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Equity portfolios generated by functions of ranked market weights


  • Robert Fernholz

    () (INTECH, One Palmer Square, Princeton, NJ 08542, USA Manuscript)


Dynamic equity portfolios can be generated by positive twice continuously differentiable functions of the ranked capitalization weights of an equity market. The return on such a portfolio relative to the market follows a stochastic differential equation that decomposes the relative return into two components: the logarithmic change in the value of the generating function, and a drift process that is of bounded variation. The method can be used to construct broad classes of stock portfolios, and has both theoretical and practical applications. Two applications of the method are presented: one offers an explanation for the size effect, the observed tendency of small stocks to have higher long-term returns than large stocks, and the other provides a rigorous analysis of the behavior of diversity-weighted indices, stock indices with weights that lie between capitalization weights and equal weights.

Suggested Citation

  • Robert Fernholz, 2001. "Equity portfolios generated by functions of ranked market weights," Finance and Stochastics, Springer, vol. 5(4), pages 469-486.
  • Handle: RePEc:spr:finsto:v:5:y:2001:i:4:p:469-486
    Note: received: November 1999; final version received: November 2000

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    Cited by:

    1. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0332-8 is not listed on IDEAS
    2. Alexander Vervuurt & Ioannis Karatzas, 2015. "Diversity-Weighted Portfolios with Negative Parameter," Papers 1504.01026,, revised Jul 2015.
    3. Robert Fernholz, 2016. "A new decomposition of portfolio return," Papers 1606.05877,
    4. Constantinos Kardaras & Scott Robertson, 2018. "Ergodic robust maximization of asymptotic growth," Papers 1801.06425,
    5. Diane Wilcox & Tim Gebbie, 2013. "Factorising equity returns in an emerging market through exogenous shocks and capital flows," Papers 1306.5302,, revised Jul 2013.
    6. Ricardo T. Fernholz & Robert Fernholz, 2017. "Universality of Zipf's Law for Time-Dependent Rank-Based Systems," Papers 1707.04285,, revised May 2018.
    7. Johannes Ruf & Kangjianan Xie, 2018. "Generalised Lyapunov Functions and Functionally Generated Trading Strategies," Papers 1801.07817,
    8. Francesco Audrino & Robert Fernholz & Roberto Ferretti, 2007. "A Forecasting Model for Stock Market Diversity," Annals of Finance, Springer, vol. 3(2), pages 213-240, March.
    9. Robert Fernholz, 2017. "Stratonovich representation of semimartingale rank processes," Papers 1705.00336,
    10. Ali Al-Aradi & Sebastian Jaimungal, 2018. "Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management," Papers 1803.05819,, revised Mar 2018.
    11. Guillaume Bernis & Nicolas Brunel & Antoine Kornprobst & Simone Scotti, 2017. "Stochastic Evolution of Distributions - Applications to CDS indices," Documents de travail du Centre d'Economie de la Sorbonne 17007, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    12. Guillaume Bernis & Nicolas Brunel & Antoine Kornprobst & Simone Scotti, 2017. "Stochastic Evolution of Distributions - Applications to CDS indices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01467736, HAL.
    13. Constantinos Kardaras & Scott Robertson, 2010. "Robust maximization of asymptotic growth," Papers 1005.3454,, revised Aug 2012.
    14. Soumik Pal & Ting-Kam Leonard Wong, 2016. "Exponentially concave functions and a new information geometry," Papers 1605.05819,, revised May 2017.
    15. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988,
    16. Ioannis Karatzas & Johannes Ruf, 2016. "Trading Strategies Generated by Lyapunov Functions," Papers 1603.08245,
    17. Alexander Vervuurt & Ioannis Karatzas, 2015. "Diversity-weighted portfolios with negative parameter," Annals of Finance, Springer, vol. 11(3), pages 411-432, November.
    18. Raouf Ghomrasni, 2005. "On Local Times of Ranked Continuous Semimartingales;Application to Portfolio Generating Functions," SFB 649 Discussion Papers SFB649DP2005-043, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    19. Attila Herczegh & Vilmos Prokaj & Mikl'os R'asonyi, 2013. "Diversity and no arbitrage," Papers 1301.4173,, revised Aug 2014.
    20. Kardaras, Constantinos & Robertson, Scott, 2012. "Robust maximization of asymptotic growth," LSE Research Online Documents on Economics 44994, London School of Economics and Political Science, LSE Library.

    More about this item


    Portfolio-generating function; local time; size effect; diversity-weighted index;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • G19 - Financial Economics - - General Financial Markets - - - Other


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