Paper: arXiv 2604.09986

Authors: Alec Kercheval, Ololade Sowunmi

Abstract

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 $F(0) \to 0$.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical analysis of long-only minimum variance portfolios under high-dimensional asymptotics, proving convergence rates and resolving an open mathematical question. However, it is purely theoretical with no empirical backtesting or real-market data application provided in the abstract.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Long-Only Minimum Variance<br>Portfolio under One-Factor Covariance Model] --> B[Input Data: Asset Betas of Arbitrary Sign<br>High-Dim Regime p &rarr; &infin; with CDF F]
    B --> C[Derive Explicit Analytical Solution<br>& Characterize Active Set]
    C --> D[Compute Asymptotic Active Ratio<br>Root Sol. &beta;* to Integral Equation]
    D --> E[Outcome 1: Resolved Open Question<br>of Qi et al. for Mixed-Sign Betas]
    D --> F[Outcome 2: Active Ratio Converges<br>to F &beta;* & Rate O F 0 ^1/3]

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