IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1710.04579.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1710.04579
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Stanislaus Maier-Paape & Qiji Jim Zhu, 2017. "A General Framework for Portfolio Theory. Part II: drawdown risk measures," Papers 1710.04818, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Schoch, Daniel, 2017. "Generalised mean-risk preferences," Journal of Economic Theory, Elsevier, vol. 168(C), pages 12-26.
    3. Gilles Boevi Koumou, 2020. "Diversification and portfolio theory: a review," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 34(3), pages 267-312, September.
    4. Leitner Johannes, 2005. "Optimal portfolios with expected loss constraints and shortfall risk optimal martingale measures," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 49-66, January.
    5. Jaehyung Choi & Hyangju Kim & Young Shin Kim, 2021. "Diversified reward-risk parity in portfolio construction," Papers 2106.09055, arXiv.org, revised Sep 2022.
    6. Barbara Glensk & Reinhard Madlener, 2018. "Fuzzy Portfolio Optimization of Power Generation Assets," Energies, MDPI, vol. 11(11), pages 1-22, November.
    7. Rockafellar, R. Tyrrell & Uryasev, Stan & Zabarankin, M., 2007. "Equilibrium with investors using a diversity of deviation measures," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3251-3268, November.
    8. Adil Rengim Cetingoz & Jean-David Fermanian & Olivier Gu'eant, 2022. "Risk Budgeting Portfolios: Existence and Computation," Papers 2211.07212, arXiv.org, revised Sep 2023.
    9. Luciano de Castro & Antonio F. Galvao & Gabriel Montes-Rojas & Jose Olmo, 2022. "Portfolio selection in quantile decision models," Annals of Finance, Springer, vol. 18(2), pages 133-181, June.
    10. Allen, D.E. & Powell, R.J. & Singh, A.K., 2016. "Take it to the limit: Innovative CVaR applications to extreme credit risk measurement," European Journal of Operational Research, Elsevier, vol. 249(2), pages 465-475.
    11. Gianfranco Guastaroba & Renata Mansini & Wlodzimierz Ogryczak & M. Grazia Speranza, 2020. "Enhanced index tracking with CVaR-based ratio measures," Annals of Operations Research, Springer, vol. 292(2), pages 883-931, September.
    12. 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.
    13. Dipankar Mondal & N. Selvaraju, 2022. "Convexity, two-fund separation and asset ranking in a mean-LPM portfolio selection framework," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(1), pages 225-248, March.
    14. Anders Johansson, 2009. "An analysis of dynamic risk in the Greater China equity markets," Journal of Chinese Economic and Business Studies, Taylor & Francis Journals, vol. 7(3), pages 299-320.
    15. Chia-Lin Chang & Jukka Ilomäki & Hannu Laurila & Michael McAleer, 2018. "Long Run Returns Predictability and Volatility with Moving Averages," Risks, MDPI, vol. 6(4), pages 1-18, September.
    16. Hany Shawky & Ronald Forbes & Alan Frankle, 1983. "Liquidity Services and Capital Market Equilibrium: The Case for Money Market Mutual Funds," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 6(2), pages 141-152, June.
    17. Malavasi, Matteo & Ortobelli Lozza, Sergio & Trück, Stefan, 2021. "Second order of stochastic dominance efficiency vs mean variance efficiency," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1192-1206.
    18. Giovanni Bonaccolto & Massimiliano Caporin & Sandra Paterlini, 2018. "Asset allocation strategies based on penalized quantile regression," Computational Management Science, Springer, vol. 15(1), pages 1-32, January.
    19. Rostagno, Luciano Martin, 2005. "Empirical tests of parametric and non-parametric Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) measures for the Brazilian stock market index," ISU General Staff Papers 2005010108000021878, Iowa State University, Department of Economics.
    20. Dimitrios G. Konstantinides & Georgios C. Zachos, 2019. "Exhibiting Abnormal Returns Under a Risk Averse Strategy," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 551-566, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1710.04579. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.