Correlation: Pitfalls and Alternatives

Correlation is a mine eld for the unwary. One does not have to search far in the literature of nancial risk management to nd misunderstanding and confusion. This is worrying since correlation is a central technical idea in nance. Correlation lies at the heart of the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT), where its use as a measure of dependence between nancial instruments is essentially founded on an assumption of multivariate normally distributed returns. Increasingly, however, correlation is being used as a dependence measure in general risk management, often in areas where the assumption of multivariate normal risks is completely untenable such as credit risk. In using correlation as an all-purpose dependence measure and transferring CAPM thinking to general risk management, many integrated risk management systems are being built on shaky foundations. This article will tell you when it is safe and unproblematic to use correlation in the way that you imagine you can use it, and when you should take care. In particular it will tell you about two fallacies that have claimed many victims. These traps are known to statisticians, but not, we suggest, to the general correlation-using public. We will help you avoid these pitfalls and introduce an alternative approach to understanding and modelling dependency copulas. The recognition that correlation is often an satisfactory measure of dependence in nancial risk management is not in itself new. For instance, Blyth (1996) and Shaw (1997) have made the point that (linear) correlation cannot capture the non-linear dependence relationships that exist between many real world risk factors. We believe this point is worth repeating and amplifying; our aim is to provide theoretical clari cation of some of the important and often subtle issues surrounding correlation. To keep things simple we consider only the static case. That is, we consider a vector of dependent risks (X1; : : : ; Xn) 0 at a xed point in time. We do not consider serial correlation within or cross correlation between stochastic processes (see Boyer, Gibson, and Loretan (1999) in this context). Nor do we consider the statistical estimation of correlation, which is fraught with diÆculty. In this paper we do not go this far because enough can go wrong in the static case to ll an entire book.