Estimating Value at Risk (VaR) and Expected Shortfall Using Normal Weighted Inverse Gaussian Distributions

This study calculates the VaR for Normal Weighted Inverse Gaussian (NWIG) distributions. Value at Risk (VaR) and Expected Shortfall (ES) is commonly used measures of potential risk for losses in financial markets. The Normal Inverse Gaussian (NIG) distribution, a particular instance of the Generalized Hyperbolic Distribution (GHD), is widely utilised in literature when discussing VaR and ES. However, there are additional specific situations of GHD called Normal Inverse Gaussian Related Distributions that can be used. The Basel Committee [1] proposed to replace Value at Risk with Expected Shortfall but concluded that the backtesting will still be done on VaR even though the capital would be based on Expected Shortfall. Therefore the two measures of risk still remain the most popular and useful in financial management. The Maximum Likelihood (ML) estimates of the suggested models for the financial data from Range Resource Corporation (RRC) have been obtained using the Expectation Maximization (EM) technique. To backtest VaR, we employed the Kupiec likelihood ratio (LR). The goodness of fit test has been conducted using the Kolmogorov-Smirnov and Anderson-Darling tests. For model selection, the Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC), and Log-likelihood have all been utilized. The outcomes unequivocally demonstrate that the NWIG distributions are good substitutes for NIG for calculating VaR and ES.