Estimation of the Value-at-Risk (VaR) Using the TARCH Model by Considering the Effects of Long Memory in Stock Investments

Value at Risk (VaR) is one of the standard methods that can be used in measuring risk in stock investments. VaR is defined as the maximum possible loss for a particular position or portfolio in the known confidence level of a specific time horizon. The main topic discussed in this thesis is to estimate VaR using the TARCH (Threshold Autoregressive Conditional Heteroscedasticity) model in a time series by considering the effect of long memory. The TARCH model is applied to the daily log return data of a company's stock in Indonesia to estimate the amount of quantile that will be used in calculating VaR. Based on the analysis, it was found that with a significance level of 95% and assuming an investment of 200,000,000 IDR, the VaR using the TARCH model approach was 5,110,200 IDR per day.


Introduction
In the capital market, almost all investments contain an element of uncertainty or risk. Investors do not know with certainty the results that will be obtained from the investment they do. Because investors face risky investment opportunities, a measurement tool is needed to test these market risks, so that investors can safely know how far they can invest (Batuparan, 2001;Jogiyanto. 2007).
Based on the description above, the purpose of this paper is to calculate the amount of Value at Risk (VaR) of a stock, which is to calculate the maximum loss at a certain position with the level of confidence that has been known in a specific horizon time. The approach used is the TARCH time series model to estimate the size of the quantile that will be used in the calculation of Value at Risk (VaR) (Sukono et al., 2019). e-ISSN: 2722-0974

Formulation of the Problem
Suppose there is a closing stock data, then the log return value will be sought as follows where P t is the share price in period t and P t -1 is the stock price in period t -1. Based on the log return data will be tested whether the data is stationary and if it is stationary then the data will be processed to get the mean equation model and the variance equation model. The mean equation model will follow the time series model, while the variance equation model will follow the TARCH (Threshold Autoregressive Heteroscedasticity) model. After getting the mean equation model and the variance equation model, the quantile size will be calculated with a known level of confidence, for example by 95% (Ruppert, 2004).
The problem that will be discussed in this paper is how to calculate the value of VaR by modeling the mean and variance equations that produce quantile values to be calculated together with a number of investments given.

Stock Returns
Return is the income that will be received if we invest money in a financial asset (stocks, bonds) or real assets (property, land). Return calculation is done as in equation (1).

Time Series Model
There are several types of time series models that can be used in modeling the mean and variance for VaR calculations as follows.
Autoregressive (AR). For r t log return series, the AR(p) model is .
Moving Average (MA). For r t log return series, the MA(q) model is .
Autoregressive Moving Average (ARMA). For r t log return series, the ARMA(p,q) model is .
Autoregressive Integrated Moving Average (ARIMA). In general, the equation for the ARIMA(p,1,q) model is .

Quantiles
Quantile can be interpreted as a value that divides a set of data into several equal parts (Ruppert, 2004).
If the cumulative distribution function of X is continuous and does not go down, then the function of X has an inverse function F -1 . For each q between 0 and 1, F -1 (q) is called q-quantile or the 100th percentile. The probability of a continuous X cumulative distribution function under q-quantile is exactly q.
Quntile that will be used in VaR calculation can be calculated as follows (10) where η is the value of the confidence coefficient chosen in normal distribution, t rˆ is the result of forecasting the mean model, and 2 t  is the volatility value resulting from forecasting variance modeling (Dowd, 2002;Herrhyanto, 2003).

Value at Risk (VaR)
Value at Risk which is usually abbreviated as VaR is generally defined as the maximum possible loss for a particular position or portfolio in the known confidence level of a specific time horizon (Redhead, 1997).
VaR calculations using the TARCH model can be calculated as follows

Data
The data used in this paper is taken from http://finance.yahoo.com/q/hp?s=MPPA.JK for Matahari Putra Prima Tbk daily shares. Observation data is the closing stock price for the last 1080 days (13 November 2000 to 31 December 2004) which lasts for five days a week except holidays. The characteristics of the analyzed data are log return (Continuously Compounded Return) of stock trading which is calculated from the closing price of the Matahari Putra Prima Tbk (MPPA) stock trading. The graph of the company data can be seen in Figure 1.

Stock Return Calculation
Using equation (1) a daily log return will be calculated from Matahari Putra Prima Tbk's closing stock price data in 1080 days.
For example, the price of Matahari Putra Prima Tbk shares on November 13, 2000, and November 14, 2000, is 600 IDR and 575 IDR, then in a time horizon of 1 day (24) hours, the 1st daily log return obtained is And so on to calculate the 2nd daily log return to the 1079th daily log return at the company Matahari Putra Prima Tbk. The log return chart of the Matahari Putra Prima Tbk stock can be seen in Figure 2.

