Interpretable Deep Learning for Stock Returns: A Consensus-Bottleneck Asset Pricing Model

TL;DR

The paper tackles the interpretability-accuracy tension in asset pricing by introducing CB-APM, a partially interpretable neural network that compresses firm and macro information into analyst-consensus-like latent states and then maps them to expected returns. By jointly training a nonlinear consensus bottleneck with a linear pricing stage, CB-APM achieves strong long-horizon prediction and provides transparent loadings that resemble priced components of analyst beliefs. Empirical results show substantial improvements in out-of-sample return forecasts, robust long-short portfolio profitability, and pricing content beyond traditional factor models, aided by macroeconomic embeddings learned via an autoencoder. The work demonstrates that integrating interpretable architecture with rational-expectations-based information aggregation can yield both predictive gains and economically meaningful insights, paving the way for more trustworthy, finance-grounded machine learning models.

Abstract

We introduce the \textit{Consensus-Bottleneck Asset Pricing Model} (CB-APM), a partially interpretable neural network that replicates the reasoning processes of sell-side analysts by capturing how dispersed investor beliefs are compressed into asset prices through a consensus formation process. By modeling this ``bottleneck'' to summarize firm- and macro-level information, CB-APM not only predicts future risk premiums of U.S. equities but also links belief aggregation to expected returns in a structurally interpretable manner. The model improves long-horizon return forecasts and outperforms standard deep learning approaches in both predictive accuracy and explanatory power. Comprehensive portfolio analyses show that CB-APM's out-of-sample predictions translate into economically meaningful payoffs, with monotonic return differentials and stable long-short performance across regularization settings. Empirically, CB-APM leverages consensus as a regularizer to amplify long-horizon predictability and yields interpretable consensus-based components that clarify how information is priced in returns. Moreover, regression and GRS-based pricing diagnostics reveal that the learned consensus representations capture priced variation only partially spanned by traditional factor models, demonstrating that CB-APM uncovers belief-driven structure in expected returns beyond the canonical factor space. Overall, CB-APM provides an interpretable and empirically grounded framework for understanding belief-driven return dynamics.

Figures (18)

• Figure 1: Disentangled representations of neural network. This schematic illustrates how a high-dimensional input is compressed into a small set of latent representations that correspond to interpretable concepts. The example shown mirrors the yield curve decomposition in fixed income, where dozens of yields can be summarized by three factors: level, slope, and curvature nelson1987parsimonious. The disentangled representation isolates these dimensions, which are then mapped by the output layer into the final prediction, here defined as the yield curve.
• Figure 2: Architecture of the Consensus-Bottleneck Asset Pricing Model (CB-APM). The model is composed of two modules, the consensus module $f(\phi)$ (left) and the prediction module $g(\theta)$ (right). The consensus module compresses firm-specific predictors $I^f_{i,t}$ and macroeconomic variables $I^m_t$ into a lower-dimensional consensus vector $\hat{C}_{i,t}$ through a feedforward neural network. This bottleneck enforces interpretability by design, as each coordinate of $\hat{C}_{i,t}$ is treated as a consensus concept. The prediction module then maps these consensus variables into expected excess returns $E_t[R_{i,t+h}]$ using a linear layer. The return loss $L_R(\phi,\theta)$ and the consensus loss $L_c(\phi)$ are optimized jointly using the weighted sum $L=\lambda L_c + L_R$, ensuring that the consensus layer is both predictive of returns and interpretable.
• Figure 3: Gaussian Error Linear Unit (GELU) activation function. GELU is a smooth nonlinear activation that combines properties of the ReLU and sigmoid functions.
• Figure 4: Autoencoder-based macroeconomic embedding. The encoder narrows horizontally to compress high-dimensional macroeconomic inputs into a latent state $\mathbf{z}_t$, concatenated with firm-level features for return prediction. The decoder is used only during training for reconstruction loss.
• Figure 5: Expanding window evaluation. This figure illustrates the expanding-window procedure used for model evaluation. At each iteration, the available data are divided into three subsets: i@ (training set), ii@ (validation set), and iii@ (test set). The training set expands over time, while the validation and test sets are fixed in length at two years and one year, respectively.