’s largest weight on the 1-day feature), not slow trend. That is also why unsmoothed GMOM churns 46% more than LinReg: fast signals cross zero often. We can smooth away the cost problem, but not into a slow strategy: the edge lives at the 1-3 day horizon.

But isn't that just timezone lookahead bias?

After publication, a reader of the published piece raised this:

“Have you tried removing non-US futures that trade in a different time zone? In my replication that reduced Sharpe to 0.7, meaning there is a lookahead bias from one timezone to another.”

So, in this new section (written after the initial publication) we will test that claim on our replication, in 4 steps:

1. Universe split: which of our 60 instruments settle outside US hours;

2. P&L decomposition: where the published GMOM Sharpe actually comes from, US vs non-US legs;

3. Mechanistic check: do same-date US closes “predict” next-date non-US closes far more than the reverse;

4. Clean rerun: re-learn every graph and regression on a US-session-only universe.

The mechanism being tested

The pipeline treats all of date ‘s closes as simultaneous. They are not. Eurex/ICE-Europe settle ~5-6 hours before the US close; Hong Kong and Sydney settle ~12-15 hours before it. So a non-US asset’s “date “ feature row is stale by the time US markets close, and (this is the harmful direction) a signal formed for a non-US asset at date uses US features observed hours after that asset’s date- close. The backtest then books the non-US close-to-close return , part of which has already elapsed when the US information arrives. Any genuine US→overnight-Asia/Europe spillover is partially untradable at those closes: in daily close data it shows up as free predictability.

Propagation through the graph is exactly a cross-asset average, so GMOM is maximally exposed; LinReg (each asset’s own features only) has no cross-asset channel and serves as the control.

Let's get started:

1. Which instruments settle outside US hours

Exchange per Norgate metadata. Two deliberate keeps:

• &NKD (Nikkei 225) stays US: the underlying is Japanese but the contract trades and settles on CME in Chicago hours.

• &CGB (Canadian 10yr) stays US-session: the Montreal Exchange runs on Eastern time and closes alongside the US session. No timezone gap, even though it is technically non-US.

That leaves 12 different-timezone instruments out of 60, the set a US-close-anchored daily bar cannot treat as simultaneous:

2. Where the published Sharpe comes from: P&L decomposition

Same cached full-universe run as the article (graphs, regressions, predictions all untouched). We only split the daily portfolio average into the legs traded on US-session symbols vs the 12 different-timezone symbols, for GMOM and for the LinReg control:

The signature is unmistakable. GMOM's 12 non-US legs run a 4.4 Sharpe on their own while its 48 US legs manage 0.88. A fifth of the headline 2.12 comes from four-fifths of the book, and the tiny non-US sleeve carries the rest. LinReg, fed the same per-asset information but with no cross-asset channel, shows the opposite pattern (US 1.17, non-US 0.58): the non-US markets are not intrinsically easier… they only become "easy" when same-date cross-asset information flows through the graph. The 0.88 US-legs figure also lands right where the commenter's US-only replication did (~0.7).

3. Mechanistic check: the lead-lag is one-directional

If the graph were harvesting a real, tradable cross-asset effect, predictability should not care about closing-time order. If it is harvesting the timezone gap, today’s US closes should “predict” tomorrow’s non-US closes (whose intervals overlap the US close) far better than non-US closes predict tomorrow’s US closes, and far better than US predicts US.

Predictability follows the clock, not the economics. Same-date correlations are similar in both directions (~0.2 each way, global markets co-move). But at one-day lag the symmetry breaks: today's US mean "predicts" tomorrow's non-US closes at +0.07 average correlation, versus +0.004 for the reverse direction and +0.02 for US→US. And within the non-US group the effect is monotone in the size of the timezone gap: +0.044 for Europe (5–6h overlap), +0.15 for Asia-Pacific (12–15h). A real momentum-spillover effect has no reason to care which market's clock closes first, let alone scale with the number of overlap hours. A close-timestamp artifact has to. The tell inside the table: for Asia-Pacific the next-date correlation (+0.15) exceeds the same-date one (+0.14), because a Sydney or Hong Kong "tomorrow" close is nearer in wall-clock time to today's US close than its own "today" close is. This is also why &YXT, &YAP and &HSI post the absurd 3.3–7.4 leg Sharpes above: an Asian "next-day" close is mostly a delayed echo of information the US close already printed.

4. The clean test: US-session-only universe, end to end

Drop the 12 different-timezone instruments before anything is computed: features, graph learning (all 5 lookbacks, every date), propagation, and the expanding-window regressions all see only the 48 US-session instruments. Same hyperparameters as published. LinReg gets the identical treatment as the control.

The comment is right. The network momentum edge in our replication is timezone lookahead:

1. The headline collapses from 2.12 to 0.97 when the 12 different-timezone instruments are removed before anything is computed, right in line with the commenter’s ~0.7 (residual difference is universe composition). Our published number was inflated ~2.2×.

2. The graph adds nothing once closes are synchronous. Clean GMOM (0.97) lands at or slightly below the clean LinReg control (1.07). The entire +1.0-Sharpe “network premium” we attributed to learned graph structure was the graph harvesting the US-close → earlier-close-market lead-lag, which is unexploitable at the closes the backtest assumes. LinReg, with no cross-asset channel, is untouched by the ablation (1.13 → 1.07), the bias lived precisely and only in the propagation step.

3. Every diagnostic in the article now reads differently, coherently. The regression’s dominant ret_1d weight, the alpha that halves under a one-month smoothing (”the edge lives at the 1-3 day horizon”), and even the placebo graphs retaining Sharpe 1.24–1.32 (uniform/random graphs also propagate same-date cross-asset information, so they were contaminated too): all are signatures of a one-day timezone artifact, not of momentum spillover. The battery of diagnostics we ran (costs, hyperparameter sensitivity, placebos, regime breakdown) could not catch this class of bug, because every variant shared the same asynchronous-close data.

4. The paper itself is exposed to the same mechanism. The Pinnacle universe includes DAX, CAC, FTSE, Hang Seng, SPI, Australian bonds, gilts, bunds. We cannot rerun their exact data, but there is no reason the bias spares them; their more modest GMOM-over-LinReg gap (1.36 vs 0.94) is consistent with the same artifact at smaller magnitude.

What a legitimate test of network momentum would need: synchronous prices (e.g. sample every instrument at one common timestamp from intraday data), or an execution-honest backtest that only lets a non-US position use information available before that market’s own close (lag non-US signals by one day, which hands the artifact zero credit but also destroys any genuine intraday spillover), or simply a single-session universe, as here.

Final thoughts

So, does network momentum hold up? Our first version of this article said: emphatically, yes. We now owe you a correction: no, not in daily close data. And we owe the correction to a sharp-eyed reader.

After publication, Maxim Edelman asked a deceptively simple question: had we tried removing the non-US futures that trade in a different time zone? In his replication, that alone cut the Sharpe to 0.7, suggesting the edge was lookahead bias leaking from one timezone to another. We ran the test. He’s right, and we’re grateful he pushed us to look.

Here’s what the ablation found:

• The alpha lives exactly where the clock is broken. Twelve of our 60 instruments settle hours before the US close (nine European, three Asia-Pacific). Those 12 legs alone run a 4.4 Sharpe in the published GMOM; the 48 US-session legs manage 0.88. LinReg (same information, no cross-asset channel) shows the opposite pattern. The bias lives precisely and only in the graph’s propagation step.

• Predictability follows the clock, not the economics. Today’s US market mean “predicts” tomorrow’s non-US closes at +0.07 correlation, versus +0.004 in the reverse direction, and the effect scales with the timezone gap (Europe +0.04, Asia-Pacific +0.15). The per-symbol leg Sharpes are a caricature of this gradient: the three Asia-Pacific contracts post 3.3, 4.8, and 7.4. A real spillover effect has no reason to care which market closes first. A timestamp artifact has to.

• The clean rerun settles it. Re-learning every graph and regression on the 48 US-session instruments: GMOM drops from 2.10 to 0.97, at or slightly below the 1.07 LinReg control. Once closes are synchronous, the network adds nothing.

The uncomfortable lesson is that our four-angle diagnostics battery (costs, hyperparameter sweeps, placebo graphs, regime breakdowns) never stood a chance, because every variant shared the same asynchronous-close data. The placebo test we were proudest of was itself contaminated: uniform and random graphs also propagate same-date cross-asset information, which is why they retained a 1.2-1.3 Sharpe. In hindsight, the artifact was hiding in plain sight in our own findings: the regression’s dominant 1-day feature, the alpha that halved under a month of smoothing, our conclusion that “the edge lives at the 1-3 day horizon.” It does, because that’s the horizon at which the timezone gap operates. And since the paper’s own Pinnacle universe contains the same DAX, CAC, FTSE, Hang Seng, SPI, gilt and bund contracts, we suspect its more modest GMOM-over-LinReg gap (1.36 vs 0.94) carries the same bias at smaller magnitude.

What survives? The pipeline replication stands: the excess-return methodology, the features, the graph learner, the reference strategies landing on the paper’s numbers. Plain individual momentum survives untouched at ~1.1. Even the most satisfying finding survives as structure: fed nothing but momentum histories, the graphs really do rediscover the wheat complex, the Treasury curve, and the gold-euro anti-dollar trade. But economically sensible structure, it turns out, does not equal tradable alpha: not when the backtest treats a 4pm Chicago close and a 4pm Hong Kong close as the same instant.

Is the work done? Differently than we thought:

• Test network momentum honestly: synchronous prices (sampling every instrument at one common intraday timestamp), or signals that only use information available before each market’s own close. That test is still worth running, genuine overnight spillover may exist; it just can’t be harvested at closes that already happened.

• Treat “too good” as a bug report. We wrote that a 2.10-vs-1.36 gap “made us suspicious,” then let a battery of same-data diagnostics reassure us. We forgot to look into the data. Garbage in, garbage out.

• Publish anyway. Four diagnostics of ours missed what another research caught. That’s not an argument against sharing results, it’s the whole argument for it.

Barabási was right that networks are everywhere. But what our graph learner found in this pile of futures prices wasn’t momentum quietly propagating from asset to asset… it was the rotation of the Earth. Thank you, Maxim, for having the eye we lacked.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

PS: That's why peer review is so important to science… Nullius in verba ;)

The idea

“An idea is nothing more nor less than a new combination of old elements.” James Webb Young

Implementing research papers can sometimes work, though a perfect replication often fails. It’s never wasted effort, though: the ideas in the paper end up feeding new ideas and good conversations with other researchers.

Here’s the 3rd edition of the Quantitativo weekly, featuring papers that caught my eye over the past week. Enjoy!

2.13% a month. A t-stat of 9.21. And not one new signal.

This paper doesn’t hunt for another anomaly. It asks the sharper question: when does each anomaly actually deserve your capital?

• Takes 212 known anomalies and groups them into 8 economic families (value, momentum, quality, liquidity…),

• Uses macro-regime memory to decide which family gets capital in each market state,

• Earns 2.13%/month, Sharpe 1.51 (t = 9.21) purely by reallocating, not discovering,

• The kicker: the Liquidity/Limits family pays +1.41%/month in calm markets and flips to 1.79% in trending ones, exactly where arbitrage capital dries up.

The lesson: an anomaly’s value isn’t fixed. It’s state-dependent, and the macro regime tells you when to hold it.

When Do Anomalies Deserve Capital? by Nima Taheri Hosseinkhani (2026)

3,636 funds. 33 years of holdings. Skilled managers time the value factor with a t-stat of 3.99! (…but can’t time momentum at all).

New paper builds a beautifully simple signal: the Fund Skill Gap (FSG).

Take the top-decile “skilled” funds. Take the bottom-decile ones. Compute the difference in factor exposure of what they hold. That’s it. No forecasts, no surveys, just the spread in portfolio tilts.

That spread predicts next month’s factor returns:

• Value (HML): corr 0.30, coefficient t = 3.99. The strongest and most robust result in the paper.

• Investment & profitability: weaker, but there.

• Momentum: nothing. Signs flip, nothing clears the noise.

• Predictability lives almost entirely in growth funds: the mandates with room to actually express a view. Cap-targeted funds show zero.

• Drop the highest- and lowest-flow funds, and it gets stronger → it’s information, not flow-induced price pressure.

The punchline: everything we call “manager skill” (active share, industry concentration, value added) is largely fundamentals-driven factor timing, not stock-picking magic.

The honest caveat: this is in-sample, and the main signal uses quarter-end holdings that arrive with a reporting lag. But the appendix builds the same signal from daily fund returns, and value and size still come through. That version is computable in real time.

Factor Timing: Evidence from Mutual Fund Skill, by Chen, Jacoby, Lu, Xia (2025)

5.6% → 15.9% a year. Sharpe 0.30 → 0.82. Nothing changed except the smoother.

And the winning smoother is an image-processing algorithm from 1990.

A new paper builds a U.S. sector-rotation strategy (9 SPDR sector ETFs, one long-only pick a month) based on four public signals: VIX, Russell 2000/S&P 500, discretionary/staples, and financials/utilities. Then it holds the portfolio rule fixed and swaps in six real-time, one-sided filters.

• Raw signal: 5.6% / 0.304 Sharpe

• Kalman 8.9% · HP 9.7% · LOESS 12.5%

• Savitzky-Golay: 13.8% / 0.707

• Perona-Malik anisotropic diffusion (yes, the edge-detection filter): 15.9% / 0.820

Why it works: edge-preserving filters smooth inside a regime but keep the jumps between regimes. Perona-Malik halves regime switches (3.0 → 1.8/yr), nearly halves turnover (8.2× → 4.4×), and leaves 82% of months with zero trades.

The honest part: against a boring equal-weight sector portfolio (10.6% / 0.739), the edge is real but only marginal (t = 1.8). The author says so out loud — and reports White RC, SPA, Deflated Sharpe and PBO to prove it.

The takeaway: the filter is part of the model, not preprocessing.

Edge-Preserving Macro-Financial Signal Extraction for Real-Time U.S. Sector Rotation, by Marc Weibel (2026)

~40 trades a year. A Sharpe up to 2.8. The only signal: whether the market rose or fell on macro-news day.

A new paper shows that investors’ appetite for lottery-type stocks (high-skew, high-vol, low-price “jackpot” names) gets switched on and off by the content of macroeconomic news.

• When the market rises on a macro-news day (FOMC, jobs, PMI, PCE), demand for lottery stocks jumps, and prices drift up for days. When it falls, they drop. Predictably.

• Long-positive / short-negative on the top lottery decile → ~45-52 bps abnormal return per window, Sharpe up to 2.8.

• Strongest on FOMC days, for firms with no earnings of their own, and in high-sentiment periods.

• Not retail hype, not high beta… It’s information-driven comovement: investors price opaque stocks off the one signal everyone shares (the market’s move).

• Fades fast: gone after ~5 days.

Conditional Demand for Lottery-Type Stocks: Information Spillovers and Asset Prices Comovement by Zhang, Kappou & Urquhart (2026)

9 out of 9 U.S. mega-caps called correctly, with ZERO machine learning.

A new paper forecasts short-term stock trends (direction, not price) by fusing two probability spaces into one:

• An ARIMA model → a normal distribution around each forecasted price,

• A parameterized random walk → a binomial distribution over where price can land,

• Stitched together via a filtration (two σ-algebras) into a single “filtered” probability.

The call is elegantly simple: if the two most-likely price zones sit side by side and the higher one wins, it’s bullish — flip it for bearish, call it undecidable if they tie.

Across 25 assets in 4 markets: 62.5% directional accuracy overall and a perfect 100% on U.S. large caps (AAPL, AMZN, GOOGL, TSLA…).

A novel predictive analytics model for forecasting short-term trends in equity asset prices by Achury-Calderón, Arredondo & Sánchez Ascanio (2025)

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The idea

“The value of an idea lies in the using of it.” Thomas Edison.

In my experience, implementing research papers can sometimes work, though a perfect replication often fails. It’s never wasted effort, though: the ideas in the paper end up feeding new ideas and good conversations with other researchers.

Here’s the 2nd edition of the Quantitativo weekly, featuring papers that caught my eye over the past week. Enjoy!

Momentum’s 10 worst months in a century. One-line tweak lost less in all 10.

A new paper proposes a factor with zero parameters: multiply classic 12-1 momentum by last month’s gross return.

B = (1 + r) · M

That’s it. And it’s an exact identity, not an approximation.

Why it works: winners glide, losers stumble:

• Winners trend up smoothly. Losers fall in fits and starts, punctuated by rebounds that fade.

• Among losers, one-month reversal is 3.5× stronger than among winners.

• (1 + r) shorts the rebounded losers harder, and the still-falling ones less.

The results (U.S. 1927-2025, plus 12 markets):

• Sharpe 0.38 → 0.78, and 100% of the gain is on the short leg

• Survives FF5 + UMD + STR (8.5%/yr, t = 6). It’s conditional reversal, not the reversal factor in disguise.

• Skewness improves 3.68 → -2.73. It mitigates momentum crashes instead of amplifying them.

• Positive in all 12 markets tested (sign test p = 0.002)

The best part? The authors bound their own claim: the big number is equal-weighted, and once you cut to a genuinely shortable universe, the harvestable premium is ~1-2% a year. Not a standalone alpha… a precise upgrade to momentum’s weakest leg.

Winners Glide, Losers Stumble: A Behavioral Reversal Tilt to Momentum

200 years of profit. A Sharpe that fell from 0.84 to 0.12. And a culprit almost no one watches: tick size.

Trend following is one of the most robust anomalies in all of finance, profitable across two centuries and nearly every liquid market. Then, around 2009, fast trends simply stopped working. A new paper from the team at CFM (including Jean-Philippe Bouchaud) finally offers a convincing “why.”

It’s not capacity. It’s not electronification. It’s not a simple crowding story.

The single variable that separates dead trends from surviving ones is the volatility-normalised tick size:

• On small-tick contracts (equity indices, FX), fast-trend PnL has completely collapsed since 2008.

• On large-tick contracts (bonds, most commodities), trend is essentially untouched.

• Neither asset class nor liquidity reproduces this clean split.

The mechanism? Trend is a self-fulfilling loop: signal → trade → price impact → reinforced signal. HFT market makers now withdraw liquidity in front of predictable CTA flow. On sparse small-tick books, that severs the loop: trend followers retreat, and the signal itself decays, not just the profits.

The kicker: even with zero execution cost, small-tick trends stay flat. So it’s not that trend got too expensive to trade: the underlying edge is gone.

Is Trend Still Your Friend?: A Microstructural Account of the Demise of Short-Term Trend-Following

$104 vs $27. A 14-factor model can’t explain the gap. Neither can a neural network.

A new paper in the Journal of Financial Economics shows that a stock’s future returns are forecast by something most models ignore: the strength of its industry peers and where it ranks within that group.

The authors compress this into a single Peer Index (PI), and the results are hard to wave away:

• A long/short PI strategy turned $1 into $104 since 1980, vs $27 for the market.

• The edge survives a 14-factor model and isn’t subsumed by gradient-boosted trees, random forests, or neural nets trained on 130 firm-level characteristics.

• It works across every market state: high/low volatility, liquidity, and sentiment, unlike most anomalies.

