Anticipatory Portfolio Optimization

Abstract

A portfolio is \emph{anticipatory} when its optimizer acts on a richer model than the myopic, price-taking estimator used to calibrate it. Enrichment may be informational, via enlarged filtrations; dynamic, via horizon forecasts; or performative, via the deployment law induced by market impact. We give a decision-theoretic definition for all three cases and measure anticipation by the realized control gap between enriched controller and restricted estimator. The same quadratic geometry separates information, planning value, impact correction, and overfitting. For log utility under initial enlargement, value is the information-drift energy $\frac12\mathbb{E} \int_0^Tα_t^2\,dt$, equivalently mutual information or relative entropy. In mean-variance form, signal value is $\frac{1}{2γ}{\rm tr}(Σ^{-1}Ω)$. Dynamic forecast anticipation gives a finite-horizon quadratic premium in the forecast stack, while permanent impact changes the price-taking allocation $θ_{\rm na} =(Λ+γΣ)^{-1}μ$ into $θ_{\rm an} = (2Λ+γΣ)^{-1}μ$ and reveals a spectral phase transition for naive recalibration. The main result is a stacked finite-horizon LQG decomposition: information, forecast, and impact combine into an information trace plus one inverse-precision norm, whose expansion yields the impact term, forecast term, and signed forecast-impact interaction. Sharp angle bounds and an orthogonal nonnegative projection identity resolve the signed term. The stationary extension endogenizes information covariance as Kalman error reduction and carries impact anticipation to an infinite-horizon Lyapunov trace with transaction costs. Finally, the penalty $\frac{1}{2}{\rm tr}(H^{-1}Σ_\varepsilon)$ shows that correctly specified anticipation creates value, vacuous anticipation has zero value, and misspecified anticipation is harmful when estimated structure is optimized as true.