Value at Risk and Expected Shortfall for Normal Weighted Inverse Gaussian Distributions

Abstract

Value at Risk (VaR) and Expected Shortfall (ES) is commonly used measures of potential risk for losses in financial markets. In literature VaR and ES for the Normal Inverse Gaussian (NIG) distribution, a special case of Generalized Hyperbolic Distribution (GHD), is frequently used. There are however, Normal Inverse Gaussian related distributions, which are also special cases of GHD that can also be used. The objective of this paper is to calculate VaR for Normal Weighted Inverse Gaussian (NWIG) distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for the Range Resource Corporation (RRC) financial data. We used Kupiec likelihood ratio (LR) for backtesting of VaR. Kolmogorov-Smirnov test and Anderson-Darling test have been used for goodness of fit test. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the NWIG distributions are good alternatives to NIG for determining VaR and ES.