# How to stress test a correlation matrix

As part of a mean variance portfolio task, I am calculating portfolio risk and optimal allocations between assets given required level of return. Input: expected returns, volatility and correlation matrix. So far so good.

As a second part, I am supposed to stress input correlation matrix by some multiplier (say 1.3) and see how to it impacts the allocations and portfolio risk.

The question I have is: can I just multiply all fields in the correlation matrix by the given multiplier? It seems wrong to me as I would end up with "self correlation" > 1 on the diagonal, which makes no sense? Should I just keep diagonal as 1s and only multiply the rest? What if the multiplier is such that correlation between i and j will be >1 anyway? Any ideas appreciated, thank you.

NOTE: No risk free asset in this scenario, but it shouldn't matter.

• As an ad-hoc fix you could set $\rho_{ij}:=\max[-1,\min[1,1.3 \rho_{ij}]]$. That is multiply by 1.3 but then limit it to be between -1 and 1. – noob2 Mar 3 '20 at 23:48
• thats what I ended up doing, thanks – NeverStopLearning Mar 6 '20 at 18:44

## 1 Answer

As the correlation matrix will most probably become non-positive-semi-definite with such an ad hoc manipulation, you may try one of the following:

1. Still run that algorithm and check that the resulting matrix is still positive (semi) definite.

2. Bootstrapp the correlation matrix, or the volatilities, or both, from your input data.

3. Manipulate the eigenvalues of your correlation matrix, e.g. by shifting more weight to the first eigenvalues (increasing correlations, overall) and then normalise the hypothetical correlation matrix obtained by multiplying your original eigenvectors with your shifted eigenvectors.

IMHO, option 2 has the most merit as it incorporates the empirical variability of the correlation matrix estimation into your portfolio selection process.