The Gumbel Copula Method for Estimating Value at Risk: Evidence from Telecommunication Stocks in Indonesia during the COVID-19 Pandemic

Abstract

The COVID-19 pandemic has had a substantial and far-reaching impact on global economic growth, extending its effects to Indonesia as well. Various sectors have witnessed a decline in stock returns as a consequence. Interestingly, the telecommunications sector has bucked this trend by experiencing an increase in stock returns, defying the negative implications of the pandemic. The relationship between returns and risk is inherently intertwined, necessitating a meticulous risk assessment. In response to this need, the Value at Risk (VaR) method has emerged as a rapidly growing and widely adopted risk measurement tool. Among the techniques employed for VaR estimation, the Monte Carlo simulation stands out due to its flexibility and comprehensiveness in accommodating factors such as time variance, volatility, returns, fat tails, and extreme scenarios. The Gumbel copula method, known for its heightened sensitivity to high-risk events, is utilized for VaR estimation on abnormal stock returns. This study aims to quantify the Value at Risk by leveraging the estimated Gumbel copula parameter for the return on the shares of PT. Indosat Ooredoo Hutchison Tbk, and PT. Smartfren Telecom Tbk during the COVID-19 pandemic. At a 90% confidence level, the VaR is determined to be 7.6%. Notably, this estimate closely aligns with the actual values, underscoring the reliability of the VaR estimation conducted using the Gumbel copula parameter estimator. Therefore, this model serves as a robust reference, particularly suitable when dealing with investment return data that deviate from the normal distribution, while considering the unique stock return characteristics within each dataset.

1. Introduction

The COVID-19 pandemic, stemming from the SARS-CoV-2 virus, emerged towards the end of 2019 and has persisted to the present day (Tinungki et al. 2022b). Its initial identification traces back to Wuhan, China, and its rapid proliferation prompted the World Health Organization to declare it a pandemic on 11 March 2020. This global pandemic status underscored the virus’s swift transmission, leaving virtually no country invulnerable (Liu et al. 2020). In response, numerous nations imposed quarantine or lockdown measures to curtail the virus’s spread and safeguard their populations. The Indonesian government, for instance, implemented Large Scale Social Restrictions as a strategic maneuver to mitigate and disrupt the COVID-19 transmission chain (Tinungki et al. 2022a). The repercussions of COVID-19 have been profound, instigating an economic crisis characterized by workforce reductions, layoffs, surging unemployment rates, and the challenge of meeting daily necessities (Hartono and Raya 2022). This pandemic has wielded substantial influence on global economic growth, with Indonesia experiencing currency depreciation and various sectors, including tourism, hospitality, aviation, food and beverages, and retail, grappling with declines. Concurrently, stock prices in several industries have plummeted. Intriguingly, amidst this widespread economic downturn, the telecommunications sector has exhibited resilience by recording increases in stock valuations (Hartono and Robiyanto 2023).

In a study conducted by Tambunan (2020), it is highlighted that investments in the telecommunications sector exhibited a notable price surge during the pandemic, suggesting the viability of investing in telecommunications stocks. This phenomenon can be attributed to the widespread implementation of preventive measures by companies and educational institutions to curtail virus transmission, emphasizing the importance of the telecommunications sector. Various policies enacted by the Indonesian government during the pandemic, such as work-from-home (WFH) arrangements for office employees and learn-from-home (LFH) programs for students at all levels, underscored the critical role of robust internet connectivity in facilitating remote work and learning. Notably, the demand for internet access and data quotas saw a substantial upswing compared to pre-COVID-19 levels (Robiyanto and Yunitaria 2022), subsequently driving up stock returns. Against this backdrop, we conducted an analysis of Indonesian telecommunications stocks, particularly PT. Indosat Ooredoo Hutchison Tbk and PT. Smartfren Telecom Tbk. It is worth noting that returns are intrinsically linked to risk, following a unidirectional and linear relationship, meaning higher expected returns entail greater associated risks (Hernandez et al. 2015). Consequently, assessing and quantifying this risk becomes imperative.

