Features

Deep learning in empirical asset pricing

By Dr Gavin Feng

Dr Gavin Feng is an Assistant Professor of Business Statistics at the Department of Management Sciences. In this article, Feng focuses on the interdisciplinary research between deep learning and asset pricing factor investing. The particular innovation is understanding the discovery and construction of asset pricing factors with a bottom-up deep learning model. This article is based on his co-authored paper "Deep learning in characteristics-sorted factor models" with Nicholas Polson and Jianeng Xu from the University of Chicago.

Models for stock returns include Nobel Prize research such as the Capital Asset Pricing Model and the Fama-French 3-factor model.

Asset pricing models study why different assets attract different expected returns. Examples of such models for stock returns include Nobel Prize research such as the CAPM (Capital Asset Pricing Model) and the Fama-French 3-factor model. In 2020, we present a bottom-up approach based on deep learning applied to the construction of asset pricing models, which include firm characteristics (inputs), risk factors (intermediate features), and security returns (outputs). The question addressed using deep learning, one special method in machine learning, is how to improve asset pricing models to explain the cross-sectional average returns.

According to ICAPM of Merton (1973), a combination of common tradable factors captures the crosssection of expected returns, and the regression intercept should be zero.

Ri,t = αi + β1,i ∗ 1,t + · · · + βk,i ∗ fk,t + Ei,t

Therefore, the model fitness for asset pricing is not about the explained variation in time series, but the magnitude of intercepts, alphas, in the cross-section. This non-arbitrage restriction on alphas implies that simply adding factors leads to statistical overfitting (time series R2) but does not cause economic overfitting (intercepts).

Researchers typically sort securities on firm characteristics and create long-short portfolios as common risk factors to build asset pricing models.

In empirical studies, researchers typically sort securities on firm characteristics and create long-short portfolios as common risk factors to build asset pricing models. The goal is to explain the time-series variation of multiple asset returns and their average returns' cross-sectional variation. For example, Fama and French (1993) add SMB (small-minusbig) and HML (high-minus-low) to CAPM. However, in the asset pricing literature, almost all proposed factor models have rejected the zeroalpha hypothesis. Therefore, we want to approach this puzzle, with a machine learning perspective, as an optimisation problem: How does one construct a factor model to minimise pricing errors or alphas?

How does one construct a factor model to minimise pricing errors or alphas?

The goal of their paper is to investigate the underlying mechanism of the characteristicssorted factor models, which includes sorting securities, generating factors, and fitting the cross section of security returns. The particular focus is the cross-sectional variation of asset average returns. They define an non-arbitrage objective function, pricing errors, for the optimization problem. They show the characteristics-sorted factor models can be dissembled as a deep learning architecture (see Figure above).

This figure provides a deep learning representation of building the Fama-French 5-factor model using firm characteristics to calculate the objective function, pricing errors, for portfolio returns. The lag characteristics are inputs. The longshort factors are hidden neurons. The portfolio returns are outputs.

(1) Inputs are firm characteristics. The neural network starts from sorting securities on firm characteristics, which is a non-linear activation to create long-short portfolio weights.

(2) Intermediate features are risk factors. The factors are linear activations (long-short portfolio weights) on realised returns from the sorting directions.

(3) Outputs are security returns. Minimising an economic objective function is equivalent to minimising pricing errors for fitting the factor model to portfolio or individual stock returns.

The focus is on "training a factor model" rather than "testing a factor or characteristic."

Distinct from the literature on stochastic discount factors, we focus on training a factor model rather than testing a factor or characteristic. Apart from the PCA literature, their innovation is to apply dimension reduction on firm characteristics (inputs) rather than the characteristics-sorted factors (intermediate features). We argue the current literature is mostly about intermediate features and outputs (security returns), whereas ours illustrates the complete channel between inputs and outputs. We adopt a non-reduced-form neural network and develop such a bottomup approach that includes security sorting, factor generation, and fitting the cross-section of security returns. The Fama-French-type characteristicssorted factor models can be shown as "shallow" learning models.

Dr Gavin Feng
Assistant Professor
Department of Management Sciences