Modeling a Time-Varying Order Statistic

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 challenging statistical problem and none of the methodologies developed so far gives a satisfactory solution. Interpreting Value at Risk as a quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation that does not require any the extreme assumptions of existing methodologies, such as normality or i.i.d. returns. The Conditional Value at Risk, or CAVIAR, model shifts attention from the distribution of returns to the behavior of the quantile. We use the regression quantile framework introduced by Koenker and Bassett to determine the parameters of the updating process. Utilizing the criterion that, each period, the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy: the Dynamic Quantile test. We use a differential evolutionary genetic algorithm to optimize an objective function that is not differentiable and hence cannot be optimized using traditional algorithms. Applications to simulated and real data provide empirical support for our methodology and illustrate the ability of these algorithms to adapt to new risk environments.