Global Minimum Variance Portfolio - 2023.2 English

Vitis Libraries

Release Date
2023.2 English

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 \(\boldsymbol{W^t.\Sigma.W}\)