The Global Minimum Variance (GMV) portfolio is defined as the portfolio that:

Question

BlackTom AI · Accepted Answer

The GMV portfolio minimizes portfolio variance subject to the budget constraint (weights sum to one), so its weights solve min_w w'Σw subject to 1'w = 1.

Relevant concepts:
- Portfolio variance: Var(R_p) = w' Σ w, where Σ is the covariance matrix and w the weight vector.
- Efficient frontier: set of portfolios that minimize variance for a given expected return; the GMV is the point on this frontier with the absolute lowest variance.
- Constrained optimization (Lagrange multipliers) yields the closed-form GMV weights when only the budget constraint is imposed.

Step-by-step derivation:
1. Form Lagrangian: L(w, λ) = w'Σw − λ(1'w − 1).
2. Take derivative w.r.t. w: ∂L/∂w = 2Σw − λ1 = 0 ⇒ 2Σw = λ1.
3. Solve for w: w = (λ/2) Σ^{-1} 1.
4. Impose budget constraint 1'w = 1:
   1'[(λ/2) Σ^{-1}1] = 1 ⇒ (λ/2)(1' Σ^{-1}1) = 1 ⇒ λ = 2 / (1' Σ^{-1}1).
5. Substitute back: w_GMV = Σ^{-1}1 / (1' Σ^{-1}1).

This w_GMV yields the minimum possible portfolio variance among all portfolios that satisfy the budget constraint. Graphically, it is the leftmost (lowest-variance) point on the mean-variance efficient frontier. If additional constraints (e.g., no short sales) are imposed, the GMV is the feasible portfolio with the smallest variance.

Why the correct answer is correct:
- By construction the GMV minimizes variance; every other efficient-frontier portfolio has equal or higher variance. Thus it “has the lowest possible risk among all portfolios on the efficient frontier.”

Why other typical choices would be incorrect:
- “Maximum return” or “highest expected return” focuses on return, not variance, so it is not the GMV.
- “Maximum Sharpe ratio” (tangency portfolio) maximizes risk-adjusted return, not absolute variance minimization.
- Any statement saying GMV maximizes return or risk-adjusted return is therefore incorrect.

Conclusion: The correct description is that the GMV portfolio has the lowest possible risk among all portfolios on the efficient frontier, which matches the provided answer.

The Global Minimum Variance (GMV) portfolio is defined as the portfolio that:单项选择题

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