Estimation of Tail Value at Risk for Bivariate Portfolio using Gumbel Copula

Fransiska Fransiska; Evy Sulistianingsih; Neva Satyahadewi

doi:10.31764/jtam.v9i2.29952

Estimation of Tail Value at Risk for Bivariate Portfolio using Gumbel Copula

Fransiska Fransiska, Evy Sulistianingsih, Neva Satyahadewi

Abstract

Investing in the stock market involves complex risks, especially under extreme and unpredictable conditions. While Value at Risk (VaR) is a widely used risk measure, it has limitations in capturing tail-end risks. This study employs Tail Value at Risk (TVaR) using the Gumbel Copula approach, which effectively models upper-tail dependence in return distributions—an aspect often overlooked by traditional linear correlation methods. This quantitative research utilizes copula-based Monte Carlo simulation. The data consists of daily closing prices of PT Adaro Energy Indonesia Tbk (ADRO) and PT Indo Tambangraya Megah Tbk (ITMG) from July 3, 2023, to July 30, 2024. The analysis begins with return calculation and tests for autocorrelation and homoskedasticity. The Gumbel Copula parameter is estimated using Kendall’s Tau, resulting in a dependence parameter of 1.7791. Based on this, 1,000 simulations are conducted to generate new return data that reflect extreme dependencies between the two stocks. An optimal portfolio is constructed using the Mean-Variance Efficient Portfolio (MVEP) method, assigning weights of 31.61% to ADRO and 68.39% to ITMG. TVaR is then calculated from the simulated portfolio returns. The results show increasing TVaR values at higher confidence levels: 2.08%, 2.64%, 3.14%, and 4.11% for 80%, 90%, 95%, and 99%, respectively. These findings demonstrate that TVaR provides more accurate insights into potential losses in extreme market conditions, supporting investors in developing more informed and risk-sensitive portfolio strategies.

Keywords

Archimedean; Kendall’s Tau; Monte Carlo; Simulation.

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References

Ardhitha, T., Sulistianingsih, E., & Satyahadewi, N. (2023). Value at Risk Analysis On Blue Chip Stocks Portfolio With Gaussian Copula. BAREKENG: Jurnal Ilmu Matematika Dan Terapan, 17(3), 1739–1748. https://doi.org/10.30598/barekengvol17iss3pp1739-1748

Badan Koordinasi Penanaman Modal. (2024). https://www.bkpm.go.id/id/info/realisasi-investasi/2024

Danielsson, J. (2011). Financial Risk Forecasting. Great Britain: British Library. http://eprints.lse.ac.uk/id/eprint/37380

Daulay, S. N. R., Halim, N. A., & Hidayana, R. A. (2022). Investment Portfolio Optimization With Mean-Variance Investment Portfolio Optimization Model Without Risk Free-Assets. International Journal of Quantitative Research and Modeling, 3(4), 113–117. https://doi.org/10.46336/ijqrm.v3i4.360

Dewick, P. R., & Liu, S. (2022). Copula Modelling to Analyse Financial Data. Journal of Risk and Financial Management, 15(3). https://doi.org/10.3390/jrfm15030104

Durante, F., & Sempi, C. (2016). Principles of Copula Theory. Chemical Rubber Company Press. https://www.routledge.com/PrinciplesofCopulaTheory/DuranteSempi/p/book/9781032098470

Duy Bui, L., Chi Le, T., Ngoc Quang, A. H., & Wong, W.-K. (2021). Determinants of the possibilities by investors’ risk-taking: Empirical evidence from Vietnam. Cogent Economics & Finance, 9(1), 1917106. https://doi.org/10.1080/23322039.2021.1917106

Handini, J. A., Maruddani, D. A. I., & Safitri, D. (2019). Frank copula on value at risk (VaR) of the construction of bivariate portfolio (Case Study: stocks of companies awarded with the IDX top ten blue with stock period of 20 october 2014 to 28 february 2018). Journal of Physics: Conference Series, 1217(1), 012078. https://doi.org/10.1088/1742-6596/1217/1/012078

Hiremath, G. S., & Kumari, J. (2014). Stock Returns Predictability and The Adaptive Market Hypothesis in Emerging Markets: Evidence From India. SpringerPlus, 3(1), 1–14. https://doi.org/10.1186/2193-1801-3-428

Hong, Y. (1997). One-sided testing for conditional heteroskedasticity in time series models. Journal of Time Series Analysis, 18(3), 253–277. https://doi.org/10.1111/1467-9892.00049

Hotta, L. K., Lucas, E. C., & Palaro, H. P. (2008). Estimation of VaR Using Copula and Extreme Value Theory. Multinational Finance Journal, 12(3/4), 205–218. https://doi.org/10.17578/12-3/4-3

Irsan, M. Y. T., & Sirait, M. L. (2020). Value at Risk (VaR) & Taile Value at Ris (TVaR): Companies Listed in LQ45. International Journal of Economics, Commerce and Management, 8(12), 560–568. https://ijecm.co.uk/wp-content/uploads/2020/12/81234.pdf

Jorion, P. (2007). Value at Risk: The New Bemchmark in Controlling Market Risk. Mcgraw-Hill.

