Reproducibility package for Exact and Approximate Component VaR and Expected Shortfall for Quadratic Normal Portfolios: Fourier, Saddlepoint, Cornish–Fisher, and Frozen–Multiplier Allocations, with Marginal Higher–Moment Attribution and a Convexity–Gradie

We study analytic Value–at–Risk (VaR) and its decomposition for portfolios whose profit and loss is a quadratic (delta–gamma) function of jointly Gaussian risk–factor changes,ΔP = δ⊤Δx + Δx⊤ΓΔx with Δx ∼ N(0,Σ). Any risk functional positively homogeneous of degree one in the sensitivity pair (δ,Γ) is Euler–additive across the deltas and the ordered gamma pairs, cross–gammas included, and we compare four allocations on this common footing. Within the quadratic–Gaussian model the exact VaR allocation consists of conditional moments of the factors given the loss, and the exact Expected Shortfall allocation of tail conditional moments, both computed by one–dimensional Fourier inversion. The saddlepoint allocation is exactly additive because the Lugannani–Rice quantile is scale–equivariant. The fourth–order Cornish–Fisher allocation reduces to trace and matrix expressions evaluable in O(n3) arithmetic, and near a linear book its quantile agrees with the exact quantile to second order in the convexity, which requires the squared–skewness term. Frozen–multiplier schemes, which scale variance shares by one portfolio multiplier, are additive but omit the marginal skewness and kurtosis terms; these form a zero–sum redistribution. Using the trace–square gradient tr(ΓΣ)Σ in place of ΣΓΣ breaks additivity by a closed–form amount, diagonal gamma matrices included. Finally, a moment constraint on the exact marginals detects allocations that add up correctly yet give hedge signals of the wrong sign.