Measuring Value at Risk with Filtered Historical Simulation Using Non-Normal Distribution for Asset Returns

Abstract

Value at Risk (VaR) is the most widely used metric to quantify risk in financial industry. However, the accuracy of the measured risk is determined by the assumptions used while measuring it. Thus, the methodology used for deducing VaR is as critical as the VaR value itself. An important concern while measuring VaR is the assumption of Gaussian distribution of asset returns which is often violated by the actual values. The phenomena of leptokurtosis and volatility clustering are widely observed in financial time series data. These add to the discrepancies in VaR figures, if not considered, while calculating it. The main objective of this report is measurement of VaR for non-normal data. We discuss various methods available for VaR measurement with their advantages and disadvantages. We then use Filtered Historical Simulation (FHS) with non-normal GARCH innovations for computation of VaR. GARCH model is used here to address the issue of heteroskedasticity and also gives the flexibility to change the distributional assumptions of returns indirectly through the innovations. Historical daily data of 10 years has been downloaded from Bloomberg for a mixed portfolio of 7 assets and 1 day VaR at 95% and 99% confidence levels is calculated using Gaussian distribution as well as Students-t, Logistic and Hyperbolic Secant distributions for innovations as an attempt to capture the fat-tailed behaviour. The parameters for each of these distributions are estimated by the method of Maximum Likelihood. Back testing data of 1 year has been generated by two methods- single fitting and repeated fitting and all the four methods have been compared using both Unconditional Coverage and Independence tests at the significance levels of 5% and 10%.