MeanVariance

Mean-variance is a framework in finance for selecting portfolios by balancing expected return and risk. It originated with Harry Markowitz's work in 1952 as part of modern portfolio theory. The approach defines risk as the variance (or standard deviation) of portfolio returns and uses expected return as the objective.

In the mean-variance model, asset returns have a mean vector μ and a covariance matrix Σ. A portfolio is described by weights w, with w_i representing the fraction invested in asset i. The portfolio's expected return is w^T μ, and its risk (variance) is w^T Σ w. The set of non-dominated portfolios forms the efficient frontier, which shows the best attainable trade-offs between return and risk.

Optimization problems are formulated to minimize risk for a target return or maximize return for a given level of risk. Common constraints include sum of weights equal to 1, and sometimes non-negativity constraints w_i ≥ 0 (no short selling). In the unconstrained case, the problem can be solved with Lagrange multipliers, yielding a portfolio with minimal variance for a specified return; the collection of such portfolios traces the efficient frontier.

Applications include asset allocation, risk management, and performance evaluation. The framework assumes returns are elliptically distributed and that μ and Σ are estimated accurately; estimation error and sensitivity to inputs are major criticisms. Extensions address constraints, alternative risk measures, and dynamic settings, but the core idea remains the same: optimize the risk-return trade-off using only the first two moments of returns.