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:

Join the Waitlist

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!

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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:

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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.

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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.

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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.

The result: t-stat = 0.51 and p-value = 0.61.

This high p-value indicates that we cannot reject the null hypothesis: the returns before and after publication are statistically indistinguishable.

In other words, while the post-publication performance looks weaker, the difference could simply be due to random variation, especially given the small number of trades after publication.

This strategy, unleveraged, generated a total return of 81% over the full period, or about 4.1% per year. The maximum drawdown was 6.0%.

Those numbers may not seem like much. But they’re actually quite impressive when you remember they’re achieved by trading only 12 times per year.

If we had traded this strategy since 2010:

• We would have had 2 negative years;

• We would have seen 61% of the months positive, with the best at +9.1% (Mar'20);

• We would have seen 39% of the months negative, with the worst at -3.7% (Mar'22);

• The longest positive streak would have been 10 months, from Jul'19 to Apr'20;

• The longest negative streak would have been 4 months, from Jul'16 to Oct'16.

Sharpe ratio

We compute the annual Sharpe ratio by scaling the daily excess return to volatility ratio by 12 trading periods, just as the authors did in the paper:

However, although we have Sharpe ratios above 1.5 in 4 out of the 16 years analyzed, our full-period Sharpe is 0.9.

This is less than half of what the authors reported in the paper.

Why? Because our sample covers only the post-2010 period, while the authors use data starting in 2003. This gives them more trades, lower estimation error, and stronger average returns.

Final Thoughts

So, did this anomaly stop working after publication? Not quite. But it clearly weakened. The performance dropped, t-stats shrank, and the beautiful pattern seen in the pre-publication charts became noisier.

Still, with such a small post-publication sample, it’s hard to say the edge is truly gone. What we can say is that it’s no longer the clean, obvious trade it once was. The crowd may have caught on.

Should we trade this anomaly? Not by itself. The edge has weakened since the paper was published. But the results don’t argue against including it as part of a broader system, alongside other strategies in a well-diversified portfolio.

We opened with Alice standing up to the Queen of Hearts… and for good reason. The internet is full of Queens of Hearts: bold claims, confident charts, and loud opinions that go unchallenged.

But sometimes, they’re just a deck of cards.

It’s our job to question, to test, and to reason for ourselves. And to do that, we have to follow our curiosity. Follow the White Rabbit.

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

“A single hair from the head of a woman is worth more than all the books of Galen and Avicenna.” Paracelsus.

Paracelsus (1493–1541) was one of the most radical and influential physicians and philosophers of the Renaissance. A restless traveler, alchemist, and fierce critic of medical orthodoxy, he believed that true knowledge came not from ancient books but from direct observation of nature and practical experience.

Paracelsus helped shift medicine from being a theoretical, bookish discipline toward an experimental, empirical one.

I want to start by mentioning Paracelsus for two reasons. First, on a very personal note, I’m thrilled that my little brother just got into medical school! He’s going to be the first doctor in our family, and I couldn’t be prouder.

Second, I still remember my German chemistry professor in engineering school speaking of Paracelsus with great respect. As someone who valued experiment over theory, and real-world observation over books. In my view, that’s the heart of science: we trust what we test.

This week, we will implement the paper Short-Term Basis Reversal, by Alberto G. Rossi, Yingguang Zhang, and Yandi Zhu. In the paper, the authors devised a strategy that delivered 18% annual returns over several decades, with a Sharpe ratio of 1.42.

Can we replicate that? Let’s find out… because, as Paracelsus taught us, the true value of any idea lies not in how good it looks on paper, but in whether it holds up in practice.

Here’s the plan we will follow:

1. First, we will review the paper and summarize its main points.

2. Then, we will implement the strategy for a single market (time-series).

3. Next, we will implement the market-neutral strategy across all markets (cross-sectional).

4. Finally, we will discuss the next steps on how to execute.

Study Group

As mentioned in the article celebrating 1 year of Quantitativo, we are starting a Study Group via Zoom to review in-depth paper implementations. The first session will take place this Wednesday, June 11th, at 11 AM ET, and will be focused on this paper. I will share the connection details at connect.quantitativo.com.

Paper Summary

The paper discovers a new and significant anomaly in commodity futures markets: the short-term basis reversal: the return spread between adjacent futures contracts (e.g., front month vs. second month) shows strong negative autocorrelation week to week.

When the first-minus-second nearby contract return is high one week, it tends to reverse the next.

Core idea

• In markets with a term structure (like futures curves), the front-month (F1) and second-month (F2) futures contracts often move together (but not perfectly).

• Each week, the spread return between F1 and F2 (i.e., Return(F1) - Return(F2)) shows a strong tendency to reverse the following week.

  • If the spread was positive this week → likely negative next week (and vice versa).

• This is called Short-Term Basis Reversal.

Why Does This Happen?

• Front-month contracts (F1) are more liquid and adjust faster to new information.

• Second-month contracts (F2) adjust more slowly (due to lower liquidity, less trading, preferred habitat of some market participants).

• This creates temporary mispricings between F1 and F2.

• Arbitrage capital is slow-moving → mispricings persist 1-2 weeks → predictable reversals.

How to Exploit It

• Each week, look at the previous week’s spread return (Return(F1) - Return(F2)):

  • If spread was strong positive → expect reversal → next week spread likely negative

  • If spread was strong negative → expect reversal → next week spread likely positive

There are two ways to trade this effect:

1. Time-Series Strategy (Single Market)

• For each commodity individually:

  • If its spread return was positive last week → trade short spread this week

  • If its spread return was negative last week → trade long spread this week

• Works in 18 of 22 commodities → high Sharpe, robust

2. Cross-Sectional Strategy (Multiple Markets)

• Each week, rank all 22 commodities based on their prior-week spread return

• Go:

  • Long spreads of the 4 commodities with most negative prior spread return (expect reversal up)

  • Short spreads of the 4 commodities with most positive prior spread return (expect reversal down)

• Portfolio holds 8 spreads (16 contracts total)

• Delivers even higher Sharpe (up to 1.42) — scalable and more diversified

This is the gist of the paper. I strongly recommend reading it, especially for those participating in the study group. Now, let’s look at the implementation results for a single market.