• Prices drift the right way and never reverse, pointing to slow underreaction to peer information, not risk or hype.

• It was even more resilient in crises, gaining 19.7% during the GFC while the market dropped 38.3%.

The takeaway: markets price stocks as if they were islands. They’re not.

Dual peer effects and cross-stock predictability

13% annual returns. Sharpe of 0.65. A 12-month-persistent signal hiding inside CFTC position data.

A new paper just dropped showing that commodity speculators extrapolate past returns, and you can systematically front-run their behavior.

The authors build a novel Degree of Extrapolation (DOX) measure from non-commercial trader positions over 26 U.S. commodities (1986–2024):

• A long/short DOX portfolio earns 13.11% per year with a Sharpe of 0.65, beating conventional momentum.

• Returns persist up to 12 months (no quick reversal: this isn’t a bubble story).

• Survives every known commodity factor: term structure, basis-momentum, hedging pressure, value, multi-horizon momentum. Alpha stays at 0.71-0.93% per month.

• Extending the same logic to the futures curve (Basis-DOX) delivers an even better Sharpe of 0.78, the highest in the paper.

The mechanism is not mispricing. It’s liquidity provision: extrapolative speculators pile in after good returns, then slowly unwind, paying commercial hedgers a premium to take the other side. The 2021 CFTC position-limit reform serves as a clean exogenous shock confirming the story.

Bonus: DOX explains why momentum and value work in commodities: momentum lives in high-DOX names, value lives in low-DOX ones.

Extrapolative Commodity Returns

321 trades. 77% win rate. Every quarter profitable.

A new paper conditions S&P 500 directional predictions on dealer gamma regimes, and the results don’t look like noise.

The setup:

• A 3-state HMM classifies each day as Pin (dealers long gamma → mean-reversion), Vacuum (neutral), or Accelerant (short gamma → momentum amplification)

• Regime-adaptive XGBoost models (one per regime, per direction) predict intraday direction

• Two-phase conviction gates filter entries; dealer-flow signals trigger early exits

• 15 months walk-forward OOS, six near-uncorrelated sleeves

The numbers:

• +3,063 SPX points over 321 trades

• Profit factor 4.40, max drawdown just 2.2% of cumulative return

• Vacuum Long: 92% WR, R/R 2.86

• Randomizing regime labels collapses performance to coin-flip → the conditioning is real

The kicker: the simplest model (Pin Long, 13 features, depth-3 trees) has the highest win rate at 81%. Compression regimes are concentrated signals.

Regime-Conditioned Directional Prediction of the S&P 500 Index: A Walk-Forward Validation Study Using Dealer Greek Exposure

A Sharpe of 2.26. Just from reading earnings call transcripts.

A new paper uses chronologically consistent LLMs (no look-ahead bias) to turn 168,139 earnings calls into a single scalar signal (EAR_AI), and the results are striking:

• 1.73% monthly long-short return spread

• Sharpe of 2.26, robust across every size group (even mega-caps)

• Subsumes classical PEAD measures (earnings & revenue surprises)

• Predicts future earnings surprises, R&D growth, CapEx, and leverage

• Distinct from 125 factor-zoo characteristics, combining it with a ridge SDF lifts Sharpe from 1.45 → 2.7

The mechanism is clean: EAR_AI is a denoised version of the earnings announcement return that isolates forward-looking content from the text. The alpha shows up specifically in firms whose future earnings confirm the signal: it’s putting you on the right side of fundamentals before they’re realized.

Even better, analysts seem to incorporate this content into their forecast revisions, and the returns aren’t explained by aggregate sentiment (so it’s not a behavioral mirage).

Large Language Models for Asset Pricing: Learning from Earnings Calls

1.5 Sharpe. 22% a year. No fundamentals, no supply chains: just prices and a hidden network.

Momentum doesn’t only persist within an asset: it spills over to connected ones. This paper asks a bold question: what if commodities, stocks, bonds, and currencies are secretly linked by their momentum, and you could map those links from price data alone?

Using graph learning, the authors build a dynamic network across 64 futures contracts in 4 asset classes, then trade the network momentum that propagates through it:

• 1.51 Sharpe and 22% annual return, out-of-sample from 2000-2022,

• Only 65% correlated with classic momentum: a genuinely new signal, not a repackage,

• Cross-class links are the edge: connecting bonds → currencies → commodities beats trading each class in isolation,

• Networks stay remarkably stable, yet visibly reshuffle during the 2008 crisis and the 2020 crash.

No economic ties. No company data. Just prices, a graph, and a regression.

Network Momentum across Asset Classes

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The idea

“I constantly see people rise in life who are not the smartest, sometimes not even the most diligent, but they are learning machines.” Charlie Munger.

As I mentioned in the piece “Two years of Quantitativo,” the newsletter went paid. In order to continue contributing to the broader community, I decided to share quick summaries of the recent papers I’ve read over the past few weeks. These weekly research emails will be open to all.

In my experience, implementing research papers can sometimes work, though a perfect replication often fails. Nevertheless, it always helps: the ideas contained in research articles inspire new ideas and conversations with other researchers.

Here goes the first issue.

$1 grows to $530, not $22. One overlay, applied to 18 anomalies.

A new paper shows that a small cluster of extreme, positively skewed stocks drives the returns of nearly every famous anomaly, and that you can predict which stocks ahead of time.

The “skewness-managed” overlay:

• Forecasts each stock’s expected skewness from simple firm characteristics (volatility, past returns, size),

• Tilts the long leg toward high-skewness stocks and the short leg away from them,

• Lifts average annual returns by 5.45 percentage points across 18 anomalies.

The kicker? It throws off large alphas even against factor models built from the same characteristics, and the premium balloons to 20.4% a year during recessions. Modern asset pricing models are quietly ignoring skewness.

Skewness Managed Portfolios by Gong, Lynch & Ogden (2026)

Skewness: -2.58 → +0.21. Max drawdown: -76% → -30%. Same momentum, minus the crashes.

Momentum is one of finance’s best anomalies, and one of its scariest. It crashed -91% in two months (1932) and -73% in three months (2009). The reason? Negative coskewness: winners tank exactly when volatility spikes.

This paper turns that bug into a signal:

• More negative coskewness predicts higher future returns, and does so up to 9 months out, far beyond the ~1-month reach of standard momentum.

• The effect lives in the tail stocks (the actual winners and losers) and is strongest in calm markets, acting as an early reversal warning.

• Bolting a coskewness overlay onto volatility-managed momentum (CS-BSC) lifts the Sortino from 2.81 → 3.56, cuts max drawdown by 61% vs. baseline, and flips skewness positive.

• Holds up across 3-12 month holding periods and in the UK, Germany, and France (just not Japan, which never had momentum to begin with).

The trade-off momentum traders always paid was a fat left tail. This one’s slightly right-skewed.

Coskewness and Reversal of Momentum Returns by Dong, Dai, Haque, Kot & Yamada (2022)

Same 60-day window. Fewer trades. 3× the Sharpe ratio.

A new paper pits the textbook Ornstein-Uhlenbeck stat-arb model against six deep time-series models on the exact same generalized pairs-trading setup, and the deep models win without trading more.

The trick is reframing pairs trading as predicting the direction of each asset’s idiosyncratic residual across the whole cross-section, not asset-by-asset mean reversion. The payoff:

• iTransformer posts a 1.94 Sharpe on the S&P 500 vs 0.57 for the classic baseline; same 60-day look-back, no extra data.

• On crypto, Scaleformer and iTransformer hit a ~2.2 Sharpe.

• The edge survives 0.5% transaction costs (and widens as costs rise) because these models trade less: lower turnover, smaller drawdowns.

• Portfolios stay market-neutral: factor betas ≈ 0, so returns come from residuals, not hidden risk premia.

The honest kicker: after multiple-testing correction, the transformers are statistically indistinguishable from each other. The win comes from the panel-prediction framing, not any single architecture.

Pairs Trading with Time-Series Deep Learning Models by Yilmaz & Sefer (2025)

3.14% a month. 35% CAGR. A market beta of basically zero.

A new paper reveals that where a stock sits in its own 52-week range predicts a striking spread in next-month returns, for both stocks and bonds.

The signal is the Price-to-52-Week-Anchor (PTA): combining the 52-week high and the long-ignored 52-week low into one cleaner “expensiveness” measure. It works because investors anchor: they read news pessimistically on expensive-looking stocks, optimistically on cheap-looking ones, and the bias corrects next month.

• Stocks: buy high-PTA past winners, short low-PTA past winners → ~3.14%/month, FF6 alpha 2.34%, ~35% CAGR, negative market beta.

• Bonds: buy bonds of high-PTA past-loser stocks, short low-PTA ones → 0.72%/month, alphas >1.2%.

• Strongest in high-sentiment periods and for high-uncertainty names.

First paper to show a stock’s 52-week anchor predicts its bond returns too.

Predicting stock returns of past-winner stocks and bond returns of past-loser stocks with a stock’s 52-week price anchor by Chen, Saha, Shafaati, Stivers & Sun (2026)

A t-stat of 9.80. Zero overfitting. Crypto has fundamentals after all.

A new paper introduces the Q-7 model, the first comprehensive factor framework for digital assets, blending price data, blockchain metrics, and derivatives sentiment. The findings flip several crypto beliefs on their head:

• On-chain quality is king: tokens with real network usage earn the most reliable premium (t = 9.80), and it works in every sentiment regime.

• Forget momentum: crypto exhibits short-term reversal, past losers beat past winners (t = 6.19), contradicting the famous Liu-Tsyvinski-Wu result.

• The size premium is fake: it’s a volatility premium in disguise; orthogonalize volatility to size and the size effect vanishes.

• Funding rates are a contrarian goldmine: crowded longs predict underperformance (t = 6.74), nearly orthogonal to everything else.

The model explains 63% of per-token returns and 85% of diversified portfolio returns out of sample, with an in/out-of-sample R² gap of -0.001. Essentially zero overfitting.

Crypto Has Fundamentals: A Seven-Factor Model for Digital Asset Returns by Babayev and Aliyev (2026)

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The idea

“Discovery consists of seeing what everybody has seen and thinking what nobody has thought.” Albert Szent-Györgyi

Albert Szent-Györgyi (1893–1986) was a Hungarian biochemist who won the Nobel Prize in 1937 for isolating vitamin C. His most famous remark, that “discovery consists of seeing what everybody has seen and thinking what nobody has thought”, describes his own achievement perfectly: he found something new in a substance that had been in plain sight all along.

This week, we will implement the paper “Slope, Strength, and Retail Extrapolation” by Chad Schmerling (2026). Imagine two stocks with the same 12-month return. One is a screaming buy, the other a short. Standard momentum can’t tell them apart: it sees only the endpoint. But this new paper shows that the alpha hides in the shape of the path between them.

Imagine how many people have implemented the standard momentum factor and never thought of using the slope of the price path and the trend strength to improve its definition. Go long smooth uptrends, short noisy downtrends, and you get:

• 7.9% annual alpha (t = 3.44), net of Fama-French 5 + momentum,

• A standalone Sharpe of 1.12 on the smoothness signal alone

If those are truly achievable numbers, that’s a discovery by Szent-Györgyi’s standards.

Here’s our plan:

1. First, we will quickly summarize the paper

2. Next, we will visualize the edge from the raw signal

3. Then, we will develop a few trading strategies based on this signal

4. Finally, we will wrap up with final thoughts and next steps

Throughout the article, we will share the Python code used in the implementation. So, if you are interested in implementation details, this is a good opportunity to follow along.

Let's get started.

Read more

The idea

“If I have seen further it is by standing on the shoulders of Giants.” Sir Isaac Newton.

Isaac Newton was arguably the greatest scientist who ever lived. He effectively discovered gravity. He showed us how to predict the motion of the planets. He had every right to brag about his genius. Yet he chose humility. Why?

His critics paraded their ideas with hubris. Newton offered his with deference to those who came before. And that humility was no pose. It came from something he understood early and deeply: knowledge builds upon itself. Each idea improves on the last, little by little, until the small gains add up to something revolutionary. That is the essence of his most famous “standing on the shoulders of Giants” metaphor.

From a young age, Newton kept a commonplace book, a gift from his father. In it, he copied passages from what he read and added his own notes, turning borrowed knowledge into original ideas. He called it his “Waste Book.” The name was a nod to the usefulness of useless knowledge and the combinatorial nature of creativity, what Einstein would later call “combinatory play.” Creating by connecting was the foundation of Newton’s mind. It was his real superpower.

This week, we will cover two articles. We will build on The Modern Spirit of Statistical Arbitrage, a great piece by SysLS. And we will implement a recent breakthrough paper that rigorously tested more than 190 signals in the US equity market.

Here’s our plan:

1. First, we will summarize the modern spirit of stat arb.

2. Next, we will construct a signal and show, in a few lines of code, how it performs across several large baskets.

3. We will then present the combined performance of the top ~20 signals and walk through them, summarizing the source paper along the way.

4. Finally, we will lay out a simple way to merge these signals into a portfolio that survives friction and costs.

Let’s get started.

What is statistical arbitrage?

At its core, statistical arbitrage is a class of strategy built from a portfolio of signals. Each signal assigns weights to instruments based on how much they outperform or underperform the rest of their basket, measured against some central point of all the other instruments. This can happen in any space: returns, price, volume, flow, dividends, and so on. The basket is then built so that its net factor exposures tend toward zero, which means most of its returns come from idiosyncratic moves rather than broad market or factor exposure. It’s a broad definition, and that’s the point. It covers every flavor of stat arb.

For more details, check the great post by SysLS.

Let’s see a concrete example. We will start with Factor 46 from the paper we are sourcing the signals.

Definition. Factor46 is the paper’s Multi-Period Mean Reversion Ratio, computed as:

(MEAN(CLOSE,3) + MEAN(CLOSE,6) + MEAN(CLOSE,12) + MEAN(CLOSE,24)) / (4 * CLOSE)

The Python code is straightforward:

The inputs are clear: both are date-indexed, point-in-time panels covering every symbol that has ever belonged to a given universe (the Norgate “… Current & Past” watchlist, so it’s survivorship-bias free):

• data: a wide DataFrame whose columns are a two-level (’Feature’, ‘Symbol’) MultiIndex, so data['Close'] slices out a single date-×-symbol price matrix. Features available are Open, High, Low, Close, Amount, Volume, and VWAP. factor46 only touches data['Close'].

• index: a same-shaped date-×-symbol integer DataFrame that is 1 on days a symbol was an actual index constituent and 0 otherwise. factor46’s final result.where(index == 1) uses it as a membership mask, blanking out the factor value for any stock/day that wasn’t in the universe at that time so it can’t be traded.

As we can see, Factor 46 takes four trailing moving averages of the closing price (3, 6, 12, and 24 days, each window roughly doubling, spanning a few days out to about a trading month), equal-weights them into a single blended reference price, and divides by today’s close. The result is a pure ratio centered near 1:

• a value > 1 means the current price sits below its own multi-horizon average (the stock has recently sold off relative to its recent history), and

• a value < 1 means it sits above it (recently run up).

It is, in essence, a smoothed, scale-free “distance from fair value” measure where “fair value” is the stock’s own recent average price rather than any fundamental anchor.

What it captures and why economically. The effect is short-horizon cross-sectional mean reversion / contrarian price correction. The paper grounds this in behavioral universality: overreaction, herding, and liquidity-seeking are cognitive traits common to all market participants, so when traders push a price away from its recent path (chasing news or dumping inventory), the dislocation tends to correct. Crucially, arbitrageurs are slow to close these gaps because of noise-trader risk and limits to arbitrage, which lets the reversal premium persist long enough to be harvested.

Now, let’s see the code that tests this signal. The most important lines are:

What is happening here?

The line signal = -(factor.subtract(factor.mean(axis=1), axis=0)) measures how far each stock sits from the center of its basket. For every day, it takes the cross-sectional mean of the factor across all stocks and subtracts it from each stock’s value. What’s left is each stock’s displacement from the group. The leading minus sign then flips the bet: stocks far below the average get positive weight, and stocks far above it get negative weight. That is the contrarian core of stat arb, betting that extremes revert toward the center.

The line signal.divide(signal.abs().sum(axis=1), axis=0) just normalizes. It divides every weight by the total gross exposure that day, so the book always sums to the same size and the longs roughly fund the shorts. This keeps the portfolio close to dollar-neutral and comparable across time.

The flip is optional. When flip_signal is true, it multiplies the whole book by -1, reversing the direction of every bet. This is useful because we don’t always know in advance which sign of a factor is the profitable one. The same formula can be tested both ways, longs and shorts swapped, without rewriting anything.

The last line runs the backtest: signal.shift(execution_lag).multiply(returns).sum(axis=1). It takes each day’s weights, shifts them forward by execution_lag days (two by default), and multiplies them by that day’s returns. The shift is the important part. It makes sure we only earn returns on positions we could have actually held, using yesterday’s signal to trade today rather than peeking at information we wouldn’t have had in time. Summing across all stocks then collapses the basket into a single daily P&L for the whole portfolio.

Let’s see the results:

Factor 46 is encouraging, but on its own it’s not a finished strategy. The same formula behaves very differently depending on where we point it: a 0.53 Sharpe on the S&P 500, a 1.46 on the S&P ASX 300. The S&P 500 version is the clear laggard, with the weakest return and a brutal 55.7% drawdown that no one would want to sit through. And these numbers are gross of costs, so the real picture is worse. That’s the honest takeaway: a single signal, however clever, is rarely tradeable alone. The edge is real but thin, and one factor gives us no diversification when it goes through a bad stretch. The fix is to stop relying on any one signal. Combine many, each with its own small edge, and the weak spots start to cancel out. That is exactly where we go next.

From a single signal to a portfolio of signals

Let’s see what happens when we combine the strongest 17 signals according to the source paper:

The effect of diversification is immediate. The single Factor 46 signal swung as deep as 55.7% on the S&P 500; the combined portfolio’s worst drawdown across all six universes is just 8.2%, and only 4.3% on the Russell 3000. Every Sharpe ratio now sits above 1.0, from 1.15 to 1.76, where the lone signal struggled to clear 0.5. The annual returns look smaller, but that’s the point: the single signal earned big numbers by running enormous risk, while the portfolio earns steadier returns with a fraction of the pain. That’s diversification doing its job. Seventeen small, imperfect edges combine into something far smoother than any one of them.

So what are these seventeen signals? Let’s look at them next.

Inside the portfolio

The list below shows the strongest signals, the most pervasive cross-sectional drivers according to the paper:

Read more

The idea

“The inches we need are everywhere around us.”

Coach Tony D’Amato.

You know the scene. A broken coach, a silent room, three minutes before kickoff. Al Pacino’s Any Given Sunday speech still gets played in locker rooms, boardrooms, and classrooms every year. Not because it’s loud, but because it’s true. The people who actually fight for things keep coming back to it. Watch it again, even if you know every line:

This month, we celebrate two years of Quantitativo! I chose to open with Tony D’Amato’s quote and speech because the past year of writing this newsletter has felt, more than anything else, like a game of inches. Let me tell you what I mean by that.