One of the increasingly popular and widely adopted tools for assessing risk in contemporary finance is Value at Risk (VaR), notably introduced by J.P. Morgan in 1994 (Jorion 1996). VaR represents the maximum anticipated loss within a specific time frame at a given confidence level (Amin et al. 2018). It has evolved into the standard internal risk assessment model, following the guidelines set by the Bank for International Settlement through the Basel Committee on Banking Supervision (BCBS) in BASEL II, and is extensively utilized by financial analysts to compute market risk for assets or portfolios (Najiha et al. 2022). The calculation of VaR primarily relies on three main methodologies: the Covariance–Variance method, Monte Carlo simulations, and historical simulations. In this research, we employ the Monte Carlo simulation method for VaR estimation. The Monte Carlo simulation method offers greater versatility, enabling the incorporation of time variances, volatilities, returns, fat tails, and extreme scenarios, making it particularly suitable for nonlinear exposures (Glasserman et al. 2000; Harsoyo 2017). Its widespread adoption is attributed to its capacity to assess diverse combinations of exposures, particularly stocks, and associated risks (Hartono et al. 2023).

When analyzing stock market returns, traditional VaR estimation approaches like variance-based methods, Monte Carlo simulations, and historical simulations prove effective when returns demonstrate stability and independence. However, when dealing with interdependencies among stock market returns driven by complex dynamics and abnormal patterns, it becomes challenging to establish a multivariate distribution for multiple sequences (Jondeau and Rockinger 2006). Consequently, the copula approach was developed for VaR estimation. Copulas, introduced by Sklar in 1959, serve as functions that effectively combine or ‘link’ multivariate distribution functions with lower-dimensional marginal distribution functions, which are typically one-dimensional (Harsoyo 2017). Copulas are widely employed in modeling joint distributions as they eliminate the need for the assumption of mutual normality and break down the n-dimensional joint distribution into marginal n-distributions and copula functions, effectively uniting them. The copula method presents a notable advantage over previous techniques as it does not rely on the assumption of a normal distribution and can capture tail dependencies among individual variables.

Previous research has explored various approaches to VaR estimation. Abad and Benito (2013) reported that their estimation results indicated the most accurate estimates were derived from a parametric model with the conditional variance estimated using an asymmetric GARCH model based on the Student’s t-distribution of returns. This finding underscores the effectiveness of parametric models in achieving successful VaR measurements when the conditional variance is accurately estimated. So and Yu (2006) investigated the RiskMetrics method and two long-memory GARCH models in VaR estimation. The specific findings highlighted the importance of considering models with fat-tailed errors when estimating VaR, even when most return series exhibit fat-tailed distributions and long memory properties. Asymmetric behavior was also detected in stock market data, where the t-error model outperformed the normal-error model in estimating 1% VaR in long positions but not in short positions. Such asymmetry was not observed in exchange rate data. Furthermore, Fan and Gu (2003) supported the proposed new time-dependent semiparametric approach for VaR estimation.

In their study, Byun and Song (2021) conducted VaR estimation using elliptical, vine, and hierarchical copulas to determine the suitable copula function for various dependency structures among asset return distributions within portfolio VaR calculation. Through simulation studies with various dependency structures and real-data analysis, the Clayton hierarchical copula demonstrated the best performance in VaR calculation for a portfolio consisting of four assets. Meanwhile, Geenens and Dunn (2022) employed a non-parametric copula approach for VaR estimation. Their findings concluded that non-parametric approaches are highly relevant, as traditionally used parametric distributions have proven to be less robust and flexible for most observed equity return datasets in practice.

Hence, this study conducts further experiments on VaR estimation using the Gumbel copula. The Gumbel copula is a member of the Archimedean family of copulas, with each sub-copula within this family having a distinct generator. Introduced by Gumbel in 1960, the Gumbel copula family is renowned for its exceptional sensitivity to high-risk events and its capability to identify upper-tail dependencies. In this research, Value at Risk (VaR) calculations were performed, utilizing the estimated Gumbel copula parameters applied to return data from PT. Indosat Ooredoo Hutchison Tbk, and PT. Smartfren Telecom Tbk., in crisis condition. The findings of this study indicate that, at a 90% confidence level, the VaR is determined to be 7.6%. Significantly, this estimation closely aligns with the actual values, underscoring the reliability of the VaR estimation conducted using the Gumbel copula parameter estimator. Consequently, this model offers a robust reference, particularly well-suited for handling investment return data that deviate from the normal distribution, while taking into account the unique characteristics of stock returns within each dataset.

