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5044
Di Asih I Maruddani 1, IJMCR Volume 13 Issue 04 April 2025
International Journal
of Mathematics and Computer Research
ISSN: 2320-7167
Volume 13 Issue 04 April 2025, Page no. – 5044-5050
Index Copernicus ICV: 57.55, Impact Factor: 8.615
DOI: 10.47191/ijmcr/v13i4.03
Modeling the Volatility-Return Relationship in the Indonesian Stock Market
using the GARCH-M Framework
Di Asih I Maruddani1, Diah Safitri2
1,2Department of Statistics, Universitas Diponegoro, Indonesia
ARTICLE INFO
ABSTRACT
Published Online:
03 April 2025
Corresponding Author:
Di Asih I Maruddani
The LQ45 Index was observed to be in the red zone, with a decline of 9.64% year-to-date
(YTD), reaching the level of 877.02. The LQ45 Index became increasingly weakened
following the announcement of Donald Trump's victory in the U.S. presidential election,
which impacted the Indonesian capital market. It was recorded that the LQ45 Index fell by
5.3% during the final trading month of 2024. Nevertheless, there remains a potential for
strengthening the stock prices of LQ45 constituent issuers in the remainder of this year,
particularly in December 2024. One of the stocks recommended by IDX is PT Indofood CBP
Sukses Makmur Tbk., which has also been one of the most liquid companies according to IDX
throughout 2024. The return volatility of stocks in emerging markets is generally much higher
than that of developed markets. High volatility reflects a higher level of risk faced by
investors, as it indicates significant fluctuations in stock price movements. Therefore, equity
investments in Indonesia carry a potentially high level of risk. A common characteristic of
financial time series data, particularly return data, is that the probability distribution of returns
exhibits fat tails and volatility clustering, often referred to as heteroscedasticity. Time series
models that can be used to model these conditions include ARCH and GARCH models. One
variation of the ARCH/GARCH models is the Generalized Autoregressive Conditional
Heteroscedasticity in Mean (GARCH-M) model, which incorporates the effect of volatility
into the mean equation. The purpose of this study is to predict volatility using the GARCH-M
model in the analysis of daily closing price return data of PT Indofood CBP Sukses Makmur
Tbk. The best model used for volatility forecasting is ARIMA(2,0,1) GARCH(1,1)-M.
KEYWORDS: Indonesian Stock Market, Stock Price Fluctuations, Risk, Heteroscedasticity
I. INTRODUCTION
The Indonesian capital market, as one of the emerging
markets, has distinct characteristics compared to developed
markets. One of its prominent features is high volatility,
which indicates a higher level of uncertainty and risk faced
by investors [1]. Volatility reflects fluctuations in stock
prices over time, which directly impacts the potential gains
or losses in stock investments. During periods of high
volatility, investors are exposed to greater risks as stock
prices tend to experience larger and more frequent changes.
Throughout 2024, the LQ45 Index, which consists of 45
stocks with the largest market capitalization and highest
liquidity on the Indonesia Stock Exchange (IDX), recorded a
significant decline. As of the end of the year, the LQ45 Index
had dropped by 9.64% year-to-date (ytd), reaching the level
of 877.02 [2]. This downturn was exacerbated by the global
market's reaction to the announcement of Donald Trump's
victory in the 2024 U.S. Presidential Election, which
triggered market uncertainty and capital outflows from
emerging markets such as Indonesia [3]. As a result, the
LQ45 Index experienced an additional 5.3% decrease during
the final trading month of 2024.
Despite these unfavorable conditions, there remains an
opportunity for price recovery among the constituent stocks
of the LQ45 Index. One of the stocks recommended by the
IDX for its strong fundamentals and high liquidity is PT
Indofood CBP Sukses Makmur Tbk. (ICBP). ICBP has been
recognized as one of the most liquid stocks on the IDX
throughout 2024 [3]. This makes ICBP a relevant subject for
further analysis, particularly in terms of return volatility and
risk assessment. Understanding the volatility of ICBP's stock
returns is crucial for investors seeking to make informed
decisions amidst uncertain market conditions.
