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Multiple Optimal Solutions in the Portfolio Selection Model with Short-Selling

Author

Listed:
  • A. Schianchi

    (Dipartimento di Economia dell' Università di Parma, via Kennedy 6, 43100 Parma, Italy)

  • L. Bongini

    (Institute Nazionale di Fisica della Materia, Largo E. Fermi 2, 50125 Firenze, Italy)

  • M. D. Esposti

    (Dipertimento di Matematica dell' Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy)

  • C. Giardinà

    (Dipertimento di Matematica dell' Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy)

Abstract

In this paper an extension of the Lintner model [1] is considered: the problem of portfolio optimization is studied when short-selling is allowed through the mechanism of margin requirements. This induces a non-linear constraint on the wealth. When interest on deposited margin is present, Lintner ingeniously solved the problem by recovering the unique optimal solution of the linear model (no margin requirements). In this paper an alternative and more realistic approach is explored: the nonlinear constraint is maintained but no interest is perceived on the money deposited against short-selling. This leads to a fully non-linear problem which admits multiple and unstable solutions very different among themselves but corresponding to similar risk levels. Our analysis is built on a seminal idea by Galluccio, Bouchaud and Potters [3], who have re-stated the problem of finding solutions of the portfolio optimization problem in futures markets in terms of a spin glass problem. In order to get the best portfolio (i.e. the one lying on the efficiency frontier), we have to implement a two-step procedure. A worked example with real data is presented.

Suggested Citation

  • A. Schianchi & L. Bongini & M. D. Esposti & C. Giardinà, 2003. "Multiple Optimal Solutions in the Portfolio Selection Model with Short-Selling," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(07), pages 703-720.
    • Handle: RePEc:wsi:ijtafx:v:06:y:2003:i:07:n:s021902490300216x
    DOI: 10.1142/S021902490300216X
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    References listed on IDEAS

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    1. V. Plerou & P. Gopikrishnan & B. Rosenow & L. A. N. Amaral & T. Guhr & H. E. Stanley, 2001. "A Random Matrix Approach to Cross-Correlations in Financial Data," Papers cond-mat/0108023, arXiv.org.
    2. B. Rosenow & V. Plerou & P. Gopikrishnan & H. E. Stanley, 2001. "Portfolio Optimization and the Random Magnet Problem," Papers cond-mat/0111537, arXiv.org.
    3. Szilard Pafka & Imre Kondor, 2001. "Noisy Covariance Matrices and Portfolio Optimization," Papers cond-mat/0111503, arXiv.org.
    Full references (including those not matched with items on IDEAS)

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