+180 shared ideas in the last 365 days

I read papers every day. A year ago, I started sharing the ones that caught my eye: short-form on Substack (to all), and long-form at connect.quantitativo.com (to the hundreds who joined the course/community). Since then, I’ve summarized and shared more than 180 of them. A few clearly hit a nerve. Here are some that resonated the most with the community (that I haven’t yet shared a detailed implementation of):

Most equity anomalies are the same trade, just badly disguised.

A new paper shows that short-term reversal, MAX, IVOL, salience, and lottery effects all come from one source: the last month of daily returns.

Instead of engineering dozens of signals, the authors let the data learn the mapping from 21 daily returns → next-month performance.

What they find:

• Timing beats magnitudeWhen returns happen matters far more than how extreme they are.

• A single factor (DRIF) earns ~1.6% per month, Sharpe > 1,

• Subsumes most short-horizon anomalies (reversal, MAX, IVOL, tail risk),

• Survives 150+ factors, multiverse tests, and modern markets.

Bottom line: The factor zoo isn’t crowded. It’s redundant. Daily returns already contain the signal: we just kept slicing them the wrong way.

Paper: A Unified Framework for Anomalies Based on Daily Returns

We can predict how big market moves will be without predicting direction at all.

This new paper shows that order-flow entropy from tick data forecasts future volatility, not returns.

• Low entropy = structured, informed trading,

• High entropy = noise trading,

• Directional accuracy stays at random chance.

In SPY, conditioning on bottom-5% entropy nearly triples the next 5-minute absolute return, with zero directional edge.

The reason is deep and elegant: entropy is sign-blind. Informed buyers and sellers look identical in entropy space.

Paper: Hidden Order in Trades Predicts the Size of Price Moves

A transformer trained on stock returns beats every factor we know.

This new paper introduces StockGPT, a GPT-style model trained directly on 70M U.S. stock returns (no text, no fundamentals, just price). The result?

• 119% annual return, Sharpe ratio of 6.5 (daily L/S portfolio),

• Still earns 69% after trading costs,

• Outperforms momentum, reversals, and all 11 major stock factors,

• Orthogonal to known signals.

A decoder-only transformer, trained once on data up to 2000, works 23 years later. No retraining needed.

Note: The paper doesn’t address short availability, which is a real challenge in small caps. However, even the long-only leg shows strong performance, making this paper a must-read.

Paper: StockGPT: A GenAI Model for Stock Prediction and Trading

Strip momentum down to its components, and a surprising thing happens: only industry and style trends persist.

A new 2025 paper decomposes each stock’s past 12-month return into its true underlying drivers, and finds that only two sources of momentum actually persist:

• Style momentum (value, quality, growth, low vol)

• Industry momentum (sector & economic underreaction)

Everything else?

• Stock-specific momentum reverses hard in the short run and is subsumed by industry & style in the long run.

• Beta and country momentum are noise: zero premium, sometimes negative.

The shocker: What we call “price momentum” is mostly stock-specific noise (the part that causes crashes). Once you strip that out, momentum becomes far more stable, especially after low-volatility regimes.

Key takeaways:

• Short-term style & industry momentum persist strongly.

• Short-term stock-specific returns reverse sharply (great hedge).

• Intermediate-term industry momentum is the real anomaly.

• Style momentum just proxies for static factor tilts (with more turnover).

• A momentum signal optimized for persistence beats classic momentum out-of-sample.

Momentum isn’t a single anomaly: it’s a hybrid of factor exposure and investor underreaction to broad economic trends.

This paper shows which trends are actually your friends.

Paper: Optimizing the Persistence of Price Momentum: Which Trends Are Your Friends?

41% annual return. Sharpe 2.51. Predicting crashes before they happen.

A new paper from Oxford introduces a hybrid ML ensemble that forecasts short-horizon market risk — and trades it.

• Combines neural nets + tree ensembles to predict 5-day SPY drawdowns (> 1%)

• Uses cross-asset features from equities, bonds, FX, commodities, and volatility

• Finds that oil, FX, and Treasury signals lead equities, warning of crashes before they hit

• A simple long/short SPY strategy earns Sharpe 2.51 with beta 0.51 over 2005–2025

The insight: systematic alpha emerges from modeling risk itself (not returns) through interpretable, causal ensembles.

Paper: Causal and Predictive Modeling of Short-Horizon Market Risk and Systematic Alpha Generation Using Hybrid Machine Learning Ensembles

These are just a few. My favorites aren’t even on the list. If we tested them all, we’d find that most don’t work. A few do… and that’s the point. Along the way, we learn enough to pull ideas from different places, recombine them, and turn them into something of our own. And the learning compounds.

Let me give you an example:

The other day, I was discussing a trade idea with a friend (based on yet another paper). The gist: devising a method to identify informed/insider trading on prediction markets, and using that data to guide option trades in traditional markets. As we worked through possible approaches, I remembered something I’d read months ago about using order-flow entropy to separate informed trading from noise trading (the second paper in the list above). My friend, a veteran with two decades of experience trading options at a big Swiss bank, loved the connection: “I love it! Let’s try it out! Can you help me implement it?” I said, “Sure! Let’s do it.”

You get the point. Building a repository of ideas to pull from, recombine, and turn into something of our own is powerful. The learning compounds.

But what does this have to do with the “life is a game of inches” speech?

D’Amato’s central metaphor is that life, like football, is “a game of inches.” The difference between winning and losing, succeeding and failing, even living and dying, isn’t usually some grand decisive moment. It’s the accumulation of tiny efforts, split-second decisions, half-steps.

But here’s what makes it useful: the inches we need “are everywhere around us.” Opportunities to fight for something better are constantly available, in every moment. Most people just don’t see them or won’t claw for them.

That’s exactly what this past year has felt like. Reading every day. Testing ideas. Folding what works into the systems we trade. Most of it goes nowhere. But every now and then, an inch. And then another. The ideas we need to improve our results really are everywhere around us.

The only question is: are we willing to fight for that inch? Or are we satisfied with the inches we already have?

The best part of last year

There’s yet another important point in Al Pacino’s speech that I think of when I reflect on last year. In the end, he says the inches aren’t won alone. D’Amato tells his players to look into the eyes of the man next to them and see someone who will go that inch with them.

Last year, thanks to the newsletter, I had the opportunity to meet and work with brilliant people all over the world. Working alongside great people was the best part of Quantitativo’s second year, no doubt about that!

Isolation is death. A team, meaning any group of people genuinely committed to each other, is what makes the fight survivable and the inches winnable.

This year, we crossed the 10,000-subscriber mark, which is no small thing. But what I find more amazing about the community is not the quantity, but the quality: we have readers from places like Morgan Stanley, Goldman Sachs, Deutsche Bank, Citigroup, Millennium Management, Point72, Wellington Management, Man Group, Millburn, Trexquant, IMC Trading… to name a few. (As well as students from schools like MIT, Columbia, Cornell, UPenn/Wharton, Carnegie Mellon, Duke, NYU, Berkeley, Georgia Tech, Imperial College London, etc.)

Truly, meeting and working with brilliant people really makes my day!

What’s next

Starting with the next article, Quantitativo is becoming a paid publication.

The plan for year three is the same shape as year two, only sharper: 12 in-depth articles, plus shorter posts on ideas that catch my eye every other day. The writing has grown a lot over two years (in depth, in rigor, in implementation detail), and I plan to keep pushing in that direction.

The most rewarding part of this newsletter has been hearing from readers who’ve actually built on the work. That’s the audience I want to keep writing for.

Final thoughts

Any Given Sunday’s speech is about the dignity of relentless effort, the necessity of brotherhood, and the redemptive power of refusing to quit… even when, especially when, we’ve already lost a lot. I find it hard not to see the connection with what we do every day.

It’s where my mind goes when I think about this second year. And it’s where my mind goes when I think about year three, too.

Thank you for being part of this journey.

I’m excited for what’s ahead. The best is yet to come.

Cheers,

Carlos

The idea

“Doubt is not a pleasant condition, but certainty is a ridiculous one.” Voltaire

Voltaire was arguably the most influential intellectual of 18th-century France. More than that, he was a provocateur. He spent his life as a one-man war against dogma, against anyone who claimed to know the truth with absolute certainty. The Age of Enlightenment didn’t begin with answers: it began with Voltaire’s refusal to accept false certainties.

After three articles on more original ideas, we return this week to paper implementations. In fact, this week we will discuss two papers with a simple yet powerful idea at their core: that embracing uncertainty (rather than fighting it) leads to better results.

We will investigate the ideas in the paper Bimodality Everywhere: International Evidence of Deep Momentum, by Chulwoo Han and Chang Qin, published in December 2025. This paper is an extension of Bimodal Characteristic Returns and Predictability Enhancement via Machine Learning, also by Chulwoo Han, published in October 2022.

Here’s our plan:

1. First, we will quickly summarize the papers

2. Next, we will create a simple trading strategy based on this idea and show its strong results

3. Finally, we will wrap up with final thoughts and next steps

Before we start, two messages.

First, the 3rd cohort of the course is still open for enrollment for the next couple of days.

I created this to respond to a request I’ve received from hundreds of readers here: access to my codebase, step by step — for those who want to develop strategies using the same approach I share here. Enrollment also includes access to the private community and an ever-growing library of study groups. More information at the link below.

Join the Course and Community

Last day for enrollment in the 3rd cohort: Apr 7.

Second, this time it took me a bit longer than the usual 3-4 weeks to write and publish. There are a few reasons why:

1. First and foremost, I implemented distinct papers, and their results came in significantly below what was reported. Instead of publishing subpar ideas, I prefer to keep searching for meaningful ones to share. That’s why I’ll never promise a rigid schedule — sometimes the search takes longer, but I’ll never compromise quality.

2. And, of course, just a lot of regular work :)

Welcome to all new subscribers — and rest assured: even if it sometimes takes a little longer than the usual 3-4 weeks to publish, the work continues non-stop.

Paper summary

Core Idea

Most investors who use momentum strategies assume a simple, intuitive relationship: past winners will likely continue to win, and past losers will likely continue to lose. If that were true, the return distribution of high-momentum stocks would look like a hill (i.e., most returns clustered around the high end).

But that’s not what the data shows.

Chulwoo Han’s paper documents a striking and uncomfortable fact: both high- and low-momentum stocks have bimodal return distributions. Past winners are most likely to yield high returns, but their second most likely outcome is to yield very low returns. The same is true, in reverse, for past losers.

This is not a hill. It’s a U-shape.

The implication is profound. The momentum strategy is not just risky in the conventional sense: it is fundamentally risky, because the stocks it bets on are precisely the ones most likely to deliver extreme outcomes in either direction.

And here’s what makes it even more interesting: this bimodality is not unique to momentum. It appears across virtually every major firm characteristic — beta, size, book-to-market, volatility, etc. It’s everywhere.

The Problem with Standard Machine Learning

Neural networks trained on momentum features do outperform the standard momentum strategy. This has already been shown by many different authors. But they don’t eliminate the bimodality. And here’s where the paper identifies a subtle but critical flaw in how machine learning is typically applied to finance.

A standard neural network predicts the most probable return class for each stock. But when return distributions are bimodal, the most probable class is not the same as the expected return.

A stock with a 55% chance of winning and a 45% chance of losing will be labeled a winner. But would you rather own that, or a stock that is 95% likely to earn a modest, predictable return? The model can’t tell the difference. You can.

The model picks one answer and discards all the doubt. That, as Voltaire warned us, is the absurd part. There’s information in the uncertainty. Discarding that is discarding alpha.

Methodology

The paper introduces the Deep Momentum (DM) framework, which fixes this in two steps:

• Step 1: Estimate the full distribution. A deep neural network classifies each stock into return deciles and outputs a probability for each decile, not just a single predicted class. This gives a complete picture of each stock’s return uncertainty.

• Step 2: Reclassify based on financial performance. Rather than selecting stocks on the most probable class, the model reclassifies stocks using their expected return, computed as a probability-weighted average across all deciles. Five reclassification methods are proposed, ranging from simple ranking adjustments to Sharpe ratio optimization.

The inputs are intentionally simple: normalized price momentum features over 1, 3, 6, 9, and 12 months, plus size dummies. No exotic data. No hundreds of factors. Models are tested on US stocks from 1975 to 2017, with all experiments repeated fifty times to account for randomness in neural network training.

Main Findings

The results presented in the paper are remarkable:

• A value-weighted long-short portfolio earns an annualized return of 35% and a Sharpe ratio above 1.6.

• An equal-weighted version earns around 50% annualized with a Sharpe ratio above 2.8.

• The standard momentum strategy, for comparison, yields a Sharpe ratio of 0.61. The best-known improvement (Barroso and Santa-Clara, 2015) reaches 1.00. The DM strategy more than doubles it.

• The monthly alpha is 2.4% (t-statistic = 6.63) after controlling for Fama-French five factors plus momentum and short-term reversal. The factors cannot explain the returns.

• No momentum crashes. The model appears to learn from past crash patterns and repositions accordingly.

• The strategy survives transaction costs. Under 30 basis points, the value-weighted portfolio still earns 23% annualized with a Sharpe ratio of 1.09.

Conclusion

The paper makes two contributions that stand on their own.

The first is empirical: the discovery that momentum stocks (and most characteristic-sorted stocks) have bimodal return distributions. A new way of understanding why these strategies are risky at the stock level, not just at the portfolio level.

The second is methodological: machine learning classifiers, when applied naively, optimize for the wrong objective. Predicting the most probable class is not the same as predicting the best financial outcome. The reclassification framework bridges that gap, and the performance improvement is dramatic.

The beauty of the result is in its simplicity. A small number of features. A clean methodology. Results that outperform models using hundreds of inputs.

The second paper

The first paper proved the concept in the US market. The latest one asks the harder question: Does it work everywhere else?

Han and Qin extend the Deep Momentum framework to 45 countries, testing it against a wide range of markets — developed and emerging, deep and shallow, high-momentum and low-momentum. The results are striking.

What’s new in this paper

• Scale. The original paper was US-only. This one covers 45 countries, using data from Refinitiv’s Datastream going back as far as 1965 for the US and UK.

• A new classifier. Instead of a deep neural network, this paper uses XGBoost (a gradient-boosted tree model). The reason is practical: training a neural network for 45 separate country models would be computationally expensive. XGBoost performs comparably with far less tuning.

• A new question: why does bimodality vary across countries? The paper doesn’t just document bimodality internationally: it tries to explain it. The key drivers turn out to be behavioral: individualism and the disposition effect (the tendency of investors to sell winners too early and hold losers too long).

• Joint learning. This is the most original methodological contribution. Instead of training one model per country, the paper trains a single global model on data from all 45 countries simultaneously. The global model outperforms country-specific models in most cases (including in the US).

Main Findings

• Bimodality is widespread and persistent across all 45 countries. It is not a US quirk.

• Bimodality is negatively correlated with momentum profits, meaning that where bimodality is highest, the standard momentum strategy works worst. This is especially visible in Asian markets like China, Japan, and Korea.

• Deep Momentum (RET) doubles the return of standard momentum on average across countries. Equal-weighted annualized mean returns: MOM 20.9%, XGBoost 31.1%, RET 41.4%. Average Sharpe ratios: MOM 0.98, XGBoost 2.13, RET 2.49.

• RET outperforms XGBoost in 43 out of 45 countries. The reclassification step (translating probabilities into expected returns) is the dominant source of improvement.

• In Asian markets where momentum historically fails, Deep Momentum shines. In China: MOM 2.0%, RET 31.5%. In Japan: MOM 2.2%, RET 22.2%. In Korea: MOM -0.2%, RET 46.1%.

• The global model is exceptional. Equal-weighted, it earns an annualized return of 52.9% and a Sharpe ratio of 8.98. Value-weighted: 33.8% return and a Sharpe ratio of 2.81. The authors note that achieving such a high value-weighted Sharpe ratio is extremely rare in recent periods.

• The global model also improves country-level portfolios in 38 out of 43 countries (including the US) by learning patterns from markets that individual countries have not yet experienced.

Why does Deep Momentum work better in some countries than others?

The paper finds that machine learning strategies outperform most in countries with:

• Higher bimodality

• Larger stock universes (cross-sectional depth matters more than time-series length)

• Higher volatility

• Higher turnover

In high-bimodality countries, one-year momentum loses predictive power, but short-term reversal becomes stronger (and the model learns to exploit that).

The bottom line: bimodality is not a statistical curiosity confined to the US. It is a global phenomenon, rooted in investor behavior, and Deep Momentum is a robust framework for turning that uncertainty into alpha — across markets, across cultures, and across history.

Implementing the idea

I will be honest: when I first read these papers, I thought the implementation would be straightforward. It wasn’t. Getting this to work required far more engineering than I thought. Detailing all of that would take an article longer than this one.

And frankly, I have been thinking for a while about shifting the format of these articles. Less “here is exactly how to replicate this,” more “here is the idea, here is the evidence, here is what I found.” The reasons for that are obvious. For those of you who want to go further, the papers are your starting point. The rest is yours to build.

Here’s what I was able to get:

The numbers speak for themselves. But let’s spell them out anyway:

• 18.5% annualized return with half the volatility of the benchmark. 9.8% vs 19.1%. This strategy doesn’t just earn more. It does it with dramatically less risk.

• Sharpe ratio of 1.78. The benchmark sits at 0.51. More than three times the risk-adjusted return.

• Max drawdown of -32.8% vs -56.8% for the benchmark. In the worst moments, this strategy gave back roughly half of what buy-and-hold did.

• Worst month: -10.6%. The benchmark’s worst month was -16.9%. When things go wrong, they go less wrong here.

• 76% positive months. Three out of every four months are winners. The benchmark manages 64%.

• Correlation to the benchmark: -0.22. Slightly negative. That’s not alpha on top of beta. That’s a genuinely different source of return.

If we had traded this strategy since 2005:

• We would have had only 1 negative year (2009);

• We would have seen 76% of the months positive, with the best at +9.8% (Jul’21);

• We would have seen 24% of the months negative, with the worst at -10.6% (May’09);

• The longest positive streak would have been 15 months, from Jan’05 to Mar’06;

• The longest negative streak would have been 3 months, from Aug’09 to Oct’09.

Looking at the Fama-French 3-Factor risk model:

• Alpha: 0.0668% per day → annualized ≈ 16.8%, highly significant (t = 8.19, p < 0.001). The strategy delivers strong abnormal returns that standard risk factors cannot explain.

• Market Beta (Mkt-RF): -0.111 → negative market exposure (t = –16.04, p < 0.001). The strategy doesn’t just avoid market risk. It actively moves against it. This is what drives that -0.22 correlation to the benchmark.

• SMB (Size): -0.107, significant (t = -8.05, p < 0.001). A tilt toward larger stocks. Consistent with an international strategy that avoids the smallest, least liquid names.

• HML (Value): +0.049, significant (t = 4.73, p < 0.001). A modest value tilt. The strategy tends to favor cheaper stocks, which is consistent with the momentum reversal dynamics documented in the papers.

• R² = 7.0% → only 7% of return variation explained by the three factors. The vast majority of performance is idiosyncratic. The Fama-French factors are nearly irrelevant here.

You can’t ask for more from a market-neutral strategy.

Final thoughts

Voltaire spent his life arguing that certainty is the enemy of truth. Two and a half centuries later, a finance paper proves him right in one of the most quantitatively rigorous ways imaginable.