2. Literature Review

2.1. Archimedean Copulas

The concept of copulas was originally introduced to the realm of mathematics and statistics by Abe Sklar and gained prominence in 1959 with the Sklar Theorem. Copula functions serve as connectors between univariate marginal distributions, transforming them into bivariate or multivariate distributions (Osmetti 2012). The copula approach offers several advantages, including the ability to handle dependencies among variables with non-normal distributions, provide richer information about the structure of dependence, and even infer the marginal distributions of dependent or unknown variables. The copula family encompasses various types, among them the Archimedean copulas. Archimedean copulas find applications in reliability theory and numerous other fields due to their capacity to capture a wide spectrum of dependence patterns and their mathematical tractability (Ayantobo et al. 2019; Zhang and Yan 2022). The Archimedean copula is defined by a single-valued function known as the φ generating function. Within the Archimedean copula family, various types exist, including the Clayton copula, the Gumbel copula, and the Frank copula. To derive the parameters of the Archimedean copula family, correlation values represented by Kendall’s tau can be employed (Susam and Hudaverdi Ucer 2018). Specifically, in the case of Archimedean copulas, the Kendall’s tau correlation value can be computed using the following equation:

$$\tau =1+4\underset{0}{\overset{1}{\int }}\frac{\phi \left(t\right)}{{\phi }^{\prime }\left(t\right)}dt$$

(1)

where, $\phi \left(t\right)$ is the generating function of the Archimedean copula family (Mai and Scherer 2014).

2.2. Gumbel Copula

The Gumbel copula, initially introduced by Gumbel in 1960, has since evolved into what is now recognized as the Gumbel copula family. This copula offers the distinctive advantage of characterizing the tail behavior associated with upper-level dependence. The Gumbel copula finds its application in modeling asymmetric relationships within the dataset (Modiri et al. 2018). The general formulations for the Gumbel copula are as follows (Sreekumar et al. 2018):

$$\mathrm{C}(u_1,{\dots ,u}_d)=\exp \left(-{\left[{\left(-\ln u_1\right)}^{\theta }+\cdots +{\left(-\ln u_d\right)}^{\theta }\right]}^{\frac{1}{\theta }}\right)$$

(2)

where these functions are derived from the generating function because the Gumbel copula belongs to the broader family of Archimedean copulas. The generating functions associated with the Gumbel copula are:

$${\phi }_{\theta }\left(t\right)={(-\ln t)}^{\theta }$$

(3)

A Gumbel copula exhibits a pronounced tail relationship at the upper extreme, as illustrated in Figure 1.

This observation reveals that the connection between independent variables and dependent variables is evident only when these independent variables attain a high level. Conversely, when independent variables are at a low level, the relationship between them is considerably weaker and, at times, may even be nonexistent (Najiha et al. 2022).

2.3. Value at Risk

The concept of Value at Risk (VaR) serves as a fundamental tool for risk assessment in the domain of risk management. In essence, VaR seeks to address the question of “to what extent (expressed as a percentage or a specific monetary value) an investor might incur losses within the investment duration $t$ while maintaining an error margin of “$\alpha$”. This inquiry highlights three critical variables: the extent of loss, the time span, and the magnitude of the error margin (Duffie and Pan 1997).

• Confidence Level

The determination of the confidence level in VaR calculations hinges on its intended application, a pivotal factor, as it denotes the company’s willingness to assume the risk of exceeding the VaR-defined loss threshold. A higher willingness to take risks corresponds to a greater degree of confidence in allocating capital to mitigate potential losses (Jorion 1996).

2.

