From Value at Risk to Stress Testing: The Extreme Value Approach
by François M. Longin of the Cergy-Pontoise Cedex
Abstract: This article presents an application of extreme value theory to compute the value at risk of a market position. In statistics, extremes of a random process refer to the lowest observation (the minimum) and to the highest observation (the maximum) over a given time-period. Extreme value theory gives some interesting results about the distribution of extreme returns. In particular, the limiting distribution of extreme returns observed over a long time-period is largely independent of the distribution of returns itself. In financial markets, extreme price movements correspond to market corrections during ordinary periods, and also to stock market crashes, bond market collapses or foreign exchange crises during extraordinary periods. An approach based on extreme values to compute the VaR thus covers market conditions ranging from the usual environment considered by the existing VaR methods to the financial crises which are the focus of stress testing. Univariate extreme value theory is used to compute the VaR of a fully aggregated position while multivariate extreme value theory is used to compute the VaR of a position decomposed on risk factors.
Keywords: Aggregation of Risks, Capital Requirements, extreme value theory, Financial Crises, Financial Regulation, Measure of Risk, Risk Management, Stress Testing, Value at Risk.
Published in: Journal of Banking & Finance, Vol. 24, No. 7, (July 2000), pp. 1097-1130.