arXiv 2604.099862026-04-11q-fin.MF

单因子市场中单向投资最小方差组合的理论与渐近性

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

Florida State University（佛罗里达州立大学）

Alec Kercheval, Ololade Sowunmi

备注：: 28 pages, 4 figures

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AI总结：

本文研究了单因子协方差模型下单向投资最小方差组合的显式解及其活跃资产集合的显式可计算刻画，揭示了在高维情况下活跃资产比例的渐近行为。

摘要：

我们研究了在单因子协方差模型下具有任意符号资产贝塔值的单向投资最小方差（LOMV）组合。我们提供了以活跃（正权重）资产集合为术语的显式解，并提供了活跃集合的显式可计算刻画。作为推论，我们解决了Qi（2021）关于混合符号贝塔值扩展的开放问题。在高维情形$p \to \infty$中，当贝塔值从具有累积分布函数$F$的分布中抽取时，我们证明LOMV组合中活跃资产比例（活跃比率）几乎总是收敛到$F(β^{*})$，其中$β^* \geq 0$是通过$F$确定的显式积分方程的根。这与Bernstein（2025）首次出现的结果相关。特别是当$F$连续且所有贝塔值为正（$F(0)=0$）时，活跃比率收敛到零。当$F(0) >0$较小，且在温和的矩条件和集中界下，我们建立了收敛速率$F(β^*)=O(F(0)^{1/3})$作为$F(0) \to 0$。

英文摘要：

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 $F(β^{*})$, where $β^* \geq 0$ 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 $F(β^*)=O(F(0)^{1/3})$ as $F(0) \to 0$.

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