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Approximating the Numeraire Portfolio by Naive Diversification

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Abstract

Estimation theory has shown, due to the limited estimation window available for real asset data, the sample based Markowitz mean-variance approach produces unreliable weights which fluctuate substantially over time. This paper proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numeraire portfolio. The numeraire portfolio is the strictly positive portfolio that, when used as benchmark, makes all benchmarked nonnegative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The paper proves for a well-securitized market that the naive equal value weighted portfolio converges to the numeraire portfolio when the number of constituents tends to infinity. This result is model independent and, therefore, very robust.The systematic construction of diversified stock indices by naive diversification from real data is demonstrated. Even when taking transaction costs into account, these indices significantly outperform the corresponding market capitalization weighted indices in the long run, indicating empirically their asymptotic proximity to the numeraire portfolio. Empirical evidence is presented that the Sharpe ratios of equi-weighted indices surpass significantly those of corresponding market capitalization weighted indices. This empirical stylized fact applies also to the market portfolio of equity markets of countries, which questions the applicability of the intertemporal capital asset pricing model. Finally, in time of financial crisis, a large equi-weighted fund carrying the investments of the major pension funds and insurance companies would provide important liquidity. It would not only dampen the extremes of a crisis but would also moderate the excesses of any asset price bubble.

Suggested Citation

  • Eckhard Platen & Renata Rendek, 2010. "Approximating the Numeraire Portfolio by Naive Diversification," Research Paper Series 281, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:281
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    References listed on IDEAS

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    1. Harry M. Markowitz, 2011. "Investment for the Long Run: New Evidence for an Old Rule," World Scientific Book Chapters,in: THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 35, pages 495-508 World Scientific Publishing Co. Pte. Ltd..
    2. Henry Allen Latane, 1959. "Criteria for Choice Among Risky Ventures," Journal of Political Economy, University of Chicago Press, vol. 67, pages 144-144.
    3. Eckhard Platen, 2004. "A Benchmark Framework for Risk Management," World Scientific Book Chapters,in: Stochastic Processes And Applications To Mathematical Finance, chapter 15, pages 305-335 World Scientific Publishing Co. Pte. Ltd..
    4. Eckhard Platen & Renata Rendek, 2009. "Simulation of Diversified Portfolios in a Continuous Financial Market," Research Paper Series 264, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
    6. Rubinstein, Mark, 1976. "The Strong Case for the Generalized Logarithmic Utility Model as the Premier Model of Financial Markets," Journal of Finance, American Finance Association, vol. 31(2), pages 551-571, May.
    7. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Hakansson, Nils H., 1971. "Capital Growth and the Mean-Variance Approach to Portfolio Selection," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 6(01), pages 517-557, January.
    9. Damir Filipović & Eckhard Platen, 2009. "Consistent Market Extensions Under The Benchmark Approach," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 41-52.
    10. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
    11. Hans Buhlmann & Eckhard Platen, 2002. "A Discrete Time Benchmark Approach for Finance and Insurance," Research Paper Series 74, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    13. Best, Michael J & Grauer, Robert R, 1991. "On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results," Review of Financial Studies, Society for Financial Studies, vol. 4(2), pages 315-342.
    14. Campbell, John Y. & Viceira, Luis M., 2002. "Strategic Asset Allocation: Portfolio Choice for Long-Term Investors," OUP Catalogue, Oxford University Press, number 9780198296942.
    15. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29.
    16. Constantinos Kardaras & Eckhard Platen, 2008. "On Financial Markets where only Buy-And-Hold Trading is Possible," Research Paper Series 213, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Cited by:

    1. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23.
    2. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2.
    3. Jan Baldeaux & Fung & Katja Ignatieva & Eckhard Platen, 2015. "A Hybrid Model for Pricing and Hedging of Long-dated Bonds," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(4), pages 366-398, September.
    4. Eckhard Platen & Renata Rendek, 2009. "Simulation of Diversified Portfolios in a Continuous Financial Market," Research Paper Series 264, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Eckhard Platen & Renata Rendek, 2017. "Market Efficiency and the Growth Optimal Portfolio," Research Paper Series 386, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Dietmar Leisen & Eckhard Platen, 2017. "Investing for the Long Run," Papers 1705.03929, arXiv.org.
    7. Jan Baldeaux & Eckhard Platen, 2015. "Credit Derivative Evaluation and CVA Under the Benchmark Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 305-331, September.
    8. Kevin Fergusson & Eckhard Platen, 2013. "Real World Pricing of Long Term Cash-Linked Annuities and Equity-Linked Annuities with Cash-Linked Guarantees," Research Paper Series 338, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Soumik Pal & Ting-Kam Leonard Wong, 2013. "Energy, entropy, and arbitrage," Papers 1308.5376, arXiv.org, revised Jan 2016.
    10. Ke Du & Eckhard Platen, 2011. "Three-Benchmarked Risk Minimization for Jump Diffusion Markets," Research Paper Series 296, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Ignatieva, Katja & Platen, Eckhard, 2012. "Estimating the diffusion coefficient function for a diversified world stock index," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1333-1349.
    12. Eckhard Platen & Renata Rendek, 2012. "The Affine Nature of Aggregate Wealth Dynamics," Research Paper Series 322, Quantitative Finance Research Centre, University of Technology, Sydney.
    13. Baldeaux Jan & Ignatieva Katja & Platen Eckhard, 2014. "A tractable model for indices approximating the growth optimal portfolio," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(1), pages 1-21, February.
    14. Baldeaux, Jan & Grasselli, Martino & Platen, Eckhard, 2015. "Pricing currency derivatives under the benchmark approach," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 34-48.
    15. Kevin Fergusson & Eckhard Platen, 2017. "Less-Expensive Valuation of Long Term Annuities Linked to Mortality, Cash and Equity," Papers 1711.02808, arXiv.org.
    16. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    17. Eckhard Platen & David Taylor, 2016. "Loading Pricing of Catastrophe Bonds and Other Long-Dated, Insurance-Type Contracts," Papers 1610.09875, arXiv.org.
    18. Ke Du & Eckhard Platen & Renata Rendek, 2012. "Modeling of Oil Prices," Research Paper Series 321, Quantitative Finance Research Centre, University of Technology, Sydney.
    19. repec:wsi:afexxx:v:12:y:2017:i:02:n:s2010495217500105 is not listed on IDEAS

    More about this item

    Keywords

    Numeraire portfolio; growth optimal portfolio; Kelly portfolio; naive diversification; equi-weighted index; market capitalization weighted index; liquidity; Sharpe ratio; intertemporal capital asset pricing model;

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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