TARCH Model
In modeling variance with the TARCH model, we must do the mean modeling, where the mean modeling and variance modeling is carried out simultaneously between the two. Analysis of the TARCH model is assisted with Eviews 4.1 software.

Modeling Equation Mean
Figure 2 shows that Matahari Putra Prima Tbk's log return data has been stationary in the mean, but this data will be performed differencing to see the long-term effect on the data. Long memory testing will be carried out using the Geweke and Porter-Hudak method. The calculation to find the value of d is by using software R, so that obtained d = -0.182. To identify an appropriate model can be seen from the ACF and PACF plots can be seen in Figure 3  Tbk log return mean data with the AR(2) or MA(2) models. Among the possibilities of the two models, the best model is AR (2), with the estimated parameters in Table 1. After estimating the parameters, a standardized residual correlogram will be seen in Table 2 to see whether or not there is a serial residual correlation effect for modeling the mean AR(1) log return data.  Table 2, it appears that there is no serial correlation in the data, this is shown by the Q-Stat probability that is greater than the tolerance level α = 5% and there is no significant ACF and PACF plot until lag-20. Therefore, the AR(2) model is suitable for modeling the mean return data log of Matahari Putra Prima Tbk.

Variance Modeling Stage
To find out whether after modeling the mean ARCH effect is still present, an ARCH LM (Lagrange Multiplier) test will be performed which is listed in Table 3.  Table 3 shows that there is an ARCH element in Matahari Putra Prima Tbk's log return data, this can be seen from the value of the statistical F probability that is smaller than the tolerance level α = 5%.
Once it is known that the log return data contains ARCH elements, the data can be identified by looking at the ACF and PACF plots of squared residuals t t t r r a  , which are found in Table 4.
The estimation results of the AR (2) -TARCH (1.1) model are in Table 5. From Table 5, it can be seen that the AR(2)-TARCH(1,1) parameter has been significantly different from zero, this can be seen from the probability value that is smaller than the tolerance level α = 5%. So the AR(2)-TARCH(1,1) model can be written as After estimating the parameters, it will be seen whether the AR(2)-TARCH(1,1) model still has the ARCH effect. From Table 6, it can be seen that there is no ARCH element in the log return data, this can be seen from the statistical F probability value greater than the tolerance level α = 5%.
To see whether there are still serial correlations in the model, look at the ACF and PACF squared residual plots standardized in Table 7.  Table 7, it can be seen that in the ACF and PACF plots, all bar graphs do not cross the dotted line (Bartlett line). Besides that the statistical probability value of Q is greater than the level of significance α = 5%. This shows that there is no serial correlation in the model.
Next will be seen whether the residual model has a nominal distribution. By using the Tools ►Input Analyzer in ARENA 7 software, the results are shown in Figure 5.  Figure 5 shows that the AR(2)-TARCH(1,1) residuals follow a bell curve, which means that the data is normally distributed.

Calculating Value at Risk (VaR)
After obtaining the results of the mean modeling stage and the variance modeling stage, the VaR amount will be calculated in the Matahari Putra Prima Tbk log return data. For example, if it is assumed to allocate funds of 200,000,000 IDR for investment in Matahari Putra Prima Tbk, the amount of VaR in Matahari Putra Prima Tbk shares will be calculated as follows.
Suppose that ε t is the normal distribution. AR (2) where the negative sign is written as the left tail of the conditional normal distribution.
Using equation (10), obtained VaR for Matahari Putra Prima Tbk shares, i.e. VaR = 200,000,000 IDR × 0.025551 = 5,110,200 IDR. So it can be concluded that with a 95% confidence level, there will be a loss of 5,110,200.00 IDR per day.

Conclussion
From the discussions that have been carried out, the following conclusions are obtained. 1) A fairly good time series model in modeling the Matahari Putra Prima Tbk log return data after differencing in the last 1080 days (13 November 2000 to 31 December 2004) is the AR(2) model.
2) The variance model that is good enough to model the Matahari Putra Prima Tbk log return data in the last 1080 days (13 November 2000 to 31 December 2004) is the TARCH(1.1) model. 3) After estimating the modeling of mean and variance equations, the results show that the appropriate model for the Matahari Putra Prima Tbk log return data is the AR(2)-TARCH(1,1) model. Using a 95% confidence level and assuming an investment of 200,000,000.00 IDR, then the value of the VaR Matahari Putra Prima Tbk is 5,110,200 IDR. So the loss obtained for Matahari Putra Prima Tbk shares is 5,110,200.00 IDR per day.