Mean-Variance Optimization

Markowitz [@markowitz1952] cast portfolio choice as a quadratic program in the mean and covariance of asset returns. Given an asset universe of \(n\) risky assets with expected return vector \(\mu \in \mathbb{R}^n\) and positive semi-definite covariance matrix \(\Sigma \in \mathbb{R}^{n \times n}\), the investor chooses portfolio weights \(w \in \mathbb{R}^n\).

The canonical problem

The classical mean-variance objective trades expected return against variance through a risk-aversion parameter \(\gamma > 0\):

\[ \max_{w}\; \mu^\top w \;-\; \tfrac{\gamma}{2}\, w^\top \Sigma w \quad \text{s.t.} \quad \mathbf{1}^\top w = 1. \tag{1} \]

With no further constraints, the closed-form solution is

\[ w^\star \;=\; \frac{1}{\gamma}\,\Sigma^{-1}\!\bigl(\mu - \lambda \mathbf{1}\bigr), \qquad \lambda \;=\; \frac{\mathbf{1}^\top \Sigma^{-1} \mu - \gamma}{\mathbf{1}^\top \Sigma^{-1} \mathbf{1}}. \tag{2} \]

The efficient frontier

Sweeping \(\gamma\) traces the efficient frontier in \((\sigma, \mu)\) space. For the fully invested, unconstrained case the frontier is the hyperbola

\[ \sigma_p^2 \;=\; \frac{1}{C}\bigl(1 + (\mu_p - A/C)^2 \cdot C / (BC - A^2)\bigr), \tag{3} \]

where \(A = \mathbf{1}^\top \Sigma^{-1} \mu\), \(B = \mu^\top \Sigma^{-1} \mu\), and \(C = \mathbf{1}^\top \Sigma^{-1} \mathbf{1}\) are the standard Merton constants [@merton1972].

With long-only and box constraints

In practice the problem is solved as a QP:

\[ \min_w\; \tfrac{1}{2} w^\top \Sigma w - \tfrac{1}{\gamma} \mu^\top w \quad \text{s.t.} \quad \mathbf{1}^\top w = 1,\; \ell \le w \le u. \tag{4} \]

Box constraints \(\ell \le w \le u\) subsume the long-only case (\(\ell = 0\)). Adding turnover constraints

\[ \|w - w_0\|_1 \;\le\; \tau, \tag{5} \]

where \(w_0\) is the current portfolio, requires auxiliary variables but remains a convex QP.

Why this is only a starting point

Problem (4) is well-posed given \(\mu\) and \(\Sigma\). The catch is that both must be estimated from finite samples. The next pages describe how estimation error propagates into wildly unstable weights, and what to do about it.

References

[@markowitz1952]; [@merton1972]. See Citations for full bibliographic details.