International Journal of Business and Economy (IJBEC) eISSN: 2682-8359 [Vol. 3 No. 4 December 2021]
Journal website: http://myjms.mohe.gov.my/index.php/ijbec
FORECASTING HEALTHCARE STOCK PRICE USING ARIMA-GARCH MODEL AND ITS VALUE AT RISK
Nur Hanis Mohd Mokhlis1*, Nur Anira Ahmad Burhan2, Nur Fatin Ainsyah Roeslan3, Siti Nur Aishah Zainal Moin4 and Ummu Aiman Mohd Nur5
1 2 3 4 5 Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara, Shah Alam Selangor,
MALAYSIA
*Corresponding author: [email protected]
Article Information:
Article history:
Received date : 8 November 2021 Revised date : 17 November 2021 Accepted date : 24 November 2021 Published date : 5 December 2021
To cite this document:
Mohd Mokhlis, N., Ahmad Burhan, N., Roeslan, N., Zainal Moin, S., & Mohd Nur, U. (2021).FORECASTING HEALTHCARE STOCK PRICE USING ARIMA-GARCH MODEL AND ITS VALUE AT RISK.
International Journal of Business and Economy, 3(4), 127-142.
Abstract: During the COVID-19 outbreak, the IHH healthcare market is prone to speculation and volatility, making it extremely difficult for investors to understand the direction of stock prices. Forecasting the IHH stock is crucial for investors to know whether they will be benefited from their investing. The objectives of this study are to explore the past trend of IHH stock price, compare the performances between ARIMA-GARCH and ARIMA-TGARCH models in forecasting the stock price. This study aims to forecast the stock price in the next six months and its Value at Risk (VaR). To achieve the first objective, time plot is plotted. The Webel-Ollech test is used to analyse the seasonality of the time series data. Also, the ADF test is used to determine its stationarity. The second objective can be realised by fitting the ARIMA model in which its orders are determined by ACF and PACF plots. The ARCH-LM test is then employed to assess the ARCH effect in the time series data before developing the GARCH models.
Consequently, the best ARIMA model will be combined with GARCH and TGARCH models. These hybrid models are utilized to forecast the IHH stock price, and their performances will be analysed using the RMSE and MAE. For the third objective, the best model selected is used to project the daily IHH stock price and finally the VaR is calculated by using the historical method. As a result, the oscillations trend is found with no seasonality, but it is stationary. Furthermore, the best hybrid model is ARIMA (4,1,5)-GARCH (1,1). Forecasting results show that the IHH stock is predicted to be traded at about $6.52. Lastly, computed VaR shows the worst loss
1. Introduction
FTSE Bursa Malaysia KLCI (FBM KLCI) has been spared by the strong demand for healthcare products and also the significant profit-taking interest among healthcare stockholders to remain sturdy throughout the worldwide pandemic crisis (Aziz, 2020). Healthcare stocks refer to stocks that deal with medical and healthcare products and services (Bakar and Rosbi, 2017).
Some of the prominent healthcare stocks include TOPGLOV, HARTA and KOSSAN, to name just a few. However, in this study, we choose to examine the trends and forecast the IHH stock since it has outperformed the healthcare industry in Malaysia, notably during the COVID-19 outbreak. IHH Healthcare Berhad is a top-tier healthcare provider and it is also one of the world's largest healthcare conglomerates by market capitalization, and it is listed on the Singapore Stock Exchange in addition to the Main Markets of Bursa Malaysia (IHH Healthcare Berhad - Overview, n.d.).
Time series forecasting is a common approach for non-stationary data whose statistical properties, such as mean and standard deviation, do not remain constant throughout time but rather change (Loukas, 2020). Thus, the Auto-Regressive Integrated Moving Average (ARIMA) model is widely employed for forecasting time series data. Despite the ARIMA model's numerous financial forecasting applications, it is unable to handle the situation where the input variance varies over time, commonly known as conditional variance (Dierckx, 2020).
