Portfolio Selection Using Fractal Statistical Measures
WU Xu1, YAN Ruzhen1, WANG Xuefei1, LI Jia2
1.School of Business,Chengdu University of Technology,Chengdu 610059,China; 2.School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China
Abstract：In order to improve the effectiveness of portfolio selection, we firstly construct the statistical measures of fractal expectation and fractal variance, and give the algorithm of two fractal statistical measures. Secondly, a fractal portfolio selection model is built based on the fractal statistical measures and an analytical solution for fractal portfolio selection model is calculated. Lastly, drawing the sample of all industrial indexes from Shanghai Stock Exchange, we verify the feasibility of constructing portfolio selection model using two fractal statistical measures. The empirical results demonstrate that fractal statistical measures make up the defect of the non-fractal statistical measure’s disability to measure the return and risks of stocks accurately, and the fractal portfolio model is more effective in diversifying risks while ensuring returns.
[1]Markowitz H. Portfolio selection[J]. Journal of Finance, 1952, 7(1): 77-91. [2]Tanaka H, Guo P, Türksen I B. Portfolio selection based on fuzzy probabilities and possibility distributions[J]. Fuzzy Sets and Systems, 2000, 111(3): 387-397. [3]Xu W J, Deng X, Li J. A new fuzzy portfolio model on background risk using MCFOA[J]. International Journal of Fuzzy Systems, 2015, 17(2): 246-255. [4]Detemple J. Portfolio selection: a review[J]. Journal of Optimization Theory & Applications, 2014, 161(1):1-21. [5]Alexander G J, Baptista A M, Yan S. Portfolio selection with mental accounts and estimation risk[J]. Journal of Empirical Finance, 2017, 41(3): 161-186. [6]Zhang J, Jin Z, An Y. Dynamic portfolio optimization with ambiguity aversion[J]. Journal of Banking & Finance, 2017, 79(6): 95-109. [7]Lim G, Kim S Y, Lee H, et al. Multifractal detrended fluctuation analysis of derivative and spot markets[J]. Physica A: Statistical Mechanics and Its Applications, 2007, 386(1):259-266. [8]Cajueiro D O, Tabak B M. Multifractality and herding behavior in the Japanese stock market [J]. Chaos Solitons & Fractals, 2009, 40(1):497-504. [9]Yuan Y, Zhuang X T, Jin X. Measuring multifractality of stock price fluctuation using multifractal detrended fluctuation analysis[J]. Physica A: Statistical Mechanics and Its Applications, 2009, 388(11): 2189-2197. [10] Wang Y D, Liu L,Gu R B. Analysis of efficiency for Shenzhen stock market based on multifractal detrended fluctuation analysis[J]. International Review of Financial Analysis, 2009, 18(5): 271-276. [11] Zunino L, Tabak B M, Figliolaf A, et al. A multifractal approach for stock market inefficiency [J]. Physica A: Statistical Mechanics and Its Applications, 2008, 387 (26):6558-6566. [12] Mandelbrot B B. Three fractal models in finance: discontinuity, concentration, risk[J]. Economic Notes, 1997, 26(2): 197-212. [13] Peters E E. Fractal Market Analysis: Applying Chaos Theory to Investment and Economics [M]. New York: John Wiley&Sons, Inc.1994. [14] Mandelbrot B B. A multifractal walk down Wall Street[J]. Scientific American, 1999, 280(2): 70-73. [15] Mandelbrot B B, Hudson R L. The (mis) Behavior of Markets: A Fractal View of Risk, Ruin, and Reward[M]. New York: Basic Books, 2004. [16] Lux T, Alfarano S. Financial power laws: Empirical evidence, models, and mechanisms[J]. Chaos, Solitons & Fractals, 2016, 88(7): 3-18. [17] Wei Y, Wang P. Forecasting volatility of SSEC in Chinese stock market using multifractal analysis[J]. Physica A: Statistical Mechanics and Its Applications, 2008, 387(7): 1585-1592. [18] Wei Y, Wang Y, Huang D. A copula-multifractal volatility hedging model for CSI 300 index futures[J]. Physica A: Statistical Mechanics and Its Applications, 2011, 390(23): 4260-4272. [19] Lévy P. Théorie de L'addition des Variables Aléatoires[M]. Paris: Gauthier- Vilars, 1937. [20] Samorodnitsky G, Taqqu M S. Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance[M]. Boca Raton: CRC Press Inc., 1994. [21] Stoyanov S V, Racheva-Iotova B, Rachev S T, et al. Stochastic models for risk estimation in volatile markets: a survey[J]. Annals of Operations Research, 2010, 176(1): 293-309. [22] Mandelbrot B B. How long is the coast of britain? statistical self-similarity and fractional dimension[J]. Science, 1967, 156(5): 636-638. [23] Falconer K. Fractal Geometry: Mathematical Foundations and Applications[M]. 2nd Edition. Chichester: John Wiley&Sons, Inc., 2003. [24] Merton R C. An analytic derivation of the efficient portfolio frontier[J]. Journal of Financial and Quantitative Analysis, 1972, 7(4): 1851-1872. [25] Gabaix X, Gopikrishnan P, Plerou V, et al. A theory of power-law distributions in financial market fluctuations[J]. Nature, 2003, 423(6937): 267-70. [26] Gabaix X. Power laws in economics and finance[J]. Quantitative Finance, 2009, 1(1): 255-293.
