Distributions or Quantiles? A Convex Variational Theory of Worst-Case Risk

Worst-case risk evaluation is central to finance and operations research, yet existing analytical results remain fragmented and largely confined to ambiguity sets with specific structures. Using spectral risk measures as the underlying class, we develop a convex variational theory that unifies these analyses and extends them to richer ambiguity models. The theory rests on a representation principle: use the quantile or distribution formulation that preserves convexity of the full ambiguity set. In both representations, the resulting optimality conditions provide a systematic route to identify the worst-case law and determine its risk value. This approach recovers classical central-moment bounds and yields closed-form or semi-closed-form solutions for practically motivated models involving Wasserstein ambiguity with tail stress, two-sided expected shortfall (ES) bands, asymmetric partial moments, geometric-moment bounds, and exponential- or Gaussian-reference Kullback--Leibler (KL) neighborhoods. Beyond these analytical results, the same theory yields projection-based algorithms with fixed-grid convergence under the stated assumptions. In a high-dimensional portfolio application, we establish an exact projection identity for mean-anchored joint KL ambiguity under a Gaussian reference. The resulting standardized worst-case ES coefficient applies to every portfolio, reducing robust portfolio selection to an exact second-order cone program after a single calibration.