A GARCH Approach to VaR Calculation in Financial Market

Value at Risk (VaR) has already becomes a standard measurement that must be carried out by financial institution for both internal interest and regulatory. VaR is defined as the value that portfolio will loss with a certain probability value and over a certain time horizon (usually one or ten days). In this paper we examine of VaR calculation when the volatility is not constant using generalized autoregressive conditional heteroscedastic (GARCH) model. We illustrate the method to real data from Indonesian financial market that is the stock of PT. Indosat Tbk.


Introduction
There are some types of financial market risk, i.e. credit risk, operational risk and market risk. Value at Risk (VaR) is mainly concerned with market risk; however the concept can be use for difference type of risks. VaR is single estimator of an institution position quantity decline of profit risk category on the market in the share period. This measure might be applied by the institution for estimating the risk and regulatory committee in this case for analyzing the investment opportunity (Jorion, 2004;Alexader, 1999).
VaR in term of the financial institution is defined as a maximum lost on period of financial position with a certain probability. VaR is considered as a lost measurement related to an extraordinary event under the standard market condition. For the regulatory committee, VaR is defined as a minimum lost under an extraordinary market condition. These two definitions has a similar based on VaR measurement, however the concept seems different (Tsay, 2005;Dowd, 2002).
In general, VaR calculation usually uses econometrics time series models. In this paper, GARCH model is used for the volatility estimating and the VaR calculation worked base on quantile. In seeking the performance of the models, we'll discuss them through the skewness and kurtosis coefficient values.

Value at Risk
Assume that at the time index t we are concerned in the risk of a financial position for the next l periods. Let be the change in value of the asset in financial position from time t to l t  . This quantity is measured in rupiah curency and is a random variable at the time index t .
Represent the cumulative distribution function (CDF) of Difined the VaR of a long position over the time horizon l with probability p as ) (1) Because the holder of a long financial position suffers a loss when 0 ) (   l V , the VaR defined in (1) naturally assumes a negative value when p is small. The negative sign signifies a loss. From the definition, the probability that the holder would run into a loss greater than or equal to VaR over the time horizon l is p . Alternatively, VaR can be interpreted as follow. With probability ( p  1 ), the potential loss encountered by the holder of the financial position over the time horizon l is less than or equal to VaR.
The holder of a short position suffers a loss when the value of the asset increases The VaR is then defined as . Therefore, it suffices to discuss method of VaR calculation using a long position (Tsay, 2005;Khindanova ) (x F l or its quantiles. 5. The amount of the financial position or the mark-to-market value of the portfolio. Among these factors, the CDF ) (x F l is the focus of econometric modeling.

The GARCH Approach
For a log return series, the time series models can be used to model the mean equation, and conditional heteroscedastic models are used to handle the volatility. In this paper, we will use GARCH models to the approach as an econometric approach to VaR calculation (Tsay, 2005;Engle & Manganelli, 2002).
Consider the log return t r of an asset. A general time series model for t r can be written as If one further assumes that t  is Gaussian, then the conditional distribution of 1 The relationship between quantiles of a Student-t distribution with v degrees of freedom, denoted by v t , and those of its standardized distribution, denoted by * t t , is (5) is a standardized Student-t distribution with v degrees of freedom and the probability is p , then the quantile used to calculate the 1-period horizon VaR at time index t is is the p th quantile of a Student-t distribution with v degrees of freedom and assumes a negative value for a small p .

The Model Parameter Estimation
Therefore the estimator is a solution for maximize the problem This estimator denoted by T ˆ and is called pseudo maximum likelihood estimator (PMLE), and the likelihood function is constructed by the assume that a normally distribution. When the distribution assumption is not true, this estimator is a normal asymptotically with the covariance matrix is where 0 E showed that the expectation taken to the true distribution (Gourieoux, 2002). Two matrix I and J is usually different. But I and J is equal if its distribution is true and appropriate with the likelihood function, that is in this case a conditional normal distribution. When the matrix J I  , the asymptotic patern becomes simple, that is On the GARCH model, notified that the MLE of ARMA model usually solved by using the backward forecast algorithm or Kalman filter. The similar case when the ARCH specification is replaced by GARCH. Let given a GARCH(p, q) model conditionally Gaussian where 2 t  given in (5).
The conditional variance patern in the parameter terms and the variable observed is where L represent the operator-lag. Hence, ) ( 2   t depend on the all previous values of t r process.
Since the time period of observation is limited, that is with truncated approximattion, where 2 t r is values that concerned with negative sign that defined equal to zero. It is equivalent with the recursive equation The initialy log-likelihood function replaced with truncated version: Continuously, the optimization is worked with numerical procedure and for the value of 1  from  that given, the conditional variance respectively calculated by using