Fortunately, the GARCH models can account for this irregularity problem, and both models can be combined to improve forecasting accuracy. Hence, this paper tries to develop the hybrid ARIMA-GARCH model in order to investigate and forecast the IHH stock price using its daily data. Correspondingly, the Value at Risk (VaR) of the IHH stock price will be determined where VaR can be understood as an investment risk measure that measures the maximum possible loss (Soeryana et al., 2019).
Therefore, the objectives of this research are to explore the past trend or pattern of IHH stock price and to compare the models’ performances between the ARIMA-GARCH and ARIMA- TGARCH in forecasting the IHH stock price. Additionally, to forecast the stock price of IHH in the next six months and finally to determine its VaR.
is projected to be at -4.067% based on the 1-day Value at Risk computed.
Keywords: forecasting, Value at Risk, ARIMA, GARCH, healthcare sector, IHH stock.
2. Literature Review
The issue of stock market volatility has attracted many scholars and market investors to address and study this issue properly to create optimum portfolios, compute Value-at-Risk (VaR), and conduct stress-testing based on how the future volatility will react to current economic conditions (Nonejad, 2017). The time-varying volatility was calculated and projected using two-time series models based on financial returns, including the generalised autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility (SV) models. In forecasting using ARIMA models, the Autoregressive Integrated Moving Average (ARIMA) model was used in a study done by Bakar and Rosbi ( 2017) to investigate the performance of Malaysia's Shariah-compliant share price of the healthcare sector. Based on the residuals diagnostics, the ARIMA (1,1,1) was a reliable model to forecast the stock price in the healthcare sector in Malaysia.
The generalized autoregressive conditional heteroskedasticity (GARCH) model is an expansion of the ARCH model. GARCH increases predicting accuracy by including all prior squared returns with lower weights corresponding to more distant volatilities (Grachev, 2017).
In his paper, Dinardi (2020) states that ARIMA models will not always be the most significant predictors of financial time series aspects. Hence, many studies use the ARCH or GARCH class of models to address concerns that ARIMA models could not handle. Next, the threshold GARCH (TGARCH) model. Gabriel (2012) researched to evaluate the forecasting performance of the GARCH model by using data from Romania (BET Index) and discovered that the TGARCH model is the most accurate in forecasting volatility. Next, in forecasting using ARIMA-GARCH models, static and dynamic forecasts will be applied. Dynamic forecast or n-step ahead forecast, for the model to compute the first forecasted value, the actual lagged value of Y will be used (Dritsaki, 2018).
Then, the danger of unanticipated changes in prices or the log-return rate within a certain period is measured by Value at Risk (VaR) (Li, 2016). Under normal market conditions, the VaR is defined as the highest predicted possible loss on a portfolio over a particular time horizon for a given confidence interval (Jorion, 2002).
2.1 Problem Statement
COVID-19 pandemic outbreaks have a significant impact on Malaysia's numerous industries, particularly the healthcare sector. In the current domestic and international economic climate, IHH stock values have been volatile, making it extremely difficult for investors to understand the direction of healthcare stock prices. Is the price of a stock predictable? Furthermore, because of the volatile nature of stock prices of IHH during COVID-19, investors need to know whether they will benefit from investing in IHH. The economic community can consider forecasting stock price fluctuation as a reference point in comprehending stock price fluctuation. It can assist the investor of IHH in gaining an ideal advantage in searching for a low price at the time of purchase and a high price at the time of a sale during this pandemic.
Aside from that, investors frequently face various real-world trading restrictions, necessitating the creation of portfolios that adhere to trading restrictions (Lwin et al., 2017). Furthermore, many risk management systems for trading operations are too expensive to implement, making them out of reach for smaller pension funds in terms of capital and personnel. This emphasizes the importance of examining the practical concerns that institutional investors must make when developing a VaR-based risk management plan (Lwin et al., 2017).
3. Method 3.1 Materials
The dataset will be historical data of IHH stock price, and the R software were utilised to construct the processes.
3.1.1 Samples, Site and Procedures
The data in this research is separated into two groups which are in-sample and out-sample set.
