Skip to main content
SpringerLink
Log in

Optimal Portfolio Projections for Skew-Elliptically Distributed Portfolio Returns

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We define the concept of optimal portfolio projection, a procedure that projects the vector of weights of the portfolio return to a lower dimension such that one can explicitly solve the problem of optimal portfolio selection for any given risk measure. We study the class of skew-elliptically distributed risks and show that following the proposed procedure, we are able to obtain explicit optimal weights for such risks, with a dramatic reduction of the complexity of such an optimization problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Aas, K., Haff, I.H.: The generalized hyperbolic skew student’st-distribution. J. Financ. Economet. 4, 275–309 (2006)

    Google Scholar 

  2. Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Financ. 26, 1487–1503 (2002)

    Google Scholar 

  3. Acemoglu, D., Ozdaglar, A., Tahbaz-Salehi, A.: Systemic risk and stability in financial networks. Am. Econ. Rev. 105, 564–608 (2015)

    Google Scholar 

  4. Adcock, C.J.: Asset pricing and portfolio selection based on the multivariate extended skew-Student distribution. Ann. Oper. Res. 176, 221–234 (2010)

  5. Adcock, C., Eling, M., Loperfido, N.: Skewed distributions in finance and actuarial science: a review. Eur. J. Finance. 21(13–14), 1253–1281 (2015)

    Google Scholar 

  6. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9, 203–228 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Azzalini, A.: A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178 (1985)

  8. Azzalini, A.: The skew-normal distribution and related multivariate families. Scand. J. Stat. 32, 159–188 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Azzalini, A., Dalla Valle, A.: The multivariate skew-normal distribution. Biometrika 83, 715–726 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Azzalini, A., Capitanio, C.: Statistical applications of the multivariate skew-normal distributions. J. Roy. Stat. Soc. B 61, 579–602 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Azzalini, A., Capitanio, A.: Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. Royal Stat. Soc. Ser. B (Statistical Methodology) 65(2), 367–389 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Bickel, P.J., Kur, G., Nadler, B.: Projection pursuit in high dimensions. Proc. Natl. Acad. Sci. 115, 9151–9156 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Black, L., Correa, R., Huang, X., Zhou, H.: The systemic risk of European banks during the financial and sovereign debt crises. J. Bank. Financ. 63, 107–125 (2016)

    Google Scholar 

  14. Borio, C.: Towards a macroprudential framework for financial supervision and regulation? Econ. Stud. 49, 181–215 (2003)

    Google Scholar 

  15. Branco, M.D., Dey, D.K.: A general class of skew-elliptical distributions. J. Multivar. Anal. 79, 99–113 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Brownlees, C., Engle, R.F.: SRISK: a conditional capital shortfall measure of systemic risk. Rev. Financ. Stud. 30, 48–79 (2016)

    Google Scholar 

  17. Duffie, D., Pan, J.: An overview of value at risk. J. Deriv. 4, 7–49 (1997)

    Google Scholar 

  18. Eini, E.J., Khaloozadeh, H.: The tail mean-variance optimal portfolio selection under generalized skew-elliptical distribution. Insur. Math. Econ. 98, 44–50 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Eling, M.: Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models? Insur. Math. Econ. 51, 239–248 (2012)

    MathSciNet  Google Scholar 

  20. Genton, M.G., Loperfido, N.: Generalized skew-elliptical distributions and their quadratic forms. Ann. Inst. Stat. Math. 57, 389–401 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Goh, J.W., Lim, K.G., Sim, M., Zhang, W.: Portfolio value-at-risk optimization for asymmetrically distributed asset returns. Eur. J. Oper. Res. 221(2), 397–406 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Goodhart, C.A.: A framework for assessing financial stability? J. Bank. Financ. 30, 3415–3422 (2006)

    Google Scholar 

  23. Goodhart, C.A., Sunirand, P., Tsomocos, D.P.: A risk assessment model for banks. Ann. Financ. 1, 197–224 (2005)

    Google Scholar 

  24. Hallin, M., Ley, C.: Skew-symmetric distributions and Fisher information—a tale of two densities. Bernoulli 18, 747–763 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Healy, M.J.R.: Multivariate normal plotting. J. Roy. Stat. Soc.: Ser. C (Appl. Stat.) 17, 157–161 (1968)

    Google Scholar 

  26. Henze, N.: Limit laws for multivariate skewness in the sense of Mòri, Rohatgi and Székely. Statist. Probab. Lett. 33, 299–307 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Huang, X., Zhou, H., Zhu, H.: A framework for assessing the systemic risk of major financial institutions. J. Bank. Financ. 33, 2036–2049 (2009)

    Google Scholar 

  28. Harvey, C.R., Liechty, J.C., Liechty, M.W., Müller, P.: Portfolio selection with higher moments. Quant. Financ. 10(5), 469–485 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Kim, H.-K.: A note on scale mixtures of skew normal distribution. Statist. Prob. Lett. 78, 1694–1701 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Landsman, Z., Makov, U., Shushi, T.: Portfolio optimization by a bivariate functional of the mean and variance. J. Optim. Theory Appl. 185, 622–651 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Landsman, Z., Makov, U., Shushi, T.: Analytic solution to the portfolio optimization problem in a mean-variance-skewness model. Eur. J. Financ. 26(2–3), 165–178 (2020)