Time-Series Strategy Results (Single Market)

We use Norgate Data, which is a different data source from the one the authors used in the paper.

Norgate provides daily data for 39 different commodity markets, across Agriculture (26), Energy (8), and Metals (5).

To test the simpler strategy (time-series in a single market), we will focus on Feeder Cattle futures (GF).

I will share the code and detailed implementation in the study group. But it is straightforward. The key is to create a table that, for each week, show the exact first-month and second-month contracts. Then, retrieving the prices, computing returns and the spread is pretty simple.

IMPORTANT: When the authors define first-nearby contract, it is the first eligible contract with at least 1 month to expiration, not simply the calendar first-month contract.

Here are the results:

Quite a strong result. This would have outperformed Buy & Hold convincingly:

• Annual return is +5.6% vs. 2.8% Buy & Hold;

• Sharpe ratio is 0.91, more than 3x better than Buy & Hold’s 0.25;

• Max drawdown is very controlled at -13.7% vs. a painful -55.1% for Buy & Hold;

• Volatility is also much lower: 6.2% vs. 16.4%;

• Finally, note the almost no correlation to Buy & Hold (0.03).

Now, let's look at the cross-sectional strategy.

Cross-Sectional Strategy Results (Multiple Markets)

Now that we've tested the strategy in a single market, we move to the full cross-sectional version. In this step, we apply the strategy across all 39 commodity markets available in Norgate Data, spanning Agriculture, Energy, and Metals. The table below lists the full set of markets considered in our implementation.

Visualizing the edge

As a simple diagnostic, before looking into the strategy backtest, let's check whether the strategy correctly predicts the direction of the spread return from week to week. The confusion matrix below shows the results (1 for positive/long weeks, 0 for negative/short weeks). The overall accuracy is 53.6%, above the 50% baseline expected from random guessing.

To visualize the edge more clearly, we classified the spread returns into quintiles based on the predicted signal and aggregated the realized returns for each group. The chart below shows the annualized mean return for each quintile. As we can see, the results display a clear monotonic pattern. The most extreme predicted spreads (Q1 and Q5) show the strongest directional returns, which highlights the predictive power of the signal.

Backtesting the strategy

Now, let’s backtest the long&short market neutral strategy. The core idea is straightforward:

• Each week, we rank all 39 commodities based on their spread return from the previous week.

• We then take positions as follows:

  • Go long the spreads of the 4 commodities with the largest negative spread returns (anticipating a rebound).

  • Go short the spreads of the 4 commodities with the largest positive spread returns (anticipating a reversal lower).

Here are the results:

We can see a strong performance for the cross-sectional strategy:

• Annual return is 19.2%, more than 2x the S&P 500’s 9.1%;

• Sharpe ratio is good at 1.45, vs. 0.62 for the benchmark;

• Max drawdown is well controlled at -24.4%, while the benchmark suffers a brutal -56.2%;

• Volatility is also lower than S&P 500: 12.6% vs. 16.6%;

• And (as expected for a market-neutral approach) the strategy has virtually zero correlation with the S&P 500 (0.01).

Now, let's see how the trading costs impact the strategy.

Impact of Trading Costs

We have assumed 10 bps of trading costs, deducted weekly. Let’s now examine how different levels of trading costs affect the results:

As expected, trading costs have a significant impact on the strategy's performance. However, the edge remains robust even with higher cost assumptions:

• With 5 bps of costs, the strategy delivers an excellent 22.3% annual return and a 1.66 Sharpe;

• At 10 bps (our base case), results are still very strong: 19.2% annual return and 1.45 Sharpe;

• Even at 15 bps and 20 bps, the strategy remains profitable (Sharpe stays above 1), though returns naturally compress: 16.1% and 13.1% annual returns, respectively;

• Importantly, max drawdown remains stable across cost levels (around -22% to -28%), indicating the strategy does not become more fragile as costs rise.

Impact of Number of Spreads

In our base case, we traded 8 spreads (4 long, 4 short). Let’s now examine how changing the number of spreads affects the results:

As we vary the number of spreads, we observe a clear trade-off between return and risk concentration:

• Using only 4 spreads maximizes returns (27.7% annually) but comes with very high volatility (21%) and large drawdowns (34.8%);

• Increasing to 6 spreads improves Sharpe (1.40) while reducing risk somewhat, though returns drop to 23.2% annually;

• The base case (8 spreads) shows the best risk-adjusted profile: Sharpe 1.45, volatility 12.6%, max drawdown 24.4%, with a still strong 19.2% return;

• Adding more spreads (10) further stabilizes the strategy (lowest volatility, lowest drawdown), but at the cost of reduced returns (16.2%), possibly due to diluting edge across weaker-ranked markets.

In short: the 8-spread configuration appears to offer the best balance between performance and robustness.

Final Thoughts

The implementation of this paper will be a great way to kick off our Quantitativo Study Group this Wednesday. It’s a clean example of how a simple empirical anomaly, when rigorously tested and thoughtfully implemented, can translate into a potentially robust trading strategy.

The Short-Term Basis Reversal strategy not only held up out-of-sample, but also performed better than in the original paper, especially when implemented as a cross-sectional market-neutral portfolio.

Starting from a simple empirical observation about spread reversals in commodity term structures, we built a scalable and robust trading strategy. Even after factoring in trading costs and varying the number of spreads, the edge remained strong: Sharpe ratios above 1, strong annual returns, and controlled drawdowns.

