MVO

MVO, or mean-variance optimization, is a mathematical framework for constructing portfolios that balance expected return and risk. Introduced by Harry Markowitz in 1952, it formalizes the trade-off between return and variability of returns. The standard problem seeks to maximize the expected portfolio return for a given level of risk, or equivalently minimize risk for a target expected return, subject to constraints such as nonnegative weights and full investment (the weights sum to one).

Risks are measured by the portfolio variance, computed from the assets’ variances and covariances. The solution is obtained by quadratic programming, yielding the efficient frontier—the set of optimal portfolios for each risk level. Typical inputs include the vector of expected returns, the covariance matrix, and constraints (no short selling, sector or budget constraints).

Limitations include sensitivity to input estimates, particularly the expected returns and covariances; small changes can produce large changes in the optimal weights. Estimation error can lead to unstable portfolios that perform poorly out-of-sample. To address these issues, practitioners use techniques such as shrinkage estimators for the covariance matrix, Bayesian or robust mean estimates, regularization, and alternative frameworks like the Black-Litterman model that blend market information with investor views.

Variants of the approach include the minimum-variance portfolio (aims to minimize risk within constraints) and the tangency or Sharpe optimization with a risk-free asset. MVO remains a foundational concept in modern portfolio theory and is widely taught and implemented in asset management.