Exercise: Cholesky Factorisation and Simulating Correlated Gaussians

Prerequisites: Covariance Matrices

Problem

You want to simulate a $3$-dimensional gaussian vector with mean $\mu = (0.05, 0.08, 0.10)$ (annual returns) and covariance matrix

(Three assets with volatilities $20\%, 30\%, 40\%$ and moderate correlations.)

1. Compute the Cholesky factor $L$ (lower triangular with $LL^\top = \Sigma$) by hand (or using np.linalg.cholesky). Verify $LL^\top = \Sigma$.

2. Explain why, if $Z \sim \mathcal{N}(0, I_3)$ and $X = \mu + LZ$, then $X \sim \mathcal{N}(\mu, \Sigma)$.

3. Simulate $N = 10{,}000$ such vectors. Compute the empirical mean and covariance and verify they are close to $\mu$ and $\Sigma$.

4. Portfolio variance check. For an equal-weighted portfolio $w = (1/3, 1/3, 1/3)$, compute (a) theoretical portfolio variance $w^\top\Sigma w$ and (b) empirical variance of $w^\top X_i$ across your $N$ samples. They should agree within sampling noise.

Hint