Blending Independent Components and Principal Components Analysis

1. Introduction and Summary

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1.1          A widely held view across the financial services industry and within relevant academic circles is that financial markets exhibit ‘fat-tails’, i.e. extreme events occur more frequently than we might expect were market behaviour to be modelled in line with the normal (i.e. Gaussian) distributions often assumed in more straightforward academic texts. Despite this, commercial risk systems often still include assumptions of normality within their underlying formulations. This is partly because such assumptions often make the mathematical formulation of the underlying risk model framework more analytically tractable. It is also partly because the risk system providers may argue that the impact of deviation from normality is insufficient (or the evidence for such deviation is insufficiently compelling) to justify the necessary refinements to that part of the underlying risk model formulation. In short, there is a trade-off between, on the one hand, complexity, practicality and presumed model ‘correctness’ and, on the other hand, simplicity, analytical tractability and practical implementability.

1.2          In these pages, we explore two methodologies that are or ought to be relevant to the design of portfolio risk systems. One of these, principal components analysis, see Section 3, is a well-known and common tool used in the creation and validation of current portfolio risk system designs. It allows the risk system designer to identify potential ‘factors’ that best seem to explain individual stock variability. Underlying its applicability to portfolio risk model design is an implicit assumption of normality (or to be more precise an investor indifference to fat-tailed behaviour, i.e. a lack of need to include in the model a characterisation of the extent to which behaviour is non-normal).

1.3          The other is independent components analysis. It, and several variants motivated by the same underlying rationale, are described in Section 2. Given that it is a less well-known technique within the financial community we introduce it first, and when doing so we explore it in greater depth than the better-known principal components technique. Independent components analysis has perhaps more commonly been applied to other signal extraction problems, e.g. image or voice recognition or differentiating between mobile phone signals. In its normal formulation it seeks to extract ‘meaningful’ signals however weak or strong these signals might be (with the remaining ‘noise’ discarded). ‘Meaningful’ here might be equated with extent of non-normality of behaviour, as is explicitly done in certain formulations of independent components analysis. In contrast, in the portfolio risk management context ‘meaningfulness’ also needs to be coupled with ‘significance’ (i.e. ‘magnitude’) for the source in question to be worth incorporating in the risk model formulation. Moreover, ‘noise’ is not to be discarded merely because it does not appear to be ‘meaningful’, because any such variability adds to portfolio risk.

1.4          Despite these apparent differences, we show in Section 4 that both of these techniques can be thought of as examples of a more generic approach in which we grade possible sources of observed behaviour according to some specified importance criterion. We show how we can use this insight to blend together the two techniques to enhance the portfolio risk model design. In particular, we can choose a blend that ought to cater better for fat-tails, i.e. extreme events, than more traditional risk model designs. However, we also highlight some of the challenges that arise when trying to cater in such a prescription for the time-varying nature of the world.

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