The insight at the core of these papers is deceptively simple: the information is in the uncertainty. Not in the most probable outcome, but in the full distribution of possibilities. A model that collapses that distribution into a single predicted class is throwing away exactly the signal it needs. The reclassification framework is, at its heart, a way of respecting doubt.

The results speak to how much edge was being left on the table. Doubling the Sharpe ratio of the best-known momentum improvements. Surviving in markets where momentum historically fails. A global model that gets better by learning from countries it has never traded. None of that happens without embracing the full probability distribution.

There is plenty left to explore:

• Expanding the feature set. The papers use only momentum and size. What happens when you add valuation, quality, or macro variables to the distribution estimation step?

• Combining with other strategies. A negative beta and near-zero correlation to the market makes this a natural diversifier. The obvious next step is blending it into a broader portfolio.

• Refining the global model. Joint learning across 45 countries is powerful, but the weighting of countries, the handling of data quality differences, and the retraining schedule all leave room for improvement.

The canvas is wide open.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The 3rd cohort of the course is still open for enrollment for the next couple of days.

I created this to respond to a request I’ve received from hundreds of readers here: access to my codebase, step by step — for those who want to develop strategies using the same approach I share here. Enrollment also includes access to the private community and an ever-growing library of study groups. More information at the link below.

Join the Course and Community

Last day for enrollment in the 3rd cohort: Apr 7.

The idea

“Casino gambling with a system where you have the edge is a wonderful teacher for elementary money management.” Ed Thorp

Ed Thorp is the money manager I admire most. Many people have never heard of him. They should have. In 1961, with Claude Shannon — the father of information theory — he built the first wearable computer to beat roulette. He wrote Beat the Dealer and proved blackjack could be conquered with math. He derived the Black-Scholes options pricing formula years before Black and Scholes — but chose to trade on it instead of publishing. Then he founded the world’s first quant hedge fund, posting 227 winning months out of 230. Finally, as he decided to end his famous hedge fund, he passed some of his learnings to the young Ken Griffin and became Citadel’s first LP. And it all started at the blackjack table.

Casinos don’t gamble. They have an edge on every game, they let everyone play, and they let the math do the rest. Over thousands of bets, the law of large numbers turns a small edge into a certainty. Thorp saw this and turned it against them. Then he took the same logic to Wall Street.

The lesson is simple. If you have a proven edge, you don’t want bigger bets. You want more bets.

This week, we take a proven mean reversion strategy — the one from Murphy’s Law — and expand it from the S&P 500 to the Russell 3000. More stocks. More signals. More bets. The challenge: doing it without diluting the edge. Here’s the plan:

1. The trap: why expanding your universe naively destroys the edge

2. The filter: dynamically avoiding names with high blow-up risk

3. The test: Does it actually work?

4. The takeaway: final thoughts and next steps

Let’s get started.

Course and Community: enrollments Open

As many of you requested, the 3rd cohort of the course is now live and open for enrollment.

It walks through my codebase step by step and is designed for readers who want to develop quant strategies using the same approach I’ve shared here.

You’ll find all the details—content, structure, pricing, and FAQs—at the link below. If you have any questions, feel free to reach out.

Join the Course and Community

Enrollment also includes access to the private community and an ever-growing library of study groups.

The trap of more

Our baseline is a mean reversion strategy on S&P 500 stocks. When a name drops more than its own history says it should, we buy. When it snaps back, we sell. Simple entry, simple exit, solid edge — a +1.2 Sharpe with only one down year in twenty-two. For the full breakdown, check the original article: Murphy’s Law.

Here’s what happens when we naively apply the same idea to the Russell 3000:

The Russell 3000 is roughly 6x larger, so we scale accordingly: 60 maximum positions instead of 10. More bets, right? Yes — but worse ones:

• Total return cut nearly in half: 4,286% vs. 8,114%

• Sharpe ratio dropped from 1.30 to 1.00

• Max drawdown nearly doubled: -31.8% vs. -18.8%

Same signal, different risk

The signal is the same. The entry logic is the same. The exit is the same. The only thing that changed is the universe. So where did the edge go?

Our hypothesis: the edge didn’t disappear. It got noisier. The Russell 3000 includes thousands of smaller, less liquid names. These stocks may mean-revert just as often, but when they don’t, they blow up harder. If that’s true, the problem isn’t signal quality — it’s tail risk. And if it’s tail risk, we can filter it out.

To test this, we compare the trade-level statistics of both universes side by side — and look specifically at three tail-risk metrics, computed on a rolling basis. At the moment a trade is triggered, the stock already carries these numbers — they describe the name as it enters the position:

• Skewness measures the asymmetry of the return distribution. Negative skew means the left tail is fatter — losses are larger and more frequent than gains. A stock that mean-reverts cleanly should have positive or near-zero skew.

• Semivariance ratio is the share of total variance that comes from below-zero returns. A ratio above 0.5 means most of the volatility lives on the downside. The higher the ratio, the more the risk is concentrated where it hurts.

• CVaR 5% (Conditional Value at Risk) is the average return in the worst 5% of trades. VaR tells you the door to the left tail. CVaR tells you what’s behind it.

Let’s see the numbers:

The edge is almost identical. Mean return per trade: 0.99% vs 0.98%. Win rate: 69% vs 68%. Median trade: 1.4% vs 1.5%. If you stopped here, you’d think expanding the universe is free money. It isn’t. Look at the risk:

• Volatility per trade jumps 37%: 6.5% vs 4.7%. Same average, much wider dispersion.

• The worst trade gets worse: -91.4% vs -77.0%. The left tail extends further.

• CVaR 5% widens: -4.1% vs -3.4%. When things go wrong, they go more wrong.

• The right tail explodes too: max winner goes from +40.5% to +146.2%. These aren’t clean mean-reversion trades. They’re lottery tickets — and lotteries have losers.

Interestingly, the median skewness and semivariance ratio are actually better in the Russell 3000. The typical name is fine. The problem isn’t the typical name.

The conclusion is clear. The signal works just as well in the broader universe. The edge per trade is virtually unchanged. But the distribution around that edge is far wilder — and in a portfolio of 60 positions, it only takes a few blow-ups to wreck the month. The fix isn’t a better signal. It’s a better filter. We need to identify the dangerous names before we enter the trade — and stay away.

Same signal, safer bets

We know the edge survives in the Russell 3000. We also know the tails are wider and more dangerous. The fix sounds simple: don’t trade names with ugly tails.

Here’s what we do. Before entering any trade, we check the stock’s rolling CVaR 5%. If the left tail is too deep, we skip it. The signal might be screaming “buy.” We don’t care. A good entry into a bad name is still a bad trade.

But filtering alone isn’t enough.

The Russell 3000 has roughly 3,000 names. Even after removing the dangerous ones, hundreds of signals fire at any given time. Taking all of them — 60 positions, like before — means diluting the portfolio into mediocre setups. The best signals get the same weight as the barely-passing ones. That’s not “more bets.” That’s more noise.

The filter tells us who not to trade. The position cap tells us how selective to be. We want both. So we filter first, then concentrate: 15 positions, drawn only from names that pass the tail-risk screen.

Let’s see what this does. Below we compare the Russell 3000 strategy with 15 positions — without and with the CVaR filter (6% threshold):

The filter works. Across the board:

• Total return jumps 57%: 13,946% vs 8,903%. Same universe, same signal, same number of positions. The only difference is which names we avoid.

• Sharpe ratio leaps from 0.99 to 1.30: not by increasing return alone, but by crushing volatility — from 22.2% down to 17.7%.

• Max drawdown improves: -32.8% vs -36.0%. The worst month shrinks from -14.8% to -10.9%.

• Best month gets smaller too: 16.4% vs 25.1%. That’s a feature, not a bug. Those +25% months were the flip side of the fat tails we just removed.

• More positive months: 69% vs 64%. The strategy wins more often when it stops stepping on landmines.

The story is simple. We didn’t change the signal. We didn’t change the entry or the exit. We just asked one question before every trade: does this name have a history of blowing up? If yes, we walk away. That single filter turned a mediocre strategy into one that rivals our S&P 500 baseline — with a 1.30 Sharpe on a completely different universe.

Which raises an obvious question. What happens when we stop choosing between the two — and combine them?

More bets, better bets

We started with a question: can we trade more without trading worse? The answer is yes — but only if we’re selective. We take our S&P 500 baseline — 10 positions, proven edge, 1.30 Sharpe — and add 15 filtered Russell 3000 names on top. Two universes. One signal. Twenty-five positions. No landmines. Here’s what that looks like:

The numbers speak for themselves. But let’s spell them out anyway:

• $100k became $12 million. The S&P 500 turned the same $100k into $760k. That’s not a comparison. That’s a different universe.

• 23.1% annualized return — with lower volatility than the benchmark. 16.0% vs 18.7%. More return, less risk. That’s the whole game.

• Sharpe ratio: 1.38. Up from 1.30 in our S&P 500-only baseline. More bets didn’t dilute the edge. They improved it.

• Max drawdown cut to less than half the benchmark: -23.5% vs -56.8%. During the worst moments in markets, this strategy gave back less than a half of what buy-and-hold did.

• Worst month: -8.0%. Compare that to the benchmark’s -16.9% — or to the unfiltered Russell 3000’s -14.8%. The filter earned its keep.

• 72% positive months. Nearly three out of four months are winners. The S&P 500-only baseline posted 65%. The benchmark: 65%.

Compared to where we started — the S&P 500 baseline with a 1.30 Sharpe — we added 2 percentage points of annual return, improved the Sharpe, and kept drawdowns in a similar range.

If we had traded this strategy since 2003:

• We would have had only one negative year (2018);

• We would have beaten the S&P 500 in 22 out of 24 years;

• We would have seen 72% of the months positive, with the best at +17.2% (Nov’20);

• We would have seen 28% of the months negative, with the worst at -8.0% (Jan’10);

• The longest positive streak would have been 12 months, from May’05 to Apr’06;

• The longest negative streak would have been 3 months, from Feb’25 to Apr’25.

Looking at the Fama-French 3-Factor risk model:

• Daily alpha of 0.069%. That compounds to roughly 17.5% annualized (×252 trading days). The t-stat of 9.09 makes it statistically bulletproof.

• Market beta of 0.65. The strategy captures about two-thirds of market moves — meaningful exposure but far from fully correlated. Lower risk than buy-and-hold.

• Positive SMB loading (0.12, significant). The strategy tilts toward small caps, which makes sense for a Russell 3000 mean reversion approach — more of the universe is small.

• HML loading is essentially zero (-0.0005, p = 0.96). No value/growth tilt at all. The edge isn’t coming from a value factor in disguise.

• R² of 0.58. The three factors explain about 58% of the strategy’s variance. The other 42% — including the alpha — is idiosyncratic. That’s the edge.

Final thoughts

We started with a simple idea from Ed Thorp: if you have an edge, you want more bets. We took a proven mean reversion strategy on the S&P 500 — a 1.30 Sharpe with one down year in twenty-two — and asked whether we could expand it to the Russell 3000 without losing the edge.

The naive answer was no. More stocks meant more noise, fatter tails, and worse drawdowns. But the edge per trade was still there — buried under blow-ups from dangerous names. One filter changed everything. By checking each stock’s rolling CVaR 5% before entering, we removed the landmines and kept the signal. The result: a 1.38 Sharpe, 23.1% annualized return, and a max drawdown less than half the benchmark’s. Not by finding a better signal. By making better bets.

There’s more to explore:

• Dynamic position sizing: scaling exposure based on volatility regimes or signal strength;

• Sector and liquidity constraints: stress-testing the filter under real-world execution limits;

• Hedging overlays: applying beta hedging to further compress drawdowns without giving up alpha.

For those who want to know more about Ed Thorp, his book A Man for All Markets is essential reading. The podcast episode below is a great companion. Worth every minute:

🎧 Listen on Spotify

Thorp’s life is the blueprint for anyone chasing more than just money.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

Course and Community: enrollments Open

As many of you requested, the 3rd cohort of the course is now live and open for enrollment.

It walks through my codebase step by step and is designed for readers who want to develop quant strategies using the same approach I’ve shared here.

You’ll find all the details—content, structure, pricing, and FAQs—at the link below. If you have any questions, feel free to reach out.

Join the Course and Community

Enrollment also includes access to the private community and an ever-growing library of study groups.

The idea

“An investor who knew future returns with certainty would invest in only one security.” — Harry Markowitz

We don’t know the future. This is why we intuitively spread our bets. Harry Markowitz turned that intuition into algebra. In 1952, he published a paper that gave diversification a rigorous mathematical foundation, proving not just that it works, but exactly how much of each asset to hold given your tolerance for risk. The Nobel committee caught up with him in 1990. His Nobel lecture is worth reading.

The idea for this article came to me while I was reading a new paper titled “Diffusion Factor Models.” The paper addresses one of the main problems of naive Mean-Variance Optimization: estimation error arising from unreliable sample means and covariances when the number of assets exceeds the number of available observations.

The authors’ proposed solution is elegant. Rather than directly estimating moments from limited data, they train a diffusion model that learns the underlying factor structure of asset returns, then generate abundant synthetic samples to obtain more stable estimates. This approach effectively trades a small modeling bias for a substantial reduction in estimation variance, achieving Sharpe ratios nearly three times higher than equal-weight benchmarks (1.36 vs. 0.49) and dramatically outperforming traditional shrinkage methods, which produced negative returns in the same test period.

In fact, the paper provides direct evidence of an important insight in portfolio optimization: accurate covariance estimation matters more than accurate mean estimation for portfolio performance. Their hybrid methods reveal this clearly: combining naive sample means with diffusion-generated covariances achieves a Sharpe ratio of 1.09, while the reverse — diffusion-generated means with naive sample covariances — yields only 0.28.

But I’m not writing about that paper this week. I’m going to test their method first, then come back to it later this year. This week is about classic Mean-Variance Optimization. We have to crawl before we walk.

Our problem is simpler: allocating capital across a handful of substrategies rather than hundreds of individual assets. In our setting, estimating means and covariances won’t be a problem; we’ll have far more observations than parameters to estimate.

Here’s the plan:

1. First, we’ll introduce the set of strategies we want to combine.

2. Next, we’ll establish a baseline using an equal-weight allocation.

3. Then, we’ll implement mean-variance optimization in Python.

4. Finally, we’ll evaluate the results and outline the next steps.

We’re building a private community for systematic traders. A place to explore ideas, exchange insights, and tackle the real technical and strategic challenges of building robust trading systems… from signal research to execution and risk.

Enrollment reopens at the end of this month. Join the waitlist below to be the first to know when new seats open:

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Speaking of great reads, there’s a new Substack I’d recommend:

Systematic Long Short

I’m an entrepreneur and systematic portfolio manager. I’ve sat in the PM seat at pod shops and managed real money at scale and seen how the sausage gets made. I want to help you understand how the game is played at the highest levels.

I’ve found several of his articles genuinely insightful. Hope you enjoy them as much as I do.

The portfolio we’re actually optimizing

In our last article (Murphy’s Law), we stress-tested a simple mean-reversion system and treated “anything that can go wrong will go wrong” as a design principle: assume failure modes exist, then engineer around them. Here’s the summary of what we did:

• Base strategy (mean reversion in liquid equities): when a stock suffers a statistically extreme short-term drop, we buy the dislocation (with simple regime filters like “in an uptrend” and a low short-term QPI), then hold briefly for the rebound.

• Better entry timing: instead of waiting for the next open, we test entering at the close to capture the earliest part of the rebound and reduce “gap risk” around the entry.

• Signal refinements: we progressively tighten what qualifies as a “good” dislocation (stronger/cleaner setups, better regime constraints, fewer low-quality events).

• Walk-forward research loop: parameters are selected out-of-sample via rolling walk-forward tests (train → validate → deploy), rather than tuned once on the full history.

• Ensemble mindset: instead of betting on one “best” parameter set, we combine a small basket of strong variants to reduce overfitting and smooth performance.

Different universes

An easy way to diversify this kind of system is to run the same playbook in other markets. Let’s see how it behaves across three universes:

1. US: S&P 500 constituents (last week’s setup, max of 10 positions)

2. Canada: S&P/TSX Composite constituents (max of 5 positions)

3. Australia: S&P/ASX 200 constituents (max of 5 positions)

In summary:

• The approach works in all three markets: ~17–21% annualized returns with ~1.0–1.3 Sharpe across US/Canada/Australia (since 2010).

• The trade-off: outside the US we earn comparable performance, but with materially deeper drawdowns (~-33% to -35% vs -19% in the US).

• Diversification benefits will be strong: cross-geo correlations are very low — US-Canada 0.08, Canada-Australia 0.10, and even the highest pair (US-Australia) is only 0.38.

I’m starting the tests in 2010 (instead of 2003) because the strategy is extremely profitable pre-2010 in Canada and Australia, and I don’t want the results to be dominated by that unusual period.

Baseline: equal-weight portfolio

Now let’s see what happens with the naive approach: equal weights across strategies, rebalanced annually:

• Volatility collapses with diversification: the equal-weight portfolio runs at 10.9% vol, versus 13.8% (US), 16.8% (Canada), and 16.4% (Australia). This is a big drop in risk despite holding three “risky” sleeves.

• Sharpe jumps meaningfully: 1.66 for the equal-weight portfolio vs 1.27 (US), 1.04 (Canada), 1.26 (Australia). “Same-ish returns, much lower variance” effect.

• Returns stay high while risk falls: annualized return is 19.1%, right in the middle of the single-country range (17.4%-21.2%), but achieved with far less volatility.

• Drawdowns become much more manageable than the worst sleeves: max DD is -21.4%, close to the US (-19.4%) but dramatically better than Canada (-35.3%) and Australia (-32.5%).

• More consistent month-to-month: 70% positive months vs 67% (US), 65% (Canada), 68% (Australia), and the worst month improves to -8.5% (vs -12.8% Canada and -19.1% Australia).

Now, let’s see how to implement Mean-Variance Optimization.

Mean-variance optimization

The Basics

The goal of mean-variance optimization is to find the portfolio weights that maximize risk-adjusted returns. Markowitz framed this as a tradeoff: we want high expected returns, but we also want to avoid volatility. The question is how to balance the two.

The standard formulation maximizes a utility function:

where:

• $\omega$ is the vector of portfolio weights

• $\mu$ is the vector of expected returns (annualized)

• $\Sigma$ is the covariance matrix of returns (annualized)

• $\eta$ is the risk aversion parameter

The first term, $\omega^\top \mu$, is the portfolio’s expected return. The second term, $\omega^\top \Sigma \omega$, is the portfolio’s variance. The risk aversion parameter $\eta$ controls how aggressively we penalize volatility. A higher $\eta$ means we’re more conservative: we’ll accept lower returns to avoid risk. Typical values range from 2 to 5.

We also impose constraints. The weights must sum to one (we’re fully invested), and we can set minimum and maximum bounds on individual weights to prevent extreme concentrations:

The implementation below uses scipy.optimize.minimize to solve this problem numerically. It also includes two optional refinements: Ledoit-Wolf shrinkage for the covariance matrix, and mean shrinkage that pulls return estimates toward their cross-sectional average. These regularization techniques can help when estimation error is a concern (though in our case, with only a handful of substrategies and years of daily data, they’re more of a safeguard than a necessity).