The in-sample data sets will be used to determine which ARIMA-GARCH or ARIMA- TGARCH model is best for forecasting the IHH stock price. IHH stock price data will be gathered from the Investing.com website and were collected in weekdays excluding public holiday and weekends, from 22nd September 2015 until 21st September 2021 and are cited in MYR (RM). The collected data of daily IHH stock prices has a total of 1468 data. The study was conducted by plotting graphs, conducting tests and determining the conclusion from hypotheses.
3.2 Data Analysis 3.2.1 Stationarity Test
Determining the number of differencing necessary to make the series stationary, as a model cannot anticipate on non-stationary time series data (Kumar G, 2021).
3.2.1.1 Augmented Dickey-Fuller (ADF) Test
A time series can be said as a stationary series if its mean, variance and covariance do not change over time (Özen, n.d.). The ADF test would check for a unit root in a given time series where a presence of unit root will indicate a linear trend in the time series, which is considered non-stationary (Khalid, 2020). The ADF test would have a null hypothesis as written as follows,
𝐻0 : 𝜑 = 1 (there is a presence of unit root in the series or the series is non-stationary) 𝐻a : 𝜑 < 1 (there is no presence of unit root in the series or the series is stationary)
Hence, a test statistic and p-value would be computed. If the computed p-value is lower than the critical value (0.05), the null hypothesis can be rejected, indicating the series to be stationary and vice versa.
3.2.2 Seasonality Test
Seasonality is a component of a time series in which the data undergoes predictable and recurring variations over the course of a calendar year (Kenton, 2020).
3.2.2.1 Webel-Ollech (WO) Test
According to Webel and Ollech (n.d.), Webel-Ollech (WO) seasonality incorporates the findings of the QS-tests and kwman-tests, which are both performed on the residuals of an automated non-seasonal ARIMA model. Thus, as long as the p-value of the QS-test and kwman-test is less than 0.01 and 0.002 respectively, the WO-test will consider the associated time series to be a seasonal time series.
3.2.3 Heteroskedasticity Test
Heteroskedasticity occurs more frequently in datasets with a wide range of observed values and when the standard deviations of a predicted variable are non-constant when compared to various values of an independent variable (Hayes, 2020). Conditional heteroskedasticity reveals nonconstant volatility that is connected to the volatility of the preceding period such as daily period.
3.2.3.1 ARCH-LM Test
Engle's (1982) Lagrange Multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH) is a commonly used specification test for univariate time series models where it tests for the non-conditional heteroskedasticity in the presence of an ARCH model. The ARCH-LM test is a two-sided alternative hypothesis test for ARCH(q) where if the null hypothesis of no ARCH is true, this statistic has an asymptotic distribution similar to that of a chi-squared random variable with q degrees of freedom.
3.2.4 Transformation
The purpose of transformation is to create stationary time-series data, which entails locating a stationary component containing a trend component.
3.2.4.1 Differencing Transformation
One of the transformations commonly employed in time series data analysis is the differencing transformation. The series of changes from one period to the next is the first difference in a time series. The first-order difference of a time series is defined as:
∆𝑍𝑡 = (1 − 𝐵)𝑍𝑡 = 𝑍𝑡 − 𝑍𝑡−1 (1)
𝑍𝑡 is the stock price explained in time t, and 𝐵 signifies the backward (lag) operator (𝐵𝑘𝑍𝑡 = 𝑍𝑡−𝑘). The nth-order difference, on the other hand, is defined as:
∆𝑛𝑍𝑡 = (1 − 𝐵)𝑛𝑍𝑡 = (1 − 𝐵)𝑛−1((1 − 𝐵)𝑍𝑡) (2) A d-degree differencing can be used to make non-stationary time series data stationary. To render the data is stationary, a new series produced out of the difference between successive stock price observations can be constructed.
∇𝑍𝑡 = 𝑍𝑡 − 𝑍𝑡−1 (3)
3.2.4.2 Logarithmic Transformation
The log transformation decreases or eliminates the skewness of our original data.
