Value at Risk and Expected Shortfall for Normal Variance Mean Mixtures of Finite Weighted Inverse Gaussian Distributions

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.


Introduction
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 generally defined as possible maximum loss over a given holding period within a fixed confidence level. An attractive feature of VaR is the backtestability of the measure. Backtesting is a method that uses historical data to gauge accuracy and effectiveness (Zhang and Nadarajah [1]). However, 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. It also lacks a mathematical property called subadditivity as stated by Wimmerstedt [2]. That is, VaR for two combined portfolios can be larger than VaR for the sum of the two portfolios independently. This implies that diversification could increase risk, a contradiction to standard beliefs in finance.
Artzner et al. [3] [4] have proposed the use of Expected Shortfall (ES) also called conditional Value at Risk (CVaR) to circumvent the problems inherent in VaR. Expected Shortfall is the conditional expectation of loss given that the loss is beyond the VaR level. Nadarajah et al. [5] have given a detailed review of VaR and ES for various distributions. One of the distributions reviewed is the Generalized Hyperbolic Distribution (GHD) introduced by Barndorff-Nielsen [6] as a Normal Variance-Mean Mixture with the Generalized Inverse Gaussian (GIG) distribution as the mixing distribution. The most common special case is Normal Inverse Gaussian (NIG) distribution introduced by Barndorff-Nielsen [7] with the Inverse Gaussian (IG) as the mixing distribution.
The objective of this paper is to determine VaR and ES for some financial data using Normal Weighted Inverse Gaussian (NWIG) distributions. In particular we consider Normal mixtures with finite mixtures of 1 as mixing distribution. We study their properties and estimate parameters using the Expectation Maximization algorithm introduced by Dempster et al. [8]. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection.
The concept of a weighted distribution was introduced by Fisher [9] and elaborated by Patil and Rao [10]. Reciprocal Inverse Gaussian and the finite mixture of Inverse Gaussian and Reciprocal Inverse Gaussian distribution are shown to be Weighted Inverse Gaussian (WIG) distributions by Akman and Gupta [11], Gupta and Akman [12], Gupta and Kundu [13]. Backtesting for value at Risk of the proposed models we use the Kupiec likelihood ratio (LR) introduced by Kupiec [14].

Value at Risk and Expected Shortfall: Mathematical Background
The most important risk measures despite their drawbacks are Value at Risk (VaR) and Expected Shortfall (ES). VaR was proposed by Till Guldimann in the late 1980s, and at the time he was the head of global research at J. P. Morgan. Value at Risk is generally defined as possible maximum loss over a given holding period within a fixed confidence level. Mathematically VaR at the (100-α) percent confidence level is defined as the lower 100α percentile of the profit-loss distribution.
In statistical terms, VaR is a quantile of distribution for financial asset returns.
and we refer to it as "Tail loss distribution".

Conditional Expectation
is the Expected Shortfall denoted as ES α . This version was used by Yamai and Yoshima [16] as presented by Zhang et al. [17].
Remarks: Equation (2.4) is the mean of the loss distribution. Equation (2.5) represents the average of the VaR between 0 and α . The loss distribution, For the purpose of VaR and ES analysis, a model for the return distribution is important because it describes the potential behaviour of a financial security in the future (Bams and Wielhouwer [18]). A Normal distribution supposedly underestimates the tail and hence VaR. Recently alternative distributions have been proposed that focus more on tail behaviour of the returns. One such candidate is IG is a special case of the Generalised Inverse Gaussian (GIG) distribution.

Generalised Inverse Gaussian
The Generalised Inverse Gaussian (GIG) Distribution is based on modified Bessel function of the third kind. Modified Bessel function of the third kind of order λ evaluated at ω denoted by ( ) K λ ω is defined as ( ) with the following properties a) ( ) ( ) which are necessary in deriving the properties and estimates of the proposed models. For more definition and properties see Abramowitz and Stegun [19].
where r can be positive or negative integers. Consider the following special weights: Case 1: Let which is also a finite mixture of which is also a finite mixture of which is also a finite mixture of which is also a finite mixture of The mean and variance for the weighted distribution are which is also a finite mixture of

Normal Variance-Mean Mixture
A stochastic representation of a Normal Variance-Mean mixture is given by letting Let ( ) Φ ⋅ are pdf and cdf of a standard normal distribution, respectively.
Thus we have a hierarchical representation as being the conditional pdf and g(z) the mixing distribution.

