The Correlation Matrix Under General Conditions: Robust Inference and Fully Flexible Stress Testing and Scenarios for Financial Portfolios
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Responsible use of any portfolio model that incorporates correlation structure requires knowledge of its sampling distribution. This is especially true of models used in stress testing, or ones requiring the specification of particular scenarios with particular correlation values (e.g. in views-based portfolio analyses a la Black-Litterman (1991) and its variants), because there is no other way to reliably associate those values with probabilities. Put differently, the correlation matrix is a model parameter like any other, and a model’s results cannot be fully understood or relied upon if the behavior of its parameters is not well defined. This makes answering the following question of central importance: given an estimated (product-moment) correlation matrix and an assumed or well-estimated data generating mechanism, what is the sampling distribution of that matrix? And how does that finite sample density relate to the densities of each of its pairwise correlation cells? We provide the solution to this question for any correlation matrix under the most general data conditions possible, requiring only the existence of the mean and variance for each marginal distribution in the portfolio and the positive definiteness of the correlation matrix. This solution explicitly connects the densities associated with each of the correlation cells to that of the entire matrix, making the latter a function of the former. One only needs to specify the cumulative distribution function (cdf) value associated with each of the cells to retrieve the corresponding, unique correlation matrix (as well as its overall probability of occurrence conditional on the estimated matrix). And in reverse, given a valid (positive definite) correlation matrix, the density provides the corresponding, unique matrix of cdf values, as well as its overall conditional probability of occurrence. Associating matrices with their 'cdf matrices' can be defined in terms of shifts from the estimated correlation matrix, providing a very convenient and flexible way to specify scenarios at the most granular level – that of every pairwise relationship between factors/variables, as opposed to merely at the level of the factors/variables. This all is accomplished within a geometric framework wherein the sampling mechanism automatically enforces positive definiteness, which not only allows for efficient enumeration of the sampling space, but also and more importantly, much more robust inference of the entire matrix compared to other more limited (spectral) approaches. Finally, unlike any other framework, the geometric approach to this problem also scales on itself: it can be applied to any submatrix of the given correlation matrix, enabling fully flexible scenario specification wherein some cells remain completely untouched, while others are appropriately affected by the scenario. This is a realistic, common need in many investment and risk settings. The foundations of the geometric approach have long been established within the relevant statistical and finance literatures, but its pieces have not previously been combined in such a way as to solve this problem under general conditions. All results are validated by well-established results from the Random Matrix Theory literature, even as the geometric approach is more robust, empirically and structurally, than those relying on spectral distributions. Lastly, the solution is scalable, having been readily implemented on a commodity laptop on matrices 100x100 and larger.
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