The code

The code is pretty straightforward:

Results

Now we can put the code above into a simple walk-forward loop: each year, we look back over the previous four years of daily returns, estimate the optimal weights, and then hold those weights fixed for the next year. We repeat this process as we roll forward through time.

Here’s what we get with minimum weight 0, maximum weight 60%:

Highlights:

• Higher return, same ballpark risk: MVO lifts annualized return to 23.1% vs 19.1% for equal-weights, while volatility stays low at 12.2% (vs 10.9%).

• Sharpe improves even with slightly higher vol: 1.76 for MVO vs 1.66 for equal-weights, i.e., we’re getting more return per unit of risk despite taking a bit more volatility.

• Much higher terminal wealth: $100k grows to $3.04M under MVO vs $1.75M under equal-weights (same 2010-2026 window).

• Drawdowns don’t get worse: max drawdown is essentially unchanged/slightly better at -20.7% vs -21.4% for equal-weights, so the return boost didn’t come from accepting materially larger drawdowns.

• Month-to-month consistency is the same: 70% positive months for both. The edge shows up mainly in the magnitude of gains, not a higher hit rate.

If we look at the monthly return distribution (2010-2026, 193 months):

• 70% of months were positive, with an average gain of +3.7%; 30% were negative, averaging -2.5%.

• The best month was +10.5% and the worst month was -9.0%.

• The “typical” month is strong: median = +2.0%; the middle 50% of outcomes sits between -0.5% (25th pct) and +4.3% (75th pct).

• Volatility at the monthly level is moderate: std = 3.6%, with an overall mean = +1.85% per month.

• Streaks: the longest positive streak was 21 months (Jun’12 → Feb’14); the longest negative streak was 3 months (Feb’25 → Apr’25).

• The distribution is mildly asymmetric: skew = -0.40 (slightly more downside tail), with kurtosis = 0.22 (not especially fat-tailed).

Looking at the Fama-French 3-Factor risk model:

• Alpha: 0.0605% per day → annualized ≈ 15%, highly significant (t = 5.70, p < 0.001): strong abnormal returns not explained by FF3 factors.

• Market beta (Mkt–RF): 0.346, highly significant (t = 35.19, p < 0.001): meaningful, but not huge, exposure to the equity market (about one-third beta).

• SMB (Size): 0.0215, insignificant (t = 1.21, p = 0.228): no reliable size tilt.

• HML (Value): 0.0129, insignificant (t = 0.95, p = 0.343): no reliable value/growth tilt.

• Explained variation: R² = 0.256: FF3 explains ~26% of daily return variation; the majority is still strategy-specific.

Final thoughts

We started with Markowitz’s 70-year-old insight — that diversification isn’t just intuition, it’s algebra — and applied it to a simple problem: allocating capital across three geographic variants of a mean-reversion strategy.

The results speak for themselves. Even the naive equal-weight portfolio delivered a +1.6 Sharpe by exploiting the remarkably low correlations across markets. But mean-variance optimization pushed it further: 23.1% annualized returns, a +1.76 Sharpe, without increasing drawdowns. The Fama-French regression confirmed what we hoped: a highly significant annualized alpha, minimal factor exposure, and only 26% of returns explained by standard risk factors. The rest is ours.

This is just the beginning. There’s plenty of room to explore:

• More sleeves: add new geographies, signals, or timeframes to increase diversification degrees of freedom.

• Smarter constraints: cap turnover, limit year-to-year weight changes, or impose risk budgets per sleeve to reduce allocation whiplash.

• Risk overlays: simple beta hedging or volatility targeting can further stabilize the equity curve.

Markowitz was right: if we knew the future, we’d bet everything on one horse. We don’t, so we diversify, optimize, and let the math do the heavy lifting.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

We’re building a private community for systematic traders. A place to explore ideas, exchange insights, and tackle the real technical and strategic challenges of building robust trading systems… from signal research to execution and risk.

Enrollment reopens at the end of this month. Join the waitlist below to be the first to know when new seats open:

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The idea

“Anything that can go wrong, will go wrong.”

— Major Edward A. Murphy Jr., aerospace engineer

A few days ago in Abu Dhabi, I met an old friend I hadn’t seen since our graduation. We were among the very few aerospace engineers trained in our home country that year. After a few years serving in the Air Force (I went straight into civilian life, he didn’t), he eventually moved on: first into quantitative hedge funds, and later back into hardcore engineering — his true passion. Today, he’s developing defense drones in the UAE.

“Why do aerospace engineers adapt well to quantitative hedge funds?” one of us asked.

For an airplane to fly, it isn’t enough to get one or two big things right: you need a thousand details to line up. A quantitative strategy is no different: profitability comes from getting a thousand details right. We settled on this hypothesis (five years of nonstop hardcore math and coding might have helped a bit, too :)).

Last week reminded me of one of the most famous aerospace engineers: Major Edward A. Murphy Jr. The man built safety-critical systems… and gifted the world the immortal line (which later became Murphy’s Law), “Anything that can go wrong, will go wrong.” One of the first strategies I wrote about here decided to take that literally — November was brutal for it.

Many people mistake him for a cosmic pessimist or a fatalist. But when he taught his students, “If something can go wrong, it will,” he meant that you should identify everything that could go wrong and fix it so that nothing does go wrong.

This week, instead of dissecting yet another research paper (my last three implementations were so far off from what the authors claimed that I’m starting to wonder who is writing these papers…), let’s talk about how to fix a strategy by getting a thousand details right. Well, maybe not a thousand, but certainly more than a few.

Here’s the plan:

1. A quick review of the base strategy and a look at what went wrong

2. A tour through the “more-than-a-few” improvements

3. Results — because that’s what matters in the end

4. And as always, a few closing thoughts and the next steps

Let’s get to it.

Study Group

Our last study group was a success. Thank you to all community members who joined live! For those who couldn’t, I will post the code and the recording on learn.quantitativo.com, as usual, in the next few days. The TAA strategy we reviewed is up almost 40% this year.

Lately, our community has shown interest in a wide range of topics on the forum. As usual, I’ll post a survey on connect.quantitativo.com so everyone can vote on the topic for the next study group.

I know I said I would reopen enrollments in December… but I am completely swamped with work, so I’ll reopen enrollments in January 2026. The waitlist already has a few hundred people. Join it below to be the first to know when new seats open:

Join the Waitlist

What is wrong with this strategy

The strategy we’re about to improve is the one described in the article A Mean Reversion Strategy from First Principles Thinking (using only the long-only version for simplicity).

The idea is straightforward: the strategy aims to capture short-term price dislocations in S&P 500 constituents. Whenever a stock experiences a drop that is statistically unlikely (below a certain quantile of its own 5-year return distribution), we open a long position. The trade is closed once the stock reverts.

Here’s the performance of the backtest + live since November last year, up-to-date:

Even at first glance, the strategy has several undesirable characteristics:

• Yes, it outperforms the S&P 500, but the Sharpe ratio isn’t even 1.0

• The maximum drawdown is less than half of the benchmark, but at 27% it’s still too high

• The average annual drawdown is 15.1% — actually higher than the S&P 500’s 14.6%

• The strategy’s volatility is 21.6%, higher than the benchmark

• The strategy has a high correlation to the market at 0.64

And November was rough: the strategy returned –7.7% while the market was essentially flat.

Why did this happen in November?

Looking at the individual trades, several were AI-related names (AMD -13%, DELL -12%, ORCL -9%). With only ~20 trades in November, these outsized losses dominated the month’s results.

It’s important to remember that this strategy is designed to profit from short-term statistical dislocations. When a stock undergoes a fundamental repricing, the signal may not mean-revert — because the price change is justified. The system cannot distinguish between:
i) non-fundamental dislocations, and
ii) fundamentally-driven moves.

It captures all statistically significant drops, regardless of their cause.

In November, we saw an unusually high share of fundamental-driven declines, which explains the deviation in performance.

What can we do to fix?

In essence, there are four levers we can pull:

1. Signal quality → Improve predictive power by separating noise-driven drops from fundamental repricings

2. Execution logic → Sharpen timing, sizing, and exit rules

3. Breadth / Diversification → Increase the number of independent bets (more slots)

4. Risk overlay → Hedge unwanted factor exposures (β, size, momentum, sector, etc.)

In this article, we will focus primarily on (1) and (2). We’ll go deeper on (1), and address (3) and (4) in subsequent pieces. Our goal is to reach a Sharpe ratio of at least 1.5 and cut the maximum drawdown in half.

Improving the edge

A) Baseline

Let’s start by improving the edge — and for that, we need a baseline to compare our experiments against. What would have happened if we had simply bought every event where a stock had:

• a price drop,

• a 3-day QPI below 15,

• a closing price above its 200-day simple moving average (i.e., the stock was in an uptrend),

• and then held the position for 5 days?

This is exactly how we measured the edge in the original article. Here are the stats:

In summary:

• Win rate: 57% of the ~95,000 trades were winners (the sample includes only S&P 500 constituents at each point in time to avoid survivorship bias).

• Expected return per trade: +0.56% overall (winners averaged +3.6%, losers averaged –3.4%).

• Distribution: The median trade was +0.6%, with fairly balanced upside/downside tails (max +69.7%, min –58.7%).

B) Trading at the close

The first improvement was actually proposed by someone in the community a while ago: entering trades at the close instead of at the next day’s open.

This can make sense because many mean-reversion signals are strongest at the moment of the dislocation. By the next morning, part of the move may have already reverted, reducing the available edge. Here’s the data:

• Win rate improved: 57% winners (vs. 57% before, but with a slightly better distribution of outcomes).

• Expected return per trade increased: +0.64% per trade (vs. +0.60% in the baseline).

• Winners were slightly stronger: average winner now +3.70% (vs. +3.60%), while losers remained roughly the same (–3.49%).

• Statistically significant improvement: a statistical test shows the new return distribution differs from the baseline (t-stat = 3.37, p = 0.00062).

C) Improving the entry

The next improvement targets the entry signal itself. In addition to the existing conditions (a statistically significant price drop and the stock trading above its 200-day SMA), we now add one more filter: IBS (Internal Bar Strength). Specifically, we require IBS < 0.10, meaning the stock closed in the bottom 10% of its intraday range.

To keep the strategy supplied with enough trading opportunities, we also relax the QPI threshold from 0.15 to 0.30. After adding the stricter IBS filter, using the old QPI cutoff would make signals too scarce; widening it ensures we still capture a meaningful number of setups.

Why might this help? Because a large daily drop isn’t always enough to guarantee a good mean-reversion setup. Sometimes a stock bounces intraday, signaling buyers have already stepped in, which weakens the edge. By requiring IBS to be low, we target days where the stock was sold off all the way into the close, indicating persistent intraday pressure. These events tend to produce cleaner, more reliable snap-back behavior in the following days.

Let’s look at the data:

• Win rate increased: 59% winners (up from 57% in the baseline).

• Expected return per trade improved: +0.77% (vs. +0.60% in the baseline).

• Winners stayed strong: average winner +3.45%, while losers averaged –3.14% (similar loss profile but with better selectivity).

• Statistically significant improvement: a statistical test confirms the distribution is different from the baseline (t-stat = 8.32, p-value = 0.0).

D) Improving the exit

So far, we’ve been analyzing this signal using a time-based exit, holding each trade for exactly 5 days. But the actual strategy exits a trade as soon as today’s close is higher than yesterday’s high.

This makes intuitive sense: if you’re trading short-term mean reversion, the moment the stock snaps back above the prior day’s high, the “bounce” you were waiting for has already happened—there’s no reason to keep holding and expose yourself to further noise or new downside risk.

These are the stats with all improvements applied so far, using this exit logic:

• Win rate jumped significantly: 71% winners (up from 57% in the baseline).

• Expected return per trade increased further: +0.90% (vs. +0.60% in the baseline).

• Loss profile remained controlled: losers averaged –3.44%, while winners averaged +2.65% — a much healthier distribution.

• Statistically significant improvement: a statistical test confirms this distribution is different from the baseline (t-stat = 14.38, p-value = 0.0).

But we can actually do better with exits. Instead of waiting for the price to break above yesterday’s high, we can test a more dynamic idea: closing the trade whenever IBS > 0.9 or RSI2 > 90.

Why might this work? Because both conditions signal exhaustion to the upside — when a stock surges into the top of its intraday range (IBS > 0.9) or becomes extremely short-term overbought (RSI2 > 90), the mean-reversion burst we were targeting has likely already played out. At that point, staying in the trade just adds noise and downside risk.

Let’s look at the data:

• Expected return per trade improved further: +1.04% (up from +0.90% with the previous exit rule).

• Win rate remained strong: 70% winners, with winners averaging +3.11% and losers averaging –3.76%.

• Distribution remains healthy: median trade at +1.4%, with similar tail behavior but slightly better upside capture.

• Statistically significant improvement: the distribution differs from the previous exit logic (t-stat = 5.37, p-value = 0.0).

Robustness

Before we move into walk-forward optimization, we first need to check whether the strategy is robust to reasonable parameter variation. To do that, we’ll test all 625 combinations from the following grid:

These ranges cover a wide and sensible portion of the parameter space—roughly 30–50% variation around each baseline value. Then,

• If the strategy is truly robust, we should see stable performance across the grid—Sharpe ratios, annual returns, and drawdowns remaining positive and reasonably consistent across all 625 parameter sets.

• If the strategy only performs well for a tiny handful of cherry-picked combinations, that’s a clear sign of fragility.

Before showing the results, it’s important to clarify the backtesting assumptions used throughout this robustness test (and in the walk-forward analysis that follows):

1. Increased diversification: we expand the maximum number of simultaneous positions from 4 to 10, as we previously discussed the importance of diversification.

2. Trading costs: each trade includes a 2 bps round-trip cost.

3. Equal-weighting: all trades are equal-weighted for simplicity and consistency across the 625 simulations.

Let’s see the data:

Robustness Results (2003–2025)

• Annual returns are tightly clustered: median ≈ 18.2%, with most parameter sets between 16.3% and 19.5%.

• Sharpe ratios remain consistently high: median ≈ 1.13, with 50% of all runs falling between 1.03 and 1.22.

• Drawdowns are stable and controlled: median max drawdown ≈ –20%, with most runs between –24% and –18%.

• No catastrophic parameter sets: even the worst combinations produce positive annual returns, Sharpe ≈ 0.70, and manageable drawdowns.

Is this a strong sign of robustness?

Yes. A strategy is robust when performance remains stable across a wide and reasonable part of the parameter space. Here, we tested 625 combinations, spanning roughly ±30–50% variation around each parameter. The results show:

• No dependence on a narrow, cherry-picked parameter set

• No sharp peaks or isolated “lucky” regions

• A broad plateau of strong performance

• Consistently attractive risk-adjusted returns (Sharpe ~1+) across the grid

• Drawdowns that remain in a tight and acceptable range

This pattern — a large plateau of good solutions rather than a single spike — is exactly what we want to see in a strategy that is truly signal-driven rather than curve-fit.

Walk-forward optimization

Now that we’ve confirmed the strategy is robust across a wide range of parameter values, we face the next challenge: choosing which parameter combination to actually trade. If we simply pick the one that performed best on the full 2003–2025 dataset, we’d almost certainly be overfitting—rewarding parameters that benefited from historical noise rather than true predictive power.

This is where walk-forward optimization (WFO) comes in. Instead of optimizing on the entire dataset at once, WFO repeatedly trains the strategy on a rolling window of past data and immediately tests it on the next, unseen period. If a parameter set survives this sequence of “train → test → roll forward” cycles, it’s far more likely to capture a genuine, persistent edge that generalizes to real trading.

Walk-Forward Procedure

In our walk-forward framework, we use the following process:

• 3-year training windows:

  For each step, we evaluate all 625 parameter combinations using only the previous 3 years of data.

• Rank by Sharpe ratio:

  Within each training window, we rank all parameter sets by their Sharpe ratio to identify the most stable performers.

• Select the top 5 sets:

  We take the five best-performing parameter sets and equal-weight them, effectively trading five independent portfolios as an ensemble.

• Trade for the next year:

  These five selected parameter sets are used to trade the following year, which serves as the out-of-sample test.

• Roll forward one year:

  After that year completes, we shift the entire window forward by one year and repeat the process:

  train → rank → select → trade → roll.

• Continue until 2025:

  This rolling sequence continues until we reach the end of the sample, producing a full walk-forward performance series.

Results

With the walk-forward framework in place, we can now look at how the strategy performs out-of-sample:

Final Strategy (after all improvements) vs. S&P 500:

• Annual returns more than double the benchmark: 19.0% vs. 9.2% per year.

• Sharpe ratio more than 2× higher: 1.27 vs. 0.56, despite slightly lower volatility (14.6% vs. 18.8%).

• Significantly smaller drawdowns: 17.6% max drawdown vs. 56.8% for the S&P 500.

• Smoother month-to-month performance: 69% positive months vs. 66% for the index.

Final Strategy vs. Original Flawed Strategy:

• Sharpe ratio improved dramatically: from 0.90 → 1.27 (a 41% increase in risk-adjusted performance).

• Volatility materially reduced: 14.6% vs. 21.6% — the improved version is much smoother.

• Max drawdown cut from 26.8% to 17.6%

• Better consistency: 69% positive months (final) vs. 62% (original).

• About the same annual returns with less risk: 19.0% vs. 18.7%, but achieved with far lower volatility and far better drawdown behavior.

Now, let’s look at monthly and annual returns:

If we had traded this strategy since 2003:

• We would have had only 1 down year in 22 years (2012);

• We would have seen 69% of the months positive, with the best at +12.1% (Mar’20);

• We would have seen 31% of the months negative, with the worst at -9.0% (Oct’08);

The table above compares the daily return distribution, and clearly shows what actually improved:

• Volatility dropped by ~32% in the improved strategy (0.92% vs. 1.36%), producing a dramatically smoother day-to-day P&L profile than the original system (and even smoother than the S&P 500 (1.18%)).

• Tail risk improved substantially: the worst daily loss shrank from -13.84% → -7.45%, a ~46% reduction. This is one of the clearest signs that the new entry/exit logic and walk-forward ensemble are filtering out dangerous setups.

• Typical losing days are much smaller: the 25th percentile improved from -0.53% → -0.28%, cutting the size of routine down days nearly in half.

• Extreme outcomes (both good and bad) were reduced: max daily gain fell (16.5% → 9.1%), but so did extreme losses. The improved strategy trades “hero days” for far better stability — exactly what long-term compounding prefers.

• Kurtosis dropped sharply (12.2 → 8.7): the distribution became significantly less fat-tailed, meaning fewer outliers and far more predictable behavior.

• Slightly lower skewness (0.4 → 0.3) supports the same point: fewer wild upside days and far fewer wild downside days.

• Despite all the risk reduction, mean daily return remained essentially the same (≈0.07% vs. 0.08%). The improved strategy delivers similar drift with meaningfully lower risk — the formula for a higher Sharpe.

Final thoughts

Murphy’s line — “Anything that can go wrong, will go wrong” — felt brutally accurate for this particular system in November. A handful of AI-related names went against it at the worst possible time, and a strategy that looked fine on paper suddenly felt fragile.

