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A General Framework for Portfolio Theory—Part I: Theory and Various Models

Author

Listed:
  • Stanislaus Maier-Paape

    (Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany)

  • Qiji Jim Zhu

    (Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008, USA)

Abstract

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the capital market pricing model Sharpe (1964), are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover and extend the results in Rockafellar et al. (2006), which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g., when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows for going beyond the “additive” performance measure to a “multiplicative” one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory Lintner (1965) and the leverage space portfolio theory Vince (2009) can also be understood and enhanced under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes far beyond. For simplicity of presentation, we phrase all for a finite underlying probability space and a one period market model, but generalizations to more complex structures are straightforward.

Suggested Citation

  • Stanislaus Maier-Paape & Qiji Jim Zhu, 2018. "A General Framework for Portfolio Theory—Part I: Theory and Various Models," Risks, MDPI, vol. 6(2), pages 1-35, May.
  • Handle: RePEc:gam:jrisks:v:6:y:2018:i:2:p:53-:d:145135
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    References listed on IDEAS

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    1. Jonathan M. Borwein & Qiji J. Zhu, 2016. "A Variational Approach to Lagrange Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 727-756, December.
    2. Stanislaus Maier-Paape, 2016. "Risk averse fractional trading using the current drawdown," Papers 1612.02985, arXiv.org.
    3. J. Tobin, 1958. "Liquidity Preference as Behavior Towards Risk," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(2), pages 65-86.
    4. Stanislaus Maier-Paape & Qiji Jim Zhu, 2017. "A General Framework for Portfolio Theory. Part II: drawdown risk measures," Papers 1710.04818, arXiv.org.
    5. Andreas Hermes & Stanislaus Maier-Paape, 2017. "Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative," Papers 1703.00476, arXiv.org.
    6. Andreas Hermes & Stanislaus Maier-Paape, 2017. "Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative," Risks, MDPI, vol. 5(3), pages 1-19, August.
    7. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    8. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    9. Rockafellar, R. Tyrrell & Uryasev, Stan & Zabarankin, Michael, 2006. "Master funds in portfolio analysis with general deviation measures," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 743-778, February.
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    Cited by:

    1. Stanislaus Maier-Paape & Andreas Platen & Qiji Jim Zhu, 2019. "A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach," Risks, MDPI, vol. 7(2), pages 1-31, June.
    2. Leonie Violetta Brinker, 2021. "Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model," Risks, MDPI, vol. 9(1), pages 1-18, January.
    3. Sagara Dewasurendra & Pedro Judice & Qiji Zhu, 2019. "The Optimum Leverage Level of the Banking Sector," Risks, MDPI, vol. 7(2), pages 1-30, May.

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