Annals of Operations Research

, Volume 282, Issue 1–2, pp 315–329 | Cite as

Managing portfolio diversity within the mean variance theory

  • Anatoly B. SchmidtEmail author
S.I.: Application of O. R. to Financial Markets


It is well documented that the classical mean variance theory (MVT) may yield portfolios (MVTP) that are highly concentrated and/or are outperformed by equal weight portfolios (EWP). In this work, it is proposed to expand the MVT minimizing objective function with an additional term that explicitly controls portfolio diversity (diversity booster DB). DB decreases with growing number of non-zero portfolio weights and has a minimum when all weights are equal. As a result, high values of DB yield EWP. For performance analysis, portfolio constructed with 12 major US equity ETFs is considered. Out-of-sample performance of maximum Sharpe portfolios is tested using statistics of bootstrapped Sharpe ratios for monthly rebalancing periods. It is found that for the 3-year calibrating window, the diversified MVT portfolio (DMVTP) outperformed both MVTP and EWP in 2012–2015. While the MVTP weights were highly concentrated and had sharp jumps between rebalancing periods, the DMVTP weights slowly changed with time.


Mean variance portfolio Portfolio diversity Performance analysis 



I am grateful to anonymous reviewers for constructive comments to my work.


  1. Bender, J., Briand, R., Nielsen, F., & Stefek, D. (2010). Portfolio of risk premia: A new approach to diversification. Journal of Portfolio Management, 36(2), 17–25.CrossRefGoogle Scholar
  2. Brandtner, M. (2013). Conditional value-at-risk, spectral risk measures and (non-) diversification in portfolio selection problems—A comparison with mean-variance analysis. Journal of Banking & Finance, 37(12), 5526–5537.CrossRefGoogle Scholar
  3. Brock, W., Lakonishok, J., & LeBaron, B. (1992). Simple technical trading rules and the stochastic properties of stock returns. Journal of Finance, 47, 1731–1764.CrossRefGoogle Scholar
  4. Bullen, P. S., Mitrinovic, D. S., & Vasic, P. M. (1988). Means and their inequalities. Dordrecht: D. Reidel Publishing Company.CrossRefGoogle Scholar
  5. Chaves, D., Hsu, J., Li, F., & Shakernia, O. (2011). risk parity portfolios vs. other asset allocation heuristic portfolios. The Journal of Investing, 20(1), 108–118.CrossRefGoogle Scholar
  6. Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification. The Journal of Portfolio Management, 35(1), 40–51.CrossRefGoogle Scholar
  7. de Vassal, V. (2001). Risk diversification benefits of multiple-stock portfolios. Journal of Portfolio Management, 27(2), 32–39.CrossRefGoogle Scholar
  8. DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal versus Naïve diversification how inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22, 1915–1953.CrossRefGoogle Scholar
  9. DeMiguel, V., Plyakha, Y., Uppal, R., & Vilkov, G. (2013). Improving portfolio selection using option-implied volatility and skewness. Journal of Financial and Quantitative Analysis, 48, 1813–1845.CrossRefGoogle Scholar
  10. Denvir, E., & Hutson, E. (2006). The performance and diversification benefits of funds of Hedge funds. Journal of International Financial Markets, Institutions & Money, 16(1), 4–22.CrossRefGoogle Scholar
  11. Duchin, R., & Levy, H. (2009). Markowitz versus the Talmudic portfolio diversification strategies. Journal of Portfolio Management, 35, 71–74.CrossRefGoogle Scholar
  12. Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis. New York: Wiley.Google Scholar
  13. Gerber, S., Markowitz, H., & Pujara, P. (2015). Enhancing Multi-Asset Portfolio Construction Under Modern Portfolio Theory with a Robust Co-Movement Measure.
  14. Green, R. C., & Hollifield, B. (1992). When will mean–variance efficient portfolios be well diversified? Journal of Finance, 47, 1785–1809.CrossRefGoogle Scholar
  15. Hirschman, A. O. (1964). The paternity of an index. The American Economic Review, 54(5), 761.Google Scholar
  16. Ibragimov, R., & Walden, J. (2007). The limits of diversification when losses may be large. Journal of Banking & Finance, 31(8), 2551–2569.CrossRefGoogle Scholar
  17. Inker, B. (2011). Dangers of risk parity. The Journal of Investing, 20(1), 90–98.CrossRefGoogle Scholar
  18. Jacobs, H., Müller, S., & Weber, M. (2013). How should individual investors diversify? An empirical evaluation of alternative asset allocation policies. Journal of Financial Markets, 19, 62–85.CrossRefGoogle Scholar
  19. James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning with applications in R. Berlin: Springer.CrossRefGoogle Scholar
  20. Kritzman, M., Page, S., & Turkington, D. (2010). In defense of optimization: The fallacy of 1/N. Financial Analysts Journal, 66, 31–39.CrossRefGoogle Scholar
  21. Lee, W. (2011). Risk-based asset allocation: A new answer to an old question? The Journal of Portfolio Management, 37(4), 11–21.CrossRefGoogle Scholar
  22. Maillard S., Roncalli, T., & Teïletche J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management, 36(4), 60–70.CrossRefGoogle Scholar
  23. Meucci, A. (2009). Managing diversification. Risk, 22(5), 74–79.Google Scholar
  24. Nadler, D., & Schmidt, A. B. (2014). Portfolio theory in terms of partial covariance. Accessed 14 June 2016.
  25. Pérignon, C., & Smith, D. R. (2010). Diversification and value-at-risk. Journal of Banking & Finance, 34, 55–66.CrossRefGoogle Scholar
  26. Qian, E. (2011). Risk parity and diversification. Journal of Investing, 20(1), 119–127.CrossRefGoogle Scholar
  27. Rudin, A. M., & Morgan, J. S. (2006). A portfolio diversification index. Journal of Portfolio Management, 32(2), 81–89.CrossRefGoogle Scholar
  28. Sankaran, J. K., & Patil, A. R. (1999). On the optimal selection of portfolios under limited diversification. Journal of Banking & Finance, 23(11), 1655–1666.CrossRefGoogle Scholar
  29. Tu, J., & Zhou, G. (2009). Markowitz meets Talmud: A combination of sophisticated and naïve diversification strategies. Journal of Financial Economics, 99, 204–215.CrossRefGoogle Scholar
  30. White, H. (2000). A reality check for data snooping. Econometrica, 68, 1097–1126.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kensho Technologies, Inc, One World Trade CenterNew YorkUSA
  2. 2.Department of Finance and Risk EngineeringNYU School of EngineeringBrooklynUSA

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