Brazilian Journal of Probability and Statistics

An optimal combination of risk-return and naive hedging

Wan-Yi Chiu

Full-text: Open access


Taking an approach contrary to the mean–variance portfolio, recent studies have appealed to an older wisdom, “the naive rule provides the best solution,” to improve out-of-sample performance in portfolio selection. Naive diversification, which invests equally across risky assets, is such an example of this simple rule. Previous studies also show that a portfolio combining naive diversification with the mean–variance strategy based on minimizing expected quadratic utility losses may show strong out-of-sample performance. Using the mean squared error, this paper derives an optimal combination of nonstochastic allocation and the mean–variance portfolio. We find that this design is equivalent to the combination of the naive rule and mean–variance strategy based on minimizing the expected utility losses. As an application of this finding, we propose a regression-based combination of maximal risk-return hedging and naive hedging. Our illustration also shows out-of-sample performance of a combined hedging that is superior to that of other methods.

Article information

Braz. J. Probab. Stat., Volume 29, Number 3 (2015), 656-676.

Received: June 2013
Accepted: February 2014
First available in Project Euclid: 11 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Optimal hedging risk-return hedging naive hedging naive rule combined forecasts


Chiu, Wan-Yi. An optimal combination of risk-return and naive hedging. Braz. J. Probab. Stat. 29 (2015), no. 3, 656--676. doi:10.1214/14-BJPS238.

Export citation


  • Bates, J. M. and Granger, C. W. J. (1969). The combination of forecasts. Operational Research Quarterly 20, 451–468.
  • Best, M. J. and Grauer, R. R. (1991). On the sensitivity of mean-variance portfolios to change in asset means: Some analytical and computation results, and the structure of asset expected returns. Review of Financial Studies 4, 315–342.
  • Black, F. and Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal 48, 28–43.
  • Chiu, W. Y. (2013). A simple test of optimal hedging policy. Statistics and Probability Letters 83, 1062–1070.
  • Clemen, R. T. (1989). Combining forecasts: A review and annotated bibliography. International Journal of Forecasting 5, 559–583.
  • Chopra, V. K. and Ziemba, W. T. (1993). The effects of errors in means, variance, and covariance on optimal portfolio choice. Journal of Portfolio Management 19, 6–11.
  • Das, S., Markowitz, H., Scheidm, J. and Statman, M. (2010). Portfolio optimization with mental accounts. Journal of Financial and Quantitative Analysis 45, 311–334.
  • DeMiguel, V., Garlappi, L., Nogales, F. and Uppal, R. (2009a). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management science 55, 798–812.
  • DeMiguel, V., Garlappi, L. and Uppal, R. (2009b). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies 22, 1915–1953.
  • Duchin, R. and Levy, H. (2009). Markowitz versus the Talmudic portfolio diversification strategies. Journal of Portfolio Management 35, 71–74.
  • Green, R. C. and Hollifield, B. (1992). When will mean-variance efficient portfolios be well diversified? Journal of Finance 47, 1785–1295.
  • Grossman, S. A. and Stiglitz, J. E. (1980). On the impossibility of information efficient markets. American Economic Review 70, 393–408.
  • Haff, L. R. (1979). An identity for the Wishart distribution with applications. Journal of Multivariate Analysis 9, 531–544.
  • Haugen, R. A. and Baker, N. L. (1991). The efficient market inefficiency of capitalization-weighted stock portfolios. Journal of Portfolio Management 17, 35–40.
  • Hirschberger, M., Steuer, R. E., Utz, S., Wimmer, M. and Qi, Y. (2013). Computing the nondominated surface in tri-criterion portfolio selection. Operations Research 61, 169–183.
  • Hwang, C. L. and Masud, A. S. M. (1979). Multiple Objective Decision Making-Methods and Applications. Berlin: Springer.
  • Jagannthan, R. and Ma, T. (2003). Risk reduction in large portfolio: Why imposing the wrong constraints helps. Journal of Finance 58, 1651–1683.
  • Jobson, J. D. and Korkie, B. (1980). Estimation for Markowitz efficient portfolios. Journal of the American Statistical Association 56, 544–554.
  • Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis 21, 279–292.
  • Kan, R. and Zhou, G. (2007). Optimal portfolio choice with parameters uncertainty. Journal of Financial and Quantitative Analysis 42, 621–656.
  • Khoury, N. T. and Martel, J. M. (1985). Optimal futures hedging in the presence of asymmetric information. Journal of Futures Markets 5, 595–605.
  • Kirby, C. and Ostdiek, B. (2012). It’s all in the timing: Simple active portfolio strategies that outperform naive diversification. Journal of Financial and Quantitative Analysis 47, 437–467.
  • Kirilenko, A. A. (2001). Valuation and control in venture finance. Journal of Finance 56, 565–587.
  • Ledoit, O. and Wolf, M. (2004). Honey, I shrink the sample covariance. Journal of Portfolio Management 30, 110–119.
  • Markowitz, H. M. (1952). Portfolio selection. Journal of Finance 7, 79–91.
  • MacKinlay, A. C. and Pástor, L. (2000). Asset pricing models: Implication for expected returns and portfolio selection. Review of Financial Studies 13, 883–916.
  • Merton, R. C. (1972). An analytic derivation of the efficient portfolio frontier. Journal of Financial and Quantitative Analysis 7, 1851–1872.
  • Okhrin, Y. and Schmid, W. (2006). Distributional properties of portfolio weights. Journal of Econometrics 134, 235–256.
  • Tu, J. and Zhou, G. (2004). Data-generating process uncertainty: What difference does it make in portfolio decisions? Journal of Financial Economics 72, 385–421.
  • Tu, J. and Zhou, G. (2011). Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics 99, 204–215.
  • Wang, Z. (2005). A shrinkage approach to model uncertainty and asset allocation. Review of Financial Studies 18, 673–705.