Converting Correlation into Covariance for Liquidation Tracking Error

# Liquidity Stress Test of Investment fund - Liquidation tracking error


# Liquidity Stress Test of Investment fund - Liquidation tracking error












It is my first message on this board, I have hesitated a few days before bothering you with my struggles, but I've seen a lot of very knowledgeable and patient people here willing to help out. I apologize in advance for my bad English as well as my naive questions, I'll try to be as precise and concrete as possible.

I am working on building a liquidity stress test framework for an investment fund in listed equities (very basic portfolio, not equally weighted).

I need to simulate a redemption shock impacting the liquidity of my fund, and to assess the consequences observed during and after the liquidation process of the assets.

I am now working on the horizontal slicing (waterfall) approach, meaning we liquidate every day as much of every asset as we can.

One of the differences with vertical slicing is that Waterfall approach implies a portfolio distortion, meaning the weights of every security in the portfolio will not be the same after the liquidation. In vertical slicing, however, we liquidate the same proportion of every asset (this proportion is equal to the redemption shock in %) so the weights remain the same.

Therefore, I need to work on the trade-off between liquidation costs and portfolio distortion in case of horizontal slicing liquidation.

I base my simulations and formulas on the ones given in the working papers published by Amundi in 2021, which I will refer to in my message. (Liquidity Stress Testing in Asset Management, Amundi Working Paper, 2021)

I need first to calculate the liquidation tracking error, to assess the portfolio distortion. The formula given by the paper is this one :

... and this is where it starts to get complicated. I wont lie, my math skills are far from great and vectors & matrixes are already a challenge. What I have already done is calculating the difference between "original weights" and weights after one day of liquidation. What I do not understand is the matrix part, I don't quite get the link between weights (vectors) and returns (matrix).

The publication gave an example and here is what I have done so far :

However, the matrix part gets me confused. The only information given by the docuument is the correlation matrix of asset returns (and not the covariance matrix of asset returns, like in the formula :

Does someone have any idea on how I can pursue with this calculation? I would like to know how I can determine this covariance matrix, in this example but also in general with other values, and how I can multiply it to the vector I calculated.

I thank you very much for reading this long post, I hope I have been clear in my explainations but if not please ask any information you'd need.

Once again thank you for your time and patience,

Kind regards

## Answer by mountshoutcap (score 1, accepted)

https://quant.stackexchange.com/a/77619

not sure if i got the gist of your question, but in general i'd say you can do a little of matrix algebra starting from the correlation matrix to get the covariance matrix. if you were to do this for just 2 assets, X and Y, the base formula is $$ \text{Cov}(X,Y) = \rho_{X,Y}\times\sigma_X\times\sigma_Y $$ now if you want to do the same with matrices you should do $$ \text{Cov} = \text{Diag}(\Sigma)\cdot\text{Corr}\cdot\text{Diag}(\Sigma) $$ where $ \text{Diag}(\Sigma)$ is a diagonal matrix with the standard deviations of each asset on the diagonal.

update from your comment.

the final result that you get from your formula, the liquidation tracking error, is indeed a measure of variation of the portfolio's returns caused by the liquidation, like you said. you can view the $D(q|\omega)$ as the standard deviation of the portfolio's excess returns due to liquidation, which is a direct measure of the risk (or variability to use your words) introduced by a shift in portfolio weights.