Application of Extreme Value Theory to Economic Capital Estimation

Abstract

We are interested in the risk of large losses of certain common financial portfolios (e.g. credit portfolios with default risk). In these cases, we would like to estimate a risk statistic called Value-at-Risk (VaRα) at an extremely high risk level α (typically 99.99%). This high risk level corresponds to rare events for which it is difficult or impossible to obtain data. We first use Monte Carlo simulation to compute a probability distribution for the portfolio loss. We then use Extreme Value Theory (EVT) to study the tail of this loss distribution. Finally we compute the VaR and associated confidence intervals using bootstrap techniques. For the portfolios under consideration, we have observed that the EVT based approach results in narrower confidence intervals and hence less sampling uncertainty in computing the VaR. We have also observed that the bootstrap replicate's distribution for the EVT based method demonstrates a better shape than the empirical method (which is typically very noisy).