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Multilayer Perceptron Neural Network Models in Asset Pricing: An Empirical Study on Large-Cap US Stocks

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that a two-hidden-layer dynamic pyramid MLP prices large-cap US stocks best out of sample, and that MLP factor investing matters mainly for downside risk control.

desk verdict Plausible model comparison, but survivorship bias and a mislabeled alpha undermine the portfolio conclusions; worth refereeing with major revision. read the letter →

arxiv 2505.01921 v3 pith:ALGJE4VT submitted 2025-05-03 q-fin.PR q-fin.CPq-fin.RM

Shanyan Lai This is my paper · ORCID
classification q-fin.PRq-fin.CPq-fin.RM
keywords assetpricingmultilayerperceptronneuralnetworkfactorinvestingout-of-sampleR-squareddownsiderisklarge-capUSstocksCOVID-19
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether multilayer perceptron networks can price individual large-cap US stocks from 182 characteristic-sorted portfolio factors, and how their structure should be chosen. Its central claim is that a two-hidden-layer MLP with dynamically sized layers (36 and 6 neurons) achieves the best out-of-sample fit, with an average R-squared of 3.66%, beating deeper networks, the fixed-shape benchmark network from the 2020 study, and linear methods such as OLS, PLS, and PCR. The second claim is that MLP factor investing does not beat buy-and-hold in absolute annual returns but is valuable for downside risk control: the winning model posts the lowest maximum drawdown and the highest risk-adjusted ratios in the full-period equal-weighted test. If the claims hold, practitioners should stop assuming 'deeper is better' for factor pricing and should evaluate neural factor strategies primarily by drawdown and downside-risk metrics.

What carries the argument

The carrying mechanism is the dynamic pyramid MLP: a fully connected network whose hidden-layer widths shrink geometrically from the 182-factor input toward the single output, with the width schedule set by a formula that depends on the total number of hidden layers. For two hidden layers this produces 36 and 6 neurons, the configuration with the best out-of-sample fit. Estimation uses MSE loss, ReLU activation, adaptive-moment gradient optimization, early stopping, batch normalization, and L1 regularization; predictions are scored by out-of-sample R-squared and a pairwise forecast-error test, then turned into long-only sign-based equal- and value-weighted portfolios.

What would settle it

Re-run the backtest on a point-in-time universe that includes stocks that later delisted or stopped reporting; if the two-layer MLP's Sharpe ratio, Sortino ratio, and maximum drawdown advantage over buy-and-hold shrinks or flips, the downside-risk claim would be refuted.

Watch

Extended reading notes

Core claim

The paper reports that, on 420 large-cap US stocks with 182 characteristic-sorted portfolio factors, a two-hidden-layer MLP whose widths are set by a dynamic pyramid rule achieves an average out-of-sample R-squared of 3.66% over 2013–2021, and 2.16% when the COVID-19 months are removed. This beats every alternative tested, including one- and three-layer dynamic networks, five-layer networks that fall to −1.04%, the fixed-shape benchmark network from the 2020 study, and OLS, PLS, and PCR. The author interprets the pattern as evidence that deeper MLPs overfit at this data size. In the long-only backtest, all models earn positive and significant alphas, yet none beats buy-and-hold annual returns; the two-layer model instead posts the lowest maximum drawdown (39.03%) and the highest Sharpe and Sortino ratios in the equal-weighted full-period test, supporting the paper's conclusion that MLP factor investing is mainly a downside-risk management tool.

Load-bearing premise

The load-bearing premise is that the 420 stocks with no missing data through the full test period are the same universe an investor could have traded from 2013, even though firms that delisted or stopped reporting were excluded.