Differentiating transformations are frequently coupled with logarithmic transformations. The equation to conduct differencing is as follows:
𝒍𝒐𝒈 ( 𝒁𝒕
𝒁𝒕−𝟏) = 𝒍𝒐𝒈(𝒁𝒕) − 𝒍𝒐𝒈 (𝒁𝒕−𝟏) (4)
3.2.5 Autocorrelation
3.2.5.1 Autocorrelation Function (ACF)
Autocorrelation is used to identify non-randomness in data. Let X is set as a stochastic process, and t is any point in time (t may be an integer for a discrete-time process or a real number for a continuous-time process). Then Xt is the value produced by a given run of the process at time t. Consider the process has mean μt and variance σt 2 at time t, for each t. Hence, stated below definition autocorrelation between times s and t:
𝐑(𝒔, 𝒕) =𝑬(𝑿𝒕− 𝝁𝒕)(𝑿𝒔− 𝝁𝒔) 𝝈𝒕𝝈𝒔
(5)
3.2.5.2 Partial Autocorrelation Function (PACF)
PACF is a conditional correlation which related between two variables under the assumption the linear dependence of one variable after eliminating other variables that influence both variables. Definition of PACF is stated as:
𝝆
𝑿𝒀∙𝒁= 𝑵 ∑𝑵𝒊=𝟏𝒓𝑿,𝒊𝒓𝒀,𝒊−∑𝑵𝒊=𝟏𝒓𝑿,𝒊∑𝑵𝒊=𝟏𝒓𝒀,𝒊
√𝑵 ∑𝑵𝒊=𝟏𝒓𝟐𝑿,𝒊−(𝑵 ∑𝑵𝒊=𝟏𝒓𝑿,𝒊)𝟐√𝑵 ∑𝑵𝒊=𝟏𝒓𝟐𝒀,𝒊−(𝑵 ∑𝑵𝒊=𝟏𝒓𝒀,𝒊)𝟐
(6)
The equation above described the partial correlation between X and Y given a set of n controlling variables 𝒁 = {𝑍1, 𝑍2, . . . , 𝑍𝑛}, written 𝜌𝑋𝑌 · 𝒁, is the correlation between the residuals 𝑅𝑋 and 𝑅𝑌 which were explained as linear regression of X with Z and of Y with Z.
3.2.6 Hybridization of Forecasting Models
To take advantage of the distinctive strengths of different models in linear and nonlinear modelling, a hybrid methodology combining both linear ARIMA and nonlinear GARCH models is developed.
3.2.6.1 ARIMA-GARCH
The formulation of ARIMA corresponds to conditional mean of stock price:
∆𝒅𝒚̂𝒕= 𝒄𝟎+ ∑ 𝝓𝒊∆𝒅𝒚𝒕−𝒊+ ∑ 𝜽𝒋∆𝒅𝝐𝒕−𝒋
𝒒
𝒋=𝟏 𝒑
𝒊=𝟏
(7) where
∆𝑑𝑦̂𝑡 = 𝑑 order differentiated data forecast of stock price 𝑐0 = constant value
𝜙𝑖 = autoregressive (AR) coefficient
∆𝑑𝑦𝑡−𝑖= 𝑑 order differentiated previous period data of stock price 𝜃𝑗 = moving average (MA) coefficient
∆𝑑𝜖𝑡−𝑗= 𝑑 order differentiated previous period forecasting errors of stock price 𝑝 = order of autoregressive (AR) term
𝑞 = order of moving average (MA) term
The GARCH (𝑟, 𝑠) model's conditional heteroskedasticity equation appears as follow:
∆𝒅𝝈𝒕𝟐= 𝝁𝟎+ ∑ 𝜶𝒊∆𝒅𝝐𝒕−𝒊𝟐
𝒓
𝒊=𝟏
+ ∑ 𝜷𝒋∆𝒅𝝈𝒕−𝒊𝟐
𝒔
𝒋=𝟏
(8) where
∆𝑑𝜎𝑡2 = 𝑑 order differentiated conditional variance forecast of stock price 𝜇0 = constant variance of stock price
𝛼𝑖 = autoregressive (AR) coefficient
∆𝑑𝜖𝑡−𝑖2 = 𝑑 order differentiated previous period forecasting errors of stock price 𝛽𝑗 = first moving average (MA) coefficient
∆𝑑𝜎𝑡−𝑖2 = 𝑑 order differentiated previous conditional variance forecast of stock price 𝑟 = order of autoregressive (AR) term
𝑠 = order of first moving average (MA) term
The estimation procedure of ARIMA and GARCH models are based on maximum likelihood method (Dritsaki, 2018). The equation for the logarithmic likelihood function is:
𝒍𝒐𝒈 𝑳 = ∑ 𝒍𝒐𝒈 (
𝟏
√𝟐𝝅∆𝒅𝝈̂𝒕𝟐 𝒆
−∆𝒅𝝐𝒕𝟐 𝟐∆𝒅𝝈̂𝒕𝟐
)
𝑻
𝒕=𝟏
(9)
where ∆𝑑𝜎̂𝑡2 is the 𝑑 order differentiated conditional variance forecast of stock price and ∆𝑑𝜖𝑡2 is the 𝑑 order differentiated forecasting errors of stock price.
