Forecasting Value-at-Risk using Maximum Entropy Density

Abstract: Despite its shortcoming, Value-at-Risk (VaR) remains as one of the most important measures of risk for financial assets. Although it is used widely by regulatory authority in assessing risk of the financial markets, the robust construction of VaR forecasts remains a controversial issue. This paper proposes a new method to construct VaR forecasts based on Maximum Entropy Density, along with the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model of Bollerslev (1986). Using the result in Ling and McAleer (2003), the Quasi-Maximum Likelihood Estimator (QMLE) with the normal density for ARMA-GARCH model is consistent and asymptotically normal under mild assumptions. This implies that it is possible to obtain consistent estimates of the standardized residuals even when the underlying distribution of returns is non-normal. Given this, the distribution of the standardized residuals can then be approximated using Maximum Entropy Density (MED) which allows different characteristics of the distribution, such as excess kurtosis, to be accommodated. The one-day-ahead VaR forecasts can then be constructed by using the estimated ARMA-GARCH model and the MED. The practical usefulness of the proposed method is evaluated empirically against ARMA-GARCH and ARMA-GJR models with different distributional assumptions using daily S&P 500 data. The empirical results show promising sign of the proposed method