Skip to main content
Article thumbnail
Location of Repository

An analytical investigation of estimators for expected asset returns from the perspective of optimal asset allocation

By Gabriel Frahm


In the present work I derive the risk functions of 5 standard estimators for expected asset returns which are frequently advocated in the literature, viz the sample mean vector, the James-Stein and Bayes-Stein estimator, the minimum-variance estimator, and the CAPM estimator. I resolve the question why it is meaningful to study the risk function in the context of optimal asset allocation. Further, I derive the quantities which determine the risks of the different expected return estimators and show which estimators are preferable with respect to optimal asset allocation. Finally, I discuss the question whether it pays to strive for the optimal portfolio by using time series information. It turns out that in many practical situations it is better to renounce parameter estimation altogether and pursue some trivial strategy such as the totally risk-free investment. --Asset allocation,Bayes-Stein estimator,CAPM estimator,James-Stein estimator,Minimum-variance estimator,Naive diversification,Out-ofsample performance,Risk function,Shrinkage estimation

OAI identifier:

Suggested articles


  1. (2008). A general approach to Bayesian portfolio optimization’,
  2. (1986). An empirical Bayes approach to efficient portfolio selection’,
  3. (2009). Asymptotic distributions of robust shape matrices and scales’,
  4. (1986). Bayes-Stein estimation for portfolio analysis’,
  5. (1991). Bayesian and CAPM estimators of the means: Implications for portfolio selection’,
  6. (2010). Dominating estimators for minimum variance portfolios’, Discussion paper,
  7. (2001). Empirical distributions of stock returns: European securities markets, 1990–95’,
  8. (2006). Estimating the global minimum variance portfolio’,
  9. (1988). For better performance: constrain portfolio weights’,
  10. (2003). Global evidence on the equity risk premium’,
  11. (1992). Global portfolio optimization’,
  12. (1979). Improved estimation for Markowitz portfolios using James-Stein type estimators’, in:
  13. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection’,
  14. (1956). Inadmissability of the usual estimator for the mean of a multivariate normal distribution’, in:
  15. (1985). International portfolio diversification with estimation risk’,
  16. (2008). Linear statistical inference for global and local minimum variance portfolios’, Statistical Papers,
  17. (1980). On estimating the expected return on the market: an exploratory investigation’,
  18. (2007). Optimal portfolio choice with parameter uncertainty’,
  19. (2009). Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy?’,
  20. (2007). Portfolio selection with parameter and model uncertainty: a multi-prior approach’,
  21. (1952). Portfolio selection’,
  22. (2005). Quantitative Risk Management,
  23. (2007). Resampling vs. shrinkage for benchmarked managers’,
  24. (2003). Risk reduction in large portfolios: why imposing the wrong constraints helps’,
  25. (2004). Schätzrisiken in der Portfoliotheorie,
  26. (1995). Stein and CAPM estimators of the means in asset allocation’,
  27. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice’,
  28. (2008). The market price of risk and the equity premium: a legacy of the Great Depression’,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.