All simulators
Covariance Ellipses & a′Σa
Correlated draws built with a Cholesky factor, their covariance ellipses, and the variance of any linear combination.
Day 22: Positive-Definite Matrices & Variance–Covariance Matricesa′Σa (theory)
2.525
var(a′X) sample
2.727
eigenvalues of Σ
2.72, 0.53
max / min of a′Σa
Cholesky L
[1.50 0; 0.60 0.80]
500 draws made from independent normals with the Cholesky factor L. The ellipses are the 1- and 2-sd contours; their axes are the eigenvectors of Σ. The orange direction a gives a combination a′X whose variance is a′Σa.