Annals of Operations Research

, Volume 262, Issue 2, pp 653–681 | Cite as

Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier

  • Hanene Ben Salah
  • Mohamed Chaouch
  • Ali Gannoun
  • Christian de PerettiEmail author
  • Abdelwahed Trabelsi
S.I.: Financial Economics


The downside risk (DSR) model for portfolio optimisation allows to overcome the drawbacks of the classical Mean–Variance model concerning the asymmetry of returns and the risk perception of investors. This model optimization deals with a positive definite matrix that is endogenous with respect to portfolio weights. This aspect makes the problem far more difficult to handle. For this purpose, Athayde (2001) developed a new recursive minimization procedure that ensures the convergence to the solution. However, when a finite number of observations is available, the portfolio frontier presents some discontinuity and is not very smooth. In order to overcome that, Athayde (2003) proposed a mean kernel estimation of the returns, so as to create a smoother portfolio frontier. This technique provides an effect similar to the case in which continuous observations are available. In this paper, Athayde model is reformulated and clarified. Then, taking advantage on the robustness of the median, another nonparametric approach based on median kernel returns estimation is proposed in order to construct a portfolio frontier. A new version of Athayde’s algorithm will be exhibited. Finally, the properties of this improved portfolio frontier are studied and analysed on the French Stock Market.


Downside risk Kernel method Mean nonparametric estimation Median nonparametric estimation Portefolio efficient frontier Semi-variance 



Christian de Peretti gratefully acknowledges financial support of the Global Risk Institute in Financial Services (Toronto).


  1. Ang, J. (1975). A note on the ESL portfolio selection model. Journal of Financial and Quantitative Analysis, 10, 849–857.CrossRefGoogle Scholar
  2. Athayde, G. (2001). Building a mean-downside risk portfolio frontier, developments in forecast combination and portfolio choice. Hoboken: Wiley.Google Scholar
  3. Athayde, G. (2003). The mean-downside risk portfolio frontier: A non-parametric approach. In: Satchell S., & Scowcroft A., (Eds.), Advances in portfolio construction and implementation (pp. 290–308). Butterworth-Heinemann.Google Scholar
  4. Berlinet, A., Cadre, B., & Gannoun, A. (2001a). On the conditional \(L_1\)-median and its estimation. Journal of Nonparametric Statistics, 13, 631–645.CrossRefGoogle Scholar
  5. Berlinet, A., Gannoun, A., & Matzner-Lober, E. (2001b). Asymptotic normality of convergent estimates of conditional quantiles. Statistics, 35, 139–169.CrossRefGoogle Scholar
  6. Estrada, J. (2004). Mean–semivariance behavior: An alternative behavioral model. Journal of Emerging Market Finance, 3, 231–248.CrossRefGoogle Scholar
  7. Estrada, J. (2008). Mean-semivariance optimization: A heuristic approach. Journal of Applied Finance, 18,(1) 57–72.Google Scholar
  8. Gannoun, A., Saracco, J., & Yu, K. (2003). Nonparametric time series prediction by conditional median and quantiles. Journal of Statistical Planning and Inference, 117, 207–223.CrossRefGoogle Scholar
  9. Harlow, V. (1991). Asset allocation in a downside risk framework. Financial Analyst Journal, 47, 28–40.CrossRefGoogle Scholar
  10. Hogan, W., & Warren, J. (1974). Computation of the efficient boundary in the ES portfolio selection. Journal of Financial and Quantitative Analysis, 9, 1–11.CrossRefGoogle Scholar
  11. King, A.-J. (1993). Asymmetric risk measures and tracking models for portfolio optimization under uncertainty. Annals of Operations Research, 45(1), 165–177.CrossRefGoogle Scholar
  12. Koenker, R. (2005). Quantile regression, Cambridge Books, Cambridge University Press, Issue 38.Google Scholar
  13. Levy, H., & Markowitz, H. M. (1979). Approximating expected utility by a function of mean and variance. American Economic Review, 69(3), 308–317.Google Scholar
  14. Mamoghli, C., & Daboussi, S. (2008). Optimisation de portefeuille dans le cadre du downside risk. Workingpaper.
  15. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77–91.Google Scholar
  16. Markowitz, H. M. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.Google Scholar
  17. Markowitz, M., Peter, T., Ganlin, X., & Yuji, Y. (1993). Computation of mean–semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45, 307–317.CrossRefGoogle Scholar
  18. Pagan, A. R., & Ullah, A. (1999). Nonparametric econometrics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  19. Pla-Santamaria, D., & Bravo, M. (2013). Portfolio optimization based on downside risk: A mean–semivariance efficient frontier from Dow Jones blue chips. Annals of Operations Research, 205(1), 189–201.CrossRefGoogle Scholar
  20. Sharpe, W. (1966). Mutual fund performance. Journal of Business, 39, 119–138.CrossRefGoogle Scholar
  21. Silverman, B. W. (1986). Density estimation for statistics and data analysis. New York: Chapman and Hall.CrossRefGoogle Scholar
  22. Sortino, F., & Price, L. (1994). Performance measurement in a downside risk framework. Journal of Investing, 3, 59–65.CrossRefGoogle Scholar
  23. Subramanian, S. (2002). Median regression using nonparametric kernel estimation. Journal of Nonparametric Statistics, 14, 583–605.Google Scholar
  24. Zenios, S.-A., & Kang, P. (1993). Mean-absolute deviation portfolio optimization for mortgage-backed securities. Annals of Operations Research, 45(1), 433–450.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Hanene Ben Salah
    • 1
    • 2
    • 3
  • Mohamed Chaouch
    • 4
  • Ali Gannoun
    • 3
  • Christian de Peretti
    • 2
    Email author
  • Abdelwahed Trabelsi
    • 1
  1. 1.BESTMOD LaboratoryLe BardoTunisia
  2. 2.Université Claude Bernard Lyon 1, Institut de Science Financière et d’Assurances, LSAF EA2429LyonFrance
  3. 3.IMAGMontpellier Cedex 05France
  4. 4.Department of StatisticsUnited Arab Emirates UniversityAl AinUAE

Personalised recommendations