A few takeaways stood out:

• The edge is surprisingly consistent across markets and over time;

• The cross-sectional version offers the best risk-adjusted profile, while also being more scalable;

• The signal shows a clear monotonic relationship with realized returns, suggesting it captures a real and persistent structural inefficiency.

Of course, this was still a baseline test. There is plenty of room for further exploration:

• Should we dynamically adjust the number of spreads traded based on market conditions?

• Could we optimize the selection and sizing of spreads (e.g., using advanced ranking, volatility scaling, or ensemble signals)?

• How well does this edge hold up with intraday data or shorter rebalancing cycles?

This is the spirit of Paracelsus: trust what you test, and always experiment further. The results so far are very encouraging.

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:

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(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

“Either write something worth reading or do something worth writing”. Benjamin Franklin

Benjamin Franklin was a printer, inventor, diplomat, and my favorite Founding Father. But more than that, he was a lifelong student of self-improvement. He believed in the power of ideas, disciplined action, and sharing knowledge to make the world better.

This week, we celebrate one year of Quantitativo! I chose to open with Franklin because, as we enter our second year, we're introducing a few innovations that reminded me of his spirit of learning and experimentation. These include:

1. Study Group via Zoom

2. Sharing what’s next on the implementation roadmap

3. Sharing live performance data from strategies in production

1. Study Group via Zoom

As many of you requested, a few weeks ago I launched a course that walks through some of my codebase step by step. It's designed for readers who want to develop quant strategies using the same approach I’ve shared here.

The feedback has been overwhelmingly positive. Thank you! I've added some spontaneous testimonials at the end of this message so everyone can get a sense of the response 🙂

The final 10 seats are still available. The cohort already has 40 students, and I’ll be closing enrollment this Monday. So if you're interested, now’s the time to check it out:

Join the Course

One of the main requests from the first students was to create a community space.

A private community for systematic traders to explore ideas, exchange insights, and discuss the technical and strategic challenges of building real trading systems.

So, we launched the community… and it’s flourishing!

It’s become a space for daily conversations about alpha research, deeper technical implementation, infrastructure design, and more. One of the first ideas that emerged was to start a Study Group via Zoom.

The idea is simple: each week (or every other week), one person reads a paper, prepares a short presentation (15 minutes max), and shares it with the group. Then we discuss how to implement it in practice.

For the kickoff session, I’ll present the paper “Short-Term Basis Reversal” by Rossi, Zhang, and Zhu (2025). The session will happen in the second week of June. I’ll share both the paper summary and my own implementation so we can discuss the practical side together.

The vision is to grow this community from the current 40 students into a broader group of people who are serious about studying, developing, and implementing quantitative trading strategies. Let’s see where this leads.

This initiative reminded me of Ben Franklin’s Junto (also known as the Leather Apron Club). In 1727, at the age of 20, Franklin founded the Junto, a club for mutual improvement. It brought together a small group of artisans and tradesmen who shared a desire for self-education, civic responsibility, and intellectual exchange. They met weekly to discuss moral, political, scientific, and economic questions.

But the Junto wasn’t just academic. It was about practical wisdom and intellectual humility. And the community he started lasted nearly half a century.

2. Sharing what’s next on the implementation roadmap

Over the past year, I’ve shared nearly 30 strategy implementations here. These started as blog posts and gradually shifted toward academic paper implementations, which have become a clear favorite within the community.

So, I’ll continue to implement and share interesting papers, along with my take on them.

I have the habit of reading 1-2 papers per week. I choose them from my personal collection, which I curate by skimming 5-10 new research papers each day and selecting up to 3 to add to the library.

Recently, someone in the community asked how I decide which papers to test. After explaining my process, I began sharing a few of my selected papers, and, to my surprise, the response was highly positive.

I’ll continue writing my implementations (“do something worth writing,” as Ben Franklin would say). And starting with today’s piece, I’ll also begin periodically sharing research ideas from my library that are currently on the implementation roadmap (“write something worth reading,” as Ben would put it).

Here are the new additions I have gathered over the past few weeks:

2.5 million headlines. A Sharpe ratio of 2.80.

A new paper just dropped that integrates pretrained language model embeddings into a stochastic discount factor (SDF) framework using adversarial networks.

The result? A model that outperforms CAPM by 471%, reduces pricing errors by 74% compared to Fama-French 5-factor, and shows that financial news holds more predictive power than macro data.

Text embeddings extracted from news lead returns by 2-3 weeks, and rank among the most important features in asset pricing.

NewsNet-SDF: Stochastic Discount Factor Estimation with Pretrained Language Model News Embeddings via Adversarial Networks, by Shunyao Wang, Ming Cheng, Christina Dan Wang.

High attention is a warning sign.

A new interesting study finds that daily investor sentiment and attention on StockTwits, Twitter, and Seeking Alpha move markets in opposite ways: high attention predicts continued losses, while high sentiment signals mean-reverting optimism. A trading strategy based on these two signals delivered a Sharpe ratio of 1.2 and outperformed traditional attention metrics.

Market Signals from Social Media (Cookson et al., 2025)

One-week holding. 17.6% annual returns and 1.42 Sharpe. Zero exposure to known risk factors.

A new paper reveals that the spread between front- and second-month futures reverses week to week. This “short-term basis reversal” is:

• Predictable,

• Unexplained by carry or momentum,

• Strongest when volatility is high.

The strategy? Go against last week’s spread. It works across commodities, bonds, and even stock index futures.

Short-Term Basis Reversal by Rossi, Zhang, and Zhu (2025)

This paper is the focus of our first Study Group via Zoom.

4.82 Sharpe. 42% better than CNNs built for stocks.

This new paper drops a clever insight: financial time series (like stocks or loans) should be treated as sets, not fixed sequences.

