The Geometry of Risk: Spectral Distance and Topological Structure in Portfolio Space
Abstract
We introduce a metric on the space of portfolio risk profiles derived from the spectral representation of loss distributions. Each portfolio's loss distribution is encoded as a vector of \(N = 128\) Fourier--cosine coefficients via the Eigen-COS method (Nagy, 2026a). The \(L^2\) distance between coefficient vectors defines a spectral risk distance \(d(P_1, P_2) = \lVert A^{(1)} - A^{(2)} \rVert_2\) that satisfies all metric axioms --- reflexivity, symmetry, the triangle inequality, and identity of indiscernibles --- with the metric axioms formally verified in Lean 4 (zero sorry, Mathlib v4.28.0). We apply three geometric tools to this metric space: (i) principal component analysis reveals that \(>90\%\) of cross-portfolio variation is captured by two axes interpretable as volatility level (PC1) and diversification structure (PC2); (ii) persistent homology detects \(4\)--\(5\) natural clusters corresponding to asset classes (bonds, equities, crypto, diversified) with no significant \(H_1\) features, indicating a simply connected risk space; (iii) stress trajectories show that volatility shocks move portfolios along PC1 while correlation shocks move them along PC2, with the two directions approximately orthogonal. The framework replaces scalar risk measures (VaR, ES) with a full shape-aware comparison of risk profiles, enables detection of risk regime transitions via trajectory analysis, and provides a formally verified mathematical foundation for geometric risk management.
Novelty
Applying the L2 distance on Fourier-cosine (COS) coefficients as a distributional metric on portfolio loss profiles is a clean repackaging, but the core idea — Euclidean distance in a spectral embedding — is standard in functional data analysis; the genuine delta is combining it with TDA and PCA in a risk-management framing with partial Lean verification.