3.2.6.2 ARIMA-TGARCH
The leverage effect can be handled using a TGARCH (r, s) model provided by Glosten et al.
(1993), but the leverage impact is expressed in quadratic form (Pahlavani & Roshan, 2015).
Equation corresponds to Threshold GARCH (TGARCH) for the conditional variance is:
∆𝒅𝝈𝒕𝟐 = 𝝁𝟎+ ∑ 𝜶𝒊∆𝒅𝜺𝒕−𝒊𝟐
𝒓
𝒊=𝟏
+ ∑ 𝜷𝒋∆𝒅𝝈𝒕−𝒊𝟐
𝒔
𝒋=𝟏
+ ∑ 𝜹𝒌𝝐𝒕−𝒌𝟐
𝒓
𝒌=𝟏
𝒅𝒕−𝒌 (10)
Where 𝑟 is the order of AR term, 𝑠 is the order of MA term, 𝛿𝑘 is the second MA coefficient and 𝑑𝑡−𝑘 is the threshold indicator.
3.2.7 Value at Risk (VaR)
According to Kenton (2021), Value at risk (VaR) is a statistical measure used by the investment and commercial banks to quantify the amount and probability of any potential losses in their institutional portfolios by quantifying potential economic losses within, over a specified time-frame. Using the historical technique, the percentage change for each risk factor on any given day is calculated using market data from the previous 250 days. Each percentage change is then used in conjunction with current market prices to generate 250 future value possibilities. Each case in the portfolio is priced using non-linear pricing models where the third worst day picked was assumed to has a 99% VaR. Therefore, the formula to calculate VaR is as follows, where 𝑣𝑖 is the number of variables on day 𝑖, 𝑣𝑖−1 is the number of variables on the day 𝑖 − 1 and 𝑚 is the number of days from which historical data is taken.
𝐕𝐚𝐥𝐮𝐞 𝒂𝒕 𝑹𝒊𝒔𝒌 = 𝒗𝒎 𝒗𝒊
𝒗𝒊−𝟏 (11)
3.3 Validity and Reliability
The best model is selected by assessing different models' performance using standard model evaluation techniques Akaike Information Criterion (AIC). AIC computed as:
𝐀𝐈𝐂 = −𝟐 𝒍𝒏(𝑳) + 𝟐𝒌 (12)
Where L denotes the value of the likelihood function at parameter estimations of stock price, and k denotes the number of estimated parameters of the model. The model with the lowest AIC value was chosen as the superior model. Then, to assess the model's performance following the forecasting, the projected values generated by the forecasting model were required to be compared to the actual values. RMSE and MAE are the two metrics selected to evaluate the performance of the ARIMA-GARCH model. RMSE restricts variance by giving greater weight to errors with large absolute values than it does to errors with small absolute values. The formula to calculate the RMSE is as follows:
𝐑𝐌𝐒𝐄 = √∑(𝑷𝒊− 𝑶𝒊)𝟐
𝒏 (13)
Where 𝑃𝑖 is the predicted value for the ith observation in the dataset, 𝑂𝑖 is the observed value for the ith observation in the dataset and 𝑛 is the sample size. Next, MAE takes the absolute error from each sample in datasets and gives the output in an averaged value where it can be calculated by using the formula below.