But that month also did us a favor: it forced us to look again into this system and ask a better question:

If this can fail, where will it fail, and how do we design it so it doesn’t?

The answer wasn’t a single clever trick. It was a list of small, unglamorous improvements: tighter entries (IBS + relaxed QPI), more intelligent exits, trading at the close, parameter robustness checks, and a walk-forward process that refuses to lock in one lucky configuration. None of these changes is spectacular on its own. Together, they turned a decent but jumpy strategy into one that doubles the S&P’s return, cuts drawdowns by two-thirds, and delivers a much cleaner return distribution.

The main lesson for me is that robustness is something we build deliberately. What worked better here was treating the idea as a prototype and then pushing on it from all sides: testing hundreds of parameter sets, forcing it through walk-forward OOS, and quietly hardening all the places where Murphy could attack — entries, exits, sizing, and risk.

There’s still plenty to do. We haven’t yet touched the other two levers we identified:

1. Breadth/diversification — scaling to more slots and possibly more universes;

2. Risk overlay — hedging factor exposures (β, size, momentum, sector) so we’re less exposed to the kind of “AI repricing month” that triggered this whole review. I also want to explore ways of explicitly separating fundamental repricings from noise-driven dislocations, so we can avoid stepping in front of real information shocks.

If there’s a broader takeaway, it’s this: Murphy’s Law doesn’t have to be a curse. Treated correctly, it’s a design principle. Assume things will go wrong, and then build systems that still behave well when they do.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

Study Group

Our last study group was a success. Thank you to all community members who joined live! For those who couldn’t, I will post the code and the recording on learn.quantitativo.com, as usual, in the next few days. The TAA strategy we reviewed is up almost 40% this year.

Lately, our community has shown interest in a wide range of topics on the forum. As usual, I’ll post a survey on connect.quantitativo.com so everyone can vote on the topic for the next study group.

I know I said I would reopen enrollments in December… but I am completely swamped with work, so I’ll reopen enrollments in January 2026. The waitlist already has a few hundred people. Join it below to be the first to know when new seats open:

Join the Waitlist

The idea

“You shall know a word by the company it keeps.” John Rupert Firth.

John Rupert Firth was a British linguist and one of the pioneers of modern semantics. In the 1950s, he proposed a simple yet revolutionary idea: “You shall know a word by the company it keeps.” More than half a century later, this insight became the foundation for a breakthrough at Google, where Tomas Mikolov and his colleagues developed the Word2Vec model in 2013.

Before Word2Vec, computers treated words as isolated symbols—mere sequences of characters with no inherent meaning or relationship to one another. Mikolov’s model changed that. By mapping words into a continuous vector space, where proximity reflected contextual similarity, Word2Vec allowed machines to grasp meaning through relationships. The model revealed that linguistic structure could be learned directly from data. This marked a paradigm shift in how machines process language—from counting words to understanding them—and laid the foundation for today’s large language models and the broader AI revolution built on distributed representations and deep learning. (Not coincidentally, Ilya Sutskever, a co-author of the original Word2Vec papers, later became the co-founder and Chief Scientist of OpenAI.)

This week, we will investigate the paper Asset Embeddings by Xavier Gabaix (Harvard), Ralph S. J. Koijen (Chicago Booth), Robert Richmond (NYU Stern), and Motohiro Yogo (Princeton), published in April 2025.

After that, we will implement an original strategy to demonstrate the power of the authors’ idea. It’s a true paradigm shift.

Here’s our plan:

1. First, we will quickly summarize the paper

2. Then, we will demonstrate the power of asset embeddings with simple experiments

3. Next, we will create a trading strategy that uses asset embeddings and show its strong results

4. Finally, we will wrap up with final thoughts and next steps

Before we start, two messages.

First, the 2nd cohort of the course is still open for enrollment for the next couple of days.

I created this to respond to a request I’ve received from hundreds of readers here: access to my codebase, step by step — for those who want to develop strategies using the same approach I share here. Enrollment also includes access to the private community and an ever-growing library of study groups. More information at the link below.

Join the Course and Community

Last day for enrollment in the 2nd cohort: Oct 14.

Second, this time it took me a bit longer than the usual 3–4 weeks to write and publish. There are a few reasons why:

1. First and foremost, I implemented three distinct papers, and their results came in significantly below what was reported. Instead of publishing subpar ideas, I prefer to keep searching for meaningful ones to share. That’s why I’ll never promise a rigid schedule — sometimes the search takes longer, but I’ll never compromise quality.

2. And, of course, just a lot of regular work :)

Welcome to all new subscribers — and rest assured: even if it sometimes takes a little longer than the usual 3–4 weeks to publish, the work continues non-stop.

Paper Summary

Core Idea

The paper introduces the concept of asset embeddings, inspired by advances in artificial intelligence such as Word2Vec and BERT. Instead of representing words by their co-occurrence in text, the authors represent assets by the portfolios that hold them.

Their central theoretical claim is that portfolio holdings contain all relevant information for asset pricing — far more than traditional firm characteristics (like book-to-market or profitability). In other words, the collective portfolio decisions of investors implicitly encode all the information used in the market to price assets.

Pause. Read that again. It is a bold paradigm shift. I loved it when I first read it a few months ago. I will repeat:

Portfolio holdings contain all relevant information for asset pricing — far more than traditional firm characteristics.

Methodology

The authors apply embedding models—recommender systems, Word2Vec-style shallow neural networks, and transformer architectures (BERT)—to financial data from 2005Q1 to 2022Q4.

Each firm and investor is represented as a point in a high-dimensional latent vector space, derived from institutional holdings data (13F filings, mutual funds, ETFs, hedge funds, etc.).

They also introduce investor embeddings, representing investors and their strategies in the same latent space.

Three empirical benchmarks test the usefulness of these embeddings:

1. Predicting relative valuations across firms.

2. Explaining stock return comovements.

3. Predicting institutional portfolio choices (the “masked portfolio” task).

Main Findings

• Asset embeddings strongly outperform firm characteristics in predicting valuation, return comovement, and investor behavior.

  • A simple 4-dimensional embedding explains over 50% of the variation in relative valuations, compared to just 15% for standard firm characteristics.

  • Larger embeddings (e.g., 128 dimensions) explain over 70% of variation.

• Transformer-based (BERT) embeddings achieve particularly high accuracy when predicting institutional portfolio decisions—showing that contextualized embeddings capture investor behavior far better than linear models or firm fundamentals.

• Text-based embeddings (from OpenAI and Cohere) perform worse, as they reflect semantic similarity rather than investor-perceived relationships.

Interpretability

To interpret these latent embeddings, the authors use large language models (LLMs) (e.g., GPT-4) to analyze earnings-call transcripts of firms that appear close in embedding space. This allows the generation of economic narratives about why investors group certain firms together—shared risks, growth prospects, ESG features, etc.

Applications

The paper proposes wide-ranging uses:

• Asset pricing and factor discovery: uncovering hidden dimensions of risk and return.

• Generative portfolios: building synthetic portfolios (e.g., “COVID exposure”) without explicit return data.

• Risk management and stress testing: simulating shocks by shifting embeddings.

• Investor analytics: classifying investor styles, detecting crowded trades, and evaluating performance.

Conclusion

The authors argue that portfolio holdings are the most information-rich data source in finance, analogous to text corpora in NLP.

Just as word embeddings revolutionized natural language processing by revealing structure in unstructured data, asset embeddings may transform finance by revealing the latent structure of markets.

This bridges modern AI representation learning and asset pricing—potentially redefining how we understand, forecast, and manage financial markets.

The Power of Asset Embeddings

Before we jump into a trading strategy, let’s first see how powerful asset embeddings really are—much like word embeddings in NLP.

The paper explores several models, but here we’ll focus on Word2Vec. Libraries such as Gensim make it straightforward to implement.

Before that, we need data. In this exercise, we’ll use 13F filings from the Nasdaq Sharadar SF3 dataset, though many other data vendors can provide similar information.

Our first step is to build a holdings matrix, with institutional investors as rows and assets as columns:

The extract_holdings method is a helper function that retrieves the holdings table described above, after applying the necessary filters. Institutional investors with less than $10 million in AUM are ignored, and stocks appearing in fewer than 20 portfolios are also excluded. The final result should look like this:

Now, let’s see how to train Word2Vec on institutional portfolios to generate asset embeddings based on co-holding relationships, as described in the paper:

The code builds portfolios from the holdings, trains a Word2Vec model where each “sentence” is an investor’s portfolio, and then extracts and normalizes the resulting embedding vectors for each ticker. The final DataFrame embeddings contains the L2-normalized representation of every stock in the latent vector space:

That’s all there is to it. You can create clusters with just a few additional lines using Scikit-learn’s KMeans module or any other clustering method. Now, let’s see how powerful these representations really are with a few examples.

First Demo: Semantic Similarity

Let’s start by computing the top 10 most similar stocks to NU, using cosine similarity on the asset embeddings we just created:

Even though the companies above come from completely different sectors, the cosine similarity in their embeddings reflects how institutional portfolios tend to group them together.

So, the similarity doesn’t come from fundamentals or business models, but from investor behavior.

Here’s a table summarizing what they have in common:

In short, these firms cluster not because they do the same thing, but because they are held by the same type of investors — global growth, EM tech, and crossover funds chasing the digitization theme.

And what’s really cool about it is that our model discovered this pattern from millions of 13F data points and just a few lines of code in a completely automatic process. Unsupervised machine learning at its best!

Second Demo: Vector Operations

If you ever find yourself in a computer science master’s or PhD program and decide to take an NLP class, I bet the first example you’ll see about Word2Vec will be the famous:

vector(”King”) − vector(”Man”) + vector(”Woman”) ≈ vector(”Queen”)

There’s a reason why they teach you this: that’s exactly the kind of demonstration that makes embeddings “click” for people. You see that the latent space encodes meaning, not just statistical co-occurrence.

Could we do the same for asset embeddings? You bet! Let’s try the following operation:

vector(”AMZN”) − vector(”WMT”) + vector(”BABA”) ≈ ?

Intuitively means:

“Take the concept of Amazon (a tech-driven online retailer), remove what makes it a traditional U.S. retailer (Walmart), and add what makes it Chinese (Alibaba).”

So we’d expect the resulting vector to point toward emerging-market e-commerce / digital consumer platforms — the BABA equivalents around the world. Here are the results:

This embedding arithmetic perfectly captured the latent theme:

“Tech-driven consumer platforms in emerging markets.”

We basically told the embedding:

“Find me the Amazon equivalent, but outside the U.S., in markets like China.”

And it delivered:

• China → PDD, JD, BABA, TCOM, YMM

• Southeast Asia → SE

• Latin America → MELI, NU

• India → IBN

That’s remarkably structured.

Even the slight outlier (CCJ) likely reflects shared institutional exposure through EM ETFs or crossover funds with a macro “growth + commodity exposure” tilt.

Alright… but how can we turn this into a profitable trading strategy? Let’s look at an example next.

A Strategy Example

Let’s review a strategy that demonstrates the power of this paradigm shift, using asset embeddings. The goal is to illustrate just one of many ways asset embeddings can be applied to build innovative systematic strategies.

I know some institutions may want to take this further and need help doing so. In that case, feel free to reach out at cs@quantitativo.com.

Important considerations

First, some important notes to keep in mind before devising a strategy:

1. Asset embeddings are learned from 13F data, which is released with a 45-day delay. For example:

   1. Embeddings from Q4’24 can be extracted and used from ~Feb 15 to May 15, 2025.

   2. Embeddings from Q1’25 can be extracted and used from ~May 15 to Aug 15, 2025.

   3. Embeddings from Q2’25 can be extracted and used from ~Aug 15 to Nov 15, 2025.

      And so on.

2. As such, any idea that aims to use asset embeddings to directly predict short-term price movements will likely not work, because we are always working with the previous quarter’s embeddings (which do change from quarter to quarter) and with a 45-day delay.

3. Nevertheless, any idea that profits from using asset embeddings to cluster assets and find inefficiencies within those clusters is more likely to work, because institutional portfolio structures and co-holding patterns tend to evolve slowly over time, making intra-cluster relationships relatively persistent.

4. The paper explores both linear (PCA) and nonlinear (Word2Vec, BERT, etc.) asset embeddings. I personally tested PCA and Word2Vec (which is a shallow neural network), and here’s what I found:

   1. Word2Vec yields superior results.

   2. However, this comes with a drawback: the coordinate system of the multidimensional vector space is not stable — it changes from quarter to quarter. For instance, imagine you trained Word2Vec models with 10 dimensions. Dimension 3 in Q1’25 is not directly comparable to Dimension 3 in Q2’25.

   3. We can use ML techniques to align or stabilize coordinate systems for linear models like PCA, but not for nonlinear embeddings.

   4. As such, this rules out any strategy that depends on inter-quarter relationships between nonlinear embeddings. For linear embeddings, however, that’s feasible.

Ok, Carlos… but what does all this mean in practice?

It means that ideas based on finding inefficiencies within groups are likely to work very well. You know all those strategies that use industries, sectors (or even pairs) to identify relative-value opportunities and trade them?

Those kinds of ideas.

Idea

We define a trading period as the ~90 days from the middle of one quarter to the middle of the next. For example:

• Feb 15 to May 15, 2025 → uses embeddings from Q4’24

• May 15 to Aug 15, 2025 → uses embeddings from Q1’25

At the beginning of each trading period:

• Use asset embeddings to cluster all assets into 50 groups.

• Select the top 10 groups based on predictive power (*).

• Each day, form long–short portfolios within each group using the volatility-adjusted residuals of daily returns relative to the return of the cluster (using quintiles).

• Rebalance daily.

(*) How are the top 10 groups selected? We assess predictive power by running an OLS regression where each group’s residuals serve as predictors of next-day returns, using the current clusters and the previous quarter’s returns. We then rank the groups by their t-statistics and select the top 10.

Friction Assumptions:

• Commissions / Costs: $0.0010 per share in trading costs.

• Shorts: Since the backtest starts in 2020, we have access to short-availability data. The model assumes that only stocks with sufficient borrow availability were shorted.

I think that’s good enough to illustrate the power of asset embeddings in a simple strategy. Let’s see the results:

The results are not bad at all:

• Annualized return is 18.5%, almost 2x the S&P 500;

• Sharpe Ratio is 2.59, nearly 4x higher than the benchmark’s 0.72;

• Volatility comes in at 6.6%, almost 1/3 of the S&P 500;

• Maximum drawdown is -5.3%, considerably smaller than the S&P 500’s -25.4%;

• Correlation to the S&P 500 is effectively zero, offering diversification benefits;

• The strategy also delivered 78% positive months, with a worst month of only -2.6% — versus the benchmark’s -9.3%.

If we had traded this strategy since 2021:

• We would have had only positive years;

• We would have seen 78% of the months positive, with the best at +7.5% (Apr’25);

• We would have seen 22% of the months negative, with the worst at -2.6% (Dec’23);

• The longest positive streak would have been 12 months, from Dec’22 to Nov’23;

• The longest negative streak would have been 3 months, from Jan’25 to Mar’25.

Looking at Fama-French 3-Factor risk model:

• Alpha: 0.0533% per day → annualized ≈ 14%, highly significant (t = 4.19, p < 0.001) — the strategy delivers strong abnormal returns not explained by standard risk factors.

• Market Beta (Mkt–RF): 0.041 → very low market exposure (t = 3.27, p = 0.001) — the strategy’s returns are largely independent from overall market movements.

• SMB (Size): 0.014, insignificant (t = 0.73, p = 0.46) — no systematic bias toward small or large caps.

• HML (Value): –0.009, insignificant (t = –0.58, p = 0.56) — no clear tilt between value and growth stocks.

• R² = 0.017 → only 1.7% of return variation explained — factor exposures are minimal; performance is almost entirely idiosyncratic.

This is what we want from a market-neutral strategy.

Final Thoughts

What we’ve seen here is not just another backtest or statistical curiosity — it’s a glimpse of the power of asset embeddings.

By reframing how we represent assets—not through accounting ratios or static fundamentals, but through the collective behavior of investors—we move closer to a model of markets that actually mirrors reality: adaptive, relational, and dynamic.

In essence, asset embeddings turn holdings into language. Each portfolio is a “sentence,” each stock a “word,” and meaning emerges from co-occurrence. From that simple shift, an entire new toolkit becomes available: clustering, similarity search, latent factor extraction, and—eventually—prediction.

Our strategy example only scratches the surface. It shows that embeddings can generate robust, market-neutral alpha, even with basic clustering and residual sorting. But the real potential lies beyond this first layer:

• Dynamic embeddings: tracking how relationships between assets evolve over time;

• Cross-domain embeddings: combining holdings, news, and price data into shared representations;

• Generative portfolios: constructing exposure to abstract concepts (“AI boom,” “energy transition”) directly from latent space.

Finance has always been about representation — how we compress the complexity of the world into variables we can reason about. Moving from firm characteristics to embeddings doesn’t replace traditional approaches, but adds a new layer of understanding — one that captures relationships, context, and behavior directly from the data. It’s an early step toward a richer, more relational way of modeling markets. And we’re just getting started.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The 2nd cohort of the course is still open for enrollment for the next couple of days.

I created this to respond to a request I’ve received from hundreds of readers here: access to my codebase, step by step — for those who want to develop strategies using the same approach I share here. Enrollment also includes access to the private community and an ever-growing library of study groups. More information at the link below.

Join the Course and Community

Last day for enrollment in the 2nd cohort: Oct 14.

The idea

"Everything should be made as simple as possible, but no simpler." Albert Einstein.

Albert Einstein had a way of capturing deep truths in simple words. His quote is a reminder, especially relevant to us when building models. Stripping away unnecessary complexity is vital, but going too far risks oversimplification: a model that looks neat but fails to capture reality.

This week, we will implement the idea from the paper Volume Shocks and Overnight Returns, by Álvaro Cartea, Mihai Cucuringu, Qi Jin, and Mungo Ivor Wilson (2025, all from Oxford).

(By the way, sorry for taking so long to publish this piece… I’m in Bali for a 3-week blockchain conference for hardcore protocol engineers. I thought it’d be a good way to finally start dipping into crypto properly… It’s been a steep but rewarding learning curve… More on that soon!)

In practice, we’ll adapt the paper’s core insight rather than copy it wholesale. On one hand, we’ll simplify a key element of their setup that we believe can be expressed more cleanly without losing its essence. On the other, we’ll add back an important piece they left out — trading frictions and costs — because ignoring these can turn an elegant academic signal into an unrealistic strategy. The goal is to strike the right balance: simple enough to be practical, but rich enough to hold up in the real world. Or, as Einstein might put it if he were a quant: make it simple, just not “fantasy-backtest simple.”

Here's the plan:

1. First, we will quickly summarize the paper

2. Then, we will run the naive, simplified implementation

3. Next, we will model friction & trading costs and compare

4. Finally, after breaking the strategy, we will make it work again

5. As usual, we will wrap up with final thoughts and next steps

Course and Community: enrollments Open

As many of you requested, the 2nd cohort of the course is now live and open for enrollment.

It walks through my codebase step by step and is designed for readers who want to develop quant strategies using the same approach I’ve shared here.