Time Period

In financial institutions like banks, VaR is typically computed over intervals of one day, one week (equivalent to five business days), or two weeks (comprising 10 business days). In contrast, enterprises with tangible assets, such as investors in real estate and property firms, often employ lengthier time frames, ranging from one month (20 days) to four months, or even up to a year, to monitor their risk exposure (Elliott 2019).

2.4. Value at Risk with Monte Carlo Simulation

The utilization of the Monte Carlo simulation technique for risk assessment was initially introduced by Boyle in 1977. Monte Carlo simulations encompass various algorithms for estimating the Value at Risk (VaR) of both individual assets and portfolios. However, the fundamental principle involves simulation through the generation of random numbers aligned with the characteristics of the data to be simulated, subsequently facilitating VaR estimation. The Monte Carlo simulation method serves as a simulation approach that leverages random numbers to address problems associated with uncertain states where conventional mathematical evaluation proves unfeasible. A Monte Carlo simulation attests to the robust size and power properties of this technique (Hurlin et al. 2017). The foundation of Monte Carlo simulations rests on probabilistic experiments conducted with random sampling. This simulation comprises two key components.

• Generation of random numbers;

random numbers play a pivotal role in simulation development and can be generated using the standard “randomize” function. This “randomize” function is designed to produce random numbers falling within the range of zero to one or greater.

2.

Generation of random variables;

the generation of these random variables employs the inverse transformation method, contingent upon the distribution pattern observed in sample data. Consequently, the sample data’s distribution should accurately represent a statistically undifferentiated distribution.

3. Results

This research exclusively utilizes secondary data in the form of daily stock investment closing prices denominated in Indonesian Rupiah. These data were meticulously extracted from http://finance.yahoo.com, accessed on 30 December 2022, and specifically consist of historical records pertaining to the performance of PT. Indosat Ooredoo Hutchison Tbk, and PT. Smartfren Telecom Tbk, spanning the timeframe between 11 March 2019 and 10 March 2020. To facilitate the comprehensive analysis of this data, the research was conducted employing RStudio software, specifically Version 3.6.3, encompassing a total dataset comprising 257 data points.

The initial phase involves the implementation of a normality assessment, a crucial step in addressing potential price volatility that may pose risks to investors. This examination aims to ascertain whether there is a risk of substantial price decline that could adversely affect investors. The assessment of normality entails an evaluation of the return data distribution for each stock, often facilitated by examining histograms. Furthermore, this study incorporates the Kolmogorov–Smirnov test to gain deeper insights into the distribution patterns of the analyzed stocks. The Kolmogorov–Smirnov test is a powerful tool employed to determine the most appropriate data distribution for a given dataset. It is recognized for its efficiency in yielding precise outcomes (Hersugondo et al. 2022). In this context, the Kolmogorov–Smirnov test was employed to deduce the distribution characteristics of the return variable for both PT. Indosat Ooredoo Hutchison Tbk. and PT. Smartfren Telecom Tbk (please see Figure 2).

Table 1. Results of the Kolmogorov-Smirnov test on the return on shares of PT. Indosat Ooredoo Hutchison Tbk., and PT. Smartfren Telecom Tbk.

Stock	Hypothesis	${\mathit{D}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}}$	p-Value	Decision
PT. Indosat Ooredoo Hutchison Tbk	${\mathrm{H}}_0$: returns are normally distributed	0.097276	0.01544	Reject ${\mathrm{H}}_0$
	${\mathrm{H}}_1$: returns are not normally distributed
PT. Smartfren Telecom Tbk	${\mathrm{H}}_0$: returns are normally distributed	0.11284	0.002875	Reject ${\mathrm{H}}_0$
	${\mathrm{H}}_1$: returns are not normally distributed

presents the outcomes of the $D_{calculate}$ values for the returns on shares of PT. Indosat Ooredoo Hutchison Tbk, revealing values exceeding the Kolmogorov–Smirnov Table benchmark of 0.097276. Simultaneously, the p-value is notably lower than the significance level $\alpha =0.05$. These combined results lead to the rejection of ${\mathrm{H}}_0$, signifying that the return variable for PT. Indosat Ooredoo Hutchison Tbk does not adhere to a normal distribution.

