Mean-Variance Optimization

Mean-variance optimization identifies the portfolio that offers the best trade-off between expected return and risk, measured by variance. The goal is to choose asset weights that minimize portfolio variance for a given level of expected return, or equivalently, maximize expected return for a given level of risk.

This approach relies on knowledge of the expected returns \( \mu_i \), variances \( \sigma_i^2 \), and covariances \( \text{Cov}(X_i, X_j) \) of the assets in the portfolio. The optimization problem can be formalized as:

Minimize:

\[\text{Var}(w^\top X) = w^\top \Sigma w\]

subject to:

\[w^\top \mu = \mu_p, \quad \sum w_i = 1\]

where \( w \) is the vector of portfolio weights, \( \Sigma \) is the covariance matrix of asset returns, \( \mu \) is the vector of expected returns, and \( \mu_p \) is the target portfolio return.