Sains Malaysiana 49(5)(2020): 1153-1164

http://dx.doi.org/10.17576/jsm-2020-4905-21

Two Stages Fitting Techniques using Generalized Lambda Distribution: Application on Malaysian Financial Return

(Teknik Penyuaian Dua Peringkat menggunakan Taburan Generalisasi Lambda: Aplikasinya ke atas Pulangan Kewangan Malaysia)

MUHAMMAD FADHIL MARSANI^1,2* & ANI SHABRI¹

¹Department of Mathematics, Universiti Teknologi Malaysia, 81310 Johor Darul Takzim, Malaysia

²School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Minden, Pulau Pinang, Malaysia

Diserahkan: 24 Jun 2019/Diterima: 24 Januari 2020

ABSTRACT

The underline distribution assumption used in the analysis of share market returns is crucial in risk management. An important aspect related to stock return modelling is to obtain accurate prediction. This paper presents an innovative fitting method called two stages (TS) method for modelling daily stock returns. The proposed approach by first establishing trend in the series, and then separately performing L-moment estimation on the generalized lambda distribution (GLD) parameter. The performance of the TS-GLD models had been evaluated using Monte Carlo simulation and Malaysian Kuala Lumpur Composite Index (KLCI) returns from year 2001 to 2015. Based on k-sample Anderson darling goodness of fit test, the two stages GLD model in location parameter (GLD.1) performed well in all studied cases. The GLD.1 model benefits risk management by providing effective distribution fitting.

Keywords: Fat-tailed distributions; generalized lambda distribution; L-moment; risk management; stock returns

ABSTRAK

Andaian taburan yang digunakan dalam analisis pulangan pasaran saham adalah penting dalam pengurusan risiko. Isu utama dalam memodelkan pulangan saham adalah untuk mendapatkan anggaran yang tepat. Kajian ini membentangkan kaedah penyuaian inovatif iaitu kaedah dua peringkat (TS) dalam memodelkan pulangan saham harian. Pendekatan ini dijalankan dengan cara mengenal pasti bentuk trend di dalam siri, kemudian melaksanakan anggaran L-momen pada parameter taburan generalisasi lambda (GLD). Prestasi model TS-GLD dinilai dengan menggunakan kaedah simulasi Monte Carlo dan data sebenar iaitu Indeks Komposit Kuala Lumpur Malaysia (KLCI) dari tahun 2001 hingga 2015. Berdasarkan ujian kebagusan k-sample Anderson darling, model dua peringkat (TS) GLD bagi parameter lokasi (GLD.1) menunjukkan prestasi yang lebih baik untuk semua kes yang dikaji. Model GLD.1 bermanfaat dalam pengurusan risiko dengan memberikan penyuaian taburan yang lebih baik.

Kata kunci: L-momen; pengurusan risiko; pulangan saham; taburan berekor tebal; taburan generalisasi lambda

RUJUKAN

Ab Razak, R. & Ismail, N. 2019. Dependence modeling and portfolio risk estimation using GARCH-Copula approach. Sains Malaysiana 48(7): 1547-1555.

Acharya, V., Engle, R. & Richardson, M. 2012. Capital shortfall: A new approach to ranking and regulating systemic risks. American Economic Review 102(3): 59-64.

Asquith, W.H. 2007. L-moments and TL-moments of the generalized lambda distribution. Computational Statistics & Data Analysis 51(9): 4484-4496.

Ben Slimane, F., Mehanaoui, M. & Kazi, I. 2013. How does the financial crisis affect volatility behavior and transmission among European stock markets? International Journal of Financial Studies 1(3): 81-101.

Calvet, L.E. & Fisher, A. 2008. Multifractal Volatility: Theory, Forecasting, And Pricing. Massachusetts: Academic Press.

Chalabi, Y., Diethelm, W. & Scott, D.J. 2012. Flexible distribution modeling with the generalized lambda distribution flexible distribution modeling with the generalized lambda distribution. Munich Personal RePEc Archive https://mpra.ub.uni-muenchen.de/43333/.

Chalabi, Y., Scott, D.J. & Würtz, D. 2009. The generalized lambda distribution as an alternative to model financial returns. Eidgenössische Technische Hochschule and University of Auckland, Zurich and Auckland. pp. 1-28.

Cont, R. 2001. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1(2): 223-236

Corlu, C.G., Meterelliyoz, M. & Tiniç, M. 2016. Empirical distributions of daily equity index returns: A comparison. Expert Systems with Applications 54: 170-192.

Corrado, C.J. 2001. Option pricing based on the generalized lambda distribution. Journal of Futures Markets 21(3): 213-236.

Cunderlik, J.M. & Burn, D.H. 2003. Non-stationary pooled flood frequency analysis. Journal of Hydrology 276(1-4): 210-223.

Ding, Z., Granger, C.W.J. & Engle, R.F. 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1(1): 83-106.

