Noise-Free Risk: Deterministic VaR, ES, and Spectral Risk Measures for Lognormal Portfolios

We present a deterministic framework for computing Value-at-Risk, Expected Shortfall, and arbitrary spectral risk measures for portfolios of correlated lognormal assets, without Monte Carlo simulation. Building on the Spectral Fenton Distribution (Nagy, 2026a) and the Eigen-COS algorithm (Nagy, 2026b), we derive closed-form Expected Shortfall from 128 Fourier coefficients and show that the resulting noise-free ES estimate maximizes the statistical power of the Acerbi-Székely ridge backtest for ES validation. For Monte Carlo with \(10^4\) paths at \(\alpha = 2.5\%\), simulation noise in the ES estimate degrades backtest power by an estimated 15–30%; the Spectral Fenton eliminates this degradation entirely. We introduce the hedge index \(H\) — a scalar diagnostic that flags cancellation-dominated portfolios where parametric VaR fails by 20% or more — and propose a skewness-based routing policy (Gaussian / NIG / Spectral Fenton) validated across 48 of 50 accuracy gym levels. Four case studies — a classic 60/40 portfolio, a crypto-bond hybrid, a long-short spread, and a 100-asset diversified book — demonstrate the framework end-to-end with numerical VaR and ES values. A \(50 \times 50\) correlation-volatility stress heatmap completes in 2.5 minutes (versus 27 minutes for Monte Carlo), and a VaR fan chart across 20 confidence levels takes 9.2 ms from cached coefficients. We characterize an extreme-volatility frontier (\(\sigma > 1.0\)) where all tested analytical methods produce \(> 10\%\) VaR error, and propose a 130-number "risk certificate" as a compact, auditable regulatory output for Basel III/FRTB compliance. Key claims — including the hedge index bound \(H \in [0,1]\), the noise-variance inequality underlying the backtest power argument, and the risk certificate reconstruction — are formally verified in Lean 4 (supplementary to Nagy, 2026a). The framework applies to portfolios of correlated lognormal assets with individual volatilities \(\sigma_i \le 1.0\); an extreme-volatility frontier beyond this limit is characterized in Section 6. The method depends on the companion papers Nagy (2026a, 2026b); a self-contained primer is provided in Section 1.3.