Similarly, Table 1 displays the $D_{calculate}$ value for the returns on PT. Smartfren Telecom Tbk, surpassing the Kolmogorov–Smirnov Table threshold of 0.11284. Furthermore, the corresponding p-value is significantly smaller than $\alpha =0.05$. Consequently, ${\mathrm{H}}_0$ is rejected for PT. Smartfren Telecom Tbk, indicating that its return variable deviates from a normal distribution.

The subsequent phase entails an autocorrelation examination. This test aims to elucidate the interplay between residual observations of one entity and those of another. The presence of robust autocorrelation can establish connections between seemingly unrelated variables, ultimately aiding in the identification of autocorrelation patterns or their absence.

As depicted in Figure 3, which illustrates the cutoff for each stock at lag 0, it is evident that there is no discernible autocorrelation effect observed in the analyzed stock. To further scrutinize the existence of autocorrelation effects, the Ljung–Box test was conducted.

Table 2. Results of the Ljung–Box test on the returns on shares of PT. Indosat Ooredoo Hutchison Tbk., and PT. Smartfren Telecom Tbk.

Stock	Hypothesis	Ljung Box		$\mathit{\alpha }$	Decision
		Lag	p-Value
PT. Indosat Ooredoo Hutchison Tbk	${\mathrm{H}}_0$: returns are not autocorrelated	1	0.1232	0.05	Accept ${\mathrm{H}}_0$
		5	0.1022
	${\mathrm{H}}_1$: returns are autocorrelated	10	0.1195
		15	0.1535
		20	0.2142
PT. Smartfren Telecom Tbk	${\mathrm{H}}_0$: returns are not autocorrelated	1	0.4939	0.05	Accept ${\mathrm{H}}_0$
		5	0.7534
	${\mathrm{H}}_1$: returns are autocorrelated	10	0.9276
		15	0.9319
		20	0.9833

presents the results of the statistical analysis, indicating that each $p\text{-}value$ exceeds the significance level $\alpha =0.05$. Consequently, the null hypothesis (${\mathrm{H}}_0$) is accepted, signifying that the variable “return on shares” for both PT. Indosat Ooredoo Hutchison Tbk and PT. Smartfren Telecom Tbk exhibits no significant correlation (Tinungki 2019).

The subsequent step involves conducting a heteroskedasticity test to assess the presence of variance inequality in the observed errors. In this study, the Lagrange multiplier (LM) test, commonly known as the ARCH LM test, was executed, aiming to identify potential heteroskedasticity elements.

The findings of the ARCH-LM test are presented in

Table 3. Results of the ARCH LM test on the return on shares of the PT. Indosat Ooredoo Hutchison Tbk.

Stock	Hypothesis	p-Value	$\mathit{\alpha }$	Decision
PT. Indosat Ooredoo Hutchison Tbk	${\mathrm{H}}_0$: there is no ARCH/GARCH effect on the return data	0.9008	0.05	Accept ${\mathrm{H}}_0$
	${\mathrm{H}}_1$: there is an ARCH/GARCH effect on the return data
PT. Smartfren Telecom Tbk	${\mathrm{H}}_0$: there is no ARCH/GARCH effect on the return data	0.9008	0.05	Accept ${\mathrm{H}}_0$
	${\mathrm{H}}_1$: there is an ARCH/GARCH effect on the return data

, revealing that the $p\text{-}value$ for each stock return surpasses the significance level α. Consequently, we accept the null hypothesis (${\mathrm{H}}_0$), signifying the absence of heteroskedasticity effects in the variable “return on shares” for both PT. Indosat Ooredoo Hutchison Ttbk and PT. Smartfren Telecom Tbk.

