The Long-Only Minimum Variance Portfolio in a One-Factor Market: Theory and Asymptotics

We study the long-only minimum variance (LOMV) portfolio under a one-factor covariance model with asset betas of arbitrary sign. We provide an explicit solution in terms of the set of active (positive weight) assets, and provide an explicit and computable characterization of the active set. As a corollary we resolve an open question of \citet{qi2021} concerning the extension to mixed-sign betas. In the high-dimensional regime $p \to \infty$ where the betas are drawn from a distribution with cdf $F$, we prove that the proportion of active assets (the active ratio) in the LOMV portfolio converges in almost all cases to $F(\beta^{*})$, where $\beta^* \geq 0$ is the root of an explicit integral equation determined by $F$. This is a variation of a result first appearing in \citet{bernstein2025}. In particular, when $F$ is continuous and all betas are positive ($F(0)=0$), the active ratio converges to zero. When $F(0)>0$ is small, under mild moment conditions and concentration bounds we establish the convergence rate $F(\beta^*)=O(F(0)^{1/3})$ as $F(0) \to 0$.