VaR PREDICTION FOR GARCH (1,1) MODEL WITH NORMAL AND STUDENT-t ERROR DISTRIBUTION

This study aims at determining the estimated parameters of the GARCH (1.1) model establishing the prediction of the VaR value, and defining the accuracy of the VaR prediction. In this study, the error in the GARCH (1,1) model uses a normal distribution and student-t distribution. The research method focuses on parameter calculation and the prediction of VaR value within two aspects regarding analytic and numeric aspects. Analytically, the prediction of the VaR value and the accuracy of the prediction of VaR value through the VaR coverage opportunity. It isn't easy to estimate the parameters for the GARCH (1.1) model analytically. Thus, the parameters are estimated numerically using the Quansi Newton optimization method. Prediction of VaR value and VaR coverage probability will be simulated numerically by using stock return data of IBM, INDF.JK and GSPC. The results show that the GARCH (1.1) model can model stock returns for IBM, INDF.JK and GSPC. There is no significant difference between the GARCH (1,1) model with a normally distributed error and GARCH (1,1) with a student-t distribution error in determining the prediction of VaR values. The numerical simulation results show that the VaR value prediction using the GARCH (1,1) model with a normally distributed error is more accurate than the student-t-distributed error.


INTRODUCTION
Nowadays, the stock market has turned into one of the exciting research objects. The trigger of stock prices is used as a country's economic health barometer [1]. The stock market itself means that it is closely related to investment. The returns obtained by an investor in the stock market are known as asset returns. Asset return is the rate of return of an investment over a certain period [2]. Asset returns in various countries generally show the phenomenon of time-varying volatility [3][4][5]. This situation indicates a relatively 'calm' return period which then changes to a 'turbulent' period. The following studies carried out to date concerning unconditional volatility from stock futures (see for example [6] and 7]).
Return behavior information is not enough to analyze financial time series data. Consequently, the term volatility appears. Volatility remains a measure of the spread of the magnitude of changes in the price of a financial instrument. In other words, volatility measures how much and how quickly the value of a financial instrument changes. Generally, the volatility of time series data is assumed to be constant over time. However, economic time series data volatility can change over a certain period. This change was caused by the financial market's reaction to various kinds of disturbances, including deteriorating political conditions, economic crises, wars, natural disasters, and others [8]. It assumes that the volatility of the financial time series data is not constant.
Volatility modeling has a significant role in the financial sector, especially in risk management.
Volatility modeling can be used to calculate the maximum loss from an asset return that can be tolerated with a certain level of significance within a certain period [9]. A good volatility model is a model that can accommodate the properties of asset returns and volatility [10][11]. One of the good volatility models in modeling stock return volatility is the Generalized Autoregressive Conditionally Heteroscedastic (GARCH) model [12]. This model states that volatility is a visual function determined by observations and the volatility of the last time by Research on GARCH volatility modeling [13][14][15].
The loss towards return is related to a measure of risk. Through the GARCH volatility model, the risk of investment loss can be estimated and measured. One measure of risk that can be used in estimating the magnitude of trouble in the GARCH model is Value at Risk (VaR). VaR is a measure of market risk that has been widely used for financial risk management, including in banking institutions, regulators, and portfolio managers [15 ].
This research determines the predictive value of value at risk analytically and numerically. Numerical calculation displays the estimated parameter GARCH (1,1) with an normally distributed error and student-t. In addition, numerical calculations will define the prediction of the VaR value for the GARCH (1,1) model with normal and student-t distribution errors and the accuracy of the prediction through the correct VaR proportion.

RESEARCH METHOD
The risk of financial assets in the form of stock investments can be measured by using volatility. A risk is a form of investor loss when investing in the stock market. The volatility model is a way to predict volatility in the stock market. The GARCH (1.1) volatility model can capture the volatility of the financial time series [12] [17][18]. Therefore, predictions of volatility models are needed to protect financial assets in the future [16]. In this study, the focus is on finding the parameter estimation of the GARCH (1,1) model that fits the definition given by Bollerslev.
The definition of Bollerslev (1986), for , is a stochastic process that states the return of asset Furthermore, to determine the accuracy of the VaR prediction obtained in equation (1), a VaR accuracy test is carried out by determining the coverage probability. This coverage opportunity is expected to be close to the given confidence level. The probability of VaR coverage for return in the GARCH (1,1) model is The accuracy of the VaR prediction in the GARCH (1,1) model that fits the data can also be seen from the correct VaR value [16]. The model is assumed to fit with the data if 0  −  

RESULT AND DISCUSSION
The results of numerical data processing from the three stock return data of International Business Machines Corporation (IBM.Inc), Indofood Sukses Makmur Tbk (INDF. JK), and the S&P 500 stock index (GSPC) are presented in the following descriptive statistical  Table 2

 +  
, then the volatility of the model will increase indefinitely from time to time. It is very rarely the case with financial data, so it must be 1 1 1

 +  
The next stage will determine the prediction of Value at Risk from the returns of the three stocks using the GARCH (1,1) model with a normal distribution error or GARCH (normal) and GARCH (1.1) with a student-t distribution error or abbreviated as GARCH (t). The VaR prediction uses α = 99%, α = 95% and α = 90%. It can be seen that the difference in VaR predictions between the GARCH(normal) and GARCH (t) models is relatively small, namely ±10−3. It shows that the difference between the two volatility models is not significant in determining the VaR value. The VaR predictions for IBM, INDF.JK and GSPC stocks at time 1, ..., T can be seen in Figure 1, Figure 2 and Figure 3.  a significance level of 99% GARCH with a student-t error distribution is more suitable to describe INDF.JK stock returns than GARCH with a normal error distribution. Meanwhile, with a significance level of 95% and 90%, GARCH with a normal error distribution is more suitable than GARCH with a student-t error distribution. In GSPC stocks, the model that best fits the data is the GARCH(1,1) model, with the error being normally distributed.

CONCLUSION
Based on the research results, it is found that the GARCH (1,1) model can be used to model the stock returns of IBM, INDF.JK and GSPC. There is no significant difference between the GARCH (1,1) model with a normally distributed error and GARCH (1,1) with a student-t distribution error in determining the prediction of VaR values. It is known from the difference between the predictions of the two models, which is relatively small, only ±10−3. The predictive accuracy of the Value at Risk value can be determined by determining the correct VaR proportion. The simulation results of the three stocks in this study indicate that, in general, the prediction of Value at Risk using the GARCH model with a normally distributed error is more accurate than the GARCH model with a Student-t distribution error. There have been many developments from experts regarding the GARCH volatility model. Research can be continued using these models, such as the integrated GARCH model, the stochastic GARCH, and others.