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A General Framework for Portfolio Theory. Part I: theory and various models

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  • Stanislaus Maier-Paape
  • Qiji Jim Zhu

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 [H. Markowitz, Portfolio Selection, 1959] and its natural generalization, the capital market pricing model, [W. F. Sharpe, Mutual fund performance , 1966] 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 the results in [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Master funds in portfolio analysis with general deviation measures, 2006] that extends 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 to go 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 [J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, 1965] and the leverage space portfolio theory [R. Vince, The Leverage Space Trading Model, 2009] can also be understood under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes much beyond.

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  • Stanislaus Maier-Paape & Qiji Jim Zhu, 2017. "A General Framework for Portfolio Theory. Part I: theory and various models," Papers 1710.04579, arXiv.org.
    • Handle: RePEc:arx:papers:1710.04579
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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. 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.
    3. Stanislaus Maier-Paape, 2016. "Risk averse fractional trading using the current drawdown," Papers 1612.02985, arXiv.org.
    4. 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.
    5. 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.
    6. 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.
    7. Stanislaus Maier-Paape & Qiji Jim Zhu, 2017. "A General Framework for Portfolio Theory. Part II: drawdown risk measures," Papers 1710.04818, arXiv.org.
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    1. Stanislaus Maier-Paape & Qiji Jim Zhu, 2017. "A General Framework for Portfolio Theory. Part II: drawdown risk measures," Papers 1710.04818, arXiv.org.

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