You’ll find all the details—content, structure, pricing, and FAQs—at the link below. If you have any questions, feel free to reach out.

Join the Course and Community

Enrollment also includes access to the private community and an ever-growing library of study groups. The code and full recording of the latest session (“Execution details of Intraday Momentum for ES & NQ”) are already available.

Paper summary

Core Idea

The paper shows that unexpected spikes in trading volume (“volume shocks”) during the day predict positive returns overnight — but not during the next trading session.

• Stocks with large volume shocks earn significantly higher close-to-open returns.

• This relationship is robust across firm sizes and persists after controlling for standard risk factors (market, size, momentum, reversal).

• Importantly, intraday returns (open-to-close) show no such effect, highlighting a unique overnight premium linked to volume shocks.

Why Does This Happen?

The usual explanation — that high volume reflects investor attention, lowers cost of capital, and should boost returns — doesn’t fit. If it did, the effect would show up during the next trading day, not just overnight.

Instead, the authors argue:

• Market frictions and constraints (liquidity, noise trading, institutional execution) delay full incorporation of information until the next morning.

• Information flow after hours (e.g., earnings, news releases, sentiment shifts) interacts with the signal from volume shocks, amplifying the overnight effect.

• Overnight risk premia: investors may demand compensation for holding positions across the less liquid, higher-risk non-trading period.

How to Exploit It

The trading challenge is that true volume shocks are only known after the close auction. The authors tackle this by predicting volume shocks during the day using linear and machine learning models (LightGBM, TabNet, etc.):

• Predicted volume shocks can be used to build long-only overnight portfolios.

• Even simple models capture ~90% of the “oracle” (perfect foresight) strategy’s Sharpe ratio.

• Best ML models (TabNet) achieve ~97% of the oracle’s performance, with Sharpe ratios around 1.1 and annualized returns near 18%.

• Crucially, trading only the high-volume side avoids liquidity traps, since low-volume stocks are harder to execute.

Naive simplified version

Instead of building complex ML models to forecast volume shocks, we’ll start with a stripped-down version that’s simple and actionable. The setup:

• Signal timing: use trading volume information up to 3:45 pm, so trades can be placed with MOC orders (data from Polygon.io)

• Portfolio construction: begin with a naive equal-weight implementation

• Universe: run the strategy across the Russell 3000 stocks (data from Norgate)

• Positioning: keep it long-only, consistent with the paper

• Sorting: classify volume shocks into 10 quantiles (instead of 5) and trade only the top decile

We will start with zero cost or friction. This is obviously unrealistic. But that's just to see what an unrealistic test looks like. Let's check the results:

The backtest looks stellar, but only because it relies on two unrealistic assumptions: no trading costs and the ability to buy unlimited size in any stock, regardless of liquidity. Trading costs are not negligible here (especially for a strategy that rebalances every single day), and market impact in less liquid names would quickly eat into performance. The result is an equity curve that looks great on paper but overstates what a tradable strategy could actually deliver.

And that’s not unusual. Academic papers often skip over these implementation details (costs, liquidity, capacity) because their focus is on documenting the signal itself. It’s our job as practitioners to stress-test those assumptions, adjust for reality, and see whether the idea survives outside the lab.

Modeling friction and trading costs

There are many different ways to model friction. As it was famously said:

“All models are wrong, but some are useful.” George E. P. Box.

To bring the strategy closer to reality, we’ll model two critical frictions using the following rules:

Commissions

We will implement Interactive Brokers’ tiered model:

• Monthly volume shares are computed using the unadjusted closed prices of all names traded

• We consider +$0.0020 in third-party fees (approx. 2x the values in every tier)

• We will consider these dollar values fixed throughout the simulation (which is a conservative estimate, as these numbers were lower 20 years ago)

Liquidity constraints

If the required order size exceeds 10% of a stock’s daily volume, we assume the trade cannot be executed and skip it entirely.

Together, these adjustments transform the “too good to be true” backtest into something that better reflects what can actually be achieved.

Let's see how they impact the strategy:

With trading costs and liquidity frictions modeled, the picture becomes more realistic. The strategy still outperforms the S&P 500 benchmark in terms of annualized return and Sharpe ratio, proving that the overnight volume-shock effect is economically meaningful. But the performance is nowhere near the “frictionless” backtest, exactly as expected. Once we factor in commissions and the inability to size up in illiquid names, a good chunk of the edge gets eaten away, leaving behind a strategy that doesn't meet our standards to go into production.

What can we do to fix that?

Changing the universe

At this point, many people would throw in the towel. They’d look at the results, dismiss the strategy as untradable, maybe even blame the academics for skipping frictions, and then move on to the next shiny idea. And that’s fair — everyone can do as they please.

But as a fan of Janis Joplin, I prefer to “try, just a little bit harder.”

This strategy is all about overnight returns — and if you’ve spent any time watching the markets, you know there’s a special corner of it that lives and breathes overnight moves: biotech stocks. They’re notorious for sharp gaps up or down, often tied to drug trial results, FDA decisions, or corporate announcements that tend to hit after hours. Big overnight moves, weak intraday drifts — it’s almost textbook for our setup.

Thanks to Norgate data, we have survivorship-bias-free constituents of the Nasdaq Biotechnology Index going back decades. So let’s narrow our focus to that universe and see whether the strategy becomes not just academically interesting, but practically powerful.

The results are striking:

• Annualized return is 36.1%, more than four times the S&P 500’s 8.4%;

• Sharpe Ratio is 1.52, nearly three times higher than the benchmark’s 0.52, showing much stronger risk-adjusted performance;

• Volatility comes in at 21.7%, slightly higher than the S&P 500’s 19.0%, but well compensated by the excess return;

• Maximum drawdown is -36.6%, considerably smaller than the S&P 500’s -56.8% (but still high for most investors);

• Correlation to the S&P 500 is effectively zero, offering diversification benefits;

• The strategy also delivered 68% positive months, with a best month of +51.9% — versus the benchmark’s +12.7%.

This is the distribution of monthly returns. Here are some interesting highlights after computing the key descriptive statistics measures:

• Mean (2.8% per month): That compounds to ~36% annualized, in line with the backtest table. Strong absolute performance.

• Std (6.8%): Moderate monthly volatility.

• Min (-14.4%) vs. Max (+51.9%): Extreme dispersion. While the downside is capped at about -14%, the upside tails are very fat (a +52% month!), likely driven by biotech gap-ups.

• Median (1.9%) vs. Mean (2.8%): Mean > Median indicates the distribution is positively skewed — the occasional explosive winners pull the average up.

• Skew (2.1): Strongly right-skewed. This is exactly what you’d hope for in a strategy: more frequent modest gains, with rare huge wins.

• Kurtosis (11.6): Very heavy tails compared to a normal distribution (which has kurtosis of 3). This means more extreme outliers — both positive and negative — though the skew suggests the extreme positives dominate.

Final thoughts

As it stands, the strategy is powerful but not for the faint of heart. Volatility runs higher than the benchmark, and while that risk is handsomely rewarded (a 1.5 Sharpe ratio is good), most investors don’t have the stomach for drawdowns north of 15–20%.

How to fix that? The naive solution is simple: don’t deploy all your capital here. Allocate a smaller slice and let the high Sharpe ratio do the compounding work. A second approach is to treat it as a component within a larger portfolio of strategies — its near-zero correlation to the market makes it an attractive diversifier. And of course, we could explore hedging overlays (like the beta hedging framework we discussed in earlier articles) to dampen volatility without giving up much alpha.

And this is only the start. Now that we’ve established a strong baseline, there’s plenty of room to push further. We could train models to predict the sign of the overnight return, using volume shock as a feature but expanding the set with price action, volatility, or news-driven variables. Both linear and non-linear methods could be tested, from simple regressions to richer ML frameworks. In short, the canvas is wide open — there’s much more we can do.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback. And if you’re looking to bring a strategy like this into production, let me know — especially if you’re running at scale, where low execution costs can make all the difference.

Cheers!

Course and Community: enrollments Open

As many of you requested, the 2nd cohort of the course is now live and open for enrollment.

It walks through my codebase step by step and is designed for readers who want to develop quant strategies using the same approach I’ve shared here.

You’ll find all the details—content, structure, pricing, and FAQs—at the link below. If you have any questions, feel free to reach out.

Join the Course and Community

Enrollment also includes access to the private community and an ever-growing library of study groups. The code and full recording of the latest session (“Execution details of Intraday Momentum for ES & NQ”) are already available.

The idea

“Basketball is a game of adjustments.” Bob Knight.

Bob Knight was the last coach to lead an NCAA team to a perfect season: 32 wins, zero losses. That record still stands nearly half a century later.

His secret? He was a masterful tactician. Obsessed with preparation, relentless on fundamentals, and unmatched in making in-game adjustments. Knight believed basketball wasn’t about memorizing plays, but about reading the moment and adapting faster than the opponent. This philosophy made him one of the greatest coaches of all time.

This week, we will implement the idea from the paper Defense First: A Multi-Asset Tactical Model for Adaptive Downside Protection, by Thomas Carlson (2025). (You can find more about Tom here: .)

Tactical Asset Allocation (TAA) takes the same principle of adaptability and applies it to investing. While Strategic Asset Allocation sets a long-term baseline mix (e.g., 60% stocks, 40% bonds), TAA introduces short- to medium-term adjustments based on market conditions. The goal? Exploit opportunities or mitigate risk by shifting weights dynamically, adding offense in favorable environments and defense when risks rise.

Lately, I’ve leaned toward more advanced ideas, which might explain why so many readers are now coming from places like Morgan Stanley, Goldman Sachs, Deutsche Bank, Citigroup, Millennium Management, Point72, Wellington Management, Man Group, Millburn, Trexquant, IMC Trading… to name a few. (As well as students from schools like MIT, Columbia, Cornell, UPenn/Wharton, Carnegie Mellon, Duke, NYU, Berkeley, Georgia Tech, Imperial College London, etc.)

That said, I often get requests to cover simple ideas. This week’s idea is exactly that… so simple that you can code it in under 40 lines.

Here's our plan:

1. First, we will summarize the paper

2. Then, we will discuss why it works and verify the edge quantitatively

3. Next, we will improve the original idea

4. Finally, we will discuss next steps

Let's get to it.

Study Group

Our next session will be on Saturday, August 9th at 11 AM ET (33% of the votes) and will focus on execution details from the Intraday Momentum for ES & NQ article (57% of the votes). Connection details will be available at connect.quantitativo.com.

We’re building a private community for systematic traders. A place to explore ideas, exchange insights, and tackle the real technical and strategic challenges of building robust trading systems… from signal research to execution and risk.

Enrollment reopens in August. Join the waitlist below to be the first to know when new seats open:

Join the Waitlist

Paper Summary

Traditional “safe haven” strategies often fail during crises because correlations rise and static allocations break down. Defense First introduces a rules-based Tactical Asset Allocation model that dynamically rotates among four defensive assets—long-term U.S. Treasuries (TLT), gold (GLD), commodities (DBC), and the U.S. dollar (UUP)—with U.S. equities (SPY) as a fallback when defenses weaken. The objective: adaptive downside protection with positive long-term returns.

How It Works

Each month, the strategy ranks four defensive assets based on multi-timeframe momentum (1, 3, 6, 12 months):

• TLT (Long-term Treasuries) — deflation hedge

• GLD (Gold) — monetary instability hedge

• DBC (Commodities) — inflation hedge

• UUP (USD Index) — liquidity crisis hedge

Allocations are 40/30/20/10 across the ranked assets if they outperform cash (T-bills). Assets that fail this screen are replaced with SPY (U.S. equities) as a fallback (not for growth, but as the “least weak” option).

It significantly reduced drawdowns during every major crash (1987, GFC, COVID, 2022), and generated 65% profitable months across decades.

Why It Works

• Multi-Asset Defense: Each asset hedges a different macro risk (deflation, inflation, currency stress, monetary instability).

• Momentum Edge: Builds on well-documented trend persistence across asset classes.

• Adaptive Risk Control: Absolute momentum filter avoids weak assets; equity fallback prevents idle cash.

Practical Perks

• All instruments are liquid ETFs.

• Simple monthly rebalancing.

• No reliance on exotic data or proprietary tools.

• Easily scalable for institutional or retail portfolios.

The Edge

Before running full backtests, let’s first examine whether the signal has a real edge. There are two ways to do this.

First, we can observe the edge by looking at the average next-month return of assets ranked by their momentum score. Each month, we rank the four defensive assets from 1 (highest momentum) to 4 (lowest momentum), then compute the average return for the following month.

When we average this across 34 years of data, the relationship is clear:

• Rank 4 (weakest momentum) delivers the lowest next-month return,

• Rank 1 (strongest momentum) delivers the highest,

  and the increase is monotonic from left to right.

This means momentum isn’t just a theory here: it shows up in the numbers.

The second way is through a Fama-MacBeth regression, a standard approach in asset pricing to test if a characteristic predicts future returns across assets and time.

Here’s how it works:

• Each month, we run a cross-sectional regression of next month’s return on the momentum score for the four defensive assets.

• We then average the slopes across all months and compute robust standard errors (Newey-West) to account for autocorrelation.

The result?

This means that a one-unit increase in the momentum score is associated with an 8.5% higher next-month return on average. Statistically significant at the 1% level, this confirms that the momentum ranking carries real predictive power.

Experiments

Wherever possible, ETF data was used. For periods prior to ETF inception, we filled the gaps with Vanguard mutual funds, asset class indices, or futures proxies, mirroring the methodology in the paper. This approach provides a consistent return history from early 1990s through 2025.

Here’s what the backtest looks like when we implement the strategy exactly as described in the paper:

The results are not bad:

• Annualized return is 10.4%, very close to the S&P 500’s 10.6%, but with much lower risk;

• Sharpe Ratio is 0.95, 50% higher than the benchmark’s 0.64, signaling far better risk-adjusted performance;

• Volatility is just 11.1% vs. 18.5% for the benchmark, a significant reduction;

• Maximum drawdown is only 19.4%, compared to a crushing 55.2% for the S&P 500;

• Correlation to equities is modest at 0.27, adding diversification benefits;

The results are also pretty close to what was presented in the paper.

And here are the key takeaways from the Fama-French 3-Factor regression:

• Alpha: 0.0262% per day → annualized ≈ 6.8%, statistically significant (t = 3.58, p < 0.001), confirming strong abnormal returns not explained by Fama-French factors;

• Market Beta (Mkt-RF): 0.16 → very low equity exposure, yet strong absolute returns;

• SMB (Size): -0.006, not significant → no clear size tilt;

• HML (Value): -0.039, significant negative → slight tilt toward growth;

• R²: 0.072 → factors explain only ~7% of variance, most return comes from strategy itself.

So, the system works: we see a good downside protection.

Now, what can we do to improve the system and go from defense to offense?

Adding leverage

The first idea is simple: add leverage… but not across the board. We’ll apply 3x leverage only to the equity portion by using the UPRO ETF. Why 3x? Because it brings the portfolio’s risk closer to that of the benchmark. See:

In summary:

• Annualized return jumps to 16.2%, a big increase from 10.4% in the unleveraged strategy, and far ahead of the S&P 500’s 10.6%;

• Sharpe Ratio remains good at 0.97, slightly above 0.95 from before, and still much better than the S&P 500’s 0.64;

• Volatility rises to 17.0%, up from 11.1%, but still slightly below the S&P 500’s 18.5%;

• Max drawdown increases to -33.9%, compared to -19.4% before, but it’s still significantly better than the S&P 500’s -55.2%;

• Average max drawdown remains controlled at -12.6% vs. -13.8% for the S&P 500;

• Correlation with the S&P 500 rises to 0.58, which is expected since we’ve leveraged the equity sleeve;

With roughly the same level of volatility as the S&P 500, the strategy now delivers significantly higher returns (16.2% vs. 10.6%) while maintaining a superior Sharpe ratio and much smaller drawdowns.

What else can we do to improve?

Portfolio of Strategies

The idea here is simple: add this strategy to a broader portfolio. Which portfolio? For example: 40% this strategy, 40% the rebalancing strategy we reviewed in the previous article, and 20% our classic mean-reversion portfolio.

This is what we get:

Those results are not bad at all:

• Annualized return climbs to 16.8%, slightly higher than the leveraged version (16.2%) and double the S&P 500’s 8.3%;

• Sharpe Ratio jumps to 1.43, a huge improvement over 0.97 in the leveraged strategy and nearly 3x better than the S&P 500’s 0.51;

• Volatility drops to 11.3%, far below both the leveraged version (17.0%) and the S&P 500 (19.0%), showing the power of diversification;

• Max drawdown improves dramatically to -14.7%, from -33.9% with leverage and -56.8% for the S&P 500;

• Average max drawdown shrinks to -7.0%, about half the S&P 500’s -14.6%;

• % of positive months rises to 73%, higher than both previous setups (65%) and the benchmark (65%);

• Best and worst month performance also improves, with the worst month limited to -6.4% vs. -16.9% for the S&P 500.

Here are the key takeaways from this Fama-French 3-Factor regression:

• Alpha: 0.0419% per day → annualized ≈ 11.1%, highly significant (t = 5.52, p < 0.001), confirming strong abnormal returns unexplained by the factors;

• Market Beta (Mkt-RF): 0.37 → moderate equity exposure, much higher than the previous strategy’s 0.16 but still well below 1, meaning returns are not just market-driven;

• SMB (Size): -0.078, significant negative → tilt toward large-cap assets;

• HML (Value): -0.108, strongly significant negative → strong preference for growth over value;

• R²: 0.386 → about 39% of variance explained by the model, leaving most performance driven by the strategy’s design.

Final Thoughts

What started as a defensive allocation model turned out to be a remarkably flexible building block. Out of the box, it offered equity-like returns with bond-like drawdowns, proving that simple rules (momentum screens, absolute thresholds, and macro diversification) can go a long way.

Then we pushed it further. By adding selective leverage, we matched the benchmark’s volatility and delivered a massive return boost without losing control of risk. Finally, by blending this strategy into a portfolio with others, we unlocked more alpha: a 1.43 Sharpe ratio, lower drawdowns, and over 70% positive months.

Does this mean the work is done? Not at all. There’s plenty of room to explore:

• Smarter leverage: dynamic sizing based on volatility or market regimes;

• Expanded asset set: more macro hedges, or even crypto for an uncorrelated sleeve;

• Adaptive weights: machine-learning or Bayesian methods for ranking confidence.

The takeaway? Defense First might be more than a crash-protection tool: a foundation for building robust, adaptive portfolios that thrive across regimes.

As always, I’d love to hear your thoughts. Feel free to reach out via Twitter or email if you have questions, ideas, or feedback.

Cheers!

The first cohort of the course was a great success. Thank you to everyone who joined! Enrollment is now closed. The next cohort opens in August, with 50 more seats. Course participants also get exclusive access to our Community and Study Group. Join the waitlist below to be notified when enrollment reopens:

Join the Waitlist

(I know, I need to update this landing page… as soon as I find time I will add more details about the course, the community, reviews from the 50 first users, etc :))

The idea

“I picked up one or two pieces and examined them attentively... I then collected four or five pieces and went to Mr. Scott... I said, ‘I believe this is gold.’” James W. Marshall.

