Portfolio opportunity set (source code)

= Portfolio opportunity set
{title2=$\{(\sqrt{w^T\Sigma w},w^T\mu):\mathbf1^Tw=1\}$}

The set of attainable return <standard deviations> and <expected returns> for budget-feasible risky <investment portfolios>. Negative weights are allowed when <short selling> is permitted. With positive definite <covariance matrix> and nonconstant means, the minimum-risk boundary is a hyperbola in standard-deviation/mean coordinates; its upper branch is the <mean-variance efficient frontier>. Adding a risk-free asset changes the budget constraint on risky holdings and produces a different attainable set.