Diagnistic Check for ARCH Effect
Rembering that the ARCH model is

Kurtosis of GARCH Models
To assess the variability of an estimated volatility, one must consider the kurtosis of a volatility model. In this section, we will derive the excess kurtosis of a GARCH(1,1) model. The same idea applies to other GARCH models. The model considered is Where  K is the excess kurtosis of the innovation t  . Based on the assumption, obtained the following: exists.
Taking the square of the volatility model (5) Taking expectation of the equation (21) and using the two properties mentioned earlier, obtained This excess kurtosis can be written in an informative expression. First, consider the case that t  is normally distributed. In this case, 0   K , and some algebra shows that Where, the superscript (g) is used to denote Gaussian distribution. This result has two important implications: (a) the kurtosis t a exists if 0 ) ( 2 1 The previous result shows that for a GARCH(1,1) model the coefficient 1  (Tsay, 2005;Shi, 2004). For a standadized Student-t distribution with v degrees of freedom, obtained . This is part of the reason that used 5 t when the degrees of freedom of a student-t distribution are prespecified. The excess kurtosis of

Cases Study
The aim of this research is to determine the Value at Risk (VaR) of the real data. As a case study, we use PT. Indosat, Tbk. stock data. In this research, we employ GARCH model to estimate this VaR value. The result of this approach can be used by investors to save their stocks. We implement MATLAB 7.1 and EView 3 for analyzing the data.
The data which will be analyzed are log return data (Continuously Compounded Return) of closing price. We observe the data from July, 4 th , 2004 till March, 3 rd 2006, or contain 396 observations. Time series plot of the data can be seen at Figure 3.1 as follow: The summary of descriptive statistics for the data is as follows: the minimum is -0.066375, the maximum is 0.058269, mean is 0.000602, variance is 0.000385, and standard deviation is 0.019634.

Normality Test
We employ MINITAB 13 to do normality test with the hypothesis test as follow H 0 : The log return data are normally distributed H 1 : The log return data are not normally distributed. The hypothesis null is rejected if p-value of the statistic test less than 0.05 (significance level). In this paper, we use Ryan-Joiner and Shapiro-Wilk Test as statistic tests. The results show that pvalue of both tests are the same, i.e. 0.081. It means we fail to reject H 0 and conclude that log return data are normally distributed.

GARCH Modeling
In this section we will estimate whether the log return data have volatility pattern following GARCH model. Identification step by using time series plot (see Figure 3.1) shows that data satisfy stationery condition in mean. Both of the ACF and PACF indicate only significant at the first lag or cut off after lag 1. Based on this result, we can propose that the appropriate ARIMA models are AR(1) or MA(1). The results of parameter estimation and diagnostic check steps show that MA(1) model is the best appropriate model. The output of MA(1) model is illustrated at   The results show that residuals model contain ARCH effect or follow heteroscedasticity pattern. The correlogram of residual indicates that lag 1 and 2 are significant. Hence, we propose that ARCH(2) and GARCH(2,2) are appropriate for this data. Parameter estimation step show that ARCH(2) model is not appropriate for modeling heteroscedasticity pattern at the residuals. It's caused one of parameters model is not statistically significant. Then, we continue to estimate GARCH(2,2) model and the result can be seen at Table  3.3 as follow:  The result at diagnostic check step is illustrated at Table 3.4. It shows that MA(1)-GARCH(2,2) model satisfies the adequacy model. Additionally, we also check the normality of the residuals model by using Jarque-Berra test, and the result obtained a probability value is 0.358359 geater than of 5% significance level. It is shows that residuals satisfy normal distribution assumption. Besides that also obtained the coefficient skewness is 0.084146 and the coefficient kurtosis is 3.310456. Hence, we can conclude that MA(1)-GARCH(2,2) is the best model for volatility estimating log return data. The estimated of this model can be seen at Figure 4.8 as follow: We only need variance model to estimate the volatility. The variance forecast at 397 th period by using GARCH(2,2) model is 0.00041. It means the daily volatility is 0.0202.

Value at Risk Calculation
In this paper, the method for calculating VaR is a variance-covariance approach model. This model has an assumption that return data are normally distributed. From the results at the previous section, we can construct a summary of VaR values at some level confidence interval, i.e.