He found gold… and died broke.

James W. Marshall unintentionally sparked one of the greatest migrations in American history. In 1848, while building a sawmill on the American River, he noticed a few glimmers in the water, triggering the California Gold Rush and forever altering the American West.

Back then, finding gold was easy. With little more than basic tools, everyday people could strike it rich. Some did (Marshall wasn’t one of them). But as more prospectors arrived, the easy gold disappeared. The shallow deposits anyone could reach with a pan were quickly depleted.

Still, despite the surge in miners and competition, gold production didn’t stop… it soared. Why? Technological advances, global exploration, and industrialization.

I find the analogy between gold mining and alpha research fascinating. Like gold, early alpha could be extracted using simple models. But as more participants entered the market, the obvious opportunities vanished. Today, finding alpha with a basic model in a crowded market is as unlikely as finding gold in the American River with a tin pan. If something glitters, it's probably pyrite… fool’s gold.

Yet alpha production has never been higher. Why? Again: technology, globalization, and industrial scale. The biggest quant firms and hedge funds have made more money in the past decade than ever before.

This week, we’ll implement the paper “The Unintended Consequences of Rebalancing” by Campbell Harvey, Michele Mazzoleni, and Alessandro Melone.

The paper explores how institutional investors are forced to rebalance frequently, and how these predictable flows create arbitrage opportunities for those willing to front-run them. The strategy is surprisingly simple.

In fact, the model is so simple it made me picture someone striking gold today with nothing but a shovel. I know, I'm very skeptical. Possible? Maybe. Likely? Well, read on and decide for yourself.

Here’s the plan:

1. We’ll begin with a quick summary of the paper

2. Then, we’ll show the results from replicating their approach

3. Next, we’ll combine this strategy with a base portfolio of mean-reversion strategies to highlight diversification benefits

4. Finally, we’ll discuss next steps

If you know basic Python and Pandas, you can replicate this paper in a few hours. It’s a great warm-up if you're thinking about joining the second cohort of the course, which opens again in August. We've had lots of interest lately… more details soon.

We’re building a private community for systematic traders. A place to explore ideas, exchange insights, and tackle the real technical and strategic challenges of building robust trading systems… from signal research to execution and risk.

Enrollment reopens in August. Join the waitlist below to be the first to know when new seats open:

Join The Waitlist

Paper Summary

“The Unintended Consequences of Rebalancing” explores how mechanical portfolio rebalancing by institutional investors, often done at predictable intervals, creates systematic, exploitable price distortions in equity and bond markets. The paper shows that these flows not only move prices but also impose hidden costs on long-term investors while offering short-term profit opportunities to arbitrageurs.

Core Idea

Institutional investors (like pension and mutual funds) regularly rebalance portfolios, selling stocks and buying bonds when equities outperform, and vice versa. This activity, often driven by either Calendar rules (e.g., monthly rebalancing) or Threshold triggers (e.g., when allocations drift by a set %), creates predictable and recurring trading patterns.

The authors construct two signals:

• Threshold Signal: Measures when equity/bond weights deviate from targets

• Calendar Signal: Captures end-of-month rebalancing pressures

Both signals significantly predict next-day cross-asset returns (S&P 500 vs. 10-year Treasuries). For example, a one standard deviation increase in these signals (indicating equities are overweight) predicts a ~17 bps decline in equities the next day.

Why Does This Happen?

Large funds must rebalance due to internal mandates, even when doing so imposes costs. These trades are:

• Mechanical, not information-driven

• Predictable, especially near month- and quarter-end

• Unavoidable, due to structural portfolio rules

This creates price pressure, which reverts within ~2 weeks, suggesting temporary inefficiencies.

The cost? Estimated at ~8 basis points per year—which adds up to $16 billion annually, or roughly $200 per U.S. household. This is more than the annual fee of passive investing.

How to Exploit It

The authors build a simple long-short strategy that front-runs rebalancers:

• If signals show equities are overweight → short S&P futures, long bonds

• If signals show equities are underweight → long S&P, short bonds

The strategy achieves a Sharpe ratio > 1 over 1997–2023. Returns are statistically significant and robust to controls for momentum, macro conditions, sentiment, and volatility.

This is one of the rare papers that documents a market-wide return anomaly caused not by alpha-seeking traders, but by the systematic behavior of the institutions themselves.

Replicating the paper

Both the Threshold Signal and the Calendar Signal are derived from the weights of a balanced equity/bond portfolio (60/40).

Over time, returns from the S&P 500 and 10-year U.S. Treasury note futures cause these weights to drift away from the target 60/40 allocation.

• In the Threshold Signal, the portfolio is rebalanced back to 60/40 whenever the drift exceeds a specified threshold (δ).

• In the Calendar Signal, the portfolio is rebalanced back to 60/40 on the last business day of each month, regardless of the drift.

Both signals are computed iteratively, tracking how portfolio weights evolve and trigger rebalancing over time:

Here, w^C_t denotes the weights for the Calendar Signal, while w^T_t denotes the weights for the Threshold Signal. The formula that updates the weights as a function of previous weights and returns is:

Coding these signals is fairly straightforward. Two questions remain:

• What threshold values (δ) should we use for the Threshold Signal?

• How many days before month-end does the Calendar effect typically materialize?

Both questions are addressed through predictive regressions using different values of δ (for the Threshold Signal) and N (the number of days before month-end for the Calendar Signal):

The calibration results guide two key modeling choices. For the Calendar Signal, we observe that the predictive power (t-stat) becomes statistically significant about 5 days before month-end, justifying the use of N = 5. For the Threshold Signal, predictability weakens beyond δ = 2.5%, so we average signals across all δ values from 0% to 2.5% in 0.1% increments to capture a representative rebalancer behavior without overfitting to a single threshold.

The strategy

The trading strategy takes a position in a S&P 500 futures contract and an opposite position in a 10-year Treasury note futures contract as follows:

where the portfolio weight $w^{\text{Strategy}}_t$ is defined as the average of modified versions of the Threshold and Calendar signals. The modifications primarily involve rescaling, inverting the signal, and averaging, among other adjustments. See the paper for full details.

Running that leads to an equity curve pretty similar to what was reported:

The Sharpe ratio we obtained was 0.94, which is close enough to the value reported in the paper. Tweaking the rescaling parameters and other small details might improve the results slightly. But that's not what we want to do next.

Adding the strategy to a portfolio

Now let’s assess the benefits of adding this strategy to a broader portfolio.

We'll use as a baseline the mean reversion strategy portfolio we've been trading (based on ideas we've shared here) for nearly a year.

Below are the baseline results before adding the new strategy:

As we've mentioned in previous articles, this is a strong portfolio, delivering more than twice the risk-adjusted returns of the S&P 500.

Now, let’s see how performance improves when we blend it 50/50 with the Front-Running strategy we just replicated:

Here are some key highlights:

• By adding the strategy to our portfolio, we increased the Sharpe ratio from 1.08 to 1.30

• Annual volatility dropped from 19% to 14%

• Maximum drawdown improved from 24% to 15%

• The average annual maximum drawdown fell from 13% to 8%

• The percentage of positive months rose from 66% to 71%

We gave up 2.5% in annual returns (from 20.6% to 18.1%) in exchange for significantly better downside protection and lower volatility.

Seems like a good trade-off. ;)

Final Thoughts

What started as a simple idea, front-running mechanical rebalancing flows, turned out to be surprisingly effective. The strategy alone delivers solid performance. But when paired with our mean reversion portfolio, it adds something even more valuable: resilience. Sharpe goes up, volatility drops, and drawdowns become shallower and shorter.

This is exactly the kind of research we like: economically intuitive, data-driven, simple to implement, and additive in a portfolio.

There’s still more to mine here. This idea could be extended across asset classes, refined with smarter execution, or combined with other signals. But even in its basic form, it looks like a real nugget of gold.

We’ll keep digging, both in future posts and in more depth inside the private community when Cohort 2 opens this August.

As always, I’d love to hear your thoughts. Questions, comments, feedback—just reach out via Twitter or email.

Cheers!

The first cohort of the course was a great success. Thank you to everyone who joined!Enrollment is now closed. The next cohort opens in August, with 50 more seats.Course participants also get exclusive access to our Community and Study Group.Join the waitlist below to be notified when enrollment reopens:

Join The Waitlist

(I know, I need to update this landing page… as soon as I find time I will add more details about the course, the community, reviews from the 50 first users, etc :))

The idea

“You're nothing but a pack of cards!” Alice, standing up to the Queen of Hearts.

I used to think Alice in Wonderland was just a goofy kids’ story.

After re-reading it as an adult, I actually think it’s brilliant.

It’s satire dressed up as nonsense. A dream disguised as a joke. A mirror hiding in plain sight.

The White Rabbit? That’s obsession and urgency. The Queen of Hearts? Power and chaos. Every character says something about the world, and about us.

That’s why I’m kicking off this week’s piece with Lewis Carroll’s trip down the rabbit hole. Because two of those symbols tie directly into what you’re about to read. You will see in a moment.

This week, I decided to implement and share my code for the paper “The Derivative Payoff Bias” by Guido Baltussen, Julian Terstegge, and Paul Whelan.

To be honest, I almost skipped it. A few smart people (people I respect) said the strategy stopped working after the paper came out. One trader even posted a backtest showing the edge had vanished. I thought, Maybe he’s right. Who am I to question it?

But then I remembered something: even the sharpest minds can be wrong. Maybe his implementation was just a pack of cards. So I dug in and ran it myself.

In fact, I’m sharing my code so you can run it too. If I made a mistake, someone will call it out. One of the perks of working in public.

Following the White Rabbit is one of the most powerful ideas in Alice in Wonderland. This means chasing curiosity down uncomfortable paths. If you're reading this, I know that instinct lives in you, too.

So consider this your invitation to follow the rabbit.

Now, let's be specific. Our curiosity demands answers to these simple questions:

Is this paper legit?
Did this anomaly stop working after publication?
Should we trade this anomaly?

Here’s the path down the hole:

1. First, we’ll summarize the paper

2. Then, we’ll replicate the chart that reveals the anomaly

3. Next, we’ll statistically compare how the market evolved after publication

4. Then, we’ll implement the strategy and share the results

5. Finally, we’ll discuss possible next steps

Let’s get to it.

We’re building a private community for systematic traders. A place to explore ideas, exchange insights, and tackle the real technical and strategic challenges of building robust trading systems… from signal research to execution and risk.

Enrollment reopens in August. Join the waitlist below to be the first to know when new seats open:

Join the Waitlist

Paper Summary

“The Derivative Payoff Bias” uncovers a persistent and predictable price distortion in U.S. equity index derivatives. It shows that the way certain derivatives are settled, using the opening price on the 3rd Friday of each month, creates a systematic bias that favors some traders and penalizes others.

Core Idea

• Many index options and futures on the S&P 500 expire using the Special Opening Quotation (SOQ), based on the opening prices of the index components on the 3rd Friday of each month.

• Since around 2003, the SOQ tends to be systematically higher than the prior day’s close, leading to inflated payoffs for certain derivative holders.

• This effect, called the “Third Friday Price Spike” (3FPS), shows prices drifting up overnight into the SOQ window, then reversing intraday.

• The result:

  • Call buyers win (higher-than-expected payoffs)

  • Put buyers lose (lower-than-expected payoffs)

• The estimated annual wealth transfer is ~$3.5 billion in favor of call buyers and against put buyers and call writers.

Why Does This Happen?

• The effect is not explained by news, earnings, macro events, or pinning to strike prices.

• The most plausible cause is dealer hedging behavior, specifically driven by a Greek called Charm (the rate of change of delta over time).

• As expiration approaches, dealers adjust their hedges based on Charm, which creates buying pressure overnight and pushes prices up into the SOQ.

• This demand imbalance is strongest on Triple Witching days and off-quarterly expirations, when hedging activity is heaviest.

• Evidence shows that the more negative dealers’ Charm, the more positive the price drift overnight.

How to Exploit It

• The pattern is highly consistent and tradable:

  • Go long at Thursday close

  • Reverse to short at Friday open

  • Close the short before Friday noon

• This creates a tent-shaped return pattern that captures the overnight drift and avoids the intraday reversal.

• The strategy produces statistically significant returns, even after transaction costs.

• The effect is isolated to a.m.-settled contracts — p.m.-settled options show no such bias.

• Traders can monitor net Charm positioning from market maker data to anticipate when the price spike is strongest.

Down the Rabbit Hole

I love insightful charts. They often say more than words ever could.

In fact, I believe that truly powerful ideas can often be captured in a single, well-crafted chart. In the case of the paper we’re exploring, that chart is the very first one on page 2:

This chart shows how S&P 500 E-mini futures behave overnight. On regular days (red line), prices move slightly. But on 3rd Fridays (black line), prices spike up overnight, and drop sharply right after the market opens at 9:30 a.m. (blue line). The pattern is clear, consistent, and striking.

I thought: what an elegant, simple, and powerful way to explain the paper’s core idea.

Can I replicate it? I wondered. That chart was my White Rabbit — and down the hole I went, chasing the anomaly. :)

Data

To replicate the chart above, we need OHLC minute-level data.

Databento is a cost-effective source. Downloading NQ and ES price data (first and second continuous contracts since 2010) costs less than $50.

But Carlos, why bother getting data with such granularity? Doesn’t the paper use open and close prices to compute the strategy’s performance? For that, isn’t end-of-day data enough?

Ah… First, it’s impossible to replicate the chart above using daily data.

But more importantly, what if trading at 9:20 a.m. turns out to be more profitable than trading at 9:30?

I can imagine less curious people settling for end-of-day data, skipping the effort of replicating the chart, and just running the strategy on low-granularity inputs. I can also imagine that path leading to inferior results.

But we’re down the rabbit hole now. And that usually means going after high-quality data. Doing things most people aren’t willing to do.

As they say, garbage in, garbage out.

This code loads and processes 1-minute OHLC data for the first and second E-mini S&P 500 futures contracts from Zstandard-compressed files stored locally. It reads the data using Databento’s Python API, concatenates it into a single DataFrame, and sorts it by symbol and timestamp.

Next, we need to decide when to use the first- or second-month contract.

We’ll use the first-month contract whenever there are more than 5 days left until expiration.

Otherwise, we’ll switch to the second-month contract.

To do this, we need the expiration date of each specific contract.

This code retrieves the expiration dates for each futures contract in the dataset. First, it identifies the start date of each unique instrument_id (i.e., when each contract first appears in the data). Then, using the Databento API client, it queries contract metadata — including the symbol and expiration date — from the GLBX.MDP3 dataset, starting from each contract’s first appearance. All responses are combined into a single DataFrame, and only the relevant columns (instrument_id, raw_symbol, and expiration) are kept for later use in determining which contract to use based on time to expiration.

The snippet above finally constructs a continuous minute-bar price series by selecting the appropriate futures contract based on time to expiration.

The result is something like this:

Next, we need to pre-process this DataFrame. What are we aiming for?

Simple: we want a table where:

• Each row represents a 5-minute interval from 4:00 p.m. ET on one day to 4:00 p.m. ET the next day;

• Each column represents a different date.

• The values are the log-returns from that moment to the first row.

Then, we’ll extract a subset of that table containing only the 3rd Thursday-to-Friday periods.

Let's do that.

The snippet above prepares the groundwork for identifying specific day-pairs and calendar-based filters. First, it creates two aligned date series: d0, a list of all unique trading days in the dataset, and d1, the next day for each d0 (i.e., the day-pair). Then, it builds a list of all 3rd Thursdays in the calendar range covered by the data.

Now, the most important code of the article:

This code builds two return matrices: one for all trading days and another for 3rd Thursday-to-Friday sessions only.

For each day-pair, it constructs a 5-minute interval time series from 4:00 p.m. ET to 4:00 p.m. the next day. It then calculates the log-returns from the start of that window and stores them in a standardized format where each column is a date and each row is a 5-minute offset from the start. Days with too much missing data are skipped.

The result is a clean, uniform dataset of intraday return curves, ready for comparing anomalies across regular days vs. 3rd Fridays.

Visualizing

We can now plot Figure 1. First mission accomplished.

The chart has approximately the same shape as the one presented by the authors.

As Josh Starmer would say, bam!

Statistical significance

Careful readers will notice a small (but important) detail:

Instead of using 9:30 a.m. as the cut-off time, I’ve chosen 9:20 a.m.

As my intuition suggested, there’s actually a better time to trade this anomaly.

Now, let’s check how many basis points we earn by being long from 4:00 p.m. to 9:20 a.m., and short from 9:20 a.m. to 4:00 p.m.

And most importantly, let’s test whether these returns are statistically different from zero.

Returns around 3rd Fridays are meaningful. Double-bam!

There’s just one small wrinkle: the t-stat on the long leg is 1.7, a bit below the threshold of 2 that we typically look for when assessing statistical significance.

I’d say that’s a minor issue, especially considering we’re analyzing data from 2010 onward, while the authors used more than twice as much data. (More data = lower standard error.)

With that said, we can confidently say that yes, the paper is legit.

Now, let’s go further down the hole and answer the second question:

Did this anomaly stop working after publication?

Before and After Publication

The paper was first published on September 20, 2023.

First, let’s visualize the intraday return patterns around the 3rd Fridays of each month:

As expected, before publication, the returns are consistent with what we’ve seen so far.

However, after publication, the return profile looks quite different.

Adjusting the cut-off time to 9:15 improved the results slightly… but it didn’t fix the problem.

The tables confirm the bad news.

After publication, the mean return on the long leg is 0.7 bps, while the mean return on the short leg is -2.6 bps. Both minimal.

The t-stats, close to zero, show that these returns are statistically indistinguishable from zero.

So, that's it. The anomaly stopped working after publication. The Queen of Hearts wins… right?

NQ to the rescue

Not so fast. The paper mentions that the anomaly also appears in other index options and futures using the Special Opening Quotation (SOQ).

Before we call it a day, let's check the numbers for NQ.

After all, we are compulsively curious. Down the rabbit hole we go.

Before publication

Here, we get a better picture than with ES: returns are stronger on both the long and short legs.

And the t-stats confirm that they’re all statistically different from zero. Double-bam!

After publication

But this is where our curiosity pays off:

Although the return chart looks odd, the mean returns are clearly different from zero on both the long and short legs: 7.8 bps and -15.1 bps, respectively.

I know, the t-stats aren’t above 2.

But here’s the key: we only have 20 samples post-publication.

So, this non-significant result may simply be due to insufficient data. Not because the edge has disappeared.

There's even another test we can run: compare the strategy's returns before and after publication.

Strategy returns

Getting the strategy's daily returns is pretty straightforward:

I am using the same 3 bps of trading cost as assumed by the authors.

Now we can check the return distributions, expected returns, win rates, and more — for all trades since 2010, before and after the publication date.

In summary:

• The strategy performed consistently well before publication, with strong average returns and good win/loss dynamics.

• After publication, results weakened, but small sample size limits our ability to draw hard conclusions.

• The edge may have decayed a little, or we may simply need more post-publication data to confirm if it’s still alive.

As a final test, we want to compare the returns before and after the paper’s publication to see if there’s a statistically significant difference between the two periods. To do that, we run a two-sample t-test with unequal variances (equal_var=False) using the returns from each period.