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Synthetic Data for Portfolios: A Throw of the Dice Will Never Abolish Chance

This paper critiques the limitations of current generative models in portfolio management, highlighting issues like the pitfalls of excessive data generation and misalignment with portfolio construction needs, while proposing a robust pipeline for synthesizing multivariate returns and introducing a "regurgitative training" method to better evaluate model identifiability and suitability for specific financial applications.

Original authors: Adil Rengim Cetingoz, Charles-Albert Lehalle

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adil Rengim Cetingoz, Charles-Albert Lehalle

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future of a chaotic, noisy world. You have a small notebook filled with observations of how things move, but you want to know what happens next. In the world of finance, this notebook contains the daily price changes of thousands of stocks. For decades, experts have tried to build "generative models"—sophisticated digital machines that learn from this notebook and then spit out endless new, fake scenarios. It's like a chef tasting a single spoonful of soup and then trying to cook a banquet that tastes exactly the same. The hope is that by feeding these fake scenarios into a computer, we can test investment strategies without risking real money. But there's a catch: if the chef only tasted a tiny spoonful, no matter how many bowls of soup they cook, the banquet will still taste wrong. This paper dives into the messy, high-stakes kitchen of financial markets to see if these digital chefs can actually cook up a meal that helps investors, or if they are just serving up a fancy illusion.

The authors, Adil Rengim Cetingoz and Charles-Albert Lehalle, start by dropping a heavy truth bomb: throwing more dice doesn't fix a bad roll. They argue that if you train a model on a small amount of real data (like 30 years of daily stock prices), generating millions of fake data points won't magically make your predictions more accurate. In fact, it might make things worse. The model learns the "flaws" of the small sample size and then confidently repeats those flaws over and over again. It's like a student who memorizes the answers to a tiny practice test; if you give them a million copies of that test, they'll get a perfect score, but they still won't know how to solve a new problem. The paper suggests that in finance, you can't just "scale up" the data like you can with images or text; the initial sample size is a hard limit that you can't cheat.

Next, the authors tackle a weird mismatch between how these models learn and how investors actually think. Most AI models are trained to get the "big picture" right—they focus on the most obvious, loud patterns in the data, like the general rise and fall of the entire market. But for building a smart investment portfolio (especially one that bets on some stocks going up and others going down), the most important clues are often the quiet, subtle, low-risk patterns that the AI tends to ignore. It's as if the AI is a student who only studies the loud, dramatic chapters of a history book and skips the quiet footnotes, yet the teacher (the investor) needs the footnotes to pass the exam. The paper shows that standard AI models are terrible at capturing these subtle, low-risk details, which are exactly what you need to build a safe portfolio.

To fix this, the authors propose a new, custom-built pipeline for generating synthetic stock data. Instead of letting the AI guess everything at once, they break the problem down into two parts. First, they separate the "loud" market movements (the factors) from the "quiet" random noise (the residuals). They use a special type of AI called a Generative Adversarial Network (GAN) to learn the complex, time-based patterns of the loud factors, but they group similar factors together so the AI doesn't get overwhelmed. For the quiet noise, they use a simpler, more flexible math trick (a mixture of Student-t distributions) to capture the weird, heavy-tailed spikes in prices that standard models miss. They tested this new system on 433 stocks from the S&P 500. The results were promising: the fake data looked surprisingly real, capturing the same strange behaviors (like volatility clustering and heavy tails) as the real market.

However, the authors don't just stop at "it looks good." They introduce a clever new way to test if the model is actually smart or just a parrot. They call it "regurgitative training." Imagine teaching a student using a textbook you wrote, then asking them to write a new textbook based on what they learned, and finally checking if that new textbook can teach the original student. If the model can't learn from its own fake data to solve the original problem, it's not a good model. Using this test, they found that many standard models fail because they can't identify the underlying rules of the market, even when those rules are hidden in the data they generated themselves.

In short, this paper suggests that while we can't just generate infinite fake data to solve our problems, we can build better, more specialized tools if we stop treating all data the same. By respecting the limits of our small real-world samples and focusing on the quiet, subtle details that matter for investing, we can create synthetic data that is actually useful for navigating the chaotic, dice-rolling world of finance. The authors suggest that the future isn't about bigger models, but smarter, more tailored ones that understand the specific quirks of the market.

Technical Summary: Synthetic Data for Portfolios: A Throw of the Dice Will Never Abolish Chance

1. Problem Statement

The paper addresses the limitations of applying state-of-the-art generative models (such as GANs, VAEs, and Diffusion models) to financial portfolio construction and risk management. While these models have succeeded in text and image generation, their adoption in finance has been hindered by unique market complexities:

Data Scarcity vs. Dimensionality: Financial datasets (e.g., 30 years of daily returns) are small compared to the high dimensionality of asset universes (thousands of stocks), making "brute-force" scaling of model parameters ineffective.
The Sample Size Paradox: The authors argue that generating excessive synthetic data from a small initial training set does not improve statistical accuracy. Instead, it introduces a bias where the estimated statistics converge to the model's learned distribution (which is already biased by the small sample) rather than the true underlying distribution.
Misalignment with Portfolio Objectives: Standard generative models minimize distributional distances (e.g., Wasserstein distance) in the "natural space," causing them to prioritize learning high-variance principal components (PCs). However, modern portfolio construction (e.g., Markowitz optimization) relies heavily on the inverse of the covariance matrix, which amplifies errors in low-variance PCs. Consequently, models that accurately capture dominant market trends may fail to generate data suitable for constructing optimal portfolios.
Evaluation Deficits: There is a lack of rigorous, application-specific evaluation metrics. Existing methods often check for qualitative "stylized facts" (like volatility clustering) without verifying if the synthetic data performs well in specific financial tasks (e.g., mean-reversion strategies).

