Long-Only Minimum Variance Optimization: Analytical Foundations, Asymptotic Estimation, and the Enhanced Active-Set Algorithm

This dissertation develops a unified theoretical and algorithmic framework for the long-only minimum variance (LOMV) portfolio problem under a one-factor covariance structure. The work begins by deriving an explicit closed-form solution to the classical LOMV problem under this structure, providing a complete analytical characterization of the optimal weights and the associated active set. Building on this foundation, the dissertation investigates the impact of factor-loading estimation in high dimensions. It establishes that the dispersion bias inherent in the PCA estimator leads to persistent over-diversification, and suggests that the James–Stein eigenvector shrinkage (JSE) estimator produces a strictly smaller asymptotic deviation from the true active-set proportion. The analysis further shows that PCA systematically underestimates portfolio risk, whereas the JSE estimator—evaluated through the Variance Forecast Ratio—does not exhibit this bias. The final part of the dissertation introduces the Enhanced Active-Set Algorithm for the One-Factor Model and proves that, under non-degeneracy assumptions, the algorithm recovers the optimal active set exactly. Numerical experiments demonstrate that the proposed algorithm reliably identifies the true solution and outperforms standard optimization routines in both accuracy and stability. Collectively, these results provide new insight into the geometry, estimation sensitivity, and computational structure of long-only minimum variance portfolios, particularly in high-dimensional regimes.