Is Causality Necessary for Efficient Portfolios? A Computational Perspective on Predictive Validity and Model Misspecification

Abstract

Portfolio optimization is increasingly argued to require causally identified return predictors to avoid signal inversion and optimization failure. This paper re-examines this claim by studying when predictive signals yield viable efficient frontiers, even under structural misspecification. We show that causal identification is not necessary for portfolio efficiency within static mean--variance and closely related quadratic portfolio optimization frameworks. Instead, efficiency is governed by geometric sufficiency conditions on predictive signals: directional alignment, ranking preservation, and calibration. We formally decompose portfolio efficiency into these three components and show that miscalibration alone attenuates Sharpe ratios even when alignment and ranking are preserved. Robustness is characterized as smooth degradation rather than collapse, with explicit attenuation behavior and continuity of performance under increasing misspecification. The theoretical results are supported by simulations and empirical analysis. Empirical validation combines equity-based illustrations with a large global bond universe spanning multiple currencies, countries, sectors, maturities along the term structure, seniority classes, and credit ratings, together with high-dimensional stress tests, nonlinear data-generating processes, rolling-window analyses, covariance regularization, realistic portfolio constraints, and bootstrap-based statistical validation. Across these settings, optimization geometry remains well-behaved whenever directional alignment is preserved. The results clarify the boundary between causality and portfolio optimization: causality may inform signal representation, but portfolio efficiency at the optimization stage is a geometric property conditional on a given representation.

Figures (13)

• Figure 1: Logical structure of the framework and the paper. Predictive models—whether causal or non-causal—produce surrogate signals whose geometric properties determine optimization behavior. Structural misspecification primarily induces attenuation through miscalibration rather than generic inversion. Portfolio efficiency is governed by alignment, ranking, and calibration within a given risk--return geometry, with robustness characterized by smooth degradation rather than frontier collapse.
• Figure 2: Portfolio Weights under Structural Cancellation. Scatter of optimal portfolio weights obtained from the true expected return vector (x-axis) against weights obtained from a structurally misspecified signal (y-axis). Each point corresponds to an asset. The preservation of sign across weights indicates directional alignment, while attenuation of magnitudes reflects miscalibration. This validates that misspecification need not invalidate diversification, but primarily reduces achievable risk-adjusted returns.
• Figure 3: Calibration and ranking under structural misspecification. Monte Carlo validation of the theoretical results in Section \ref{['sec:optundermiss']}. Panel (a) shows that monotone transformations preserve asset ranking, ensuring feasibility and efficient-set membership (Lemma \ref{['lem:decomposition']}). Panel (b) shows that miscalibration alone suffices to attenuate risk-adjusted performance, with Sharpe ratios contracting smoothly as calibration deteriorates (Proposition \ref{['prop:sharpe_cosine']}). Together, the panels demonstrate that ranking secures validity of optimization, while calibration determines the attainable position along the efficient frontier.
• Figure 4: Directional robustness under nonlinear misspecification.Scatter of portfolio weights implied by the true return signal (x-axis) versus weights implied by a misspecified predictive signal (y-axis) under a nonlinear confounded data-generating process. Each point corresponds to an asset. While nonlinear misspecification compresses weight magnitudes, the dominant diagonal structure indicates preservation of sign and relative direction. This confirms that nonlinear confounding does not generically induce signal inversion and that directional alignment can persist even when structural correctness fails.
• Figure 5: Efficient frontier viability under misspecified signals. Mean--variance efficient frontier constructed using misspecified predictive signals. The x-axis reports portfolio volatility and the y-axis expected return implied by the surrogate signal. Despite reduced slope and attainable Sharpe ratios relative to the correctly specified case, the frontier remains smooth, convex, and well defined. This illustrates that structural misspecification degrades efficiency quantitatively through calibration effects but does not collapse optimization geometry.

Theorems & Definitions (28)

• Lemma 1: Structural Cancellation
• proof
• Corollary 1: Omitted-Variable Bias Does Not Necessarily Imply Inefficiency
• Theorem 1: Directional Agreement Does Not Imply Efficiency
• proof
• Lemma 2: Ranking Does Not Guarantee Efficient Sizing
• proof
• Corollary 2: Calibration Is Necessary for Quantitative Efficiency Within the Efficient Set
• proof
• Proposition 1: Sharpe Ratio with Surrogate Signals