Empirical Issues in Value at Risk Estimation: Time Varying Volatility, Fat Tails and Parameter Uncertainty - Proceedings AFIR 2000 - Tromsø, Norway

This paper describes alternative approaches to estimate the Value at Risk (VaR) of a position. Four methods are compared: the unconditional case, the model with time varying drift (modeled as an AR(l) process), the model with time varying drift and time varying volatility (modeled as a GARCH(I,l) process) with error terms that are normally distributed, and the model with time varying drift and time varying volatility with error terms that are Student-t distributed. Two issues are important. First, different specifications for mean, variance and fat tailness lead to different point estimates for the associated distribution function and hence to other VaR measures. Second, uncertainty in parameter estimates implies that the VaR also is uncertain. The model with error terms that are t-distributed is the preferred model, since: (I) the time varying volatility incorporates that recent volatility is a better predictor for the future, (2) the time varying volatility makes it possible to use a longer time series which implies less uncertain VaR estimates and (3) the fat tail of the distribution is taken care of by the t-distributed error terms. An important contribution of the paper lies in the fact that we explicitly take account for parameter uncertainty and propose ways to deal with it.