Mean-Variance Optimization

Notation

• -- Vector of expected returns for assets
• -- Covariance matrix of asset returns (symmetric, positive semi-definite)
• -- Vector of portfolio weights
• -- Target minimum portfolio return
• -- Risk-free rate of return

Portfolio Return

The expected return of a portfolio is the weighted sum of individual asset expected returns:

Portfolio Variance

The portfolio variance captures both individual asset variances and all pairwise covariances, weighted by the portfolio allocation:

where is the covariance between assets and . When , this reduces to the variance of asset .

Full Optimization Problem

Textbook expositions usually state MVO as a variance minimization subject to a target return. The problem this platform solves for the MVO method sits at a different point on the same frontier: it maximizes the Sharpe ratio, so it returns the tangency portfolio. The request carries no target-return field, and the program carries no target-return row:

The first constraint ensures full investment (weights sum to one). The second constraint prevents short selling. Both are fixed for this method. The risk-free rate comes from the request. Because the objective is the Sharpe ratio, MVO and the Maximum Sharpe Ratio method describe the same tangency point on the efficient frontier.

The inputs come from the daily returns of the run window. The expected-return vector is the historical mean return, annualized as a compound growth rate. The covariance matrix is the Ledoit-Wolf shrunk estimate, annualized over 252 trading days. PyPortfolioOpt's EfficientFrontier.max_sharpe solves the program.

The tangency objective needs at least one asset whose expected return is above . When no asset is, the run lowers the rate it optimizes against to just below the highest expected return, marks the result as degraded, and records both the requested and the applied rate. Under constraints the run also checks that some portfolio which satisfies them has an expected return above . When none does, the method does not run, and the run states the reason.

Sharpe Ratio

The Sharpe Ratio measures the risk-adjusted return of a portfolio, quantifying the excess return per unit of total risk:

where is the portfolio return, is the risk-free rate, and is the portfolio standard deviation. The tangency portfolio on the efficient frontier maximizes this ratio.

Capital Market Line

When a risk-free asset is available, the Capital Market Line (CML) represents the set of optimal portfolios that combine the risk-free asset with the tangency portfolio:

where and are the return and standard deviation of the market (tangency) portfolio. The slope of the CML equals the Sharpe Ratio of the market portfolio.

Portfolio with 60% Stock, 40% Bond

Expected return:

Portfolio variance:

Portfolio standard deviation:

Notice that the portfolio standard deviation (13.3%) is less than the weighted average of individual standard deviations (0.6 x 20% + 0.4 x 8% = 15.2%). This reduction is the diversification benefit, arising because the correlation is less than 1.