TABLE OF CONTENTS Correlation risk is one of the most dangerous and least-analyzed risks. In order to come to grips with correlation risk with many variables, we need to be able to deal with the problem of stressed correlation matrices. Stressed correlations are needed for robust risk management because correlations are notoriously unstable [1]. In particular, during stressed markets, correlations often increase dramatically [i]. Therefore, for risk assessment, we need to stress the correlations.
Unfortunately, the procedure of stressing the correlations is problematic. Often, stressed correlation matrices are non-positive definite ( NPD) matrices because they arise from inconsistent stressing of the various correlation matrix elements that are interdependent and constrained [2].
In this section, we discuss the representation of correlations in N dimensions, i.e. correlation matrices with elements as correlations between pairs of N variables. One of the main problems with correlations is that they are dependent. By reformulating the problem geometrically, we recast the correlations into functions of independent angular variables [3]. These angles are the natural spherical co-ordinates for an 
sphere where 
. In this way, theproblem of describing positive definite (PD) correlation matrices becomes tractable. With this, we are then able to deal with stressed correlation matrices.
In general, an arbitrarily stressed correlation matrix will be non-positive-definite (NPD). The idea that we will present in Ch. 24 is to get a best PD matrix fit to the NPD matrix, using least-squares fit in the angular...