“Modeling the Volatility-Return Relationship in the Indonesian Stock Market using the GARCH-M Framework”
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Di Asih I Maruddani 1, IJMCR Volume 13 Issue 04 April 2025
The concept of volatility plays a vital role in modern
financial theory and practice. Volatility is often used as a
measure of market risk [4], and its accurate estimation is
essential for asset pricing, risk management, and portfolio
allocation [5]. In emerging markets such as Indonesia, stock
return volatility tends to exhibit unique patterns, including
volatility clustering—a phenomenon where periods of high
volatility are followed by similar periods, and low volatility
periods are followed by low volatility [6, 7]. Additionally,
financial time series data, particularly stock returns,
frequently display fat tails in their probability distributions.
This indicates a higher likelihood of extreme price
movements compared to what is predicted by a normal
distribution [8].
Given these empirical characteristics, traditional time series
models such as Autoregressive Integrated Moving Average
(ARIMA) are insufficient for modeling financial time series
data with conditional heteroscedasticity [9]. To address this
limitation, Engle [4] introduced the Autoregressive
Conditional Heteroskedasticity (ARCH) model, which
explicitly models time-varying volatility. Bollerslev [7] later
generalized this model into the Generalized Autoregressive
Conditional Heteroskedasticity (GARCH) model, which has
become widely adopted in volatility modeling due to its
flexibility and accuracy.
A further development of the GARCH model is the GARCH-
in-Mean (GARCH-M) model proposed by Engle, Lilien, and
Robins [10]. The GARCH-M model allows the conditional
variance (volatility) to directly affect the conditional mean of
stock returns. This implies that investors demand higher
returns as compensation for taking on higher risk, aligning
with modern portfolio theory. The incorporation of risk into
the return equation makes the GARCH-M model particularly
relevant for studying the risk-return tradeoff in financial
markets [10].
The Indonesian stock market, characterized by high volatility
and sensitivity to both domestic and global economic events,
presents a compelling case for the application of the
GARCH-M model. PT Indofood CBP Sukses Makmur Tbk.
(ICBP), as a leading consumer goods company with strong
market presence, offers an ideal case study for analyzing
stock return volatility using this model. By modeling ICBP's
daily stock returns with a GARCH-M approach, it is possible
to gain insights into the volatility-return relationship and
assess the compensation investors require for bearing risk.
some empirical studies showed that the GARCH-M and GJR
GARCH models work well, e.g., in [11–15].
The objective of this study is to predict the volatility of daily
stock returns of PT Indofood CBP Sukses Makmur Tbk.
using the GARCH-M model. The empirical analysis
conducted in this study identifies the ARIMA(2,0,1)-
GARCH(1,1)-M model as the best-fit model for forecasting
volatility in ICBP's stock returns throughout 2024. The
results of this analysis are expected to provide valuable
information for investors, portfolio managers, and
policymakers in understanding the behavior of stock returns
and the associated risks in the Indonesian capital market.
Furthermore, the findings contribute to the existing body of
literature on volatility modeling in emerging markets and
highlight the importance of adopting appropriate econometric
models, such as GARCH-M, for more accurate risk
measurement and investment decision-making [16, 17].
Accurate volatility forecasts are essential for optimizing
investment strategies, determining appropriate risk
premiums, and designing effective hedging techniques [18].
In conclusion, the volatility of stock returns in emerging
markets, including Indonesia, remains a critical area of
research in financial economics. The application of advanced
time series models, such as the GARCH-M framework,
provides a comprehensive approach to understanding and
managing market risks. As the global investment landscape
continues to evolve, incorporating volatility modeling into
investment practices is crucial for achieving long-term
financial stability and sustainable portfolio performance.
II. THEORETICAL FRAMEWORK
2.1. ARCH and GARCH Model
In general, time series data modeling is expected to fulfill the
assumption of constant variance (homoscedasticity).
