Abstract
Estimation theory has shown, owing to the limited estimation window available for real asset data, that the sample-based Markowitz mean-variance approach produces unreliable weights that fluctuate substantially over time. This article proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numéraire portfolio (NP). The NP is the strictly positive portfolio that, when used as benchmark, makes all benchmarked non-negative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The article proves for a well-securitized market that the naive equal value-weighted portfolio converges to the NP 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 NP. Finally, in the time of financial crisis, a large equi-weighted fund carrying the investments of major pension funds and insurance companies would provide important liquidity. It would not only dampen the drawdown of a crisis, but would also moderate the excesses of an asset price bubble.
Similar content being viewed by others
References
Bajeux-Besnainou, I. and Portait, R. (1997) The numeraire portfolio: A new perspective on financial theory. The European Journal of Finance 3: 291–309.
Becherer, D. (2001) The numeraire portfolio for unbounded semimartingales. Finance Stochastics 5: 327–341.
Best, M.J. and Grauer, R.R. (1991) On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies 4 (2): 315–342.
Breiman, L. (1961) Optimal gambling systems for favorable games. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, pp. 65–78.
Browne, S. (1999) The risks and rewards of minimizing shortfall probability. Journal of Portfolio Management 25 (4): 76–85.
Bühlmann, H. and Platen, E. (2003) A discrete time benchmark approach for insurance and finance. Astin Bulletin 33 (2): 153–172.
Campbell, J. and Viceira, L. (2002) Strategic Asset allocation, Portfolio Choice for Long Term Investors. Oxford: Oxford University Press.
Cover, T. (1991) Universal portfolios. Mathematical Finance 1: 1–29.
DeMiguel, V., Garlappi, L. and Uppal, R. (2009) Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies 22 (5): 1915–1953.
Filipović, D. and Platen, E. (2009) Consistent market extensions under the benchmark approach. Mathematical Finance 19 (1): 41–52.
Hakansson, N.H. (1971) Capital growth and the mean-variance approach to portfolio selection. Journal of Financial and Quantitative Analysis 6 (1): 517–557.
ICB. (2008) Industry classification benchmark, http://rcbenchmark.com/index.html, accessed 5 March 2011.
Karatzas, I. and Kardaras, C. (2007) The numeraire portfolio in semimartingale financial models. Finance Stochastics 11 (4): 447–493.
Kardaras, C. and Platen, E. (2008) On Financial Markets Where Only Buy-and-hold Trading Is Possible. Sydney, Australa: University of Technology, Sydney. QFRC Research Paper no. 213.
Kelly, J.R. (1956) A new interpretation of information rate. Bell System Technical Journal 35: 917–926.
Latané, H. (1959) Criteria for choice among risky ventures. Journal of Political Economy 38: 145–155.
Litterman, B. and the Quantitative Research Group (2003) Modern Investment Management: An Equilibrium Approach. Wiley Finance.
Long, J.B. (1990) The numeraire portfolio. Journal of Financial Economics 26: 29–69.
Luenberger, D.G. (1998) Investment Science. Oxford: Oxford University Press.
MacLean, L.C., Thorp, E.O. and Ziemba, W. (2011) The Kelly Capital Growth Investment Criterion. Singapore: World Scientific.
Markowitz, H. (1952) Portfolio selection. Journal of Finance VII (1): 77–91.
Markowitz, H. (1976) Investment for the long run: New evidence for an old rule. Journal of Finance XXXI (5): 1273–1286.
Merton, R.C. (1973) An intertemporal capital asset pricing model. Econometrica 41: 867–888.
Platen, E. (2002) Arbitrage in continuous complete markets. Advances in Applied Probability 34 (3): 540–558.
Platen, E. (2004) A benchmark framework for risk management. In: S. Watanabe, (ed.) Stochastic Processes and Applications to Mathematical Finance, Proceedings of the Ritsumeikan Intern. Symposium: World Scientific, pp. 305–335.
Platen, E. (2006) A benchmark approach to asset management. Journal of Asset Management 6 (6): 390–405.
Platen, E. and Heath, D. (2006) A Benchmark Approach to Quantitative Finance. Springer Finance. Berlin: Springer.
Platen, E. and Rendek, R. (2010) Simulation of Diversified Portfolios in a Continuous Financial Market. Sydney, Australia: University of Technology, Sydney. QFRC Research Paper no. 264.
Rubinstein, M. (1976) The strong case for the generalized logarithmic utility model as the premier model of financial markets. Journal of Finance 31: 551–571.
Shiryaev, A.N. (1984) Probability. Berlin: Springer.
Simon, H.A. and Bonini, C.P. (1958) The size distribution of business firms. American Economic Review 48 (4): 607–617.
Stutzer, M.J. (2000) A portfolio performance index. Financial Analysts Journal 56 (3): 52–61.
Thorp, E.O. (1972) Portfolio choice and the Kelly criterion. In: Proceedings of the 1971 Business and Economics Section of the American Statistical Association, Vol. 21, pp. 5–224.
Ziemba, W.T. and Mulvey, J.M. (1998) Worldwide Asset and Liability Modeling. Cambridge: Cambridge University Press.
Acknowledgements
The Industry Classification Benchmark system ICB has been applied, jointly developed by the FTSE and Dow Jones. The data utilized were provided by Thomson Reuters Datastream. The authors thank Harry Markowitz, Stephen Satchell and Adrian Pagan for their interest and stimulating discussions on diversified portfolios, as well as, Hardy Hulley, Vikram Kuriyan and Anthony Tooman for suggestions on the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
1holds the Chair in Quantitative Finance at the University of Technology, Sydney. Before this appointment, he was the Founding Head of the Centre for Financial Mathematics at the Institute of Advanced Studies at the Australian National University and is Adjunct Professor of this University. He has a PhD in Mathematics from the Technical University in Dresden and obtained his Dr Sc from the Academy of Sciences in Berlin, where he was heading the Sector Stochastics at the Weierstrass Institute. He is co-author of three leading books on numerical methods for stochastic differential equations and a book on his innovative benchmark approach at Springer Verlag, the latter highly relevant to quantitative investing. He has authored more than 150 papers in finance, insurance and applied mathematics and serves on the editorial boards of five international journals including Mathematical Finance and Quantitative Finance. His current research interests cover areas ranging from quantitative investing, financial market modelling, portfolio optimization, insurance, pension modelling, long-term derivative pricing and dynamic risk analysis to numerical methods in finance and insurance.
Rights and permissions
About this article
Cite this article
Platen, E., Rendek, R. Approximating the numéraire portfolio by naive diversification. J Asset Manag 13, 34–50 (2012). https://doi.org/10.1057/jam.2011.36
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1057/jam.2011.36