Abstract
We present an explicit closed-form solution to the problem of minimizing the combination of linear functional and a function of quadratic functional, subject to a system of affine constraints. This is of interest for solving important problems in financial economics related to optimal portfolio selection. The new results essentially generalize previous results of the authors concerning optimal portfolio selection with translation invariant and positive homogeneous risk measures. The classical mean-variance model and the recently introduced and investigated tail mean-variance model are special cases of the problem discussed here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In this equation and many others, a bold index 1 corresponds to the first \(n-m\) elements. Similarly, a bold index 2 corresponds to the remaining m elements.
References
Steinbach, M.C.: Markowitz revisited: mean-variance models in financial portfolio analysis. SIAM Rev. 43(1), 31–85 (2001)
Elton, E.J., Gruber, M.J.: Modern Portfolio Theory and Investment Analysis. Wiley, New York (1987)
Zenios, S.A.: Financial Optimization. Cambridge University Press, Cambridge (1992)
Ziemba, W.T., Mulvey, J.M.: Worldwide Asset and Liability Modeling. Cambridge University Press, Cambridge (1998)
McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management. Princeton University Press, Princeton (2005)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Boyle, P.P., Cox, S.H., et al.: Financial Economics: with Applications to Investments, Insurance and Pensions. Actuarial Foundation, Schaumburg (1998)
Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77–91 (1952)
Landsman, Z.: Minimization of the root of a quadratic functional under an affine equality constraint. J. Comput. Appl. Math. 216(2), 319–327 (2008)
Landsman, Z.: On the tail mean-variance optimal portfolio selection. Insur. Math. Econ. 46, 547–553 (2010)
Landsman, Z., Makov, U.: Translation-invariant and positive-homogeneous risk measures and optimal portfolio management. Eur. J. Finance 17(4), 307–320 (2011)
Landsman, Z.: Minimization of the root of a quadratic functional under a system of affine equality constraints with application to portfolio management. J. Comput. Appl. Math. 220(1–2), 739–748 (2008)
Luenberger, D.G.: Linear and Nonlinear Programming. Addison-Wesley, Stanford (1984)
Merton, R.C.: An analytic derivation of the efficient portfolio frontier. J. Financ. Quant. Anal. 7, 1851–1872 (1972)
Furman, E., Landsman, Z.: Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bull. 36(2), 433–452 (2006)
Neumark, S.: Solution of Cubic and Quartic Equations. Pergamon Press, Oxford (1995)
Acknowledgments
We are grateful to two reviewers and the Editor-in-Chief for their useful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10957-015-0856-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0856-z