meanvarianceporteføljeoptimering

Mean-variance portfolio optimization is a mathematical framework for constructing investment portfolios that balance expected return against risk. It is a core component of modern portfolio theory, introduced by Harry Markowitz in the 1950s. In this framework, assets are described by their expected returns vector μ and their covariance matrix Σ, which captures how asset returns move together. A portfolio is represented by weights w that sum to one (and may be subject to additional constraints, such as no short selling).

There are two common objective formulations. One minimizes portfolio variance w^T Σ w subject to a target expected return μ^T w = r_p and the constraint 1^T w = 1. The other maximizes expected return μ^T w for a given level of risk, often expressed as limiting variance. The set of optimal portfolios obtained by varying the target return forms the efficient frontier, a curve of non-dominated options that offer the best trade-off between return and risk.

Solutions can be obtained by quadratic programming or, in unconstrained cases, via analytical expressions. The optimal weights typically take a form involving the inverse covariance matrix Σ^{-1} applied to μ and to the vector of ones, with coefficients chosen to satisfy the constraints. In practice, covariances and expected returns are estimated from data, and numerical methods are used to compute the portfolio.

Limitations include sensitivity to estimation error, which can degrade out-of-sample performance. Extensions address these issues with transaction costs, no-short-selling bounds, regularization, robust or Bayesian estimation, and factor models. Despite limitations, mean-variance optimization remains a foundational tool for quantitative asset allocation and risk management.