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. The Author 2007. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

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Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?

Mean-variance efficient portfolios constructed using sample moments often involve taking extreme long and short positions. Hence practitioners often impose portfolio weight constraints when constructing efficient portfolios. Green and Hollifield (1992) argue that the presence of a single dominant factor in the covariance matrix of returns is why we observe extreme positive and negative weights. If this were the case then imposing the weight constraint should hurt whereas the empirical evidence is often to the contrary. We reconcile this apparent contradiction. We show that constraining portfolio weights to be nonnegative is equivalent to using the sample covariance matrix after reducing its large elements and then form the optimal portfolio without any restrictions on portfolio weights. This shrinkage helps reduce the risk in estimated optimal portfolios even when they have negative weights in the population. Surprisingly, we also find that once the nonnegativity constraint is imposed, minimum variance portfolios constructed using the monthly sample covariance matrix perform as well as those constructed using covariance matrices estimated using factor models, shrinkage estimators, and daily data. When minimizing tracking error is the criterion, using daily data instead of monthly data helps. However, the sample covariance matrix without any correction for microstructure effects performs the best.

Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps

Green and Hollifield (1992) argue that the presence of a dominant factor would result in extreme negative weights in mean-variance efficient portfolios even in the absence of estimation errors. In that case, imposing no-short-sale constraints should hurt, whereas empirical evidence is often to the contrary. We reconcile this apparent contradiction. We explain why constraining portfolio weights to be nonnegative can reduce the risk in estimated optimal portfolios even when the constraints are wrong. Surprisingly, with no-short-sale constraints in place, the sample covariance matrix performs as well as covariance matrix estimates based on factor models, shrinkage estimators, and daily data. MARKOWITZ'S (1952, 1959) PORTFOLIO THEORY is one of the most important theoretical developments in finance. Mean-variance efficient portfolios play an important role in this theory. Such portfolios constructed using sample moments often involve large negative weights in a number of assets. Since negative portfolio weights (short positions) are difficult to implement in practice, most investors impose the constraint that portfolio weights should be nonnegative when constructing mean-variance efficient portfolios. Green and Hollifield (1992) argue that because a single factor dominates the covariance structure, it would be difficult to dismiss the observed extreme negative and positive weights as being entirely due to the imprecise estimation of the inputs. They consider the global minimum variance portfolio to avoid the effect of estimation error in the mean on portfolio weights. They note that when returns are generated by a single factor model, minimum variance portfolios can be constructed in two steps. First, naively diversify over the set of high beta stocks and

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Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that onlybounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.

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Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach

We develop a model for an investor with multiple priors and aversion to ambiguity. We characterize the multiple priors by a "confidence interval" around the estimated expected returns and we model ambiguity aversion via a minimization over the priors. Our model has several attractive features: (1) it has a solid axiomatic foundation; (2) it is flexible enough to allow for different degrees of uncertainty about expected returns for various subsets of assets and also about the return-generating model; and (3) it delivers closed-form expressions for the optimal portfolio. Our empirical analysis suggests that, compared with portfolios from classical and Bayesian models, ambiguity-averse portfolios are more stable over time and deliver a higher out-of sample Sharpe ratio. (JEL G11) Copyright 2007, Oxford University Press.

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Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach

(1992). Global Portfolio Optimization. Financial Analysts Journal: Vol. 48, No. 5, pp. 28-43.

BlackFischer

Papers

Global Portfolio Optimization