Extreme Value Theory

I begin by applying extreme value theory (EVT) to model fat-tailed return distributions, specifically, the distribution of losses exceeding a prespecified lower threshold. Extreme value theory has received much attention in the insurance industry to predict rare events such as floods, earthquakes, and other natural disasters and has a large devoted literature. See for example, the text by Embrechts, Kluppelberg, and Mikosch (1997). Our interest centers not only on applying extreme value theory to predict rare market events but, more importantly, monitoring changes in the parameters of the extreme value distributions themselves, which may signal a fundamental shift in downside risks. There is also a deep and rich literature on applying EVT to financial markets. See, for example, Cotter (2006), Longin and Solnik (2001), LeBaron, Blake, and Samanta (2004), Malevergne, Pisarenko, and Sornette (2006).

There is a related literature on extreme outcomes, specifically, the distribution of order statistics (see, for example, Mood et al. 1974 for an introduction, and the related quantile regression theory developed by Koenker and Basset [1978], which models specific extreme quantiles, for example, the fifth percentile of the distribution of returns). EVT, on the other hand, models the likelihood function for the tail density and not a quantile, which in our case, is the set of minimum returns found in the left-hand tail. The challenge with EVT is to estimate the tail ...

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