Extreme Value Theory

2. Caveats

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In practice life is not as simple as is suggested in the Introduction. A particularly important issue is that extrapolation into the tail of a probability distribution isn’t challenging because it is difficult to identify possible probability distributions that might fit the observed data. Instead the challenge is that the range of answers that can plausibly be obtained can be very wide. Extrapolation of any sort (including, as here, extrapolation into the tail of a distribution) is an intrinsically uncertain exercise, much less reliable than interpolation, as is explained in Press et al (2007).


Three other important caveats are relevant when EVT is applied to financial data:


(a)          EVT relies on the tail of the distribution in question actually converging in some suitable sense. This generally occurs for smooth distributions commonly used by statisticians like the normal distribution, the Student’s t distribution, the Pareto distribution etc. However, these sorts of distributions are very ‘well behaved’ in a mathematical sense and also form an infinitesimal proportion of the totality of possible distributions that might apply. So it is by no means obvious that convergence of the sort required for EVT to apply will actually take place in practice. It is relatively straightforward to construct distributions where convergence doesn’t occur, although whether they are plausible is again a matter of opinion. Fundamentally, extrapolation involves exercise of judgement, and what one person thinks is reasonable someone else may think is not.


(b)          EVT is usually developed from a univariate, i.e. single series, perspective. Some important financial problems, in particular portfolio construction, are intrinsically multivariate in nature. For example, most practical portfolio construction problems require selection between asset categories, so require an understanding of the joint behaviour of different return series. It is possible to develop a multivariate extreme value theory (including results for multivariate maxima and multivariate threshold exceedances), but the mathematics is quite complicated, perhaps best analysed using copulas, and is not very easily aligned to the portfolio construction problem.


(c)           The conceptual appeal of EVT may encourage researchers to leap in with the technique without first trying to understand what might be causing the observed tail behaviour. An important point here is that financial data often exhibits time-varying volatility (also known as volatility clustering). If this point is given insufficient weight then EVT results, even if theoretically applicable, may be easily misinterpreted or misapplied.


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