However, financial sector time series data often exhibit very
high volatility. This is indicated by the presence of non-
constant variance, known as heteroscedasticity. To address
this issue, the ARCH model was introduced by Engle (1982),
and later the GARCH model was developed by Bollerslev
(1986) as a generalized form of the ARCH model.
The general form of the ARCH(p) model is as follows [9]:
The general form of the ARCH(p) model is as follows [9]:
2.2. GARCH in Mean (GARCH-M) Model
If the conditional variance or standard deviation is
incorporated into the mean equation, the resulting model is
referred to as the GARCH in Mean (GARCH-M) model
[10]. The GARCH(p, q)-M model can be defined as follows:
where
Here, μ and c are constants. A positive c indicates that
returns are positively affected by past volatility.
Other specifications for the risk premium commonly used in
the literature include [9]:
and
“Modeling the Volatility-Return Relationship in the Indonesian Stock Market using the GARCH-M Framework”
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and
The formulation of the GARCH-M model suggests the
existence of serial correlation in the return series. This serial
correlation is demonstrated by the volatility process.
The existence of a risk premium implies that the historical
returns of a stock exhibit serial correlation.
2.3. Quasi Maximum Likelihood Estimation
Tsay [9] offers the application of the Quasi Maximum
Likelihood (QML) method for time series analysis in cases
where the error terms do not follow a normal distribution.
The QML approach still utilizes the maximum likelihood
method as its foundation, so the computation of quasi
covariance variance relies on values obtained from the
maximum likelihood estimation (MLE) method.
Within the ARCH/GARCH specification, it is still possible
to produce a valid model and consistent parameter estimates
by using the QML method, which maximizes the log-
likelihood function through linear forecasting of the squared
residuals. With this method, the consistency of standard
errors is maintained even if the distributional assumptions
are violated. The parameter estimation model using QML is
given by:
III. RESEARCH METHODS
The data used in this study are secondary data, specifically
stock data of PT Indofood CBP Tbk. (ICBP.JK), obtained
from www.finance.yahoo.com covering the period from
January 1, 2024, to December 31, 2024. This study uses
return data of the stock, comprising a total of 236
observations.
The data in this study are processed using R software.
The steps taken to analyze the data are as follows:
1. Convert the stock price data of ICBP.JK into return data.
2. Identify the ARIMA model based on the time series plot
to determine whether the data are stationary. Once
stationarity is achieved, the Autocorrelation Function
(ACF) and Partial Autocorrelation Function (PACF)
plots are generated to identify the appropriate model.
3. Estimate the parameters of the ARIMA model.
4. Perform model verification by conducting residual
independence tests and residual normality tests. If
necessary, other models can be considered through
underfitting and overfitting analysis.
5. Perform the Lagrange Multiplier (LM) test to determine
whether there is an ARCH effect in the model.
6. Identify the appropriate ARCH and GARCH models.
7. Identify the GARCH-M model.
8. Estimate the model parameters using the Quasi
Maximum Likelihood (QML) method.
9. Conduct another Lagrange Multiplier (LM) test to verify
if there is still an ARCH effect remaining in the model.
10. Verify the GARCH-M model to select the best-fit model.
11. Forecast the volatility of PT Indofood CBP Tbk.’s stock
using the best model.
IV. RESEARCH FINDINGS AND DISCUSSION
The observational data consist of the daily closing stock
prices of PT Indofood CBP Tbk. from January 1, 2024 to
December 31, 2024, using active trading days from
Monday to Friday. The data analyzed are the return data of
the closing stock prices. The chart of the company’s stock
prices can be seen in Figure 1, while the stock returns are
shown in Figure 2.
In Figure 2, it can be observed that the return plot of PT
Indofood CBP Tbk.’s stock is stationary in the mean. This
is indicated by the fact that the average of the observation
series over time remains constant (fluctuating around a
central value).
Fig 1. Stock Prices Plot
Fig 2. Stock Returns Plot
To provide an initial overview of the data used in this
study, a descriptive statistical analysis was conducted. This
analysis includes key measures such as the mean, standard
deviation, minimum, maximum, skewness, and kurtosis,
which offer insights into the distribution and variability of
the return series. The complete descriptive statistics of the
research data are presented in the Table 1.
