Feasible closed form moment matching for scenario generation through covariance rotations

Abstract

Closed form moment matching constructs small scenario sets without numerical optimisation. We study two generators that match the mean, the covariance matrix, and the marginal third and fourth moments. Both place their scenarios along the columns of a covariance factor, usually the symmetric or Cholesky root. Although every valid factor reproduces the same covariance matrix, its choice can determine whether the resulting weights are nonnegative. By viewing each column as a univariate mixture component, Pearson's inequality gives the minimum probability needed to reproduce its assigned variance, skewness, and kurtosis, which leads to exact feasibility conditions whenever the relevant entrywise powers are nonsingular. The effect of the factor is governed by its entrywise fourth powers, whose sum can vary by a factor equal to the dimension. This explains why standard roots fail in high dimension. We combine these results into a rotated moment matching procedure that searches for feasible factors without altering the target moments. Applied to stock returns and synthetic data, the procedure produces valid probabilities for both generators in nearly every instance considered, and it outperforms the original constructions, whose standard factors give invalid weights in most of them. Finally, the two asset case explains when such a rotation exists. With symmetric marginals, a suitable rotation reaches Pearson's bound except in one degenerate case, whereas with skewness the factor must balance the allocation of third and fourth moments, and some attainable targets admit no feasible rotation. We also establish a sufficient condition for arbitrary correlation and equal marginal kurtosis.