Stochastic Discount Factors with Cross-Asset Spillovers

This paper develops a unified framework that links firm-level predictive signals, cross-asset spillovers, and the stochastic discount factor (SDF). Signals and spillovers are jointly estimated by maximizing the Sharpe ratio, yielding an interpretable SDF that both ranks characteristic relevance and uncovers the direction of predictive influence across assets. Out-of-sample, the SDF consistently outperforms self-predictive and expected-return benchmarks across investment universes and market states. The inferred information network highlights large, low-turnover firms as net transmitters. The framework offers a clear, economically grounded view of the informational architecture underlying cross-sectional return dynamics.

💡 Research Summary

**
This paper introduces a unified framework that simultaneously incorporates firm‑level predictive signals and cross‑asset spillover effects into a single stochastic discount factor (SDF) optimized for the Sharpe ratio. Traditional asset‑pricing literature largely assumes self‑predictability—each firm’s own characteristics forecast its own returns—while a growing body of work documents cross‑predictability, where characteristics or returns of one asset help predict another. The authors bridge these strands by defining a linear trading strategy ωₜ = Λ′Sₜ′Ψ, where Sₜ is an N × M matrix of standardized firm signals, Λ is an M‑dimensional vector of signal loadings, and Ψ is an N × N “spillover matrix” that captures how signals from asset i affect positions in asset j.

The objective is to maximize the Sharpe ratio of the resulting portfolio. By normalizing Λ and the vectorized Ψ (Φ = vec(Ψ′)) to have unit Euclidean norm, the problem reduces to a generalized eigenvalue problem. The authors employ ridge‑type regularization with a single tuning parameter chosen by five‑fold cross‑validation, ensuring numerical stability in high‑dimensional settings. In contrast, an expected‑return‑maximization benchmark leads to a bilinear problem with closed‑form solutions but tends to concentrate on a single predictor, whereas Sharpe‑ratio maximization naturally diversifies across signals.

Empirically, the methodology is applied to two extensive US equity universes: (1) 138 univariate “spread” portfolios constructed from the Jensen et al. (2023) dataset, and (2) 544 bivariate portfolios sorted by size and a secondary characteristic. Using a rolling 10‑year estimation window, the Sharpe‑optimal SDF delivers annualized Sharpe ratios of 2.21 for the spread portfolios and 3.32 for the bivariate portfolios. By comparison, a self‑predictive Sharpe‑ratio benchmark achieves only 0.60, and a maximum‑expected‑return strategy yields Sharpe ratios between 0.5 and 1.0.

Robustness checks split the out‑of‑sample period by investor sentiment and VIX‑based volatility regimes. The Sharpe‑optimal strategy maintains Sharpe ratios above 2 in all sub‑samples, whereas the expected‑return benchmark exhibits pronounced state‑dependence. Alpha tests against a comprehensive set of factor models—including the Fama‑French five‑factor, Hou‑et‑al. q‑factors, liquidity, behavioral, and a fourteen‑factor specification—show statistically significant alphas of about 0.25 % per month (t‑statistics > 11). This indicates that the cross‑asset spillover information captured by the SDF is not priced by existing models.

Analysis of the estimated Λ reveals that signals related to investment, value, and profitability receive the highest weights, while return‑based signals such as momentum, short‑term reversal, and seasonality receive negligible weight. The average off‑diagonal element of Ψ exceeds the average diagonal element, suggesting that cross‑asset predictive linkages contain more information than self‑predictive signals. Network visualizations identify large, low‑turnover firms as net transmitters of predictive information, whereas smaller, high‑turnover firms act as net receivers. This pattern aligns with economic mechanisms such as supply‑chain relationships, peer effects, and institutional trading pressure.

Temporal analysis shows peak performance in the 1990s (Sharpe > 2 for spreads, > 4 for bivariate portfolios) with a gradual decline after 2000, mirroring the documented erosion of self‑predictability in the literature. Nonetheless, from 2000 to 2023 the strategy still outperforms traditional benchmarks by a factor of 3–5 in Sharpe terms.

In sum, the paper makes three major contributions: (1) it proposes a tractable, analytically transparent SDF that jointly models multiple firm signals and cross‑asset spillovers; (2) it demonstrates that Sharpe‑ratio maximization yields a diversified, robust portfolio that substantially outperforms both self‑predictive and expected‑return‑maximizing strategies; and (3) it provides empirical evidence that cross‑asset predictive networks contain economically meaningful information not captured by standard factor models. Future extensions could explore nonlinear signal transformations, real‑time updating of the spillover matrix, and application to international markets.