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Table 1. Statistics Descriptives of Price and Return
Statistics
Price
Return
Number of Data
237
236
Mean
11263.71
0.00027905
Standard Deviation
691.9917
0.01682765
Variance
478852.5
0.00028317
Skewness
0.1012407
-0.20530410
Kurtosis
-0.6779047
1.67403100
The descriptive statistics of ICBP’s stock prices provide an
initial understanding of the characteristics of the dataset,
which consists of 237 observations. The mean value of
11263.71 indicates the average level of the observed data,
suggesting a relatively high central tendency in the series.
The standard deviation of 691.99, along with the variance
of 478,852.5, reflects a moderate level of dispersion around
the mean, implying that the data exhibit noticeable
fluctuations over the observed period.
The skewness value of 0.10 suggests that the distribution of
the data is approximately symmetric, with a slight positive
skew. This means that the dataset has a minor tendency for
values to be concentrated on the left side, with a few larger
observations on the right. Meanwhile, the kurtosis value of
-0.68 indicates that the distribution is platykurtic, which
implies that the data have lighter tails and a flatter peak
compared to a normal distribution. In other words, extreme
values (outliers) are less likely to occur within this dataset.
Overall, the descriptive statistics of ICBP’s stock prices
demonstrate a dataset with moderate volatility, near-
symmetrical distribution, and relatively fewer outliers,
providing a stable basis for further time series modeling
and volatility analysis.
The descriptive statistics of the return series, based on 236
observations, provide valuable insights into the distribution
and characteristics of the data. The mean return is
0.00027905, indicating a very small average daily return,
which is typical for financial time series data, particularly
when using daily stock returns. Although the average return
is positive, its magnitude suggests minimal daily gains over
the observed period.
The standard deviation of 0.01682765 reflects the level of
volatility present in the return series. This indicates a
moderate degree of fluctuation around the mean, which is
consistent with the nature of stock returns in emerging
markets, where volatility tends to be higher compared to
developed markets. The variance is 0.00028317, further
confirming the variability in returns.
The skewness value of -0.20530410 suggests a slight
negative skew in the distribution of returns. This implies
that the return series exhibits a tendency for more extreme
negative returns compared to positive ones, although the
skewness is relatively small and close to zero, indicating
near-symmetry.
The kurtosis value of 1.67403100 is lower than the normal
distribution kurtosis of 3, indicating a platykurtic
distribution. This means the return series has lighter tails
and a flatter peak than a normal distribution, implying
fewer occurrences of extreme returns or outliers.
Overall, the descriptive statistics indicate that the return
series of the stock exhibits low average returns, moderate
volatility, slight negative skewness, and lower tail risk,
suggesting the data is relatively stable but still exhibits
characteristics typical of financial time series.
The first step in ARIMA modeling is model identification
to determine the most appropriate model for the stock
return data. This identification process can be observed
through the ACF and PACF correlograms. The preliminary
results suggest that the stock return data follows an
ARIMA(2,0,1) with zero mean model.
After obtaining the preliminary model, the next step is
parameter estimation. The results of the parameter
estimation are presented in Table 2.
Table 2. ARIMA Parameter amd Signicance Test
Parameter
Estimation
p-value
Result
AR1
-0.6975
0.000733
Significant
AR2
-0.1859
0.004741
Significant
MA1
0.5507
0.006419
Significant
All parameters in the ARIMA model are statistically
significant, as each p-value is below the 5% significance
level (α = 0.05). AR(1) and AR(2) terms show negative
coefficients, indicating inverse relationships with past
observations. The MA(1) term is positive, reflecting a
direct impact of past error terms on the current value. This
suggests the ARIMA model has well-defined dynamics and
can capture the time series' autocorrelation structure
effectively.
Table 3. ARCH LM and Box-Ljung Test Results for
ARIMA Model Residuals
Test
p-value
Result
ARCH LM Test
0.04697
There is evidence of
ARCH effects.
