Three Numbers for Risk: A Data-Driven Spectral Basis for Portfolio Loss Distributions

The Spectral Fenton Distribution represents a portfolio loss distribution with 128 Fourier coefficients. Using simulated market data (5 assets, 250 trading days, stochastic volatility and correlation dynamics), we show that the time series of these coefficients is almost entirely one-dimensional: a single principal component captures 97.8\% of all distributional variation, and three components capture 99.998\%. This means the daily change in a portfolio's risk profile is described by three numbers --- \(z_1(t)\), \(z_2(t)\), \(z_3(t)\) --- instead of 128 coefficients, a 43\(\times\) compression with VaR reconstruction error below 0.05\%. The result is robust: out-of-sample holdout error remains below 0.1\%, and the 3-mode structure persists across portfolio sizes from 3 to 20 assets. The dominant mode \(z_1\) correlates with VaR at \(r = 0.998\), confirming it captures the overall risk level. We construct a real-time risk dashboard where each day's entire distributional change is a point in \(\mathbb{R}^3\), and anomalous days (regime shifts) appear as outliers in this space. The learned basis is the natural state space for the Bayesian Live Risk, Dynamic URRT, and Schr\"{o}dinger Bridge frameworks proposed in companion papers: once the basis is extracted, all of these become three-dimensional problems.