Understanding the Long-Only Minimum Variance Portfolio

For a covariance matrix coming from a factor model of returns, we investigate the relationship between the long-only global minimum variance portfolio and the asset exposures to the factors. In the case of a 1-factor model, we provide a rigorous and explicit description of the long-only solution in terms of the parameters of the covariance matrix. For $q>1$ factors, we provide a description of the long-only portfolio in geometric terms. The results are illustrated with empirical daily returns of US stocks.

💡 Research Summary

The paper investigates the long‑only global minimum‑variance (LOMV) portfolio when the asset return covariance matrix is generated by a factor model. The authors first distinguish the unconstrained (long‑short) minimum‑variance problem, whose solution is the familiar closed‑form w_LS = Σ⁻¹1 / (1ᵀΣ⁻¹1), from the long‑only problem, which adds non‑negativity constraints on the weights. The central difficulty of the long‑only problem is identifying the set K of assets that receive positive weight in the optimal portfolio.

Theorem 1 provides a general solution once K is known: by extracting the sub‑matrix Σ_K (the rows and columns of Σ corresponding to K) and computing its Moore‑Penrose inverse, the optimal long‑only weights are exactly the long‑short solution for the reduced universe of k = |K| assets, padded with zeros for the excluded assets. Thus the problem reduces to determining K.

In the single‑factor case, where Σ = σ²ββᵀ + Δ with β the vector of factor loadings and Δ a diagonal matrix of idiosyncratic variances, the authors derive an explicit, constructive method for K. After sorting the loadings β₁ ≤ … ≤ β_p, they define a sequence

R_i = 1/σ² + ∑_{j=1}^{i} β_j δ_j⁻² (β_j − β_i), i = 1,…,p.

The sequence is initially non‑decreasing, reaches a maximum at some index s, and then becomes non‑increasing. The largest index k for which R_k > 0 determines the active set K = {1,…,k}. The corresponding weights are obtained from (Σ_K)⁻¹1_k / (1_kᵀ(Σ_K)⁻¹1_k). This algorithm runs in O(p) time, or O(log p) with a bisection search, and requires only the factor loadings and idiosyncratic variances. The authors also present Corollary 1, which restates the positivity condition as β_i <