Correlation Memory and Allocation-Invariant Calibration

AbstractComparisons of covariance models often hold marginal volatilities fixed and match averagecorrelation. We show that such matching need not change a dollar-neutral allocation. Forpositive-definite correlation matrices and fixed positive marginal volatilities, we characterizeevery matrix that preserves the gross-normalized mean–variance allocation for every nonconstant forecast. The admissible matrices form a one-parameter affine family. A prescribedaverage correlation selects a unique candidate, and two scalar generalized eigenvalues giveits exact feasibility interval. This yields an explicit diagnostic for separating changes in amatrix summary from changes in portfolio decisions. We apply the diagnostic in a reproducible reanalysis of 40 US industry portfolios using a September 2026 data vintage. In1963–2010, the current 126-day correlation estimate has a before-cost Sharpe ratio of 0.999,compared with 0.630 for historical correlation and 0.667 for a positive-definite, mean-matchedhistorical-shape control. The corresponding advantages are not distinguished from zero bythe reported simultaneous bootstrap bands in 2011–2025, and costs can reverse rankings.Allocation-invariant matching is feasible on 54.6% of decision dates. The theoretical resultsidentify a limitation of average-correlation controls; the empirical results describe a conditionalhistorical comparison, not universal superiority or an optimal correlation-memory horizon.Keywords: constrained quadratic optimization; correlation matrices; allocation equivalence;covariance estimation; portfolio evaluation.