Value at Risk Calculations, Extreme Events, and Tail Estimation

Abstract

Value at risk has become a standard approach for estimating and expressing a firm9s exposure to market risk. Unlike the traditional risk measure, standard deviation, VaR focuses only on the tail of the distribution of outcomes - the extreme events. This makes a lot of sense in theory, but a major problem arises in practice, because empirical returns distributions tend to have tails that look quite different from those of the normal and lognormal distributions that we typically assume in finance. Extreme value theory offers an elegant solution. Like the familiar central limit theorem, which proves that under general conditions the distribution of the average of n identically distributed random variables will converge to the normal as n grows large, extreme value theory provides similar convergence results for the extreme tails of distributions. In this article, Neftci gives a clear and intuitive introduction to extreme value theory and demonstrates how to use it. In an example applying it to forecasting the tails of the distributions of interest rate and exchange rate changes, extreme value theory is shown to be much more accurate than the standard value at risk calculated from the normal distribution.

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