Power Laws and Extreme Value Theory

The commonly used risk measures of VaR and CVaR almost always deal with the tails of a distribution. A risk manager often needs to report the 99% and 99.9% VaR and CVaR. He or she rarely needs to find the 60% VaR or CVaR. This indicates that much of the full distribution is ignored for risk purposes, even though a lot of effort may have gone into creating the whole distribution of future gains and losses of some asset. This prompts the question, "Why not simply have a methodology to create only the tail of a distribution and ignore everything else?" Extreme value theory (EVT) is a field of probability that studies the distribution of extreme realizations of a given distribution function. The fundamental result of EVT is that the distribution of extreme values of independent and identically distributed (IID) random samples from a given distribution essentially converges to one out of three EVT-type distributions. This means that the asymptotic nature of extreme values does not depend on the exact nature of the parent distribution. This is particularly useful for risk purposes, as there is no general agreement on which fat-tailed distribution is the ideal one to use for an asset class. EVT tells you that the precise type of distribution may not matter for risk purposes. Throughout this chapter, losses will be seen as positive numbers—e.g., a loss of $100MM. Using this nomenclature, the right side of the tail will be of interest to risk management.