Klugman, S. A., Panjer, H. H., & Willmot, G. E. (2019). From Data to Decisions, 5th Edition (Fifth). John Wiley & Sons, Inc. https://www.wiley.com/en-hk/Loss+Models%3A+From+Data+to+Decisions%2C+5th+Edition-p-9781119523789

Lu, S. (2023). Empirical Analysis of Value at Risk (VaR) of Stock Portfolio Based on Python. In MSEA 2022, ACSR 101 (pp. 559–567). Antlatis Press. https://doi.org/10.2991/978-94-6463-042-8_80

Manurung, A. H., Machdar, N. M., Sijabat, J., & Manurung, A. (2024). The Construction of a Portfolio Using Varying Methods and the Effects of Variables on Portfolio Return. International Journal of Economics and Financial Issues, 14(1), 233–241. https://doi.org/10.32479/ijefi.15314

Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91. https://doi.org/10.2307/2345307

Miskolczi, P. (2017). Note on Simple and Logarithmic Return. Applied Studies in Agribusiness and Commerce, 11(1–2), 127–136. https://doi.org/10.19041/apstract/2017/1-2/16

Molino, A., & Sala, C. (2020). Forecasting value at risk and conditional value at risk using option market data. Journal of Forecasting, 40(7), 1190–1213. https://doi.org/10.1002/for.2756

Najiha, A., Herdiani, E. T., & Tinungki, G. M. (2023). the Application of Gumbel Copula To Estimate Value At Risk With Backtesting in Telecommunication Stock. BAREKENG: Jurnal Ilmu Matematika Dan Terapan, 17(1), 0245–0252. https://doi.org/10.30598/barekengvol17iss1pp0245-0252

Najjari, V. (2018). On Generators in Archimedean Copulas. Sahand Communications in Mathematical Analysis, 12(1), 155–166. https://doi.org/10.22130/scma.2017.28401

Nelsen, R. B. (2006). An Introduction to Copulas. In The Elements of Statistical Learning (Second). Springer New York. https://doi.org/10.1007/0-387-28678-0

Nurrahmat, M. H., Noviyanti, L., & Bachrudin, A. (2017). Estimation of value at risk in currency exchange rate portfolio using asymmetric GJR-GARCH Copula. American Institute Physics Conference Proceedings, 1827(1), 1–9. https://doi.org/10.1063/1.4979422

Palaro, H. P., & Hotta, L. K. (2006). Using Conditional Copula to Estimate Value at Risk. Journal of Data Science, 4(1), 93–115. https://doi.org/10.6339/jds.2006.04(1).226

Pasieczna, A. H. (2019). Monte Carlo simulation approach to calculate Value at Risk: application to WIG20 and mWIG40. Financial Sciences, 24(2), 61–75. https://doi.org/10.15611/fins.2019.2.05

Rönkkö, M., Holmi, J., Niskanen, M., & Mättö, M. (2024). The Adaptive Markets Hypothesis: Insights Into Small Stock Market Efficiency. Applied Economics, 56(25), 3048–3062. https://doi.org/10.1080/00036846.2024.2326039

Sahamkhadam, M. (2021). Copula-based Portfolio Optimization. In Linnaeus University Dissertations. Linnaeus University Press.

Salmah, & Harahap, K. (2023). The Effect of Knowledge Investment, Perceived Return and Risk and Technological Advances on Investment Interest of Medan City Students in the Capital Market. Journal of Finance and Business Digital, 2(3), 415–428. https://doi.org/10.55927/jfbd.v2i3.6217

Soeryana, E., Halim, N. B. A., Sukono, Rusyaman, E., & Supian, S. (2017). Mean-Variance portfolio optimization by using non constant mean and volatility based on the negative exponential utility function. American Institute Physics Conference Proceedings, 1827, 020042. https://doi.org/10.1063/1.4979458

Subekti, R., Ratna Sari, E., Kusumawati, R., Pintari, H. O., & Renggani, P. (2019). Value at risk in the black litterman portfolio with stock selection through cluster analysis. Journal of Physics: Conference Series, 1320(1), 012004. https://doi.org/10.1088/1742-6596/1320/1/012004

Sulistianingsih, E., Rosadi, D., & Abdurakhman. (2019). Delta normal and delta gamma normal approximation in risk measurement of portfolio consisted of option and stock. American Institute Physics Conference Proceedings, 2192. https://doi.org/10.1063/1.5139181

Sulistianingsih, E., Rosadi, D., & Abdurakhman. (2022). Measuring Risk utilizing Credible Monte Carlo Expected Tail Loss. International Journal of Applied Mathematics, 52(1).

Syuhada, K., Puspitasari, R., Arnawa, I. K. D., Mufaridho, L., Elonasari, E., Jannah, M., & Rohmawati, A. (2024). Enhancing Value-at-Risk with Credible Expected Risk Models. International Journal of Financial Studies, 12(3), 80. https://doi.org/10.3390/ijfs12030080

Tinungki, G. M., Siswanto, S., & Najiha, A. (2023). The Gumbel Copula Method for Estimating Value at Risk: Evidence from Telecommunication Stocks in Indonesia during the COVID-19 Pandemic. Journal of Risk and Financial Management, 16(10), 424. https://doi.org/10.3390/jrfm16100424

Virginia, E., Ginting, J., & Elfaki, F. A. M. (2018). Application of GARCH Model to Forecast Data and Volatility of Share Price of Energy (Study on Adaro Energy Tbk, LQ45). International Journal of Energy Economics and Policy, 8(3), 131–140.

ZongMing, T., Prince, K., & Guoping, D. (2020). Investment Risk and Returns : The Relationship between a Stock and an Index Using the Modern Portfolio Theory Investment Risk and Returns : The Relationship Between A Stock and An Index Using the Modern Portfolio Theory. 8(4), 160–171. https://doi.org/10.2139/ssrn.3014223

Zuhra, F., Noviyanti, L., & Bachrudin, A. (2014). Estimasi value at risk return portofolio menggunakan metode copula. Bandung: Universitas Padjadjaran. https://pustaka.unpad.ac.id/archives/133361

DOI: https://doi.org/10.31764/jtam.v9i2.29952