Dong, Y. & Wang, J. 2013. Fluctuation behavior of financial return interval series model for percolation on Sierpinski carpet lattice. Fractals 21(03n04): 1350023.

Fang, W. & Wang, J. 2012. Statistical properties and multifractal behaviors of market returns by Ising dynamic systems. International Journal of Modern Physics C 23(03): 1250023.

Fournier, B., Rupin, N., Bigerelle, M., Najjar, D. & Iost, A. 2006. Application of the generalized lambda distributions in a statistical process control methodology. Journal of Process Control 16(10): 1087-1098.

Gabaix, X., Gopikrishnan, P., Plerou, V. & Stanley, H. E. 2003. A theory of power-law distributions in financial market fluctuations. Nature 423(6937): 267.

Gagniuc, P.A. 2017. Markov Chains: From Theory to Implementation and Experimentation. John Wiley & Sons.

Gettinby, G.D., Sinclair, C.D., Power, D.M. & Brown, R.A. 2006. An analysis of the distribution of extremes in indices of share returns in the US, UK and Japan from 1963 to 2000. International Journal of Finance and Economics 11(2): 97-113.

Hasan, H., Radi, N.F.A. & Kassim, S. 2012. Modeling of extreme temperature using Generalized Extreme Value (GEV) distribution: A case study of Penang. Proceedings of the World Congress on Engineering 2012 89: 82-89.

Hussain, S.I. & Li, S. 2015. Modeling the distribution of extreme returns in the Chinese stock market. Journal of International Financial Markets, Institutions and Money 34: 263-276.

Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A. & Stanley, H.E. 2002. Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications 316(1-4): 87-114.

Karian, Z.A. & Dudewicz, E.J. 2000. Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods.London: Chapman and Hall/CRC.

Longin, F.M.F.M. 1996. The asymptotic distribution of extreme stock market returns. Journal of Business 69(3): 383-408.

Mandelbrot, B.B. 2013. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. Springer Science & Business Media.

Mantegna, R.N. & Stanley, H.E. 1995. Scaling behaviour in the dynamics of an economic index. Nature 376(6535): 46-49.

Marsani, M.F. & Shabri, A. 2019. Random walk behaviour of Malaysia share return in different economic circumstance formula. MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics DOI: https://doi.org/10.11113/matematika.v35.n3.1105

Marsani, M.F., Shabri, A. & Jan, N.A.M. 2017. Examine generalized lambda distribution fitting performance: An application to extreme share return in Malaysia. Malaysian Journal of Fundamental and Applied Sciences 13(3): 230-237.

Niu, H. & Wang, J. 2013a. Volatility clustering and long memory of financial time series and financial price model. Digital Signal Processing 23(2): 489-498.

Niu, H. & Wang, J. 2013b. Power-law scaling behavior analysis of financial time series model by voter interacting dynamic system. Journal of Applied Statistics 40(10): 2188-2203.

Ramberg, J.S. & Schmeiser, B.W. 1974. An approximate method for generating asymmetric random variables. Communications of the ACM 17(2): 78-82.

Rizvi, S.A.R., Dewandaru, G., Bacha, O.I. & Masih, M. 2014. An analysis of stock market efficiency: Developed vs Islamic stock markets using MF-DFA. Physica A: Statistical Mechanics and its Applications 407: 86-99.

Scholz, F.W. & Stephens, M.A. 1987. K-sample Anderson–Darling tests. Journal of the American Statistical Association. doi:10.1080/01621459.1987.10478517.

Sen, P.K. 1968. Estimates of the regression coefficient based on Kendall’s tau. Journal of the American Statistical Association 63(324): 1379-1389.

Stošić, D., Stošić, D., Stošić, T. & Stanley, H.E. 2015. Multifractal properties of price change and volume change of stock market indices. Physica A: Statistical Mechanics and its Applications 428: 46-51.

Suárez-García, P. & Gómez-Ullate, D. 2014. Multifractality and long memory of a financial index. Physica A: Statistical Mechanics and its Applications 394: 226-234.

Tolikas, K. 2014. Unexpected tails in risk measurement: Some international evidence. Journal of Banking and Finance 40(1): 476-493.

Tolikas, K. 2011. The rare event risk in African emerging stock markets. Managerial Finance 37(3): 275-294.

Tolikas, K. 2008. Value-at-risk and extreme value distributions for financial returns. Journal of Risk 10(3): 31-77.

Tolikas, K. & Gettinby, G.D. 2009. Modelling the distribution of the extreme share returns in Singapore. Journal of Empirical Finance 16(2): 254-263.

Yu, Y. & Wang, J. 2012. Lattice-oriented percolation system applied to volatility behavior of stock market. Journal of Applied Statistics 39(4): 785-797.

*Pengarang untuk surat-menyurat; email: fadhilmarsani@gmail.com