Bivariate Distributions and Copula-Tvar Estimates: A Comparative Study Based on The Selected Financial Returns and Marginal Distributions
Abstract
This study explores the joint distribution of bivariate financial returns on DJIA-S&P500 and SSE-SZSE, employing copulas and model selection criteria to identify the most suitable distribution. The aim is to estimate Conditional Tail Value at Risk (C-TVaR) at various confidence levels for portfolio risk management. Unlike previous studies, which typically focus on univariate analysis, this research examines into the joint distribution of bivariate financial returns. Additionally, it introduces the application of copulas and model selection criteria to determine the optimal joint distribution for portfolio risk assessment, offering valuable insights for financial decision-makers. Several copulas and model selection criteria are employed to assess the joint distribution of bivariate financial returns. By evaluating the minimum values of model selection criteria such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), the Student’s t copula is identified as the most appropriate copula. C-TVaR estimates are then obtained at different confidence levels using the selected copula and various combinations of marginal distributions, namely, normal, Student's t, Cauchy, and alpha power transformed logistic (APTL) marginal distributions. Empirical results demonstrate that Student's t copula models with APTL-Student's t and APTL-APTL marginals gave the smallest expected portfolio losses for the DJIA-S&P500 and SSE-SZSE portfolios, respectively. These insights contribute to enhancing portfolio risk management strategies, particularly in assessing tail risk at different confidence levels.
Keywords: Alpha power transformed logistic distribution, bivariate copula, C-TVaR, model selection criteria, portfolio loses.
Abstrak
Studi ini mengeksplorasi distribusi bersama dari keuntungan finansial pada DJIA-S&P500 dan SSE-SZSE, dengan menggunakan kopula dan kriteria pemilihan model untuk mengidentifikasi distribusi yang paling cocok. Tujuannya adalah untuk memperkirakan Conditional Tail Value at Risk (C-TVaR) pada berbagai tingkat kepercayaan untuk manajemen risiko portofolio. Berbeda dengan penelitian sebelumnya, yang umumnya berfokus pada analisis univariat, penelitian ini menyelidiki distribusi bersama dari keuntungan finansial. Selain itu, penelitian ini memperkenalkan aplikasi kopula dan kriteria pemilihan model untuk menentukan distribusi bersama optimal untuk penilaian risiko portofolio, memberikan wawasan berharga bagi pengambil keputusan finansial. Beberapa kopula dan kriteria pemilihan model digunakan untuk menilai distribusi bersama dari keuntungan finansial. Dengan mengevaluasi nilai minimum dari kriteria pemilihan model seperti Akaike Information Criterion (AIC) dan Bayesian Information Criterion (BIC), kopula Student’s t diidentifikasi sebagai kopula yang paling cocok. Estimasi C-TVaR kemudian diperoleh pada tingkat kepercayaan yang berbeda menggunakan kopula yang dipilih dan berbagai kombinasi distribusi marginal, yaitu distribusi marginal normal, Student's t, Cauchy, dan alpha power transformed logistic (APTL). Hasil empiris menunjukkan bahwa model kopula Student's t dengan distribusi marginal APTL-Student's t dan APTL-APTL memberikan ekspetasi kerugian portofolio terkecil untuk masing-masing portofolio DJIA-S&P500 dan SSE-SZSETemuan ini berkontribusi untuk meningkatkan strategi manajemen risiko portofolio, khususnya dalam menilai risiko ekor pada tingkat kepercayaan yang berbeda.
Kata Kunci: distribusi Alpha power transformed logistic, kopula bivariat, C-TVaR, kriteria pemilihan model, kehilangan portofolio.
2020MSC: 62H05
Keywords
References
M. Y. T. Irsan and M. L. Sirait, "Value-at-risk (VaR) & Tailed value-at-risk (TVaR): Companies listed in LQ45," International Journal of Economics, Commerce and Management, vol. 8, no. 12, pp. 560-568, (2020.
Z. Shen, L. Yukun and W. Chengguo, "Nonparametric inference for VaR, CTE and expectile with high-order precision," North American Actuarial Journal, vol. 23, p. 364–85, 2019.
C. Bolance, M. Guillen and A. Padilla, Risk estimation using copulas. U.B. Risk centre working papers series 2015-01, Universitat de Barcelona, 2015.
J. E. Methni, G. Laurent and G. Stephane, "Non-parametric estimation of extreme risk measures from conditional heavy-tailed distributions," Scandinavian Journal of Statistics, vol. 41, p. 988–1012, 2014.
