Deterministic Portfolio VaR Without Monte Carlo: The Eigen-COS Method
Abstract
We present the Eigen-COS method, a deterministic algorithm that computes exact Value-at-Risk, closed-form Expected Shortfall, and the full CDF/PDF for weighted sums of correlated lognormal assets — without Monte Carlo simulation. The method conditions on the eigenvalues of the correlation matrix — provably the optimal rank-\(K\) conditioning strategy (Nagy, 2026a) — and applies Fourier-cosine inversion to produce a 130-parameter distributional summary, the Spectral-Fenton Distribution, available in a one-time precomputation of 15–175 ms. Convergence in \(K\) is exponential under a spectral gap condition, explaining why \(K = 1\)–\(3\) factors suffice in practice.
We benchmark against Monte Carlo, Gaussian VaR, and Cornish-Fisher across 60 portfolio configurations (\(n = 5\) to \(100\) assets): sub-basis-point accuracy for uncorrelated portfolios, a mean error of 3.4% across the full grid (driven by COS domain truncation in high-volatility tails), and 10–570\(\times\) the speed of Monte Carlo. A three-regime error analysis identifies the factor-count, domain-truncation, and quadrature-budget boundaries of the method.
The analytic quantile function unlocks the complete space of spectral risk measures (Acerbi, 2002), including Basel III/FRTB Expected Shortfall. The mathematical foundations — optimality, convergence, error decomposition, and all four Acerbi coherence axioms — are formally verified in Lean 4 (59 files, 120+ theorems, 0 sorry) and appear in the companion paper (Nagy, 2026a).
Novelty
The genuine intellectual delta is the eigenvalue-conditioned reduction of the multivariate lognormal sum CDF to a sequence of 1D COS inversions, yielding a portable 130-parameter distributional object; the individual ingredients (COS method, eigen-decomposition, Gauss-Hermite quadrature) are all known, but their specific composition for this problem is new.