Box-Ljung Test
on Residuals
0.00706
Significant
autocorrelation in
squared residuals,
indicating
heteroskedasticity
Both the ARCH LM Test and the Box-Ljung Test on the
squared residuals indicate the presence of
heteroskedasticity in the residuals of the ARIMA model.
The ARCH LM Test yields a p-value of 0.04697, leading to
the rejection of the null hypothesis of no ARCH effects at
the 5% significance level.
Similarly, the Box-Ljung Test on squared residuals results
in a p-value of 0.007065, confirming the presence of
significant autocorrelation in the variance of residuals.
These findings strongly suggest that the residuals exhibit
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time-varying volatility, which is not captured by the
ARIMA model alone. Therefore, it is appropriate to fit a
GARCH or GARCH-M model to account for the changing
variance over time and to better model the volatility
dynamics of the time series. Then, we have to proceed with
estimating a GARCH(1,1) or GARCH-M model on the
residuals or return.
Table 4 shows a summary of the ARIMA-GARCH-M
estimation results for ICBP.JK returns in 2024. And Table
5 shows the assumptions diagnostics of the ARIMA-
GARCH-M for ICBP.JK returns in 2024.
Table 4. ARIMA(2,0,1)-GARCH(1,1)-M Model
Estimation
Parameter
Estimation
p-value
Result
0.000051
0.00947
Significant
0.469173
0.04918
Significant
0.440526
0.02823
Significant
Table 5. Diagnostic Test Results for ARIMA-GARCH-
M Model
Test
p-value
Result
Ljung-Box Test on
Standardized
Residuals (Lag 14)
0.4623
No serial correlation
Ljung-Box Test on
Standardized
Squared Residuals
(Lag 9)
0.2804
No ARCH effects
remaining
ARCH LM Test
0.1030
No further ARCH
effects
The returns are modeled with an ARIMA(2,0,1) process.
Both AR(1), AR(2), and MA(1) terms are statistically
significant (p < 0.01), indicating a strong autoregressive
and moving average structure in return dynamics. Volatility
is captured by a standard GARCH(1,1) model, with both
the ARCH (α1) and GARCH (β1) terms being significant (p
< 0.05). This shows that volatility clustering exists in ICBP
returns, where large changes are likely followed by large
changes. The coefficient of the ARCH-M term is negative
and statistically significant (p = 0.0934). This implies that
the conditional variance has a significant direct impact on
expected returns in this model.
Ljung-Box Tests on both residuals and squared residuals
indicate no significant autocorrelation remaining in the
model (p-values > 0.05). This suggests that the ARIMA-
GARCH-M specification has successfully captured the
dynamics in the mean and variance. ARCH LM Test shows
no remaining ARCH effects at lags up to 7 (p = 0.1030),
confirming that conditional heteroskedasticity has been
adequately modeled.
The sum of α1 (0.469) and β1 (0.441) is approximately 0.91,
implying high volatility persistence. This means volatility
shocks take time to dissipate, which is consistent with
financial market behavior.
The ARIMA(2,0,1)-GARCH(1,1)-M model is an
appropriate specification for modeling and forecasting the
volatility of ICBP returns in 2024. Despite the ARCH-M
term being insignificant, the GARCH component captures
volatility clustering effectively. Residual diagnostics
confirm no remaining autocorrelation or ARCH effects.
The high volatility persistence suggests that risk
management strategies should consider the prolonged
impact of market shocks. The model can be used for
forecasting volatility, Value-at-Risk (VaR) calculations, or
as an input in portfolio optimization.
After successfully estimating the ARIMA(2,1,1)-
GARCH(1,1)-M model for the ICBP stock returns, we
proceed with generating out-of-sample forecasts for the
next 10 trading days. The forecast includes both the
expected returns (mean equation forecast) and the
conditional volatility (variance equation forecast), which
represents the predicted risk level or uncertainty in the
market.
