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

Theoretical Characterization and Asymptotics of the Long-Only Minimum Variance Portfolio in One-Factor Markets

Problem Formulation and Motivation

This work rigorously investigates the characterization and high-dimensional asymptotics of the long-only minimum variance (LOMV) portfolio within the context of a one-factor covariance model. Unlike the classic global minimum variance (GMV) solution, which allows both positive and negative (long/short) positions and is analytically tractable due to the unconstrained quadratic program structure, the LOMV introduces binding non-negativity constraints on asset weights. The presence of these constraints fundamentally alters the sparsity and diversification profile of the resulting portfolios and makes the active set identification nontrivial, especially in large $p$ regimes with potentially mixed-sign factor loadings (“betas”). The study fills a gap left open by prior literature on the generality of mixed-sign betas and asymptotic active set characterization.

Explicit Solution for the LOMV in One-Factor Models

The authors first generalize the explicit construction for the LOMV under arbitrary positive definite covariance, leveraging the Karush-Kuhn-Tucker (KKT) conditions to show that the LOMV solution on the active set $K$ coincides with the unconstrained GMV solution restricted to those assets. However, the crux is the selection of $K$, i.e., indices of positive weights.

For the single-factor model with

$$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$

with arbitrary (mixed-sign) $\beta_i$, the work provides an explicit, computational criterion: Letting $R_1 = \frac{1}{\sigma^2}$ and recursively

$$R_i = R_{i-1} + \frac{\beta_{i-1}}{\delta_{i-1}^2}(\beta_{i-1} - \beta_i),$$

the active set is the largest contiguous group $\{1,\ldots, \ell\}$ for which $R_i > 0$. The monotonicity properties of $R_i$ ensure this is well-defined, and in practical terms, $K$0 can be rapidly searched (e.g., bisection, $K$1) for large $K$2.

This resolves the open problem posed in Qi (2021) regarding portfolios with mixed-sign betas, extending prior results which either imposed positivity assumptions (CST 2011; Qi 2021) or lacked constructive methods.

Figure 1: Comparison of long-short (blue) and long-only (black) minimum variance portfolio weights versus beta for $K$3 assets.

In (Figure 1), the sparsity-inducing effect of the long-only constraint is illustrated: under a normal beta distribution, the LOMV portfolio concentrates all weight on a small subset of assets with the lowest betas (including those with negative beta), whereas the GMV portfolio allocates positive weights much more broadly.

High-Dimensional Asymptotics and Active Set Proportion

In the high-dimensional limit ($K$4), the study investigates the "active ratio", $K$5, where $K$6 is the number of nonzero weights in the LOMV portfolio for size $K$7. When $K$8 are iid from a distribution $K$9, it is proven that under mild moment and regularity conditions, $K$0 converges almost surely to $K$1, where $K$2 is the positive root of the integral equation

$K$3

Continuity properties and explicit expressions for $K$4 lead to a sharp trichotomy of possible asymptotic regimes:

• Mixed-sign betas, positive mean: $K$5 unless $K$6 is an atom of $K$7 (in which case the limit does not exist and oscillates on the atom’s mass band).
• All non-negative betas: $K$8 unless $K$9 has atomic mass at its minimum support.
• All negative betas, zero mean: $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$0.

This result quantifies the interplay between systematic exposure in the asset universe and the sparsity implicit in long-only risk minimization.

Figure 2: Empirical distribution of $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$1 for LOMV with Normal beta distribution, illustrating the convergence to its theoretical limit as $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$2 grows.

Quantitative Bounds and Rate of Decay

A major theoretical contribution is the formal rate at which the active set vanishes as the prevalence of non-positive betas decreases. Provided mild moment and concentration bounds on $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$3, if the probability $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$4 that $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$5 vanishes, the limiting active ratio satisfies $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$6. Thus, for asset universes with almost all positive market exposure, the LOMV portfolio becomes extremely sparse.

Figure 3: Distribution of $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$7 under decreasing negative-beta mass, visualizing rapid sparsification predicted by the $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$8 rate.

Non-Convergence Phenomenon

The work also constructs and analyzes "pathological" cases in which the limiting active ratio does not exist. If the distribution $$\Sigma = \sigma^2 \boldsymbol{\beta}\boldsymbol{\beta}^\top + \Delta$$9 has an atom at the critical threshold $\beta_i$0, $\beta_i$1 can oscillate between $\beta_i$2 and $\beta_i$3, as shown both analytically and via simulation. Such scenarios are statistically rare but impossible to exclude in discrete or empirical universes.

Figure 4: Simulation example demonstrating non-convergence of $\beta_i$4 when $\beta_i$5 has an atom at $\beta_i$6. Two modes correspond to realizations where all, or none, of the atom's block is active.

Implications and Future Directions

Practical implications: The results provide operationally efficient algorithms for constructing the LOMV portfolio in high-dimensional settings, especially relevant to index fund design, risk premia strategies, and regulatory compliance where shorting is prohibited. The sparsity induced by the long-only constraints suggests that naive expectations of diversification are not met under factor-dominant regimes—most assets receive zero allocations unless sufficiently de-risked by negative betas.

Theoretical implications: The precise relationship between the distributional properties of factor exposures and sparsity/selection in constrained quadratic optimization deepens understanding of convex program behavior under linear inequality constraints. These asymptotics may inform regularization strategies in high-dimensional factor models and motivate further study of alternative shrinkage estimators and relaxation of covariance assumptions.

Future research: Addressing extensions to multi-factor models, adaptive shrinkage for empirical covariance estimation, and robustness to sample estimation error remain outstanding. The general methodology for active set identification may translate to broader classes of portfolio and signal selection problems under convex constraints.

Conclusion

The paper provides an explicit, computationally efficient solution for the LOMV portfolio under a one-factor model with mixed-sign betas and delivers a rigorous asymptotic theory for the active ratio in large universes. The active set is driven by the lower tail of the beta distribution, and, in most realistic regimes, sparsifies rapidly as negative betas vanish. The results clarify both the structure and the limitations of long-only minimum variance allocations relative to unconstrained benchmarks and establish a foundation for further work in high-dimensional constrained optimization in finance and statistics.