Monte Carlo Simulation by Cholesky or PCA?-PartII
Jun 03, 2006
Continued from Part I ............................
Why Cholesky Decomposition method is unstable?
The Cholesky decomposition method is very easy to implement, especially on an Excel spreadsheet. However, a Cholesky matrix is not very stable and the decomposition may break down from time to time.
A Cholesky matrix can exist (i.e. its value can be found) if and only if the variance-covariance (VCV) matrix is positive definite. Now what do we mean by this? This means that all eigenvalues of the VCV matrix must be positive. In practical terms this implies that no risk factor (or an asset) can have a perfect correlation (+1 or -1) with another risk factor (or an asset). The condition of positive definiteness often breaks down when constructing the covariance matrices due two main reasons: (1) there is not enough historical data to show that the assets of the risk factors are independent or have imperfect correlation and/or (2) there errors in the data, or the data for different assets is taken using different time periods. It is generally observed that beyond a 20 asset portfolio the Cholesky decomposition method breaks down.
JP Morgan /RiskMetricsTM in their landmark technical document on Value at Risk (VaR) talk about "cleaning a correlation matrix" at length. Remember, it was after the publication of RiskMetricsTM VaR technical document in mid nineties that made the Cholesky decomposition method popular amongst the risk managers worldwide.
What is a Nonsensical Correlation Matrix and How do we clean such a Matrix?
If correlations between assets (or risk factors) are nonsensical for whatever reasons (fudged or invented correlation, error in data sample, etc.) then the Cholesky Matrix will not exist because nonsensical correlations will prevent the VCV matrix from becoming positive definite, or at least positive semi-definite. The necessary condition for a covariance matrix and a correlation matrix to be positive semi-definite is that all eigenvalues must be positive which in turn necessitates that:
Very simply put, one can see that the equation (identity) above is simply the variance (or variance-covariance) of the portfolio. Therefore, for the correlation matrix to be valid the variance of the portfolio cannot be negative.
In the event that the Cholesky matrix breaks down due to nonsensical correlation matrix then there is a simple (to understand but slightly complicated to implement) procedure to clean it:
- Set all the negative eigenvalues to zero;
- Calculate the new correlation matrix which is then used in place of the original correlation matrix;
- Generate new eigenvalues and eigenvectors to test the matrix.
The new correlation matrix is calculated as:
The Alternative is to go for the PCA method
Due to the reasons mention above, despite its simplicity, a lot of practitioners go for the eigenvalue decomposition method (PCA) right from the start. It requires more numerical recipes to be written (in Excel/VBA macros or C++) to calculate the eignevalues and the eigenvectors (usually it is done using Jacobi method) but it much more stable. However, one must remember that even the eigenvalues decomposition method will fail if the correlation matrix (and thereby the VCV matrix) is built from data from different time periods because data inconsistency will cause the variances to become negative.
Source: Risk Latte Company has an extensive library of numerical recipes of the Cholesky decomposition and the eigenvalue decomposition matrices in Excel/VBA and C++ and it uses them extensively in developing models, pricing tools and during the training sessions. There are two excellent sources (books) for the theoretical foundations of above concepts on which the above articles I & II are based. These books are "Implementing Value at Risk" by Philip Best (John Wiley & Sons) and "The Fundamentals of Risk Management" by Chris Marrison (McGraw Hill books).
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