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Approximating the Numeraire Portfolio by Naive Diversification

Listed author(s):
  • Eckhard Platen


    (Finance Discipline Group, UTS Business School, University of Technology, Sydney)

  • Renata Rendek

    (Finance Discipline Group, UTS Business School, University of Technology, Sydney)

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.

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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 281.

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Length: 28 pages
Date of creation: 01 Aug 2010
Publication status: Published as: Platen, E. and Rendek, R., 2012, "Approximating the Numeraire Portfolio by Naive Diversification", Journal of Asset Management, 13(1), 34-50.
Handle: RePEc:uts:rpaper:281
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  1. 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..
  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, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
  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. 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.
  6. Damir Filipović & Eckhard Platen, 2009. "Consistent Market Extensions Under The Benchmark Approach," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 41-52.
  7. 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.
  8. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29.
  9. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
  10. Campbell, John Y. & Viceira, Luis M., 2002. "Strategic Asset Allocation: Portfolio Choice for Long-Term Investors," OUP Catalogue, Oxford University Press, number 9780198296942.
  11. Markowitz, Harry M, 1976. "Investment for the Long Run: New Evidence for an Old Rule," Journal of Finance, American Finance Association, vol. 31(5), pages 1273-1286, December.
  12. 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.
  13. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
  14. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
  15. 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.
  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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