Where 𝑥𝑡 is the actual value, and 𝑥𝑖 is observed or forecasted value. The mean of absolute errors calculated by just dividing them with 𝑛 number of datasets.
𝑴𝑨𝑬 =∑𝒏𝒊=𝟏|𝒙𝒊− 𝒙𝒕|
𝒏 (14)
4. Results and Discussion 4.1 Data Description
Table 1: Summary of Variables
Date Price
Length:1468 Min. : 4.540
Class: character 1st Qu. : 5.490
Mode: character Median : 5.790
Mean : 5.830 3rd Qu. : 6.173 Max. : 6.750
From the table, the minimum stock price of IHH is RM4.54, the maximum stock price is RM6.75 and the average price of date within the date of collected data is RM5.83.
4.2 Analysis of Trend of Stock Price
Figure 1: Daily IHH Healthcare Stock Price Sept 2015 - Sept 2021
The dataset has been split into training and validation data for the purpose of this study. The training dataset is taken from 22nd September 2015 until 31st December 2021 while the validation dataset consists of data from 1st January 2021 until 21st September 2021. The plot shows that the series has a downward trend from 2016 to the end of the month of 2018, then fluctuates around RM 5.00 until RM 6.00 until the end of 2019. It can also be observed that the stock prices show an upward tendency from 2020 until 2021.
4.3 Seasonality
The Webel-Ollech overall seasonality test incorporates the results of many seasonality tests.
The results of the QS-test and the kw-test, both derived on the residuals of an automatic non- seasonal ARIMA model, are combined by default in the WO-test. The test yielded a p-value of 0.3004236, indicating that the series does not exhibit seasonality.
Table 2: Webel-Ollech Test Result Test Used: Webel-Ollech, Test Data: stock_ts_t
Test Statistics P-value
0 0.3004236
4.4 Stationarity
The p-value in the first ADF test conducted is 0.02082, thus, the null hypothesis saying that the series is not stationary will be rejected. Therefore, the time series data have to be log- transformed and differenced appropriately to obtain stationarity. Then, the ADF was being conducted again. The results of the second ADF test performed are demonstrated in the following table where the p-value is then indicating the null hypothesis to be rejected and thus conclude that the series is stationary.
Table 3: Augmented Dickey-Fuller (ADF) Test Results for diff_log_stock_ts_t Test: Augmented Dickey–Fuller (ADF) test
Time Series Data: diff_log_stock_ts_t
Dickey-Fuller Lag order P-value
-11.993 10 0.01
Finally, the critical value at 5% significance level is greater than the test statistic, and thus, suggesting that the null hypothesis saying that the series is not stationary is failed to be rejected.
To conclude, the series were found to be stationary in the unit root test after the log- transformation and first-differencing.
4.5 Determining the Best Model 4.5.1 Fitting ARIMA Models
Since this series went through first order difference to be stationary, the ARIMA model's order of differencing (I) was set to 1. In general, the ACF and PACF demonstrate the potential of selecting an acceptable order for autoregressive (AR) and moving average (MA) functions.
Figure 2: ACF Plot of Transformed Stock Price Data
Figure 3: PACF Plot of Transformed Stock Price Data
According to the PACF plot, the possible orders of AR, p are 1, 2, 4 and 8 while the ACF plot display the possible orders of MA, q are 1, 2, 4, 5 and 8. The order is determined by the substantial lag, which is defined as latency that exceeds the blue line. Hence, the best model was determined by computing AIC for all possible combinations of model. ARIMA (4,1,5) is picked as the best ARIMA model since it has a slightly lower AIC.
