The Constrained Minimum

The global minimum variance portfolio minimizes risk without any return target — it asks only “what combination of assets has the least volatility?” Without constraints, the answer involves shorting overexposed assets to hedge factor risk. With a long-only constraint — no short positions — the answer changes structurally, not just quantitatively.

For covariance matrices arising from factor models, the relationship between the long-only minimum variance portfolio and the factor exposures has an explicit characterization. In the single-factor case, the solution can be written directly in terms of covariance parameters — closed-form expressions for which assets receive weight and which are excluded. In the multi-factor case, the structure is geometric: the constraint boundary shapes the portfolio through the interaction of factor exposures, and the solution lives on specific facets of the weight simplex determined by the factor structure.

The through-claim: the long-only constraint doesn’t just truncate the unconstrained solution at zero — it changes what the portfolio is doing. The unconstrained minimum variance portfolio hedges across all factors by shorting assets with high exposure and buying assets with low exposure. The long-only version cannot hedge — it can only select. The assets it includes are not the ones with the lowest variance but the ones whose factor exposures happen to cancel through positive-weight combinations. The constraint transforms an optimization over all positions into a selection problem among assets whose exposures naturally diversify. The geometry of the factor model determines which assets are even candidates, before any optimization begins.