The proposed Set-Sequence model learns cross-sectional dependencies without handcrafted features. It crushes benchmarks on:

• Stock return prediction (36,600 equities),

• Mortgage risk (5M loan-months),

• Synthetic contagion (1,000-unit default simulation).

Efficient. Interpretable. Exchangeable. The future of multi-series forecasting might look like this.

A Set-Sequence Model for Time Series, by Elliot L. Epstein, Apaar Sadhwani, and Kay Giesecke.

A simple word count model beats BERT and sentiment dictionaries. 1.86 Sharpe. Long Lasso, short LLMs.

This new paper builds a bottom-up dictionary, selecting the most predictive words in 10-Ks using lasso regressions.

The result? A factor portfolio with:

• 1% monthly returns,

• 1.86 Sharpe (equal-weight),

• Minimal overlap with Fama-French 5.

Outperforms:

• Top-down sentiment models,

• Large language model embeddings (BERT),

• Even neural nets with the same input.

The winning signal? Words like currency, oil, restructuring, and research.

Bottom Up vs Top Down: What Does Firm 10-K Tell Us?, by Landon J. Ross, Jim Horn, Mert Pilanci, Kaihong Luo, and Guofu Zhou.

3. Sharing live performance data from strategies in production

If I had to summarize the strategy development process I follow, it would look like this:

After a year of testing, only a handful of strategies made it through. These are now in production, with a few million dollars allocated to them.

Starting this year, I also plan to begin sharing live performance data for these strategies. This will roll out later in the year.

Final thoughts

It’s been an incredible first year.

Since launching Quantitativo, we’ve connected with nearly 5,000 people: readers, students, and fellow traders from all over the world. More important than any strategy I’ve shared, what truly stood out were the relationships built along the way: new friends, thoughtful conversations, and meaningful collaborations that helped shape many of the ideas shared.

What began as a simple idea (sharing the process of developing trading strategies) has evolved into a growing community of people who are serious about building conviction through research, iteration, and implementation.

At the core of Quantitativo is a simple belief:

The more strategies we test, the higher our odds of achieving above-average returns.

Our goal isn’t to hand out plug-and-play systems. It’s to spark curiosity, build skill, and help each of us become the kind of person who’s always learning, questioning, and creating.

Because strategies come and go. But the habit of developing and testing ideas, and the relationships we build along the way, are what truly endure.

Thank you for being part of this journey.

I'm excited for what’s ahead, and I look forward to meeting more of you, exchanging ideas, and continuing to build great things together.

Cheers,

Carlos

P.S. As mentioned earlier, here are a few spontaneous testimonials from early users:

Thanks again for the incredible feedback! It really makes my day. :)

The idea

"If you're not thinking about risk, then you're not thinking." William Sharpe.

William Sharpe is a Nobel Prize-winning economist renowned for his work on the Capital Asset Pricing Model (CAPM) and the Sharpe Ratio, both of which highlight the central role of risk in pricing and evaluating assets. While the quote “If you're not thinking about risk, then you're not thinking” may not be his exact words, it captures the essence of his legacy: that risk must be understood, measured, and deliberately managed.

This week, we’ll show how to systematically hedge a long-biased portfolio of strategies to reduce its market exposure to near zero. It will be a simple yet practical example, with code included. The basic model can be further extended or refined as needed.

Here's the plan:

1. First, we’ll review a portfolio of strategies, examine its performance, and highlight the issue of having a moderately high beta.

2. Then, we’ll outline a hedging approach designed to bring the portfolio’s overall beta close to zero.

3. Next, we’ll run experiments to evaluate how this hedging impacts the portfolio’s overall returns.

4. Finally, we’ll share some concluding thoughts and discuss possible next steps.

Let's get to it.

As many of you requested, 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

Now, to the article!

The problem

We’ve been trading a portfolio of mean reversion strategies—based on ideas we’ve shared here—for over six months. The live results are promising: a return of +10.1% versus -2.8% for the S&P 500. Here’s an overview of the portfolio’s performance:

A portfolio with a 1.1 Sharpe Ratio is what I’d consider OK. Someone asked me over email what I consider a good Sharpe ratio, and here’s the heuristic I use:

• < 1.0: Subpar or not investable

• 1.0–1.5: OK strategy/portfolio

• 1.5–2.0: Good strategy/portfolio

• 2.0–3.0: Great strategy/portfolio

• > 3.0: Amazing or outlier strategy/portfolio

I always interpret these numbers in the context of capacity. For niche or capacity-constrained strategies, higher Sharpe ratios may be possible—but they’re typically not scalable. Since this portfolio trades only S&P 500 constituents and therefore has very high capacity, a 1.1 Sharpe is reasonable. It's over twice the S&P 500.

The correlation to the S&P 500 is 0.57, which I consider moderately high.

Now, let's use the Fama-French 3-Factor Model to analyze how much of the strategy's performance can be attributed to common risk factors like the market, size, and value. By running an OLS regression on the strategy's excess returns using the Fama-French factors, we can break down the sources of performance into:

1. Market Risk (Mkt-RF): The sensitivity of the portfolio to market movements, which reflects general market exposure.

2. Size (SMB): The sensitivity of the portfolio to the size factor, indicating whether it leans towards small or large-cap stocks.

3. Value (HML): The sensitivity to the value factor, showing whether it favors value or growth stocks.

Here are the results:

The OLS regression on the strategy’s excess returns using Fama-French factors shows a statistically significant alpha of 0.0534, alongside a market beta (Mkt-RF) of 0.57, which is moderately high. Exposure to SMB is small but positive, while the negative HML loading suggests a tilt away from value stocks.

