Mean-variance optimization

Given $n$ assets with expected returns $\boldsymbol{\mu}$ and covariance matrix $\Sigma$, a portfolio $\mathbf{w}$ (with $\mathbf{1}^\top\mathbf{w} = 1$) has:

• Expected return: $\mu_p = \mathbf{w}^\top\boldsymbol{\mu}$
• Variance: $\sigma_p^2 = \mathbf{w}^\top\Sigma\mathbf{w}$

The minimum variance frontier solves:

\[\min_{\mathbf{w}} \mathbf{w}^\top\Sigma\mathbf{w} \quad \text{s.t.} \quad \mathbf{w}^\top\boldsymbol{\mu} = \mu^*, \quad \mathbf{1}^\top\mathbf{w} = 1\]

This is a quadratic program with a closed-form solution via Lagrange multipliers.

The efficient frontier and Sharpe ratio

The efficient frontier is the upper half of the minimum-variance frontier — portfolios with maximum return for each risk level. The tangency portfolio (market portfolio under CAPM) maximises the Sharpe ratio:

\[SR = \frac{\mu_p - r_f}{\sigma_p}\]

Practical challenges

Mean-variance optimization is notoriously error-amplifying — small errors in $\boldsymbol{\mu}$ lead to extreme, unstable allocations. Remedies:

• Shrinkage estimators: blend sample $\hat\Sigma$ toward a structured prior (Ledoit–Wolf)
• Robust optimization: optimise worst-case over an uncertainty set for $\boldsymbol{\mu}$
• Risk parity: weight inversely to risk contribution rather than optimising returns
• Black-Litterman: blend market equilibrium with investor views