Robert F. Engle and Simone Manganelli

U.C.S.D.

January 1999

Value at Risk has become the standard measure of market risk employed by financial institutions for both internal and regulatory purposes. Despite its conceptual simplicity, its measurement is a very challenging statistical problem and none of the methodologies developed so far give satisfactory solutions. Interpreting Value at Risk as a quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation which does not require any of the extreme assumptions invoked by existing methodologies (such as normality or i.i.d. returns). The Conditional Value at Risk or CAVIAR model moves the focus of attention from the distribution of returns directly to the behavior of the quantile. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, and postulating a variety of dynamic updating processes, we use GMM estimation to determine the parameters of the updating process and tests of model adequacy. We use a differential evolutionary genetic algorithm to optimize an objective function which is horizontal almost everywhere and hence cannot be optimized using traditional algorithms based on differentiation. Applications to simulated and real data provide empirical support to our methodology and illustrate the ability of these algorithms to adapt to new risk environments.