How do I construct a global minimum variance portfolio (GMVP)?

Question

AcadiFi · Accepted Answer

The global minimum variance portfolio (GMVP) is the portfolio with the lowest total variance achievable from a set of risky assets, with no return target constraint. It's the unique solution to min w'Σw subject to w'1 = 1.

Closed-form solution via Lagrangian:
w_GMVP = (Σ⁻¹ × 1) / (1' Σ⁻¹ × 1)

where Σ is the covariance matrix and 1 is a vector of ones.

Example — 3 assets:
Variances: σ₁² = 0.04, σ₂² = 0.09, σ₃² = 0.16
All pairwise correlations = 0.2

Σ = [[0.04, 0.012, 0.016], [0.012, 0.09, 0.024], [0.016, 0.024, 0.16]]

Computing Σ⁻¹, summing rows, normalizing gives approximately:
w_GMVP ≈ (0.62, 0.29, 0.09) — 62% in lowest-variance asset.

Key properties:

1. GMVP depends only on covariance matrix — expected returns are irrelevant. This makes GMVP robust to return estimation errors, which plague mean-variance optimization.

2. Low-beta bias: GMVP typically overweights low-vol stocks, creating a structural low-beta exposure.

3. Concentration risk: unconstrained GMVP often places large weights in few assets with low correlations. Many practitioners impose upper bounds (e.g., 5% max per asset).

4. Defensive characteristics: historically GMVP has similar returns to cap-weighted benchmarks with 20-30% lower volatility — the source of the low-vol anomaly.

5. Sensitivity: small changes in Σ produce large shifts in weights. Shrinkage estimators (Ledoit-Wolf) stabilize.

```mermaid
flowchart LR
    A[Covariance Matrix] --> B[Invert Sigma]
    B --> C[Compute Weights]
    C --> D[GMVP]
```

Commercial minimum-variance ETFs (USMV, SPLV) target GMVP-like characteristics with constraints for sector balance and tracking error.

How do I construct a global minimum variance portfolio (GMVP)?

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