Editorial extensions

If this is right

  • The optimal depth for MLP factor models at this data scale is two or three hidden layers; networks with five layers go to negative out-of-sample R-squared.
  • Practitioners should evaluate MLP factor strategies by Sharpe ratio, Sortino ratio, and maximum drawdown rather than annual return, since none of the tested models beat buy-and-hold in absolute return.
  • Including the COVID-19 months in the test window improves the proposed models' out-of-sample fit, indicating the dynamic pyramid networks stay usable in extreme market moves.
  • Value-weighting the portfolio lowers maximum drawdown further, with the three-layer model reaching 33.04% in the full testing period.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: a point-in-time tradable universe that includes delisted stocks would likely reduce the reported Sharpe and Sortino ratios, so the downside-risk advantage is best read as an upper bound.
  • Inference: the dynamic pyramid width rule should be portable to other markets; the depth result implies the best architecture depends on the data-to-parameter ratio, so smaller samples should favor even shallower networks.
  • Inference: adding a signal filter that requires a minimum predicted return before opening a long position may recover some of the shortfall against buy-and-hold in strong uptrends, where the paper finds unfiltered sign signals fail.
  • Inference: because Announcement Return, Earnings Forecast Disparity, and Size dominate variable importance across models, an ablation study using fewer than 182 factors could test whether the remaining factor zoo adds predictive value.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper applies multilayer perceptron (MLP) models with a dynamic pyramid structure to 182 firm-characteristic-sorted portfolio factors to forecast excess returns of 420 large-cap US stocks over 2013-2021, extending the GKX (2020) framework. The main empirical claims are (i) a two-hidden-layer MLP achieves the highest out-of-sample R2 (3.66%) and outperforms deeper MLPs, OLS, PLS, PCR, and fixed-shape GKX models; (ii) the COVID-19 period improves MLP OOS fit; and (iii) MLP factor investing is more valuable for downside-risk control than for absolute returns, based on backtests of a long-only signal strategy. The paper also reports variable importance and compares equal- and value-weighted portfolios.

Significance. If the results held, the paper would provide a useful practical data point for ML asset pricing: it shows that with a limited cross-section and a relatively short OOS window, shallow dynamic MLPs can be more effective than deeper fixed architectures, and that factor-sorted portfolio characteristics can substitute for a much larger predictor set. The within-universe OOS R2 comparison and DM tests are appropriate tools, and the paper is transparent about its data sources and provides pseudocode for its optimization procedures. However, the backtest and downside-risk conclusions rest on a survivorship-biased universe, the alpha measure is mislabeled and does not measure risk-adjusted performance, and the architecture is selected on the same test set used for reporting, so the headline claims require substantial revision before they can be accepted as stated.

