Annals of Operations Research

, Volume 266, Issue 1–2, pp 223–253 | Cite as

On robust portfolio and naïve diversification: mixing ambiguous and unambiguous assets

  • A. Burak Paç
  • Mustafa Ç. PınarEmail author
Analytical Models for Financial Modeling and Risk Management


Effect of the availability of a riskless asset on the performance of naïve diversification strategies has been a controversial issue. Defining an investment environment containing both ambiguous and unambiguous assets, we investigate the performance of naïve diversification over ambiguous assets. For the ambiguous assets, returns follow a multivariate distribution involving distributional uncertainty. A nominal distribution estimate is assumed to exist, and the actual distribution is considered to be within a ball around this nominal distribution. Complete information is assumed for the return distribution of unambiguous assets. As the radius of uncertainty increases, the optimal choice on ambiguous assets is shown to converge to the uniform portfolio with equal weights on each asset. The tendency of the investor to avoid ambiguous assets in response to increasing uncertainty is proven, with a shift towards unambiguous assets. With an application on the \(\textit{CVaR}\) risk measure, we derive rules for optimally combining uniform ambiguous portfolio with the unambiguous assets.


Naïve diversification Robust portfolio optimization Ambiguous and unambiguous assets Conditional Value-at-Risk Worst-case risk measures 

Mathematics Subject Classification

91G10 90C15 90C90 



Many thanks to our anonymous refrees. The manuscript greatly benefited from their constructive comments regarding both presentation of ideas and depth of content.


  1. Behr, P., Guettler, A., & Miebs, F. (2013). On portfolio optimization: Imposing the right constraints. Journal of Banking & Finance, 37(4), 1232–1242.CrossRefGoogle Scholar
  2. Benartzi, S., & Thaler, R. H. (2001). Naive diversification strategies in defined contribution saving plans. American Economic Review, 91(1), 79–98.CrossRefGoogle Scholar
  3. Bertsimas, D., Brown, D. B., & Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464–501.CrossRefGoogle Scholar
  4. Bessler, W., Opfer, H., & Wolff, D. (2017). Multi-asset portfolio optimization and out-of-sample performance: An evaluation of Black–Litterman, mean-variance, and naïve diversification approaches. The European Journal of Finance, 23(1), 1–30.CrossRefGoogle Scholar
  5. Black, F., & Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28–43.CrossRefGoogle Scholar
  6. Brown, S. J., Hwang, I., & In, F. (2013). Why optimal diversification cannot outperform naive diversification: Evidence from tail risk exposure. Available at SSRN 2242694.Google Scholar
  7. Calafiore, G. C. (2007). Ambiguous risk measures and optimal robust portfolios. SIAM Journal on Optimization, 18(3), 853–877.CrossRefGoogle Scholar
  8. Cesarone, F., Moretti, J., & Tardella, F. (2016). Optimally chosen small portfolios are better than large ones. Economics Bulletin, 36(4), 1876–1891.Google Scholar
  9. DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009a). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science, 55(5), 798–812.CrossRefGoogle Scholar
  10. DeMiguel, V., Garlappi, L., & Uppal, R. (2009b). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22(5), 1915–1953.CrossRefGoogle Scholar
  11. Erdoğan, E., & Iyengar, G. (2006). Ambiguous chance constrained problems and robust optimization. Mathematical Programming, 107(1–2), 37–61.CrossRefGoogle Scholar
  12. Fabozzi, F. J., Huang, D., & Zhou, G. (2010). Robust portfolios: Contributions from operations research and finance. Annals of Operations Research, 176(1), 191–220.CrossRefGoogle Scholar
  13. Fabozzi, F. J., Kolm, P. N., Pachamanova, D., & Focardi, S. M. (2007). Robust portfolio optimization and management. New York: Wiley.Google Scholar
  14. Fletcher, J. (2011). Do optimal diversification strategies outperform the 1/n strategy in uk stock returns? International Review of Financial Analysis, 20(5), 375–385.CrossRefGoogle Scholar
  15. Frahm, G., Wickern, T., & Wiechers, C. (2012). Multiple tests for the performance of different investment strategies. AStA Advances in Statistical Analysis, 96(3), 343–383.CrossRefGoogle Scholar
  16. Fugazza, C., Guidolin, M., & Nicodano, G. (2015). Equally Weighted vs. Long-Run Optimal Portfolios. European Financial Management, 21(4), 742–789.CrossRefGoogle Scholar
  17. Gibbs, A. L., & Su, F. E. (2002). On choosing and bounding probability metrics. International Statistical Review, 70(3), 419–435.CrossRefGoogle Scholar
  18. Guidolin, M., & Rinaldi, F. (2013). Ambiguity in asset pricing and portfolio choice: A review of the literature. Theory and Decision, 74(2), 183–217.CrossRefGoogle Scholar
  19. Haley, M. R. (2016). Shortfall minimization and the naive (1/n) portfolio: An out-of-sample comparison. Applied Economics Letters, 23(13), 926–929.CrossRefGoogle Scholar
  20. Jacobs, H., Müller, S., & Weber, M. (2014). How should individual investors diversify? An empirical evaluation of alternative asset allocation policies. Journal of Financial Markets, 19, 62–85.CrossRefGoogle Scholar
  21. Kan, R., Wang, X., & Zhou, G. (2016). Optimal portfolio selection with and without risk-free asset. Available at SSRN.Google Scholar
  22. Kan, R., & Zhou, G. (2007). Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(03), 621–656.CrossRefGoogle Scholar
  23. Kirby, C., & Ostdiek, B. (2012). It’s all in the timing: Simple active portfolio strategies that outperform naive diversification. Journal of Financial and Quantitative Analysis, 47(02), 437–467.CrossRefGoogle Scholar
  24. Murtazashvili, I., & Vozlyublennaia, N. (2013). Diversification strategies: Do limited data constrain investors? Journal of Financial Research, 36(2), 215–232.CrossRefGoogle Scholar
  25. Owen, J., & Rabinovitch, R. (1983). On the class of elliptical distributions and their applications to the theory of portfolio choice. The Journal of Finance, 38(3), 745–752.CrossRefGoogle Scholar
  26. Paç, A. B., & Pınar, M. Ç. (2014). Robust portfolio choice with cvar and var under distribution and mean return ambiguity. Top, 22(3), 875–891.CrossRefGoogle Scholar
  27. Pachamanova, D. A. (2013). Robust portfolio selection. Wiley Encyclopedia of Operations Research and Management Science, 1–12.Google Scholar
  28. Pflug, G. C., Pichler, A., & Wozabal, D. (2012). The \(1/n\) investment strategy is optimal under high model ambiguity. Journal of Banking & Finance, 36(2), 410–417.CrossRefGoogle Scholar
  29. Pınar, M. Ç., & Paç, A. B. (2014). Mean semi-deviation from a target and robust portfolio choice under distribution and mean return ambiguity. Journal of Computational and Applied Mathematics, 259, 394–405.CrossRefGoogle Scholar
  30. Romisch, W., & Pflug, G. C. (2007). Modeling, measuring and managing risk. Singapore: World Scientific.Google Scholar
  31. Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.CrossRefGoogle Scholar
  32. Tu, J., & Zhou, G. (2011). Markowitz meets talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics, 99(1), 204–215.CrossRefGoogle Scholar
  33. Villani, C. (2003). Topics in optimal transportation. Providence: American Mathematical Soc.CrossRefGoogle Scholar
  34. Villani, C. (2008). Optimal transport: Old and new. New York: Springer.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Industrial EngineeringBilkent UniversityAnkaraTurkey

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