Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Portfolio selection based on asymmetric Laplace distribution, coherent risk measure, and expectation-maximization estimation

1 School of Mathematics and Systems Science, Beihang University, Beijing, China
2 Department of Statistics, Chonnam National University, 500-757, Gwangju, Korea
3 Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

Special Issues: Systemic Risk Measurement

In this paper, portfolio selection problem is studied under Asymmetric Laplace Distribution(ALD) framework. Asymmetric Laplace distribution is able to capture tail-heaviness, skewness, andleptokurtosis observed in empirical financial data that cannot be explained by traditional Gaussiandistribution. Under Asymmetric Laplace distribution framework, portfolio selection methods basedon di erent risk measures are discussed. Moreover, we derived the Expectation-Maximization (EM)procedure for parameter estimation of Asymmetric Laplace distribution. Performance of the proposedmethod is illustrated via extensive simulation studies. Two real data examples are complemented toconfirm that the Asymmetric Laplace distribution based portfolio selection models are effcient.
  Article Metrics

Keywords portfolio selection; Asymmetric Laplace distributions; EM procedure; tail-heaviness; Skewness

Citation: Yue Shi, Chi Tim Ng, Ka-Fai Cedric Yiu. Portfolio selection based on asymmetric Laplace distribution, coherent risk measure, and expectation-maximization estimation. Quantitative Finance and Economics, 2018, 2(4): 776-797. doi: 10.3934/QFE.2018.4.776


  • 1. Arslan O (2010) An alternative multivariate skew Laplace distribution: properties and estimation. Stat Pap 51: 865–887.    
  • 2. Artzner P, Delbaen F, Eber JM, et al. (1999) Coherent measures of risk. Math financ 9: 203–228.    
  • 3. Ayebo A, Kozubowski TJ (2003) An asymmetric generalization of Gaussian and Laplace laws. J Probab Stat Science 1: 187–210.
  • 4. Barndorff-Nielsen OE (1995) Normal inverse Gaussian distributions and the modeling of stock returns. 24: 1–13.
  • 5. Bingham NH, Kiesel R (2001) Modelling asset returns with hyperbolic distributions. In Return Distributions in Finance, 1–20.
  • 6. Behr A, Ptter U (2009) Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models. Ann Financ 5: 49–68.    
  • 7. Dias JG, Vermunt JK, Ramos S (2015) Clustering financial time series: New insights from an extended hidden Markov model. European J Oper Res 243: 852–864.    
  • 8. Eberlein E (2001) Application of generalized hyperbolic Lvy motions to finance. In Lvy processes, Birkhuser, Boston, MA, 319–336.
  • 9. Eberlein E, Keller U (1995) Hyperbolic distributions in finance. Bernoulli 1: 281–299.    
  • 10. Hellmich M, Kassberger S (2011) Effcient and robust portfolio optimization in the multivariate generalized hyperbolic framework. Quant Financ 11: 1503–1516.    
  • 11. HuW, Kercheval A (2007) Risk management with generalized hyperbolic distributions. In Proceedings of the Fourth IASTED International Conference on Financial Engineering and Applications, ACTA Press, 19–24.
  • 12. Hu W, Kercheval AN (2010) Portfolio optimization for student t and skewed t returns. Quant Financ 10: 91–105.    
  • 13. Hrlimann W (2013) A moment method for the multivariate asymmetric Laplace distribution. Stat Probabil Lett 83: 1247–1253.    
  • 14. Iorio C, Frasso G, DAmbrosio A, et al. (2018) A P-spline based clustering approach for portfolio selection. Expert Syst Appl 95: 88–103.    
  • 15. Kollo T, Srivastava MS (2005) Estimation and testing of parameters in multivariate Laplace distribution. Commun Stat-Theor M 33: 2363–2387.    
  • 16. Kotz S, Kozubowski TJ, Podgrski K (2001) Asymmetric multivariate Laplace distribution. In The Laplace Distribution and Generalizations, Birkhuser, Boston, MA, 239–272.
  • 17. Kotz S, Kozubowski TJ, Podgrski K (2002) Maximum likelihood estimation of asymmetric Laplace parameters. Ann I Stat Math 54: 816–826.    
  • 18. Kotz S, Kozubowski T, Podgorski K (2012) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer Science & Business Media.
  • 19. Kozubowski TJ, Podgrski K (1999) A class of asymmetric distributions. Actuarial Research Clearing House 1: 113–134.
  • 20. Kozubowski TJ, Podgrski K (2001) Asymmetric Laplace laws and modeling financial data. Math Comput Model 34: 1003–1021.    
  • 21. Lai TL, Xing H (2008) Statistical models and methods for financial markets. New York: Springer.
  • 22. Mandelbrot BB (1997) The variation of certain speculative prices. In Fractals and scaling in finance, Springer, New York, NY, 371–418.
  • 23. Markowitz H (1952) Portfolio selection. The journal of finance 7: 77–91.
  • 24. Punathumparambath B (2012) The multivariate asymmetric slash Laplace distribution and its applications. Statistica 72: 235.
  • 25. Seneta E (2004) Fitting the variance-gamma model to financial data. J Appl Probab 41: 177–187.    
  • 26. Socgnia VK, Wilcox D (2014) A comparison of generalized hyperbolic distribution models for equity returns. J Appl Math.
  • 27. Stacy EW (1962) A generalization of the gamma distribution. Ann math stat 1187–1192.
  • 28. Surya BA, Kurniawan R (2014) Optimal portfolio selection based on expected shortfall under generalized hyperbolic distribution. Asia-Pacific Financ Mark 21: 193–236.    
  • 29. Visk H (2009) On the parameter estimation of the asymmetric multivariate Laplace distribution. Commun StatTheor M 38: 461–470.
  • 30. Yiu KFC (2004) Optimal portfolios under a value-at-risk constraint. J Economic Dyn Control 28: 1317–1334.    
  • 31. Zhao S, Lu Q, Han L, et al. (2015) A mean-CVaR-skewness portfolio optimization model based on asymmetric Laplace distribution. Ann Oper Res 226: 727–739.    
  • 32. Zhu Y (2007) Application of Asymmetric Laplace Laws in Financial Risk Measures and Time Series Analysis. Doctoral dissertation, University of Florida.


This article has been cited by

  • 1. Gaoke Liao, Zhenghui Li, Ziqing Du, Yue Liu, The Heterogeneous Interconnections between Supply or Demand Side and Oil Risks, Energies, 2019, 12, 11, 2226, 10.3390/en12112226

Reader Comments

your name: *   your email: *  

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved