The Global Minimum Variance Portfolio is the asset weight distribution that minimizes the variance (risk) of the overall portfolio. It can be calculated as: \(\boldsymbol{A_mz_m = b}\)
Where \(A_m\) is \(= \begin{bmatrix} 2\boldsymbol{\Sigma} & \boldsymbol{1} \\ \boldsymbol{1^t} & 0\end{bmatrix}\)
\(\boldsymbol{\Sigma}\) is the covariance matrix, \(\boldsymbol{1}\) is an all one’s vector and \(\boldsymbol{1^t}\) its transpose.
\(\boldsymbol{z_m}\) is the asset weights (plus a Lagrange Multiplier).
\(\boldsymbol{b}\) is a zero vector with the last entry a one.
This equation is solved for \(\boldsymbol{z_m}\) using LU decomposition and back substitution.
Expected Portfolio Return is calculated as: \(\boldsymbol{W^t.\mu}\)
Where \(\boldsymbol{W^t}\) is the transpose of the GMVP weights vector and \(\boldsymbol{\mu}\) is the asset mean returns vector.
Portfolio Variance is calculated as \(\boldsymbol{W^t.\Sigma.W}\)