Analysing the Determinant of a Variance-Covariance or a Correlation Matrix
September 22, 2009
In quantitative finance the determinant of a variance-covariance (VCV) matrix or a correlation matrix should be strictly positive. If it is negative or zero then we cannot use that VCV or correlation matrix in our calculations. Correlation (and VCV) matrices with negative determinant are called nonsensical and they have to be cleaned before that can by input in any model.
Why does the sign of the Determinant of a VCV matrix or a Correlation matrix matter? This is important when we are dealing with multi-asset Monte Carlo simulation to value financial derivatives or compute the value at risk (VaR) of a portfolio. If the Determinant of the VCV matrix or the Correlation matrix is zero or negative then the VCV or the Correlation matrix is said to be nonsensical. This means that the VCV matrix or the Correlation matrix is not positive semi definite.
When is a matrix positive semi definite? A symmetric matrix, like a VCV or a Correlation matrix, is positive semi definite when all its eigenvalues are non-zero (positive). For a valid correlation or a VCV matrix all eigenvalues of the matrix have to be positive. By valid we mean that the VCV or the correlation matrix is positive semi definite. If we do not have a positive semi definite correlation (or a VCV) matrix then we cannot use this as an input to the problems of multi-asset Monte Carlo simulation or risk analysis.
But the how is the determinant of a VCV matrix related to the property of positive semi-definiteness?
This is because the determinant of a square, symmetric matrix is related to the eigenvalues of the matrix as:
Where, is a square, symmetric matrix with dimension ( rows and columns), is the determinant of and are eigenvalues of .
Therefore, if all eigenvalues of a matrix are non-zero, then the determinant of that matrix has to be non-zero as well. This is a quick and dirty check for validity of correlation matrix. It is a much easier to calculate the determinant of a matrix - for example in ExcelTM spreadsheet, there is a simple command for this - than calculating the eigenvalues of a matrix, and especially so, if the matrix is of higher dimensions (such as a 5 asset or 10 asset case). Whenever we have a correlation matrix or a VCV matrix we should calculate the determinant of the matrix and see if it is non-zero. If it is then we can go ahead and use it in our calculations (if not, then we have to clean the correlation or the VCV matrix).
What does all this mean at a more fundamental level?
Let's look at a simple 2 x 2 correlation matrix. The geometrical interpretation of determinant is that - in a 2 x 2 framework (2 x 2 matrix) - it measures the area that is spanned by the two column vectors of the 2 x 2 correlation matrix. If both the vectors are aligned, which means one of the vectors is linearly dependent on the other, then the determinant is zero. The value of the determinant becomes maximum when the angle between the vectors is 90 degrees which is equivalent to the area under a rectangle formed by the two vectors.
If the determinant is a measure of the area then how can the area of a surface outlined by two vectors be negative? Therefore, a negative determinant, though possible for any arbitrary square matrix, is geometrically incomprehensible for a symmetric, correlation or covariance matrix.
Actually, a determinant of zero is also not allowable - for any arbitrary matrix - for the simple reason that when the determinant is zero the matrix is non-invertible, i.e. the inverse of the matrix does not exist. Such matrices are called singular matrix (think of dividing a number by zero; the result is undefined or indeterminate).
The determinant of a VCV matrix is a measure of "generalized variance", which is in essence a measure of the spread of the observations in the data set. The determinant for a 2 x 2 VCV measures the overall information in the matrix, i.e. the total variance of each variable less whatever correlation (covariance) there is between the variables. Here again, a negative generalized variance cannot be defined and therefore, if the value of the determinant of a VCV (or correlation) matrix is negative then we are in the realm of the nonsense. And a zero variance will only apply to constants, i.e. no spread in the observations in the data set. Therefore, zero variance is meaningless because this would mean all observations in the data set are same.
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