Portfolio Risk Bounds without Cross-Asset Return Covariances: Distributional Fields from Language-Model Representations

Abstract

Portfolio risk normally begins from cross-asset return covariances, which are difficult to estimate in short and high-dimensional panels. This paper asks what observable differences between firms' information distributions can rule out before those covariances are estimated.

Under maintained links from information to systematic exposure and from exposure to returns, multi-firm Wasserstein-2 dispersion yields a sharp one-sided upper bound on systematic portfolio variance. A weighted pairwise relaxation gives a practical allocation objective requiring marginal volatility scales but not cross-asset return covariances.

Mathematical sequence

1. Use observed Wasserstein-2 separation to place lower floors on latent exposure separation.
2. Require one coherent joint law for all exposures; independently optimal pairwise couplings need not coexist.
3. Apply weighted Hilbert-space polarization to write systematic variance as a marginal benchmark minus cross-firm dispersion.
4. Deduct information-certified dispersion from the perfect-alignment benchmark.
5. Choose portfolio weights by minimizing the certified upper bound rather than an estimated covariance objective.

The certificate

For normalized long-only risk weights, the pairwise form has the structure

V_sys(q) ≤ Σᵢ qᵢvᵢ − ½ ΣᵢΣⱼ qᵢqⱼℓ²ᵢⱼ.

The second term is the amount of perfect alignment ruled out by observable separation floors. The sharp construction replaces the sum of pairwise relaxations with coherent multi-firm transport dispersion, equivalently represented through a free Wasserstein centre.