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Large and Deep Factor Models

Author

Listed:
  • Bryan T. Kelly

    (Yale SOM; AQR Capital Management, LLC; National Bureau of Economic Research (NBER))

  • Boris Kuznetsov

    (Swiss Finance Institute)

  • Semyon Malamud

    (Ecole Polytechnique Federale de Lausanne; Centre for Economic Policy Research (CEPR); Swiss Finance Institute)

  • Teng Andrea Xu

    (AQR Capital Management, LLC; École Polytechnique Fédérale de Lausanne (EPFL))

  • Yuan Zhang

    (Shanghai University of Finance and Economics)

Abstract

We show that a deep neural network (DNN) trained for stochastic discount factor (SDF) estimation admits a sharp additive decomposition that separates characteristics discovery from pricing rule estimation. The pricing-relevant component of this decomposition is governed by a new object, the Portfolio Tangent Kernel (PTK), which summarizes the features learned by the network and yields a linear factor representation for the SDF. In population, the PTKimplied SDF converges to a ridge-regularized version of the true SDF, where the effective degree of regularization is determined by the spectral complexity of the PTK. Using U.S. equity data, we show that the PTK representation delivers economically and statistically significant gains in SDF performance, but its spectral complexity has increased sharply over time-by roughly a factor of six since the early 2000s-coinciding with a deterioration in finite-sample pricing performance.

Suggested Citation

  • Bryan T. Kelly & Boris Kuznetsov & Semyon Malamud & Teng Andrea Xu & Yuan Zhang, 2026. "Large and Deep Factor Models," Swiss Finance Institute Research Paper Series 26-20, Swiss Finance Institute.
    • Handle: RePEc:chf:rpseri:rp2620
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    File URL: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6225778
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    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C45 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Neural Networks and Related Topics
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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