\( \newcommand{\Frac}[2]{\displaystyle\frac{#1}{#2}} \newcommand{\eins}{{\mathbf 1}} \newcommand{\var}{\text{var}} \newcommand{\cov}{\text{cov}} \newcommand{\w}{{\mathbf w}} \newcommand{\p}{{\mathbf p}} \newcommand{\R}{\mathbb R} \newcommand{\r}{{\mathbf r}} \newcommand{\bmu}{{\boldsymbol \mu}} \newcommand{\x}{{\mathbf x}} \)

The Mean-Variance Portfolio Problem

This page gives geometric intuition for the global minimum-variance portfolio and its orthogonality properties.

The Global Minimum-Variance Portfolio


The global minimum-variance portfolio, \(\w^{\text{gmv}}\), is the portfolio with the lowest possible return variance. It is the solution of the problem

\(\begin{eqnarray*} \frac{1}{2}\var(\w)&=&\frac{1}{2}\,\w'\Sigma \w \to \min_w\\ \text{s.t.}\quad \w'\eins &=& 1 \end{eqnarray*} \)

  • The iso-variance ellipsoid corresponding to \(\w^{\text{gmv}}\) is the "smallest" among all ellipsoids that intersect the portfolio plane (i.e., representing the lowest possible variance).
    This ellipsoid has only one unique intersection with the portfolio plane, \(\w^{\text{gmv}}\).
  • The global minimum-variance portfolio, \(\w^{\text{gmv}}\), is orthogonal to all possible portfolio transactions (= covariance orthogonal).
    This means that the return of any transaction from one portfolio to another portfolio is uncorrelated to the return of the global minimum-variance portfolio.
    Thus, for two arbitrary portfolios \(\w_1\) and \(\w_2\) we have

    \( (\w_1-\w_2)'\, \Sigma\, \w^{\text{gmv}} = 0 \)

Interactive Chart.

The Lagrange Problem

  • Let us now use the Lagrange formalism to determine the global minimum-variance portfolio.
  • The Lagrange function of the global minimum-variance portfolio problem is given by

    \( L(\lambda,\w) = \frac{1}{2} \w'\Sigma\w + \lambda(1-\w'\eins) \)

  • First-order optimality conditions of the Lagrange problem are

    \(\begin{eqnarray*} \Sigma\w-\lambda\eins&=&0,\quad(\text{i})\\ \w'\eins&=&1.\quad(\text{ii}) \end{eqnarray*} \)

  • Equation (i) states that at the optimum \(\w^{\text{gmv}}\), the Euclidean normal vector of the iso-variance ellipsoid at (\(\Sigma\w^{\text{gmv}}\)) must be parallel to the vector \(\eins\), with \(\lambda\) as the constant of proportionality.

    \(\begin{eqnarray*} \Sigma\w^{\text{gmv}}&=&\lambda\eins,\\ \Rightarrow\quad \w^{\text{gmv}}&=&\lambda\Sigma^{-1}\eins. \end{eqnarray*} \)

  • Since the global minimum-variance portfolio is a portfolio, its weights must sum to one,

    \(\begin{eqnarray*} {\w^{\text{gmv}}}'\eins&=& 1,\\ \lambda\eins'\Sigma^{-1}\eins&=& 1,\\ \lambda &=& \frac{1}{\eins'\Sigma^{-1}\eins}. \end{eqnarray*} \)

    • Weights of the global-minimum variance portfolio

    The unique portfolio that has minimum variance among all possible portfolios is characterized by the weight vector

    \( \w^{\text{gmv}}=\Frac{1}{\eins'\Sigma^{-1}\eins}\Sigma^{-1}\eins. \)


  • What is the minimum portfolio variance among all portfolios? Lower portfolio variance than the variance of \(\w^{\text{gmv}}\) cannot be attained.

    • Variance of the global-minimum variance portfolio

    The global minimum-variance portfolio has a return variance of

    \( \var(\w^{\text{gmv}}) = {\w^{\text{gmv}}}'\Sigma\w^{\text{gmv}}=\Frac{1}{(\eins'\Sigma^{-1}\eins)^2}\eins'\Sigma^{-1}\Sigma\Sigma^{-1}\eins=\frac{1}{\eins'\Sigma^{-1}\eins}. \)