IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2402.06635.html

Large and Deep Factor Models

Author

Listed:
  • Bryan Kelly
  • Boris Kuznetsov
  • Semyon Malamud
  • Teng Andrea Xu
  • Yuan Zhang

Abstract

We show that a deep neural network (DNN) trained to construct a stochastic discount factor (SDF) admits a sharp additive decomposition that separates nonlinear characteristic discovery from the pricing rule that aggregates them. The economically relevant component of this decomposition is governed by a new object, the Portfolio Tangent Kernel (PTK), which captures the features learned by the network and induces an explicit linear factor pricing representation for the SDF. In population, the PTK-implied SDF converges to a ridge-regularized version of the true SDF, with the effective strength of regularization determined by the spectral complexity of the PTK. Using U.S. equity data, we show that the PTK representation delivers large and statistically significant performance gains, while its spectral complexity has risen sharply-by roughly a factor of six since the early 2000s-imposing increasingly tight limits on finite-sample pricing performance.

Suggested Citation

  • Bryan Kelly & Boris Kuznetsov & Semyon Malamud & Teng Andrea Xu & Yuan Zhang, 2024. "Large and Deep Factor Models," Papers 2402.06635, arXiv.org, revised Feb 2026.
    • Handle: RePEc:arx:papers:2402.06635
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2402.06635
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Martin Lettau & Markus Pelger & Stijn Van Nieuwerburgh, 2020. "Factors That Fit the Time Series and Cross-Section of Stock Returns," The Review of Financial Studies, Society for Financial Studies, vol. 33(5), pages 2274-2325.
    2. Serhiy Kozak & Stefan Nagel, 2023. "When Do Cross-Sectional Asset Pricing Factors Span the Stochastic Discount Factor?," NBER Working Papers 31275, National Bureau of Economic Research, Inc.
    3. Doron Avramov & Si Cheng & Lior Metzker, 2023. "Machine Learning vs. Economic Restrictions: Evidence from Stock Return Predictability," Management Science, INFORMS, vol. 69(5), pages 2587-2619, May.
    4. Mark Britten‐Jones, 1999. "The Sampling Error in Estimates of Mean‐Variance Efficient Portfolio Weights," Journal of Finance, American Finance Association, vol. 54(2), pages 655-671, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Fernández-Avilés, Gema & Mattera, Raffaele & Scepi, Germana, 2024. "Factor-Augmented Autoregressive Neural Network to forecast NOx in the city of Madrid," Socio-Economic Planning Sciences, Elsevier, vol. 95(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lin William Cong & Guanhao Feng & Jingyu He & Xin He, 2022. "Growing the Efficient Frontier on Panel Trees," NBER Working Papers 30805, National Bureau of Economic Research, Inc.
    2. Cong, Lin William & Feng, Guanhao & He, Jingyu & He, Xin, 2025. "Growing the efficient frontier on panel trees," Journal of Financial Economics, Elsevier, vol. 167(C).
    3. Wu, Hongxu & Wang, Qiao & Li, Jianping & Deng, Zhibin, 2025. "Enhancing stock return prediction in the Chinese market: A GAN-based approach," Research in International Business and Finance, Elsevier, vol. 75(C).
    4. Yilie Huang & Yanwei Jia & Xun Yu Zhou, 2024. "Mean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study," Papers 2412.16175, arXiv.org, revised Mar 2026.
    5. Doron Avramov & Xin He, 2026. "Stochastic Discount Factors with Cross-Asset Spillovers," Papers 2602.20856, arXiv.org.
    6. Muhinyuza, Stanislas & Bodnar, Taras & Lindholm, Mathias, 2020. "A test on the location of the tangency portfolio on the set of feasible portfolios," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    7. Mishra, Anil V., 2016. "Foreign bias in Australian-domiciled mutual fund holdings," Pacific-Basin Finance Journal, Elsevier, vol. 39(C), pages 101-123.
    8. Andrew F. Siegel & Artemiza Woodgate, 2007. "Performance of Portfolios Optimized with Estimation Error," Management Science, INFORMS, vol. 53(6), pages 1005-1015, June.
    9. Sebastiano Michele Zema & Giorgio Fagiolo & Tiziano Squartini & Diego Garlaschelli, 2025. "Mesoscopic structure of the stock market and portfolio optimization," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 20(2), pages 307-333, April.
    10. Chai, Bailin & Jiang, Fuwei & Lin, Yihao & You, Tian, 2025. "Predicting bond risk premiums with machine learning: Evidence from China," Pacific-Basin Finance Journal, Elsevier, vol. 93(C).
    11. Lee, King Fuei, 2021. "An Anomaly within an Anomaly: The Halloween Effect in the Long-term Reversal Anomaly," MPRA Paper 110859, University Library of Munich, Germany.
    12. José Afonso Faias & Juan Arismendi Zambrano, 2022. "Equity Risk Premium Predictability from Cross-Sectoral Downturns [International asset allocation with regime shifts]," The Review of Asset Pricing Studies, Society for Financial Studies, vol. 12(3), pages 808-842.
    13. Cong Wang, 2024. "Stock return prediction with multiple measures using neural network models," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-34, December.
    14. Liu, Yunting & Zhu, Yandi, 2025. "Good idiosyncratic volatility, bad idiosyncratic volatility, and the cross-section of stock returns," Journal of Banking & Finance, Elsevier, vol. 170(C).
    15. Juan-Pedro Gómez & Fernando Zapatero, 2001. "Asset pricing implications of benchmarking: A two-factor CAPM," Economics Working Papers 693, Department of Economics and Business, Universitat Pompeu Fabra.
    16. Andrew Y. Chen, 2019. "The Limits of p-Hacking : A Thought Experiment," Finance and Economics Discussion Series 2019-016, Board of Governors of the Federal Reserve System (U.S.).
    17. Maillet, Bertrand & Tokpavi, Sessi & Vaucher, Benoit, 2015. "Global minimum variance portfolio optimisation under some model risk: A robust regression-based approach," European Journal of Operational Research, Elsevier, vol. 244(1), pages 289-299.
    18. Viet Anh Nguyen & Fan Zhang & Shanshan Wang & Jose Blanchet & Erick Delage & Yinyu Ye, 2021. "Robustifying Conditional Portfolio Decisions via Optimal Transport," Papers 2103.16451, arXiv.org, revised Apr 2024.
    19. Chen, Andrew Y. & McCoy, Jack, 2024. "Missing values handling for machine learning portfolios," Journal of Financial Economics, Elsevier, vol. 155(C).
    20. Alejandro Rodriguez Dominguez, 2025. "Is Causality Necessary for Efficient Portfolios? A Computational Perspective on Predictive Validity and Model Misspecification," Papers 2507.23138, arXiv.org, revised Feb 2026.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2402.06635. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.