Following a series of examinations, the subsequent step involves parameter estimation. The parameter of interest, denoted as $\widehat{\theta }$, is determined through the utilization of Kendall’s tau correlation method. Initially, this involves calculating the correlation coefficient, denoted as $\tau$, for the return variables associated with the daily closing prices of stock investments in PT. Indosat Ooredoo Hutchison Tbk and PT. Smartfren Telecom Tbk, as follows:

$$\tau =0.09037397$$

(4)

after which, the acquired correlation coefficient value is employed in the estimation of the parameter $\widehat{\theta }$, as delineated below:

$$\widehat{\theta }=\frac{1}{1-\tau }=\frac{1}{1-0.09037397}=1.099353$$

(5)

and subsequently, the estimated parameter $\widehat{\theta }$ is incorporated into the Gumbel copula model in the following manner:

$$C(u_1,u_2)=\mathit{exp}\left(-{\left[{\left(-\mathit{ln}{u}_1\right)}^{1.2678}+{\left(-\mathit{ln}{u}_2\right)}^{1.099353}\right]}^{\frac{1}{1.099353}}\right)$$

(6)

$$C(u_1,u_2)=\mathit{exp}\left(-{\left[{\left(-\mathit{ln}{u}_1\right)}^{1.2678}+{\left(-\mathit{ln}{u}_2\right)}^{1.099353}\right]}^{0.9096259}\right)$$

(7)

The underlying pattern represented by the Gumbel copula demonstrates the occurrence of extremes at high values and a correlation between the two variables when both exhibit high values. Conversely, as the observation values on the variable decrease, the association weakens, as the Gumbel copula primarily focuses on relationships when there is a concentration at the upper end. A Gumbel copula simulation involving a dataset of $n=1000$ reveals that as stock returns increase, specifically the returns of PT. Indosat Ooredoo Hutchison Tbk, there is a corresponding rise in the returns of PT. Smartfren Telecom Tbk.

With the Gumbel copula model in place, the final phase of this study involves the Value at Risk (VaR) calculation using the Monte Carlo simulation method, enabling the estimation of risk. VaR is computed at a 90% confidence level. It is important to note that VaR values vary across simulations due to the randomness of the data generated. However, these values generally remain consistent since they are derived from the same model and parameters. To enhance precision, conducting multiple simulations and calculating the average value is a viable approach to mitigate this variability.

Table 4. Estimated VaR returns on shares of PT. Indosat Ooredoo Hutchison Tbk and PT. Smartfren Telecom Tbk.

VaR	$\mathit{\alpha }$	Confidence Level
−0.076 1	0.05	90%

¹ VaR estimates that are negative indicate the value of risk.

presents the Value at Risk (VaR) outcomes for the returns on shares of PT. Indosat Ooredoo Hutchison Tbk and PT. Smartfren Telecom Tbk, which amount to 0.076 at a 90% confidence level. This confidence level signifies a 10% probability that the losses will fall below the reported VaR.

In Figure 4, the chart displays the actual data alongside the Value at Risk (VaR) projections for the subsequent 80 days. This graphical representation reveals a strikingly similar pattern for each data point, indicating that the VaR estimations, conducted utilizing the Gumbel copula parameter estimator, can serve as a reliable reference model, particularly when dealing with investment return data that does not conform to a normal distribution. This conclusion holds while considering the unique stock return patterns within each dataset employed.

4. Conclusions

The results generated through Value at Risk (VaR) calculations utilizing the estimated parameters of the Gumbel copula provide valuable insights for practical applications. Let us consider an investor who allocates IDR 1,000,000.00 into a portfolio consisting of PT. Indosat Ooredoo Hutchison Tbk, and PT. Smartfren Telecom Tbk. At a 90% confidence level, the VaR is computed as −0.076. This implies that, with a 90% confidence level, the maximum potential loss for the investor in the analyzed stock portfolio amounts to IDR 76,000. When this VaR estimate is juxtaposed with actual values, a noticeable resemblance in patterns becomes evident. This observation underscores the reliability of VaR estimates obtained through the Gumbel copula parameter estimator, particularly when dealing with investment return data that deviate from a normal distribution. It is worth noting that this conclusion remains applicable even when accounting for the distinct stock return patterns within each dataset employed. However, it is essential to acknowledge the limitations of this study, primarily focused on estimations using the Gumbel copula method. Further research avenues may involve exploring asymmetric copula models and empirical copulas to enhance the depth of analysis in this domain (Liebscher 2008).