major comments (4)
  1. [Section 3, Tables 6-7] The stock selection criterion 'having no missing data in the testing period' is an ex post survivorship filter: an investor at the start of the test period cannot know which stocks will survive through 2021. This biases the backtest metrics (Sharpe, Sortino, MDD, alpha) in Tables 6 and 7, because stocks that crashed or delisted during the test window are mechanically excluded. The paper's justification using Lehman Brothers is temporally misplaced: Lehman failed in 2008, which falls in the validation window (2/2003-12/2012), not the 2013-2021 test window. Consequently, the Section 6 conclusion that MLP factor investing is mainly useful for downside-risk control is not established for a point-in-time tradable universe. The R2 ranking in Table 3 is less affected because all models share the same survivor universe, but the backtest conclusions need to be re-run on an investable universe or explicitly reframed as conditional on survival.
  2. [Section 5.2, Table 3] The best architecture (fw2, two hidden layers) is selected on the basis of the highest OOS R2 computed on the same 2013-2021 test period used to report results and to run DM tests. This creates a selection-on-the-test-set bias: the reported R2 gap for fw2 and the DM test significances against other models are inflated because the same data were used to choose the architecture. The paper should either use a separate validation period for architecture selection, or honestly report that the R2 values are conditional on in-sample selection and adjust the inference accordingly.
  3. [Section 5.3, Equation (43)] Equation (43) defines alpha as the difference between the expected out-of-sample excess return and the expected predicted excess return. This is not Jensen's alpha, which is the intercept from a time-series regression of portfolio excess returns on factor exposures. As a result, the alpha values and t-statistics in Tables 6 and 7 do not measure risk-adjusted performance or factor profitability; they merely say that the average predicted return is lower than the average realized return, which is not an economically meaningful performance metric. The interpretation that 'all models have significant positive alphas, which indicates the extra gain from factors' is therefore unsupported.
  4. [Sections 4.1-4.4] The empirical results are not reproducible because the hyperparameter values are not reported. The paper mentions L1 regularization (Equation (34)), early stopping, batch normalization, and Adam, and Appendix A gives pseudocode, but the actual values used (learning rate, batch size, maximum epochs, early-stopping patience, regularization strength lambda, and any hyperparameter tuning procedure) are missing. Given that the central claim is a comparative empirical evaluation, the absence of these details prevents verification and makes the results sensitive to unspecified choices.
minor comments (6)
  1. [Equation (21)] The dynamic pyramid formula appears to contain a typo: 'O(l0)' is not clearly defined, and the neuron counts in Table 2 (e.g., 36 and 6 for two hidden layers) do not obviously follow from the formula as printed. Please correct the notation and verify the computation.
  2. [Section 5.1, trading signal rule] The description of the trading signal is ambiguous: 'if the sign of the actual excess return and predicted excess return are both positive, then long the stock from the next value.' Please clarify whether the actual return used is the realized return at time t (a momentum filter) or the contemporaneous return at time t+1 (which would introduce look-ahead bias).
  3. [Section 5.3, value-weighted portfolio] The statement that 'portfolio weighting has no impact on extra gain alpha' and that the value-weighted alpha equals the equal-weighted alpha is incorrect for a weighted average of stock-level alphas; the value-weighted alpha is a weighted average, not generally equal to the equal-weighted alpha. This should be corrected or clarified.
  4. [Table 3] The reported average R2 and average MSE are non-monotonic across models (e.g., fw2 and pcr share the same average MSE but differ greatly in average R2). The text explains this with the multiple-labels caveat, but it would aid the reader to also report the cross-sectional dispersion of the underlying variances or per-stock R2 distributions.
  5. [Section 3] The date at which the top-15% market-capitalization sort is applied is not specified; please state whether the sort is point-in-time or uses full-sample information, because this affects the interpretation of the selected universe.
  6. [Throughout] There are numerous typos and formatting issues, including 'Univiersity', 'gamil.com', 'Rresearch', 'Shape Ratio' for 'Sharpe Ratio', and inconsistent author formatting for the Chen-Zimmermann reference. A careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's empirical comparisons, model architectures, and backtests are self-contained evaluations against external benchmarks; the best-depth finding is a test-set result, not a fitted input renamed as a prediction.

full rationale

The paper's derivation chain is empirical rather than definitional. The factor model is specified as ri,t+1 = g(fi,t; θ) + εi,t+1 (Eq. 5), with the MLP function form defined by standard forward-propagation equations and a pyramid width formula taken from Coqueret and Guida [19]; the output (one stock's excess return) and inputs (182 firm characteristic-sorted factors from Chen et al. [18]) are distinct quantities, and no equation defines one in terms of the other. Out-of-sample R2 (Eq. 35) uses the training-period mean as the benchmark, so it is a standard predictive accuracy measure and does not embed the test outcomes it evaluates. The DM test compares held-out absolute forecast errors between models, and the portfolio metrics (Sharpe, Sortino, MDD, alpha) are computed directly from predicted and realized returns; none of these quantities is fitted to a subset and then relabeled as a prediction. The central finding that the 2-hidden-layer MLP has the highest average OOS R2 is a horse-race result on the fixed testing window, not a parameter calibrated on that window and then called a forecast; selecting the best performer after seeing the test results may raise overfitting or multiple-comparisons concerns, but that is a validity issue, not circularity. The survivor-universe filter in Section 3 is an ex post selection bias that affects the generalizability of the backtests, yet all models are evaluated on the same universe and the paper's comparative claims are not derived from the filter by construction. Citations to GKX2020 and Coqueret and Guida are external references, and the single odd phrase 'As one of the developers of the GKX2020's work' is not load-bearing for any result. No step reduces by the paper's own equations to its inputs, so the appropriate finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper is an empirical ML study with no theoretical derivation. The central claim depends on the survivor-biased stock universe, the borrowed dynamic pyramid architecture, and several unreported hyperparameters; these are listed as free parameters and assumptions. No new entities are introduced.

