arxiv: 2604.09986 · v1 · submitted 2026-04-11 · 💱 q-fin.MF

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

Alec Kercheval, Ololade Sowunmi

Pith reviewed 2026-05-10 16:30 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords long-only minimum variance portfolioone-factor modelactive sethigh-dimensional asymptoticsbeta distributionportfolio optimization

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The pith

In a one-factor market the long-only minimum variance portfolio is supported exactly on assets whose betas exceed a positive threshold solving an integral equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a closed-form description of the long-only minimum variance portfolio when returns obey a strict one-factor model, allowing betas of either sign. The solution is expressed directly in terms of the active set of assets that receive positive weight, together with an explicit, computable rule that identifies precisely which assets belong to that set. In the high-dimensional limit where betas are drawn i.i.d. from a fixed distribution F, the fraction of active assets converges almost surely to F(β*), with β* the nonnegative root of a single integral equation whose integrand is determined by F. As a direct corollary the result settles an earlier open question on the mixed-sign case and shows that the active ratio vanishes when all betas are positive.

Core claim

Under the one-factor covariance model the long-only minimum-variance portfolio is given explicitly once the active set is known; the active set itself is characterized by a threshold condition on the betas that can be computed from the factor loadings and the target. In the p → ∞ regime with betas drawn from cdf F the active ratio converges to F(β*), where β* ≥ 0 solves the integral equation determined by F. When F is continuous and F(0) = 0 the limit is zero; when F(0) is small the rate is O(F(0)^{1/3}) under mild moment and concentration assumptions.

What carries the argument

The explicit characterization of the active asset set, which selects assets whose betas lie above the threshold β* solving the integral equation fixed by the beta distribution F.

If this is right

The portfolio can be constructed by first solving for the threshold β* and then assigning weights only to assets with betas strictly larger than β*.
When every beta is positive the limiting active ratio is zero, so the portfolio becomes sparse in the high-dimensional limit.
The same integral-equation threshold governs the mixed-sign case, resolving the extension left open by earlier work.
The convergence rate O(F(0)^{1/3}) supplies a quantitative bound on how many negative-beta assets survive in the active set when their proportion is small.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sparsity result suggests that, for equity universes whose betas are mostly positive, long-only minimum-variance portfolios will concentrate on a vanishing fraction of names as the universe grows.
The explicit threshold rule could be used to derive approximate closed-form risk formulas for large portfolios even when the exact active set is not enumerated.
One could test whether real-market active ratios track the predicted F(β*) by sorting stocks on estimated betas and comparing the empirical active fraction to the theoretical limit.

Load-bearing premise

The covariance matrix is exactly that of a one-factor model, the betas are drawn independently from a fixed distribution F, and the stated mild moment and concentration conditions hold.

What would settle it

A large-scale simulation or empirical check in which the realized fraction of positive-weight assets in the long-only minimum-variance portfolio fails to approach the predicted F(β*) for a known F would refute the high-dimensional convergence claim.

Figures

Figures reproduced from arXiv: 2604.09986 by Alec Kercheval, Ololade Sowunmi.

**Figure 2.** Figure 2: Distribution of the active ratio kp/p for the LOMV portfolio, with βi ∼ N (1.0, s2 ) for the values s = 0.4, 0.25, 0.1. For each trial, the simulated covariance matrix is Σ = σ 2ββ⊤ + δ 2 I where δ 2 = 0.5 and σ 2 = 1. For each p, 400 simulation trials were performed and the resulting active ratio statistics across trials summarized in boxplots. The dashed line is the theoretical limit F(β ∗ ) as p → ∞. Lo… view at source ↗

**Figure 3.** Figure 3: Distribution of active ratio kp/p with β ∼ N (1.0, s2 ) for s = 0.4, 0.25, 0.1, this time with δ 2 = 0.1 and otherwise the same as in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Non-convergence example of the LOMV active ratio kp/p (Theorem 2, Case 1). The beta distribution is discrete with four atoms: P(β = −1) = 0.05, P(β = 1) = 0.15, P(β = 2) = 0.30, P(β = 5) = 0.50. Bottom panel: The population G function, G(y) = P i pi βi(βi − y) 1βi≤y (shaded green where G > 0 and pink where G < 0), has its unique zero at β ∗ = 2, which is an atom of F with mass 0.30. Because β ∗ is an atom,… view at source ↗

read the original 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 $F(\beta^{*})$, 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 $F(0) \to 0$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They give an explicit active-set rule for long-only min-var under one-factor with mixed betas and a sharp high-dim limit with rate O(F(0)^{1/3}).