A. Castaner, M. M. Claramunt and M. Marmol, Tail value at risk: An analysis with the normal-power approximation. In Statistical and Soft Computing Approaches in Insurance Problems, pp. 87-112, Nova Science Publishers. ISBN 978-1-62618-506-7, 2013, pp. 87-111.
M. Barges, H. Cossette and E. Marceau, "TVaR-based Capital Allocation with Copulas," Insurance: Mathematics and Economics, vol. 45, no. 3, pp. 348-361, 2009.
V. Brazauskas, L. J. Bruce, L. P. Madan and Z. Riˇcardas, "Estimating Conditional Tail Expectation with Actuarial Applications in View," Journal of Statistical Planning and Inference, vol. 138, no. 3, p. 590–604, 2008.
T. Kaiser and B. Vytaras, "Interval Estimation of Actuarial Risk Measures," North American Actuarial Journal, vol. 10, p. 249–268, 2006.
R. Wang and Y. Wei, "Characterizing Optimal Allocations in Quantile-based Risk Sharing," Insurance: Mathematics and Economics, vol. 93, p. 288–300, 2020.
R. Bairakdar, C. Lu and M. Melina, "Range Value-at-risk: Multivariate and Extreme Values," arXiv .12473, 2020.
C. Bernard, K. Rodrigue and V. Steven, "Range Value-at-risk Bounds for Unimodal Distributions Under Partial Information," Insurance: Mathematics and Economics, vol. 94, p. 9–24, 2020.
K. Syuhada, O. Neswan and B. P. Josaphat, "Estimating Copula-based Extension of Tail Value-at-risk and its Application in Insurance Claim," Risk, vol. 10, no. 113, pp. 1-26, 2023.
S. Dutta and B. Suparna, "Nonparametric estimation of 100(1 - p)% expected shortfall: p as sample size is increased," Communications in Statistics-Simulation and Computation, vol. 47, p. 338–352, 2018.
A. C. Iwuji, E. W. Okereke, B. Oruh and J. C. Nwabueze, "The Alpha Power Transformed Distribution: Properties, Application and VaR Estimation," Indonesian Journal of Pure and Applied Mathematics, vol. 5, no. 1, p. 82–98, 2023.
M. Haugh, An Introduction to Copulas in IEOR E4602: Quantitative Risk Management, Lecture Notes. New York: Columbia University, 2016.
M. Sklar, "Functions de répartition à n dimensions et leurs marges," 1959, vol. 8, p. 229–231, Publ. Inst. Stat. Univ. Paris.
B. Beare, "Copulas and Temporal Dependence.," Econometrica, vol. 78, p. 395–410, 2010.
K. Byun and S. Song, "Value at Risk of Portfolios using Copulas," Communications for Statistical Applications and Methods, vol. 28, no. 1, pp. 59-79, 2021.
R. A. Razak and N. C. Ismail, "Portfolio Risks of Bivariate Financial Returns using Copula-Var Approach: A Case Study on Malaysia and U.S. Stock Markets," Global journal of pure and applied mathematics, vol. 12, no. 3, pp. 1947-1964, 2016.
M. Sahamkhadam, Copula-based portfolio optimization., Unpublished PhD thesis, Linnaeus University dissertations, Vaxjo. ISBN: 978-91-89283-79-4, 2021.
N. Trabelsi and A. K. Tiwari, "Market Risk Optimization among Developed and Emerging Markets with CVaR Measure and Copula Simulation," Risks, vol. 7, no. 3, p. 78, 2019. https://doi.org/10.3390/risks7030078.
P. H. Ferreira, R. L. Fiaccone, J. S. Lordelo, S. O. L. Sena and V. R. Duran, "Bivariate Copula-based Linear Mixed Effects Model: An Application to Longitudinal Child Growth Data," Tendendas em matematica aplicada e computacional, vol. 20, no. 1, p. 37–59, 2019.
N. Salleh, F. Yusof and Z. Yusop, "Bivariate Copula Functions for Flood Frequency Analysis.," in AIP conference proceedings 1750, 060007-1 – 060007-13, 2016.
M. Kemp, "VaR and Tail VaR Mindset: Presentation to VaR Open Forum," in The Actuarial profession, 2009 (March).
D. L. Oslon and D. Wu, "The Impact of Distribution on Value-at-Risk Measure," Mathematical and Computer Modelling, vol. 58, pp. 1670-1676, 2013.
DOI: 10.15408/inprime.v6i1.37158
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