The forecasted values are essential for investors and risk
managers to understand potential price movements and
volatility dynamics in the short-term future. Higher
conditional volatility signals increased risk, while the
expected returns give insight into the potential direction of
asset prices. Table 6 and Figure 3 are the 10-day ahead
forecast, showing the predicted returns and their
corresponding conditional standard deviations (volatility).
Table 6. Forecast: Expected Returns & Conditional
Volatility
Time
Forecasted Return
Forecasted Volatility
T+1
0.0033999
0.01330
T+2
-0.0005439
0.01456
T+3
0.0008605
0.01561
T+4
0.0005958
0.01651
T+5
0.0004635
0.01729
T+6
0.0006126
0.01796
T+7
0.0005031
0.01856
T+8
0.0005424
0.01909
T+9
0.0005215
0.01955
T+10
0.0005171
0.01997
Fig 3. Forecast: Expected Returns
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The expected returns are relatively close to zero over the
next 10 days. The first forecasted return (T+1) shows a
slight positive return of 0.0034, followed by fluctuations
around zero. These results are typical for financial return
series, which often display low and stationary expected
returns in the short term. The conditional volatility is
increasing over time. It starts at 0.01330 (or 1.33%) at T+1.
It rises to 0.01997 (or 1.997%) by T+10. This gradual
increase in volatility suggests the presence of volatility
clustering, a common feature in financial time series, where
past high volatility tends to lead to future high volatility.
The GARCH(1,1) process forecasts that volatility will
increase slightly in the forecast horizon before stabilizing,
consistent with the persistence of volatility estimated in
your model.
Higher forecasted volatility implies greater risk in future
periods, especially towards the end of the 10-day horizon.
The mean forecasted returns are small, suggesting neutral
or slightly positive expectations, but the increasing
volatility warns of heightened uncertainty.
The 10-step ahead GARCH-M forecast for ICBP returns
shows relatively stable and low expected returns, with an
increasing pattern of conditional volatility. This indicates
rising risk levels over the forecast horizon, potentially due
to volatility clustering. Investors should remain cautious, as
higher predicted volatility may lead to larger fluctuations in
returns.
V. CONCLUSION
The modeling and analysis of PT Indofood CBP Sukses
Makmur Tbk (ICBP.JK) returns for the year 2024, using the
ARIMA-GARCH-M approach, provide several insightful
findings regarding the stock's return behavior and volatility
dynamics. The best-fitted mean equation was identified as
an ARIMA(2,0,1), indicating that the return series is
influenced by two autoregressive terms and one moving
average component. All estimated parameters for the AR
and MA terms were found to be statistically significant,
suggesting the presence of autocorrelation in the return
series. This highlights the predictive power of past returns
and error terms in explaining the current return movements
of ICBP.
In terms of volatility modeling, the GARCH(1,1)-M model
with a Student-t distribution was selected due to its ability to
capture the heavy-tailed nature of financial return
distributions. The model estimates show that both ARCH
(α1 = 0.4691) and GARCH (β1 = 0.4405) parameters are
positive and statistically significant under conventional
standard errors. The sum of these parameters, approximately
0.91, indicates high volatility persistence, meaning that
shocks to the volatility process have long-lasting effects.
However, the ARCH-in-Mean (ARCH-M) parameter was
not statistically significant, suggesting that volatility does
not directly influence the expected return of ICBP during the
analyzed period.
The residual diagnostics, including Ljung-Box and ARCH
LM tests, confirm that the standardized residuals and
squared residuals exhibit no significant autocorrelation or
remaining ARCH effects, implying that the model
adequately captures the volatility clustering present in the
data. Additionally, the 10-day-ahead volatility forecast
indicates a gradual increase in conditional standard
deviation, reflecting a potential rise in market uncertainty.
Overall, the ARIMA-GARCH-M model demonstrates its
effectiveness in modeling and forecasting ICBP stock
returns and volatility. These findings provide valuable
insights for investors and risk managers by offering a robust
tool to anticipate future volatility and manage investment
risks in the Indonesian stock market, particularly in the
consumer goods sector.
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