2. Methodology

Theoretical Foundations

The authors establish two key theoretical constraints:

Bias in Synthetic Estimation: Using U-statistics, they prove that if the learning accuracy ( $a_n$ ) of a model trained on $n$ samples is non-zero, increasing the synthetic sample size ( $\tilde{n}$ ) to infinity drives the probability of the synthetic statistic being close to the true statistic to zero. The error is dominated by the initial sample bias, not the variance of the synthetic sample.
Eigenvector Sensitivity: They demonstrate that portfolio weights are highly sensitive to perturbations in low-variance eigenvectors. Since standard generative losses focus on high-variance directions, they inherently neglect the components most critical for portfolio optimization.

Proposed Generative Pipeline

To address these issues, the authors propose a structured pipeline for generating multivariate asset returns, applied to a universe of 433 S&P 500 stocks:

Factor Decomposition (PCA):
- Standardize asset returns and compute the covariance matrix.
- Use Random Matrix Theory (Marcenko-Pastur distribution) to determine the number of significant factors ( $m$ ). Eigenvalues exceeding the theoretical noise threshold ( $\lambda_+$ ) are treated as factors; the rest are residuals.
- Decompose returns into factor-based returns ( $F$ ) and residual returns ( $Z$ ).
Modeling Factor Returns (Memory & Clustering):
- Rescaling: Factor returns are rescaled by their eigenvalues to have unit variance, preventing the model from being biased toward high-variance factors.
- Clustering: Factors are grouped into clusters based on statistical properties (skewness, kurtosis, eigenvalue, volatility clustering, leverage effect).
- GAN Training: Instead of training one GAN per factor (which is data-hungry), a single GAN is trained per cluster. The GANs use Temporal Convolutional Networks (TCNs) to capture time-series memory (e.g., volatility clustering and leverage effects).
Modeling Residual Returns (Marginal Distribution):
- Residuals are treated as independent and identically distributed (i.i.d.) across time and assets, as they lack significant cross-asset correlation or temporal dependence.
- However, to capture heavy tails and skewness, residuals are modeled using a mixture of Student-t distributions (or a single Student-t for simplicity), estimated via Maximum Likelihood.
Synthesis:
- Synthetic factor returns are generated by the cluster-specific GANs, rescaled by eigenvalues, and projected back to the asset space using the eigenvectors.
- Synthetic residuals are sampled from the fitted parametric distributions.
- The final synthetic returns are reconstructed by combining the projected factors and the residuals, then re-scaled by the original asset means and volatilities.

Evaluation Framework

The paper proposes a two-step evaluation methodology:

Stylized Fact Verification: Checking if the synthetic data reproduces cross-sectional and inter-temporal dependencies (volatility clustering, leverage effects) across different time scales.
Task-Specific Identifiability (Regurgitative Training): A novel test where the model is retrained on its own generated data. If the model cannot identify the underlying structure of the data it generated (i.e., it fails to converge to the same parameters), it indicates the model is not capturing the true data-generating process. This is applied to mean-reversion strategies to assess the Sharpe ratio stability across time scales.

3. Key Results

Theoretical Validation: The analysis confirms that generating more data than the initial sample size does not reduce estimation error; it merely concentrates the error around the biased model estimate.
Factor vs. Residual Characteristics: Empirical analysis of S&P 500 data reveals that:
- Factor-based returns exhibit strong volatility clustering and leverage effects, and negative skewness.
- Residual returns are largely uncorrelated and i.i.d. in terms of linear/non-linear dependence but retain heavy tails and some skewness.
Pipeline Performance:
- The proposed pipeline successfully generates 433-dimensional time series that match the marginal distributions (Wasserstein distance) of historical data, outperforming simple parametric fits (like Student-t) in out-of-sample scenarios.
- The model captures the non-linear temporal dependencies (volatility clustering and leverage effects) observed in the original data.
- Portfolio Relevance: Unlike standard GANs that fail to reproduce the structure needed for minimum variance portfolios (due to poor learning of low-variance PCs), the proposed pipeline, by decoupling factors and residuals and rescaling, aims to preserve the covariance structure relevant for portfolio construction.
Identifiability Test: The paper demonstrates that standard generative models often fail the "regurgitative training" test, suggesting they do not truly learn the underlying data generation process, whereas the proposed architecture shows promise in maintaining consistency.

4. Significance and Claims

The paper claims to provide a critical re-evaluation of the role of synthetic data in finance, moving beyond the "more data is better" paradigm. Its primary contributions are:

Theoretical Warning: It formally establishes that in high-dimensional, low-sample regimes, generating excessive synthetic data is counterproductive for statistical estimation due to inherent learning biases.
Architectural Alignment: It argues that generative models must be designed specifically for their downstream application. A model optimized for visual realism or general distributional fit is insufficient for portfolio construction, which requires precise modeling of low-variance components.
Practical Pipeline: It offers a concrete, modular architecture that separates the modeling of systematic risk (factors) from idiosyncratic risk (residuals), using appropriate tools (GANs for memory, parametric models for marginals) for each.
Evaluation Standard: It advocates for "identifiability" tests and task-specific metrics (like the Sharpe ratio of mean-reversion strategies) over generic distributional checks, arguing that a model's value is defined by its utility in specific financial applications rather than its ability to mimic historical data superficially.

The authors conclude that while synthetic data is a powerful tool, its utility in finance is bounded by the quality of the initial sample and the alignment of the model's objective function with the specific financial task at hand.