Forecasting value at risk allowing for time variation in the variance and kurtosis of portfolio returns
A common approach to forecasting the value at risk (VaR) of a portfolio is to assume a parametric density function for portfolio returns, and to estimate the parameters of the density function by maximum likelihood using historical data. In order to allow for volatility clustering in short horizon returns, this approach is typically combined with a conditional variance model such as EWMA or GARCH. However, these models implicitly assume that while the volatility of returns may be time-varying, the kurtosis of the return distribution is constant, at least over the estimation sample. In this paper, we show that the EWMA variance estimator can be obtained as a special case of a more general, exponentially weighted maximum likelihood (EWML) procedure that potentially allows for time-variation not only in the variance of the return distribution, but also in its higher moments. We use EWML to forecast VaR allowing for time-variation in both the variance and the kurtosis of daily equity returns. Our results show that the EWML based VaR forecasts are generally more accurate than those generated by both the EWMA and GARCH models, particularly at high VaR confidence levels.