The performance of value-at-risk models during the crisis

Using stock data that covers the period from April 6, 2001 to June 17, 2009, including data for the recent crisis period, we perform value-at-risk (VaR) risk model validation by backtesting the performance of VaR models in predicting future losses of a portfolio of stocks, futures and options. The stock returns are modeled using univariate generalized autoregressive conditional heteroskedasticity (GARCH) models for the stock returns with different assumptions on the distribution of the univariate model residuals such as normal distribution, t-distribution, and empirical distribution with t-distributed tails. To capture co-dependence between financial return series we model dependence using the normal copula, the t copula and the discrete normal mixture copula. Backtesting evaluations of different models indicate that GARCH models for estimating and forecasting volatility make a significant contribution to the accuracy of VaR models. The contribution is both in terms of making VaR models match the expected number of VaR exceedances of losses and the VaR model exceedances having expected durations between them. However, while stochastic volatility is an important component of VaR model specifications it is not the only component. In particular, the choice of univariate model for the GARCH residuals and the choice of copula also affect the VaR model performance. Standard practice models used in VaR estimation such as the delta-normal model, the covariance simulation model, with or without RiskMetrics covariance matrix, and the historical simulation model are shown to underestimate risk quite severely while models such as the discrete normal variance mixture copula with univariate GARCH perform well.