While the positive alpha is encouraging (annualizes to 14.4%), the moderately high beta indicates that a large portion of the returns is explained by general market movements, rather than pure strategy edge. This is not ideal—especially in a multi-strategy or market-neutral framework—because it means the portfolio is still exposed to systematic market risk. Our goal is to hedge this market exposure to isolate the strategy’s true performance and reduce drawdowns during broader market downturns.

This is the problem we are solving. Let's see how.

Beta hedging

Beta hedging is the process of neutralizing a portfolio’s exposure to broad market movements by taking an offsetting position in a market index, typically using futures or ETFs. The goal is to isolate the strategy’s alpha by removing the influence of systematic risk.

In our case, the portfolio has a market beta of 0.57, meaning it tends to move 0.57% for every 1% move in the S&P 500. To offset this exposure, we’ll short S&P 500 futures (ES contracts) in proportion to the portfolio’s beta and current notional value. This allows us to bring the net beta close to zero, reducing sensitivity to market direction.

We’re not attempting to hedge other factors like SMB or HML (their betas are not as high). Our focus here is purely on market exposure. Because beta and portfolio value can fluctuate daily, we’ll recalculate the hedge each day and adjust the number of ES contracts accordingly. This ensures the portfolio stays close to beta-neutral over time.

This is meant to be a simple, focused example to demonstrate the concept. Extending the approach to hedge exposure to other risk factors—such as size (SMB) or value (HML)—follows the same logic: estimate the factor loading, identify a tradable proxy, and size the hedge accordingly.

We start with a DataFrame containing three columns:

• daily_rets_pre_hedge – the portfolio’s daily returns before applying the hedge

• total_value_pre_hedge – the portfolio’s total dollar value each day

• es_close – the daily closing price of the S&P 500 futures (ES)

Now, we need to compute the rolling beta of this portfolio:

The code above estimates the portfolio’s rolling beta to the market using a 63-day window. For each window, it applies the get_alpha_and_betas function to compute the market beta (Mkt-RF), which is then forward-filled to handle any initial NaNs. To smooth short-term fluctuations, the resulting beta series is further filtered using an exponential moving average (EMA) with a 20-day span. Additionally, the portfolio’s daily notional value is scaled to an initial capital of $1,000,000, and rows with missing values are dropped to prepare the data for hedging calculations.

The method get_alpha_and_betas is something like this:

The get_alpha_and_betas function runs an OLS regression of a strategy’s excess returns against the Fama-French factors (either 3 or 5), in order to estimate the strategy’s alpha and factor loadings (betas). It first downloads and preprocesses the relevant factor data using the external helper get_ff_factor_model, which pulls data directly from Kenneth French’s website (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/) and returns it ready for regression (this helper is included in the course but not shown here for brevity). The strategy returns are joined with the factor dataset, and if there are enough data points (default: 42), a regression is run using statsmodels.OLS.

Then, we can see the rolling beta with the following snippet:

Now that we’ve estimated the portfolio’s rolling beta, we can compute the number of ES contracts to hold each day and track the impact of the hedge on both PnL and overall portfolio returns:

Here's what the code above is doing, step-by-step:

1. Hedge Sizing:

   The number of ES futures contracts to short is calculated based on the portfolio’s dollar value, its rolling beta, and a dampening beta_factor (set to 0.5). The hedge is not fully sized to neutralize all beta—this allows for some market exposure to remain, which may improve Sharpe or reduce over-hedging. The hedge is scaled by the ES contract multiplier (50), and the result is rounded to an integer and negated to reflect a short position.

2. Futures PnL Calculation:

   The PnL from holding the ES futures is computed as the change in futures price multiplied by the previous day’s position size and the contract multiplier. This simulates daily rebalancing, where the position is adjusted at each step based on the updated beta.

3. Post-Hedge Portfolio Value:

   The PnL from the futures hedge is added to the original (pre-hedge) portfolio value to compute the total value of the post-hedge portfolio. This reflects how the hedge offsets market-driven moves in the unhedged portfolio.

4. Post-Hedge Returns:

   Finally, the daily returns of the post-hedge portfolio are calculated as the percentage change in total value. The first day's return is manually set to match the pre-hedge return to avoid a missing value at the start of the series.

Results

Now, we can see the new results:

Here are the main highlights and conclusions from the pre- and post-hedge performance comparison:

• Market exposure was significantly reduced:

  Correlation to the S&P 500 dropped from 0.57 pre-hedge to 0.29 post-hedge, indicating effective removal of systematic market risk.

• Sharpe ratio remained stable:

  The Sharpe ratio stayed nearly unchanged (1.09 → 1.08), suggesting that the hedge did not dilute the portfolio's risk-adjusted performance.

• Return and volatility slightly decreased:

  Annualized return decreased marginally (21.0% → 19.8%) along with volatility (19.1% → 18.2%), consistent with a dampened beta.

• Max drawdown slightly increased:

  Maximum drawdown worsened slightly (-24.4% → -26.0%), which may reflect the cost of daily rebalancing or increased short-term fluctuations introduced by the hedge.

• Reduced dependence on market direction:

  Despite lower beta, the strategy preserved almost all of its long-term return, which reinforces the presence of true alpha.

Re-running the Fama-French 3-Factor Model allows us to assess how much of the strategy’s performance can be explained by common risk factors. The results are as follows:

Highlights:

• Market beta dropped substantially:

  The loading on Mkt-RF fell from 0.57 pre-hedge to 0.27 post-hedge, confirming that the hedge significantly reduced market exposure.

• Alpha increased after hedging:

  The annualized alpha rose from 14.4% to 16.4%, suggesting that isolating the portfolio from market movements revealed even stronger idiosyncratic performance.

• Model explanatory power decreased:

  The R-squared dropped from 0.33 to 0.09, indicating that a much smaller portion of the portfolio’s returns is now explained by the Fama-French factors—consistent with a more market-neutral profile.

