Skip to main content
SpringerLink
Log in

Portfolio optimization with two coherent risk measures

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We provide analytical results for a static portfolio optimization problem with two coherent risk measures. The use of two risk measures is motivated by joint decision-making for portfolio selection where the risk perception of the portfolio manager is of primary concern, hence, it appears in the objective function, and the risk perception of an external authority needs to be taken into account as well, which appears in the form of a risk constraint. The problem covers the risk minimization problem with an expected return constraint and the expected return maximization problem with a risk constraint, as special cases. For the general case of an arbitrary joint distribution for the asset returns, under certain conditions, we characterize the optimal portfolio as the optimal Lagrange multiplier associated to an equality-constrained dual problem. Then, we consider the special case of Gaussian returns for which it is possible to identify all cases where an optimal solution exists and to give an explicit formula for the optimal portfolio whenever it exists.

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

Similar content being viewed by others

References

  1. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MathSciNet  Google Scholar 

  2. Barahona, F., Anbil, R.: The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87(3), 385–399 (2000)

    Article  MathSciNet  Google Scholar 

  3. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. 57(1), 15–48 (1992)

    Article  Google Scholar 

  4. Capiński, M., Zastawniak, T.: Mathematics for Finance, 2nd edn. Springer, London (2011)

    Book  Google Scholar 

  5. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. De Gruyter Textbook Series, 2nd edn. De Gruyter, Berlin (2016)

    Book  Google Scholar 

  6. Glück, J., Weber, M.R.: Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups. arXiv:1901.03306 (2019)

  7. Hale, J.K., Magalhães, L.T., Oliva, W.: Dynamics in Infinite Dimensions, 2nd edn. Springer, New York (2002)

    Book  Google Scholar 

  8. Kaina, M., Rüschendorf, L.: On convex risk measures on \(L^p\)-spaces. Math. Methods Oper. Res. 69, 475–95 (2009)

    Article  MathSciNet  Google Scholar 

  9. Landsman, Z.: Minimization of the root of a quadratic functional under an affine equality constraint. J. Comput. Appl. Math. 216, 319–327 (2008)

    Article  MathSciNet  Google Scholar 

  10. Landsman, Z., Makov, U.: Minimization of a function of a quadratic functional with application to optimal portfolio selection. J. Optim. Theory Appl. 170, 308–322 (2016)

    Article  MathSciNet  Google Scholar 

  11. Larsson, T., Patriksson, M., Strömberg, A.: Ergodic, primal convergence in dual subgradient schemes for convex programming. Math. Program. 86(2), 283–312 (1999)

    Article  MathSciNet  Google Scholar 

  12. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  13. Merton, R.C.: An analytic derivation of the efficient portfolio frontier. J. Financ. Quant. Anal. 74, 1851–1872 (1972)

    Article  Google Scholar 

  14. Owadally, I.: An improved closed-form solution for the constrained minimization of the root of a quadratic functional. J. Comput. Appl. Math. 236, 4428–4435 (2011)

    Article  MathSciNet  Google Scholar 

  15. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

    Article  Google Scholar 

  16. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  17. Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)

    Article  MathSciNet  Google Scholar 

  18. Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  Google Scholar 

Download references

Acknowledgements

We are grateful to Oya Ekin Karaşan for her continuous support at the early stages of the project. In addition, we would like to thank two anonymous referees whose comments motivated us for much of the developments in Sect. 3.

Author information

Authors and Affiliations

  1. Booth School of Business, The University of Chicago, Chicago, IL, USA

    Tahsin Deniz Aktürk

  2. Department of Industrial Engineering, Bilkent University, Ankara, Turkey

    Çağın Ararat

Authors
  1. Tahsin Deniz Aktürk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Çağın Ararat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Çağın Ararat.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aktürk, T.D., Ararat, Ç. Portfolio optimization with two coherent risk measures. J Glob Optim 78, 597–626 (2020). https://doi.org/10.1007/s10898-020-00922-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00922-y

Keywords

Mathematics Subject Classification

Search

Navigation