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.

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 Normal Weighted Inverse Gaussian (NWIG) distributions are good alternatives to Normal Inverse Gaussian (NIG) for determining VaR and ES

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Summary

Introduction

The most popular measures for financial risk are Value at Risk (VaR) and Expected Shortfall (ES). The main shortcoming of VaR is that it ignores any loss beyond the value at risk level. That is, it fails to capture tail risk. Heath ([3] [4]) have proposed the use of Expected Shortfall (ES) called conditional Value at Risk (CVaR) to circumvent the problems inherent in VaR. The most common special case is Normal Inverse Gaussian (NIG) distribution introduced by Barndorff-. There are other special cases of GHD which are related to the NIG distribution which have not been considered for VaR an ES. Corporation (RRC) financial data using Normal Weighted Inverse Gaussian (NWIG) distributions.

Value at Risk and Expected Shortfall
Generalised Inverse Gaussian Distribution
Weighted Inverse Gaussian Distribution
Generalized Hyperbolic Distribution
Special Cases of Interest
Parameter Estimation via EM Algorithm
Iterations
E-Step
Fitting of the Proposed Models
Risk Estimation and Backtesting
Conclusions
Full Text
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