• Alpha remained statistically significant in both cases (p-value < 0.001), reinforcing the robustness of the strategy’s non-factor-driven edge.

These results validate that the hedge effectively reduced systematic risk while preserving the portfolio’s alpha characteristics.

To wrap up the analysis, the chart below offers a visual comparison of the portfolio’s daily returns versus the S&P 500 before and after hedging:

The left panel shows a clear positive relationship between the portfolio and the market, with a beta of approximately 0.57. After applying the hedge (right panel), the slope of the linear fit drops to 0.27, confirming that the portfolio’s sensitivity to market movements has been significantly reduced. The post-hedge cloud is more centered and less tilted, illustrating how the hedge successfully dampened systematic exposure while retaining the core return profile.

Final thoughts

This was a simple but effective demonstration of how to implement beta hedging in practice. Starting from a long-biased portfolio of strategies with a moderately high market beta, we systematically reduced its exposure by shorting S&P 500 futures, recalibrating the hedge daily. The results were encouraging: we cut the beta nearly in half, maintained the Sharpe ratio, and preserved nearly all of the original alpha—while significantly lowering the portfolio's correlation to the market.

While the model we used was intentionally simple, the core concept generalizes well. Hedging other factor exposures—such as size or value—follows the exact same playbook. The point wasn’t to overfit, but to show that a practical, transparent beta hedge can be built with just a few lines of code and some clean assumptions.

There’s room to explore further. One direction is to turn this static hedge into a more dynamic framework, where the dampening factor or even the hedge asset itself adjusts based on regime shifts or higher-order factor interactions. Another is to combine beta hedging with volatility targeting or cross-asset overlays to create a fully adaptive market-neutral engine.

But that’s a topic for another day.

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!

As many of you requested, 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

The idea

“The biggest lesson that can be read from 70 years of AI research is that general methods that leverage computation are ultimately the most effective, and by a large margin.” Richard Sutton.

Richard Sutton is one of the greatest minds of our time. He is a founding figure in modern AI and a pioneer of reinforcement learning, the framework behind many of today’s most advanced intelligent systems. His work laid the foundations for agents that learn not from instruction, but from experience. A recipient of the Turing Award (the “Nobel Prize of Computing”), Sutton's influence is profound, reshaping how we think about learning, intelligence, and progress in AI.

In his landmark 2019 essay The Bitter Lesson, Sutton delivers a hard truth distilled from 70 years of AI research: methods that scale with computation, like search and learning, consistently outperform approaches that rely on built-in human knowledge. While it may be tempting to design systems that reflect how we think we think, history has shown that progress comes instead from embracing general-purpose algorithms and the exponential growth in computing power. It's a lesson many resist… until reality beats them over the head with it.

Human intuition may offer short-term gains, but brute-force computation ultimately wins.

This week, I’m going to write about this idea—and a failed experiment. But don’t be misled: the absence of a successful first result doesn’t mean the idea should be dismissed. On the contrary, this is the most important concept I’ve shared here so far. In fact, the core of The Bitter Lesson is one of the most important ideas in the history of computer science.

You’re probably already seeing where this line of reasoning leads. Many quantitative systems are built around rule sets that mimic human thinking. But Sutton’s core argument is that, in the long run, general-purpose algorithms that scale with compute will always outperform hand-crafted, human-informed approaches. He’s essentially saying that trying to manually craft trading rules is a waste of time. It might have worked in the past, but it won’t keep working much longer.

It’s a controversial point, I know. Let’s dive in.

As many of you have requested, I’ll be sharing some of my codebase in a course. It’s almost ready and should be finished in the next week or so. I’ve created the page below to explain the course content, who it’s for, and to answer the main questions about it. If you have any additional questions, feel free to send me an email.

Course Content

Now, to the article!

Thoughts from an ex-Senior Quant from Citadel

Richard Sutton is also a personal hero. I devoured his Reinforcement Learning book—the definitive guide in the field—while taking the Reinforcement Learning and Decision Making class in grad school. In that class, under the guidance of the great professors Charles Isbell and Michael Littman, I developed the habit of reimplementing influential academic papers to deepen my understanding. It was my favorite course during my Master’s.

A few weeks ago, while watching the presentation "Deep Order Flow Imbalance: Extracting Alpha at Multiple Horizons from the Limit Order Book”, three minutes in, I started listening to Nicholas Westray make Sutton's Bitter Lesson point.

Westray, an ex-Senior Quant at Citadel (now at Point72, I think), explains Sutton's article and states the idea of his research: to forecast returns directly from raw limit order book (LOB) data, leveraging computation over domain knowledge.

His research shows that yes, using order flow data derived from raw limit order book data, a massive dataset, and a lot of compute power can indeed result in a model that can predict short-term returns.

It's a great talk worth listening to. Here are the paper and the slides for those who are more curious.

My first thought was to reimplement Westray’s paper as a learning exercise—something I picked up from Isbell and Littman in my Reinforcement Learning class. (Coincidence or not, one of the first papers I ever reimplemented was Sutton’s seminal work on temporal difference learning.)

I also wondered whether a simpler version of the idea might have profitable applications in less competitive corners of the market. There's only one way to find out: test it.

But as I read the paper more carefully, I realized this is a much larger project. Unfortunately, I don’t have the free time to tackle it in just a few weeks. It’ll take longer.

That led to my second thought: can we build a toy project instead? The idea is the same: throw a lot of data and compute at a return prediction problem to put Sutton’s Bitter Lesson to the test. Toy problems are also great learning tools.

Back in the day when OpenAI was still non-profit, I spent my fair share of time solving OpenAI Gym environments to learn deep reinforcement learning. I know the value of implementing toy projects.