free parameters (3)
  • Number of hidden layers L = 2 (selected by highest OOS R2)
    The paper compares L=1 to 5 and chooses 2 based on the same test set used for evaluation. This data-driven selection is a free choice that affects the central result.
  • L1 regularization strength λ = Not reported
    Equation (34) includes the penalty λ||θ||_1, but the paper does not state λ's value or tuning procedure, making the model non-replicable.
  • Batch size, epoch count, early-stopping patience = Not reported
    Training details beyond the Adam defaults (learning rate 0.001, beta1 0.9, beta2 0.999, epsilon 1e-8) are missing; these affect converged model quality and are effectively free parameters.
assumptions (4)
  • domain assumption The 420 selected stocks must have no missing data over the full testing period (the 'going concern' condition), so firms that delisted are excluded.
    Section 3 imposes this data-availability filter to define the universe. The backtest conclusions depend on this survivor-biased sample.
  • domain assumption The 182 characteristic-sorted portfolio factors from Andrew Y.; Zimmermann (2020) are valid, point-in-time predictors of future excess returns.
    Section 3 uses these factors as features without independent verification of look-ahead bias or alignment with the return timing.
  • ad hoc to paper The dynamic pyramid architecture formula (Equation 21) taken from Coqueret and Guida is an appropriate way to set hidden-layer sizes.
    The paper borrows this heuristic from prior work and uses it for all models; it is not derived or justified from first principles.
  • domain assumption The additive prediction error model with E_t[r_{i,t+1}]=0 (Equation 4) is assumed before introducing predictors.
    This standard conditional-mean assumption underlies the loss function and evaluation; it is invoked in Section 4.

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Cite this review

Pith. "Pith review of Multilayer Perceptron Neural Network Models in Asset Pricing: An Empirical Study on Large-Cap US Stocks." pith.science (2026). https://pith.science/paper/ALGJE4VT

@misc{pith2026250501921,
  author       = {Pith},
  title        = {Pith review of: Multilayer Perceptron Neural Network Models in Asset Pricing: An Empirical Study on Large-Cap US Stocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALGJE4VT}},
  note         = {Machine review of arXiv:2505.01921}
}
read the original abstract

In this study, MLP models with dynamic structure are applied to factor models for asset pricing tasks. Concretely, the MLP pyramid model structure was employed on firm characteristic-sorted portfolio factors for modelling the large-cap US stocks. It was further developed as a practical factor investing strategy based on the predictions. The main findings were evaluated from 2 angles: model predictive power and backtesting performance, which were compared for the periods with and without COVID-19. The empirical results indicated that, given the constraints of the data size, the MLP models no longer perform 'deeper, better' in terms of predictive power, whereas the proposed MLP models with 2 and 3 hidden layers have greater flexibility in modelling the factors in this case. This study also verified the idea from previous work that MLP models for factor investing are more meaningful for downside risk control than for pursuing absolute annual returns.

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Forward citations

Cited by 1 Pith paper

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  1. Is attention truly all we need? An empirical study of asset pricing in pretrained RNN sparse and global attention models

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    Pre-trained RNN models with global self-attention or sliding-window sparse attention deliver the highest value-weighted Sortino ratios (2.0 and 1.80) in a COVID-era backtest of 420 large-cap US stocks.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.