    MATH  Google Scholar 

  32. Landsman, Z., Makov, U., Shushi, T.: A generalized measure for the optimal portfolio selection problem and its explicit solution. Risks (2018). https://doi.org/10.3390/risks6010019

    Article  MATH  Google Scholar 

  33. Ley, C., Paindaveine, D.: On the singularity of skew-symmetric models. J. Multivar. Anal. 101, 1434–1444 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Ley, C., Paindaveine, D.: On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models. METRONLXVIII, 235-250(2010b)

  35. Li, X., Qin, Z., Kar, S.: Mean-variance-skewness model for portfolio selection with fuzzy returns. Eur. J. Oper. Res. 202(1), 239–247 (2010)

    MATH  Google Scholar 

  36. Liese, F., Vajda, I.: On divergences and informations in statistics and information theory. IEEE Trans. Inf. Theory 52, 4394–4412 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Liu, M., Lin, T.I.: Skew-normal factor analysis models with incomplete data. J. Appl. Stat. 42(4), 789–805 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Loperfido, N.: Canonical transformations of skew-normal variates. TEST 19, 146–165 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Loperfido, N.: Linear transformations to symmetry. J. Multivar. Anal. 129, 186–192 (2014)

    MathSciNet  MATH  Google Scholar 

  40. Loperfido, N.: Singular value decomposition of the third multivariate moment. Linear Algebra Appl. 473, 202–216 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Loperfido, N.: Skewness-based projection pursuit: a computational approach. Comput. Stat. Data Anal. 120, 42–57 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Loperfido, N.: Finite mixtures, projection pursuit and tensor rank: a triangulation. Adv. Data Anal. Classif. 31, 145–173 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Loperfido, N.: On a vector-valued measure of multivariate skewness. Symmetry 13, 1817 (2021)

    Google Scholar 

  44. Loperfido, N.: Kurtosis removal for data pre-processing. Adv. Data Anal. Classif. 17, 239–267 (2023)

    MathSciNet  MATH  Google Scholar 

  45. Markowitz, H.: The utility of wealth. J. Polit. Econ. 60, 151–158 (1952)

    Google Scholar 

  46. Martins-Filho, C., Yao, F., Torero, M.: Nonparametric estimation of conditional value-at-risk and expected shortfall based on extreme value theory. Economet. Theor. 34, 23–67 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Mardia, K.V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530 (1970)

    MathSciNet  MATH  Google Scholar 

  48. Mòri, T.F., Rohatgi, V.K., Székely, G.J.: On multivariate skewness and kurtosis. Theory Probab. Appl. 38, 547–551 (1993)

    MathSciNet  MATH  Google Scholar 

  49. Oh, S., Kee, H., Park, K.: Tail risk under price limits. Econ. Model. 77, 113–123 (2019)

    Google Scholar 

  50. Rao, C.R., Rao, M.B.: Matrix Algebra and its Applications to Statistics and Econometrics. World Scientific Co. Pte. Ltd., Singapore (1998)

    MATH  Google Scholar 

  51. Serfling, R.J.: Multivariate symmetry and asymmetry. In: Kotz, S., Read, C.B., Balakrishnan, N., Vidakovic, B. (eds.) Encyclopedia of Statistical Sciences, 2nd edn. Wiley, New York (2006)

    Google Scholar 

  52. Simaan, Y.: Portfolio selection and asset pricing–three-parameter framework. Manage. Sci. 39, 568–577 (1993)

    MATH  Google Scholar 

  53. Sun, Q., Yan, Y.: Skewness persistence with optimal portfolio selection. J. Bank. Financ. 27, 1111–1121 (2003)

    Google Scholar 

  54. Shushi, T.: Skew-elliptical distributions with applications in risk theory. Eur. Actuar. J. 7, 277–296 (2017)

    MathSciNet  MATH  Google Scholar 

  55. Vernic, R.: Multivariate skew-normal distributions with applications in insurance. Insur. Math. Econ. 38, 413–426 (2006)

    MathSciNet  MATH  Google Scholar 

  56. Yamai, Y., Yoshiba, T.: Value-at-risk versus expected shortfall: a practical perspective. J. Bank. Financ. 29, 997–1015 (2005)

    MATH  Google Scholar 

  57. Yuanyuan, L.U., Jiaming, L.I.: Mean-variance-skewness portfolio selection model based on RBF-GA. Manag. Sci. Eng. 11(1), 47–53 (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Economia, Società e Politica (DESP), Università degli Studi di Urbino “Carlo Bo”, Via Saffi 42, 61029, Urbino, PU, Italy

    Nicola Loperfido

  2. Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Tomer Shushi

Authors
  1. Nicola Loperfido
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomer Shushi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomer Shushi.

Additional information

Communicated by Arvind Raghunathan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Loperfido, N., Shushi, T. Optimal Portfolio Projections for Skew-Elliptically Distributed Portfolio Returns. J Optim Theory Appl 199, 143–166 (2023). https://doi.org/10.1007/s10957-023-02252-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02252-x

Keywords

Search

Navigation