read the letter

The main takeaway is that this paper works out exactly which assets get positive weight in the long-only minimum variance portfolio when returns follow a one-factor model and betas can be negative. It also shows that in the high-dimensional limit the active fraction converges to F(beta*) for an explicit integral equation in F, with the rate bound when F(0) is small. That directly answers the open question left in qi2021 and adds a concrete convergence rate that earlier work did not have.

Referee Report

0 major / 3 minor

Summary. The paper studies the long-only minimum-variance portfolio under an exact one-factor covariance model that permits betas of either sign. It supplies an explicit solution expressed via the active (positive-weight) asset set together with a computable characterization of that set, thereby resolving an open question of Qi et al. (2021) on the mixed-sign case. In the high-dimensional regime p→∞ with betas drawn i.i.d. from a fixed cdf F, the paper proves that the active ratio converges almost surely to F(β*), where β*≥0 solves an explicit integral equation determined by F; when F is continuous and F(0)=0 the limit is zero, while for small F(0)>0 a rate F(β*)=O(F(0)^{1/3}) is obtained under mild moment and concentration conditions.

Significance. If the derivations are correct, the work supplies a precise theoretical account of the sparsity induced by the long-only constraint in one-factor markets. The explicit active-set rule and the high-dimensional limit (including the rate result) extend earlier findings and furnish falsifiable predictions for large portfolios. These contributions are valuable for both the mathematical-finance literature on constrained optimization and for practical high-dimensional portfolio construction.

minor comments (3)

The qualifier “in almost all cases” appearing in the abstract and high-dimensional theorem should be replaced by an explicit probability statement (e.g., “with probability 1 under the stated concentration hypotheses”) once the precise measure-theoretic setting is introduced.
The integral equation satisfied by β* is described as “explicit and determined by F”; a short displayed equation in the introduction or the statement of the main limit theorem would improve immediate readability.
The bibliography entries for qi2021 and bernstein2025 should be completed with full publication details (journal, volume, pages) if they are not already present.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our contributions, and recommendation for minor revision. The report correctly identifies the explicit active-set characterization and the high-dimensional asymptotic results for the long-only minimum-variance portfolio under the one-factor model with mixed-sign betas.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives an explicit active-set characterization for the long-only minimum-variance quadratic program directly from the KKT stationarity and complementarity conditions applied to the exact one-factor covariance model. The high-dimensional limit is obtained by substituting the empirical measure of i.i.d. betas drawn from F into the resulting threshold rule and passing to the integral equation whose root β* yields the limiting active proportion F(β*). Both steps are self-contained consequences of the model assumptions, moment conditions, and concentration hypotheses; the integral equation is solved from F rather than fitted to data or defined in terms of the target quantity. The reference to bernstein2025 is noted as a related prior result but is not load-bearing for the present derivation, which independently resolves the mixed-sign extension of qi2021. No self-definitional, fitted-input, or self-citation reduction appears in the claimed chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claims rest on the one-factor covariance model (standard in the field) and the high-dimensional i.i.d. beta assumption; no new entities or fitted constants are introduced beyond the input distribution F.

axioms (2)

domain assumption Asset returns follow a one-factor model with covariance determined by betas of arbitrary sign.
Explicitly stated as the setting for the LOMV portfolio study.
domain assumption In the p to infinity limit, betas are i.i.d. draws from a distribution F satisfying the listed continuity or moment conditions.
Required for the convergence statements and rate result.

pith-pipeline@v0.9.0 · 5539 in / 1477 out tokens · 74333 ms · 2026-05-10T16:30:34.785784+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

SIAM, 2nd edition, 2023

Amir Beck.Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with Python and MATLAB. SIAM, 2nd edition, 2023. A. Bernstein and A. Shkolnik. Asymptotics of quadratic forms on a simplex. preprint, September 2025. M. J. Best and R. R. Grauer. Positively weighted minimum-variance portfolios and the structure of asset expected returns...

2023