Abstract
The Normal Inverse Gaussian (NIG) distribution, a special case of the Generalized Hyperbolic Distribution (GHD) has been frequently used for financial modelling and risk measures. In this work, we consider other normal Variance mean mixtures based on finite mixtures of special cases for Generalised Inverse Gaussian as mixing distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for some financial data. We estimate Value at risk (VaR) and Expected Shortfall (ES) for the fitted models. The Kupiec likelihood ratio (LR) has been applied for backtesting of VaR. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the proposed models are good alternatives to NIG for determining VaR and ES.
Highlights
The most popular measures for financial risk are Value at Risk (VaR) and Expected Shortfall (ES)
Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection
The results clearly show that the proposed models are good alternatives to Normal Inverse Gaussian (NIG) for determining VaR and ES
Summary
The most popular measures for financial risk are Value at Risk (VaR) and Expected Shortfall (ES). These risk measures are based on return distributions. VaR is that it ignores any loss beyond the value at risk level That is, it fails to capture tail risk. Artzner et al [3] [4] have proposed the use of Expected Shortfall (ES) called conditional Value at Risk (CVaR) to circumvent the problems inherent in. The objective of this paper is to determine VaR and ES for some financial data using Normal Weighted Inverse Gaussian (NWIG) distributions. Backtesting for value at Risk of the proposed models we use the Kupiec likelihood ratio (LR) introduced by Kupiec [14]
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