( )
, , we obtain the Generalized Hyperbolic Distribution (GHD) introduced by Barn- dorff-Nielsen [6]. In general the integral formulation for constructing the Weighted Inverse Gaussian (WIG) distributions is presented as We now construct six (6) models based on the special cases developed above. Model 1: Assuming case 1 presented by Formulation (3.12) the mixed model becomes With the following properties Model 2: Assuming case 2 presented by Formulation (3.16) the mixed model becomes With the following properties Model 3: Assuming case 3 presented by Formulation (3.21) the mixed model becomes With the following properties δγ δ γ δ γ δ µ β γ δγ γ δ With the following properties Model 5: Assuming case 5 presented by Formulation (3.30) the mixed model becomes With the following properties With the following properties

Maximum Likelihood Estimation via Expectation-Maximization (EM) Algorithm
EM algorithm is a powerful technique for maximum likelihood estimation for data containing missing values or data that can be considered as containing missing values. It was introduced by Dempster et al. [8].
Assume that the true data are made of an observed part X and unobserved part Z. This then ensures the log likelihood of the complete data ( ) Karlis [21] applied EM algorithm to mixtures which he considered to consist of two parts; the conditional pdf is for observed data and the mixing distribution is based on an unobserved data, the missing values.

M-Step for Conditional pdf
Since the conditional distribution for the six models is normal distribution as presented in Formula

E-Step
Since Then the k-th iterations are as follows: We therefore have the following iterative scheme  and the k-th iteration is given as Note that the iterative for β and µ is the same for all the models considered below. The posterior estimates differentiate the values obtained.

M-Step
( ) ( ) Maximizing with respect to δ and γ we have the following representation ( )

E-Step
The posterior expectations for the k-th iteration are: We therefore have the following iterative scheme    (5.13) and the k-th iteration for the log likelihood is given by Maximizing with respect to δ and γ we have the following representation Both equations are quadratic in γ and δ respectively.

E-Step
The posterior expectations for the k-th iteration are:  Similarly, define and the k-th iteration for the loglikelihood is given by Maximizing with respect to δ and γ we have the following representation ( ) ( )

E-Step
The posterior expectations for the k-th iteration are: From Equations (5.27) and (5.28), we obtain the following iterative scheme and the k-th iteration for the loglikelihood is given by

E-Step
The posterior expectations for the k-th iteration are: These can be used to obtain the (k + 1)-th values as follows Maximizing with respect to δ and γ we have the following representation

E-Step
The posterior expectations for the k-th iteration are: Remarks: 1) For all the proposed models, the β , µ and α parameters of the conditional distribution are updated as follows: 2) The stopping criterion is when

Fitting of the Proposed Models
The data used in this research is the Shares of Chevron (CVX) weekly returns for the period 3/01/2000 to 1/07/2013 with 702 observations. The histogram for the weekly log-returns in shows that the data is negatively skewed and exhibiting heavy tails. The Q-Q plot shows that the normal distribution is not a good fit for the data especially at the tails. Table 1 provides descriptive statistics for the return series in consideration. We observe that the excess kurtosis of 2.768252 indicates the leptokurtic behaviour of the returns. The log-returns have a distribution with relatively heavier tails than the normal distribution. We observe skewness of −0.1886714 which indicates that the two tails of the returns behave slightly differently.
The initial values were used in all the proposed models to obtain the maximum likelihood estimates as shown in Table 2 below The parameter estimates from Table 2 are now fitted to RRC weekly logreturns. Figures 1-6 show the histogram and Q-Q plots of the RRC returns fitted with the proposed models. Figures 1-6 show that the proposed model fit the data well.

Risk Estimation and Backtesting
We use the parameter estimates for our proposed model to determine the VaR and ES at levels { } 0.001, 0.01, 0.05 α ∈ as given in Table 4 and Table 5 respectively. The three level are used to measure the risk of long position, We apply the Kupiec Likelihood Ratio (LR) test (Kupiec, [14]) which test the hypothesis that the expected proposition of violations is equal to α . The method consist of calculating ( ) τ α the number of times the observed returns, t x falls below the VaR α estimates at level α as given in Table 6; i.e., t x VaR α < , and compare the corresponding failure rate to α . Model 3 has the highest VaR and ES value indicating that it perform well than the other models at the tails. Table 7 gives the P-value for the Kupiec Test for Each Distribution. Remark: At 5 percent level of significant, the Normal distribution is rejected at levels at the level 0.001. The Normal weighted Inverse Gaussian distributions were all effective and well specified on all levels of VaR.