Table 4: AIC for ARIMA Models
Model AIC
ARIMA (1,1,1) -7528.240
ARIMA (1,1,2) -7526.404
ARIMA (1,1,4) -7532.500
ARIMA (1,1,5) -7530.565
ARIMA (1,1,8) -7532.04
ARIMA (2,1,1) -7526.383
ARIMA (2,1,2) -7532.227
ARIMA (2,1,4) -7530.542
ARIMA (2,1,5) -7528.552
ARIMA (2,1,8) -7530.737
ARIMA (4,1,1) -7530.629
ARIMA (4,1,2) -7529.951
ARIMA (4,1,4) -7537.945
ARIMA (4,1,5) -7538.214
ARIMA (4,1,8) -7532.521
ARIMA (8,1,1) -7532.626
ARIMA (8,1,2) -7530.725
ARIMA (8,1,4) -7536.996
ARIMA (8,1,5) -7535.016
4.5.2 Fitting GARCH and TGARCH Models
The ARCH effect can be realized by employing the following ARCH-LM test.
Table 5: Results for ARCH-LM Test
Test Used: ARCH-LM Test, Data: squared residuals of ARIMA (4,1,5) Null hypothesis: There is no ARCH effect
Chi-Squared Df P-value
122.49 12 < 2.2e-16
For both tests, p-value of <2.26e-16 is obtained, which is less than 0.05, hence reject both hypotheses. As a result, autocorrelation is present on the squared residuals of the model and ARCH effect is present in the ARIMA (4,1,5) model fitted. Hence, GARCH and TGARCH models can be used.
Figure 4: ACF Plot for Squared Residuals of Transformed Stock Price Data
Figure 5: PACF Plot for Squared Residuals of Transformed Stock Price Data
The results obtained from the plots show that the possible r orders are 1,2,3,4 and 5 meanwhiles s orders are 1,2,4 and 5. However, the best model has been chosen according to past study made by other researchers. For GARCH model, the GARCH (1,1), references were made from Babu and Reddy (2015) and Farah and Mohd (2019) stated they are using GARCH(1,1) as best model to forecast stock prices. Hence, the initial hybrid models obtained are ARIMA (4,1,5)- GARCH (1,1) and ARIMA (4,1,5)-TGARCH (1,1) respectively.
4.5.2.1 ARIMA-GARCH
To determine the best model for ARIMA-GARCH, the AIC for models in the neighbourhood of ARIMA (4,1,5)-GARCH (1,1) which are ARIMA (4,1,5)-GARCH (0,1) and ARIMA (4,1,5)-GARCH (1,0) was compared to the AIC of ARIMA (4,1,5)-GARCH (1,1). Hence, ARIMA (4,1,5)-GARCH (1,1) was chosen as best model because of the lowest AIC value.
Table 6: AIC for ARIMA-GARCH Models
MODEL AIC
ARIMA (4,1,5)-GARCH (1,1) -6.0034
ARIMA (4,1,5)-GARCH (0,1) -5.8113
4.5.2.2 ARIMA-TGARCH
To find the T-GARCH fitting model, the AIC for models in the neighbourhood of ARIMA (4,1,5)-TGARCH (1,1), such as ARIMA (4,1,5)-TGARCH (0,1) and ARIMA (4,1,5)- TGARCH (1,0) was compared to the AIC of ARIMA (4,1,5)-TGARCH (1,1). The model ARIMA (4,1,5)-TGARCH (1,0) was dropped due to the convergence problem. Thus, only the ARIMA (4,1,5)-TGARCH (1,1) and ARIMA (4,1,5)- TGARCH (0,1) AICs were compared.
Table 7: AIC for ARIMA-TGARCH Models
MODEL AIC
ARIMA (4,1,5)-TGARCH (1,1) -6.0384
ARIMA (4,1,5)-TGARCH (1,0) -5.8423
ARIMA (4,1,5)-TGARCH (1,1) was determined to be the best ARIMA-TGARCH model since it has the lowest AIC.
4.5.3 Comparing Accuracy of the Models
All the hybrid models produced from the previous procedures have to be compared based on its accuracy in order to identify the best model to be used in forecasting the IHH stock price.
