# Why do we need correlated random variables in a Monte Carlo simulation?

Question: I don't understand why a Monte Carlo simulation needs correlated random variables. Isn't each simulation thread independent?

Background:

Specifically, I'm referring to the below example on pg 319 in the Malz text (http://ca.wiley.com/WileyCDA/WileyTitle/productCd-0470481803.html).

He describes a Monte Carlo simulation with 1,000 simulation threads to calculate credit losses on a CDO with 100 underlying credits.

### #1.

In the simulation, we setup a matrix of 1,000 draws from a 100 dimension joint normal distribution.

### #2.

We posit 4 separate assumptions for pairwise correlation 0, 0.3, 0.6, 0.9

### #3.

For each correlation assumption, the matrix of 1,000 random normals is transformed into matrix of 1,000 correlated random normals (which of course are still 100 dimensional normals).

I don't understand why we need to transform the matrix of uncorrelated random normals into correlated ones? Isn't each simulation thread independent of the previous ?

## 2 Answers

Surely what is meant is that the 100 components are pairwise correlated but the 1000 draws are independent.

• I was thinking that's what made sense, but wasn't sure. Thanks for clarifying. – AfterWorkGuinness May 12 '16 at 0:18

Correlated (simulated) features or variables are used when the source data has correlated features. If the original features in the source data are not correlated and are orthogonal, then there is no reason to use correlation when simulating. Many assets are correlated, mostly through volatility clustering and sentiment. Secular bull and bear markets can also cause many assets to correlate. If there was no correlation, then you could merely build a portfolio by randomly selecting the assets, and then weighting the amount owned by historical returns -- which is never a good idea.

Bank portfolio stress testing involves use of copulas (simulations) to artificially introduce greater levels of correlation between portfolio assets to determine Variance at Risk, Expected Tail Loss, etc. Basically, when times are bad (e.g., subprime mortgage crisis), many assets begin to become strongly correlated. Thus, if there are large market corrections (loss), when many assets start correlating, the chances of greater loss are larger -- because you are not only losing net worth via one asset, but losing a lot more via loss among many correlated assets. (a particular type of portfolio which can safeguard against high correlation during increased volatility with correlation is called a collar portfolio -- which is a form of hedging). When correlation is also greater, there is less diversity -- which is less sustainable.