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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Nonlinear Programming: A Unified Approach

Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small- to medium-scale convex test examples of the Maros-Meszaros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

qpOASES: a parametric active-set algorithm for quadratic programming

Tbis paper investigates the sensitivity of meanvariance(MV)-efficientportfolios to changes in the means of individual assets. When only a budget constraint is imposed on the investment problem, the analytical results indicate that an MV-efficient portfolio's weights, mean, and variance can be extremely sensitive to cbanges in asset means. When nonnegativity constraints are also imposed on the problem, the computational results confirm that a positively weigbted MV-efficient portfolio's weigbts are extremely sensitive to cbanges in asset means, but the portfolio's returns are not. A surprisingly small increase in the mean of Just one asset drives balf the securities from the portfolio. Yet the portfolio's expected return and standard deviation are virtually uncbanged. This paper supersedes "Should Investors Hold Well-Diversified Portfolios?" Financial support from the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. We also acknowledge the most capable assistance of Ruth Cornale, Simon Ng, Jean-Marc Potier, and Frederick Shen. The paper was presented at Simon Fraser University and the Western Finance Association meetings in San Diego. We thank the participants, especially Stephen Brown, John Herzog, John Heaney, Ray Koopman, and Bruce Lehmann, and the editors, Jon Ingersoll and Chester Spatt, for

On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results

In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.

Active set algorithms for isotonic regression: a unifying framework

This paper shows how to perform sensitivity analysis for Mean-Variance MV portfolio problems using a general form of parametric quadratic programming. The analysis allows an investor to examine how parametric changes in either the means or the right-hand side of the constraints affect the composition, mean, and variance of the optimal portfolio. The optimal portfolio and associated multipliers are piecewise linear functions of the changes in either the means or the right-hand side of the constraints. The parametric parts of the solution show the rates of substitution of securities in the optimal portfolio, while the parametric parts of the multipliers show the rates at which constraints are either tightening or loosening. Furthermore, the parametric parts of the solution and multipliers change in different intervals when constraints become active or inactive. The optimal MV paths for sensitivity analyses are piecewise parabolic, as in traditional MV analysis. However, the optimal paths may contain negatively sloping segments and are characterized by types of kinks, i.e., points of nondifferentiability, not found in MV analysis.

Sensitivity Analysis for Mean-Variance Portfolio Problems

We present an ”active set” algorithm for the solution of the convex (but not necessarily strictly convex) parametric quadratic programming problem. The optimal solution and associated multipliers are obtained as piece-wise linear functions of the parameter. At the end of each interval, the active set is changed by either adding, deleting, or exchanging a constraint. The method terminates when either the optimal solution has been obtained for all values of the parameter, or, a further increase in the parameter results in either the feasible region being null or the objective function being unbounded from below. The method used to solve the linear equations associated with a particular active set is left unspecified. The parametric algorithm can thus be implemented using the linear equation solving method of any active set quadratic programming algorithm.

Michael J. Best

Papers

On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results

Active set algorithms for isotonic regression: a unifying framework

Sensitivity Analysis for Mean-Variance Portfolio Problems

An Algorithm for the Solution of the Parametric Quadratic Programming Problem

Linear Programming: Active Set Analysis and Computer Programs