Table 8: RMSE and MAE for Hybrid Models
MODELS ERROR MEASURES
RMSE MAE
ARIMA (4,1,5)-GARCH (1,1) 0.02289412 0.01672682
ARIMA (4,1,5)-TGARCH (1,1) 0.02289852 0.01674321
Based on the results presented in the above table, the ARIMA (4,1,5)-GARCH (1,1) hybrid model was observed to be the model with smallest RMSE and MAE values as compared to the other hybrid model. Therefore, the ARIMA (4,1,5)-GARCH (1,1) model will be used as the predictive model to forecast the IHH stock price.
4.6 Forecasting Price of Stock
Subsequently, as ARIMA (4,1,5)-GARCH (1,1) model will be employed in forecasting the IHH stock price for the period of six months starting from 22nd September 2021 until 21st March 2022.
Figure 6: Forecasted Daily IHH Stock Price (September 2021 – March 2022)
The daily forecasted values for IHH stock price were generated with several different outputs which include the point forecast, the 80% and 95% prediction intervals. The point forecast is expected to be within the range of computed upper and lower limit. Thus, the results demonstrate that the highest forecasted stock price for IHH is at $6.71 which is on 18th January 2022 and the lowest is at $6.28 which is on 9th November 2021.
4.7 Value at Risk (VaR)
The Value at Risk (VaR) is also calculated to aid in risk assessment and management, particularly for investors and other parties who interested in IHH stocks.
Figure 7: Histogram of Daily Distribution of IHH Stock Returns
Based on the histogram above, there is about 317 days where the IHH daily stock return was between 0% and 1%. Additionally, the red line on the left tail of the histogram marks the few worst 1% returns from other daily returns including the daily losses of around 4% to 9%. This could also be interpreted as; the worst daily loss will not go beyond 4% with 99% confidence.
Specifically, based on the 1-day value at risk at 99% confidence interval, the daily worst-case loss calculated is -4.067%. Explicitly, if $100,000 is invested in this IHH stock with a projected annual return of 7% and 100% of the portfolio is invested, the investors stand a 1% risk of losing $199.67 of their total IHH stock investment on a daily basis if no trading occurs. Hence, investors may employ this method to reduce losses and maximise profits by calculating the possible losses associated with each transaction.
5. Conclusion
In studying the trend of IHH stock, the plotted graph of the stock price performance against time reveals that the series has a negative tendency from 2016 to the end of 2018, then fluctuates around RM 5.00 to RM 6.00 until the end of 2019. It is also worth noting that stock prices are trending upward from 2020 to 2021. The series moves in a pattern called oscillations, which implies that the trend has many ups and downs. There is no detectable seasonality in the seasonality tests conducted. In addition, after the log-transformation and first-differencing performed on the time series, the series was found to be stationary in both unit root tests.
Then, the result for the best model for each fitted hybrid models were determined by the lowest value of AIC. For the first hybrid model, the best model is ARIMA (4,1,5)-GARCH (1,1) and ARIMA (4,1,5)-TGARCH (1,1). The result shown ARIMA (4,1,5)-GARCH (1,1) hybrid model has the higher model accuracy with smaller RMSE and MAE value, which is 0.02289412 and 0.016726822, respectively. Then, this best model is used to estimate the daily IHH stock price for the period of 6 months. The highest stock price predicted falls on 18th January 2022 at $6.71. Whereas the lowest predicted stock price was observed to be at $6.28 which is on 9th November 2021. On average, the IHH stock is predicted to be traded at about
$6.52.
Following that, its Value at Risk (VaR) is also being computed by using the historical method on the past data from 22nd September 2015 until 21st September 2021. The computation reveals that the the daily loss incurred on IHH stock would not go beyond -4.067% with 99%
confidence. Additionally, if an amount of $100,000 was invested on the IHH stock, the predicted loss to be incurred is at least RM199.67 daily.
6. Acknowledgement
This research is funded by Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (UiTM), Shah Alam, Selangor, Malaysia.
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