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W e provide a general framework for finding portfolios that perform well out-of-sample in the presence of estimation error. This framework relies on solving the traditional minimum-variance problem but subject to the additional constraint that the norm of the portfolio-weight vector be smaller than a given threshold. We show that our framework nests as special cases the shrinkage approaches of Jagannathan and Ma (Jagannathan, R., T. Ma. Rev. Financial Stud. 22 1915-1953. We also use our framework to propose several new portfolio strategies. For the proposed portfolios, we provide a momentshrinkage interpretation and a Bayesian interpretation where the investor has a prior belief on portfolio weights rather than on moments of asset returns. Finally, we compare empirically the out-of-sample performance of the new portfolios we propose to 10 strategies in the literature across five data sets. We find that the norm-constrained portfolios often have a higher Sharpe ratio than the portfolio strategies in Jagannathan and Ma (2003), Wolf (2003, 2004), the 1/N portfolio, and other strategies in the literature, such as factor portfolios.

We evaluate the out-of-sample performance of the sample-based mean-variance model, and its extensions designed to reduce estimation error, relative to the naive 1/N portfolio. Of the 14 models we evaluate across seven empirical datasets, none is consistently better than the 1/N rule in terms of Sharpe ratio, certainty-equivalent return, or turnover, which indicates that, out of sample, the gain from optimal diversification is more than offset by estimation error. Based on parameters calibrated to the US equity market, our analytical results and simulations show that the estimation window needed for the sample-based mean-variance strategy and its extensions to outperform the 1/N benchmark is around 3000 months for a portfolio with 25 assets and about 6000 months for a portfolio with 50 assets. This suggests that there are still many "miles to go" before the gains promised by optimal portfolio choice can actually be realized out of sample. (JEL G11) In about the fourth century, Rabbi Issac bar Aha proposed the following rule for asset allocation: "One should always divide his wealth into three parts: a third in land, a third in merchandise, and a third ready to hand." 1 After a "brief" We wish to thank Matt Spiegel (the editor), two anonymous referees, andĽuboš Pástor for extensive comments; John Campbell and Luis Viceira for their suggestions and for making available their data and computer code; and Roberto Wessels for making available data on the ten sector portfolios of the S&P 500 Index. We also gratefully acknowledge the comments from

In this paper, we extend the mean-variance portfolio model where expected returns are obtained using maximum likelihood estimation to explicitly account for uncertainty about the estimated expected returns. In contrast to the Bayesian approach to estimation error, where there is only a single prior and the investor is neutral to uncertainty, we allow for multiple priors and aversion to uncertainty. We characterize the set of priors as a confidence interval around the estimated value of expected return and we model aversion to uncertainty via a minimization over the set of priors. The multi-prior model has several attractive features: One, just like the Bayesian model, the multi-prior model is firmly grounded in decision theory; Two, it is flexible enough to allow for uncertainty about expected returns estimated jointly for all assets or different levels of uncertainty about expected returns for different subsets of the assets; Three, we show how in several special cases of the multi-prior model one can obtain closed-form expressions for the optimal portfolio, which can be interpreted as a shrinkage of the mean-variance portfolio towards either the risk-free asset or the minimum variance portfolio. We illustrate how to implement the multi-prior model using both international and domestic data. Our analysis suggests that allowing for parameter uncertainty reduces the fluctuation of portfolio weights over time and, for the data set considered, improves the out-of sample performance.

In this paper, we extend the mean-variance portfolio model where expected returns are obtained using maximum likelihood estimation to explicitly account for uncertainty about the estimated expected returns. In contrast to the Bayesian approach to estimation error, where there is only a single prior and the investor is neutral to uncertainty, we allow for multiple priors and aversion to uncertainty. We characterize the set of priors as a confidence interval around the estimated value of expected return and we model aversion to uncertainty via a minimization over the set of priors. The multi-prior model has several attractive features: One, just like the Bayesian model, the multi-prior model is firmly grounded in decision theory; Two, it is flexible enough to allow for uncertainty about expected returns estimated jointly for all assets or different levels of uncertainty about expected returns for different subsets of the assets; Three, we show how in several special cases of the multi-prior model one can obtain closed-form expressions for the optimal portfolio, which can be interpreted as a shrinkage of the mean-variance portfolio towards either the risk-free asset or the minimum variance portfolio. We illustrate how to implement the multi-prior model using both international and domestic data. Our analysis suggests that allowing for parameter uncertainty reduces the fluctuation of portfolio weights over time and, for the data set considered, improves the out-of sample performance.