# Multivariate normal when Cholesky decomp fails on Sigma

I'm trying to do multivariate distributions of returns on buckets where all the returns are at least 0.6 correlated at a 95% confidence level. I have the buckets, but their Sigmas cannot be decomposed and my random number generator fails.

What alternatives do I have? If I break the buckets up into smaller ones, I lose the correlation in the simulation. Or am I completely wrong?

• Is your matrix $\Sigma$ singular? – Bob Jansen Jul 6 '16 at 18:44
• I can only assume that it is. I'm running smaller buckets and using rolling means to try to add in some of the historical correlation. – milkmotel Jul 6 '16 at 19:02
• I don't believe I understand your setup but you can check whether the matrix is singular and whether that is the cause of the problem. – Bob Jansen Jul 6 '16 at 19:04
• It wasn't singular. My setup was to calculate the covariance matrix for say ACWI, SP500, MSCI EM, HFRI Direct Hedged Equity, HFRI FOF, etc. I broke them into smaller buckets to simulate the returns over time, so I lose the correlation from period to period between the buckets, but since I'm only sampling the noise, the rolling means I use for each asset class of returns should add back in some of the correlation. – milkmotel Jul 6 '16 at 19:21

If you're going to go down this route though, it works better to obtain your $\Sigma^\frac{1}{2}$ matrix using the EVD rather than cholesky.