Choosing a paper

If not Nicholas Westray's paper, which paper should I put to test? That was the next question. I have a library with over 100 papers I want to implement. It will take me many years, and I don't believe I will ever finish, because the library keeps growing.

Navigating through the library is hard, so I turned to Google and searched for “deep learning momentum stocks”. Why these terms? Simple: “deep learning” accounts for the lots of compute power & learning algorithms from Sutton's Bitter Lesson, and “momentum” is one of the most persistent anomalies in “stocks”. The top three results:

1. Slow Momentum with Fast Reversion: A Trading Strategy Using Deep Learning and Changepoint Detection, from Kieran Wood, Stephen Roberts, and Stefan Zohren

2. Dissecting Momentum: We Need to Go Deeper, from Dmitry Borisenko

3. Applying Deep Learning to Enhance Momentum Trading Strategies in Stocks, from Lawrence Takeuchi and Yu-Ying (Albert) Lee

The first two papers look far more interesting, but are also longer-term projects to implement. On top of that, they involve too much feature engineering for my taste, especially Borisenko’s work, which somewhat defeats the purpose of using deep learning. Feature engineering depends heavily on human expertise, which contradicts the very lesson Sutton emphasized.

Now, the third paper has a different issue in my view: it seems too simple, almost too good to be true. The dataset is straightforward, the features are easy to derive… so where’s the catch? Still, simplicity means speed, so let’s start by testing this one.

The paper

Here are the key points for implementing the paper. For all details, check the original.

Data

Sample Selection

• Universe: U.S. stocks on NYSE, AMEX, NASDAQ.

• Exclude stocks priced below $5 to avoid microstructure noise.

• Training: 1965–1989 (848,000 samples), Testing: 1990–2009 (924,300 samples).

Input Variables and Preprocessing

• Input (33 features):

  • 12 monthly returns from months t−13 to t−2.

  • 20 daily returns from month t.

  • 1 January dummy variable.

• Convert returns into cumulative returns and z-scores for cross-sectional normalization.

• Target labels:

  • Class 1: returns below the monthly median.

  • Class 2: returns above the monthly median.

The model

Model and Training

• Use stacked RBMs to form an autoencoder, pretrained layer-wise.

• Compresses input to a low-dimensional feature space, then passes it to a feedforward neural network (FFNN) for classification.

• Final model is fine-tuned using backpropagation.

Architecture Specification

• Conducted grid search with hold-out validation (1965–1982 train / 1983–1989 validation).

• Final architecture: 33-40-4-50-2:

  • 33 inputs → compressed to 4 → classified into 2 classes.

• Justification: balance performance and complexity. No gain from deeper models.

Implementation

The full implementation is also available on GitBook for a better code browsing experience.

1. Data

In our replication, we will use daily stock prices from January 1, 1990, to today, obtained from Norgate data. Norgate provides a high-quality survivorship bias-free daily data for the US stock market that is very affordable. For more information on how to acquire a Norgate data subscription, please check Norgate website.

The first step is to retrieve the data for a given symbol:

The function get_data(symbol) retrieves daily price data for a given stock symbol using Norgate, then constructs a feature-rich DataFrame indexed by (date, symbol) for use in predictive models. It calculates monthly cumulative returns (ret-m1 to ret-m12) based on end-of-month closes and daily cumulative returns (ret-d1 to ret-d20) over the last 20 trading days. It also includes the unadjusted close price, a binary flag is_next_jan indicating if the next month is January, and the forward return for the next month (next_month_ret). The function ensures no missing or infinite values and returns a clean, multi-indexed dataset ready for modeling.

We can check it with a call like get_data('AAPL'):

Now, retrieving all data is straightforward:

In about 30 minutes, we are ready to continue.

2. Pre-processing

Before training the model, we apply several pre-processing steps to clean the data, standardize features, and define the target variable:

We first filter out low-priced stocks (unadjusted close ≤ $5), which are often illiquid and noisy. Then, we apply cross-sectional z-score standardization to all features (excluding the last two columns) on each date to normalize them. The target variable is defined as a binary label: 1 if the next month’s return is above the median return for that date, and 0 otherwise. We also preserve the raw (unstandardized) version of the last feature (is_next_jan) and the original forward return for later analysis. All components are then combined into a single DataFrame for model input.

3. Cross-validation splits

To evaluate model performance over time, we implement a rolling-window cross-validation framework tailored for time series data:

The train_val_test_split function generates chronological train/validation/test splits by sliding a multi-year window forward one year at a time. For each iteration, it defines a training period, a validation period, and a test year—ensuring no data leakage. We can choose whether to place the validation set before or after the training set using the validation_first flag. This approach mimics a realistic backtesting setup and provides out-of-sample evaluation across multiple time periods.

We can verify that the datasets are being generated correctly with the following code:

Now, we are ready to work on the model.

4. The model

We define a simple feedforward neural network (FFNN) architecture to classify the pre-processed features into two classes, as specified in the paper:

The FFNN model consists of three hidden layers with ReLU activations, followed by a final output layer that produces logits for binary classification. The input layer expects 33 features, which are transformed through progressively deeper representations: from 40 units, to 4, then to 50, before reaching the final output layer with 2 neurons. The model returns raw logits, which are suitable for use with CrossEntropyLoss, a standard loss function for multi-class classification tasks.

In the original 2013 paper, Takeuchi and Lee used a stacked autoencoder built from restricted Boltzmann machines (RBMs) for pretraining, which was common at the time. They pre-trained the encoder (33→40→4) to extract features and then attached a feedforward classifier (4→50→2) on top. Our model directly implements the final architecture they found optimal through cross-validation, bypassing the pretraining phase.

