How is the efficient frontier computed in mean-variance optimization?

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AcadiFi · Accepted Answer

The efficient frontier is the set of portfolios that minimize variance for each target expected return (or equivalently, maximize return for each target variance). Computationally, solve a family of quadratic programs.

Problem formulation: min ½ w'Σw subject to w'μ = E[R]_target and w'1 = 1, with optional w ≥ 0.

Closed-form frontier equation (no short-sale constraint):
σ²(E[R]) = (C × E[R]² − 2B × E[R] + A) / D

where:
- A = 1' Σ⁻¹ 1
- B = μ' Σ⁻¹ 1
- C = μ' Σ⁻¹ μ
- D = AC − B²

The frontier is a parabola in (σ², E[R]) space, or a hyperbola in (σ, E[R]) space.

Numerical procedure:

1. Choose grid of target returns: say 20 points from min(μ) to max(μ).
2. For each target, solve QP via Lagrangian or numerical solver (cvxopt, quadprog).
3. Plot (σ*, E[R]*) pairs to form frontier.

With short-sale constraints (w ≥ 0), closed form is lost — use QP solvers. The constrained frontier lies inside/below the unconstrained one.

Example — 2 assets:
μ = (0.08, 0.12), σ = (0.15, 0.25), ρ = 0.3

For target E[R] = 10%: solve w₁ × 0.08 + w₂ × 0.12 = 0.10, w₁ + w₂ = 1. Gives w₁ = 0.5, w₂ = 0.5. Variance = 0.25 × 0.0225 + 0.25 × 0.0625 + 2 × 0.5 × 0.5 × 0.3 × 0.15 × 0.25 = 0.02687. σ = 16.4%.

Efficient region: upper half of parabola, from GMVP upward. Lower half is dominated (more risk for less return).

```mermaid
flowchart LR
    A[Inputs: Mu, Sigma] --> B[Compute A B C D]
    B --> C[Grid Target Returns]
    C --> D[Solve QP]
    D --> E[Frontier Curve]
```

Most practitioners add: cardinality limits, turnover constraints, tracking error bounds. These convert the frontier into a feasible investable region.

How is the efficient frontier computed in mean-variance optimization?

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