The reason we can now skip RBMs and pretraining is due to major advances in hardware (GPUs/TPUs), software (PyTorch, TensorFlow), initialization methods, and optimization algorithms (like Adam). These allow deep neural networks to be trained end-to-end from scratch efficiently, even on noisy financial datasets. In contrast, a decade ago, training deep nets from scratch often resulted in poor convergence, which is why unsupervised layer-wise pretraining with RBMs was widely used to help the network learn useful representations before supervised fine-tuning.

We can verify that our architecture works using the following code:

The code builds a PyTorch DataLoader from a DataFrame, converting features and labels to tensors for training. It retrieves one mini-batch, checks the shapes, and runs a forward pass through the FFNN to verify the model works as expected. We should see something like this:

Now, let's move to training.

5. Training

We now define the training loop used to optimize the model parameters using the training and validation sets:

The train function performs mini-batch training of the FFNN model using the Adam optimizer and cross-entropy loss. It tracks performance on both the training and validation sets, saving the model weights that yield the lowest validation loss. After training for a specified number of epochs, it returns the best-performing model along with key metrics such as validation loss, accuracy, and the epoch at which the best performance was achieved.

We now generate predictions and rank stocks by model confidence:

The method above generates model predictions on the test set by applying the trained network to each mini-batch and collecting the predicted classes and class probabilities. It appends these outputs to the original test DataFrame and assigns each example to a quantile bucket based on its predicted probability of belonging to class 1. This enables cross-sectional ranking of stocks by confidence level, which is useful for evaluating strategy performance by signal strength.

We now run rolling-window training and evaluation to collect predictions and track model performance over time:

Running the code above will take a bit over an hour. We should see something like this:

The DataFrame that tracks the performance should record something like this:

Now, we are ready to see the results.

Results

Let's start reviewing the model's classification performance by analyzing the confusion matrix:

This code compares the true labels and the model’s predicted classes across the entire test set to compute a confusion matrix, which is then column-normalized to show the distribution of predictions per class. This helps assess whether the model is biased toward one class and how well it distinguishes between them. The results are visualized using ConfusionMatrixDisplay, with values shown as proportions for easier interpretation.

We should see something like this:

Highlights:

• The model is slightly better than random guessing, with ~52% accuracy for both predicted classes.

• It shows no strong bias toward one class—performance is symmetrical.

• This aligns with expectations for noisy financial datasets with low signal-to-noise ratios, where even a small edge (e.g., >50% accuracy) can be economically meaningful when scaled properly in a trading strategy.

We now compute the monthly long-short returns based on the spread between the top and bottom quantile predictions:

This snippet computes the average forward return for each quantile bucket on a monthly basis by grouping predictions by date and quantile. It then unstacks the results into a table where each column represents a quantile. Finally, it prints how many months are missing one or more quantiles, which can occur due to duplicates='drop' in qcut, highlighting potential data sparsity in certain months.

We now visualize the annualized average return by quantile to assess the relationship between predicted confidence and realized performance:

This code calculates the mean return for each quantile across all months, annualizes it, and plots it as a bar chart. The chart highlights how returns vary by model confidence:

Highlights:

• Returns increase monotonically from quantile 1 to quantile 10.

• Quantile 1 has a negative return (-0.64%), indicating the model is correctly identifying poor-performing stocks.

• Quantile 10 shows the highest return at 13.0%, suggesting strong predictive power in selecting top-performing stocks.

• The spread between quantile 10 and quantile 1 is approximately 13.6 percentage points, which is economically meaningful.

• The results are not as good as described in the paper, but still, worth looking into it.

This pattern strongly supports the idea that the model’s confidence scores are informative, as shown by the authors in the paper. A long-short strategy (long top quantile, short bottom) would likely be profitable and consistent, as we can verify:

The snippet above generates the cumulative returns for the strategy. We can visualize it with one line, cumulative_rets.plot(logy=True), or if we add a bit more formatting and the benchmark:

To get a summary of the backtest main stats, here's what we can do:

We should see something like this:

Overall, the strategy outperforms the benchmark across all key dimensions:

• Annualized return is 12.8%, nearly double the S&P 500’s 7.0%;

• Sharpe Ratio is 1.03, more than twice that of the benchmark, indicating superior risk-adjusted performance;

• Maximum drawdown is limited to 24%, less than half of the S&P 500’s 52.6%, reflecting strong downside protection;

• Correlation to the market is negative, offering valuable diversification benefits in a broader portfolio.

Final thoughts

While our implementation successfully demonstrated the viability of applying deep learning to momentum-based stock prediction, the results we obtained were a far cry from those reported by Takeuchi and Lee (2013). There are several likely reasons for this discrepancy. First, we used a different and significantly smaller dataset, both in terms of the number of stock-month observations and market breadth. Second, our time horizon was more recent and extended into a different market regime, potentially less favorable to momentum strategies. Third, unlike the authors’ original approach which involved unsupervised pretraining using stacked RBMs, we trained the feedforward network end-to-end using modern initialization and optimization techniques. Additionally, we adopted a rolling-window training and evaluation protocol, whereas the original study trained once on a large historical sample and tested on a fixed out-of-sample window. These methodological and structural differences, combined with evolving market dynamics, likely account for the performance gap.

Nevertheless, replicating the exact results reported by Takeuchi and Lee was never the true goal. The purpose of this project was to explore, in a simplified and more accessible setting, the core insight behind Richard Sutton’s Bitter Lesson: that general methods powered by computation outperform systems infused with human-crafted rules and domain expertise. Our toy example, despite its simplicity and the constraints we faced, served as a proof of concept for this idea. By throwing a clean pipeline, raw price data, and compute at the return prediction problem—without relying on hand-engineered financial indicators—we were able to extract signal and generate meaningful returns. In that sense, the experiment achieved what it set out to do: demonstrate that scaling learning systems and leaning into computation-first approaches can produce viable results even in challenging, noisy domains like financial markets.