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?

Investor Psychology and Security Market Under- and Overreactions

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

We 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. 2003. Risk reduction in large portfolios: Why imposing the wrong constraints helps. J. Finance58 1651--1684) and Ledoit and Wolf (Ledoit, O., M. Wolf. 2003. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Empirical Finance10 603--621, and Ledoit, O., M. Wolf. 2004. A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Anal.88 365--411) and the 1/N portfolio studied in DeMiguel et al. (DeMiguel, V., L. Garlappi, R. Uppal. 2009. Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Rev. Financial Stud.22 1915--1953). We also use our framework to propose several new portfolio strategies. For the proposed portfolios, we provide a moment-shrinkage 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), Ledoit and Wolf (2003, 2004), the 1/N portfolio, and other strategies in the literature, such as factor portfolios.

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A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms

Forecast combinations have frequently been found in empirical studies to produce better forecasts on average than methods based on the ex ante best individual forecasting model. Moreover, simple combinations that ignore correlations between forecast errors often dominate more refined combination schemes aimed at estimating the theoretically optimal combination weights. In this chapter we analyze theoretically the factors that determine the advantages from combining forecasts (for example, the degree of correlation between forecast errors and the relative size of the individual models' forecast error variances). Although the reasons for the success of simple combination schemes are poorly understood, we discuss several possibilities related to model misspecification, instability (non-stationarities) and estimation error in situations where the number of models is large relative to the available sample size. We discuss the role of combinations under asymmetric loss and consider combinations of point, interval and probability forecasts.

Chapter 4 Forecast Combinations

The 52-week high momentum strategy in international stock markets

Recurring changes in the institutional and ideological structures of Central and Eastern Europe have made it difficult to quantify and interpret its long-term economic development. This article provides the foundations for a unified picture of economic growth in the region since 1870. We construct a consistent time series of GDP per capita for the present day states of Central and Eastern Europe over the period 1870–1989, and then draw on the literature of the ‘new’ growth theory in economics to assess their long-term economic performance in the wider European setting.

The economic growth of Central and Eastern Europe in comparative perspective, 1870–1989

We develop a jackknife estimator for the conditional variance of a minimum tracking error variance portfolio constructed using estimated covariances. We empirically evaluate the performance of our estimator using an optimal portfolio of 200 stocks that has the lowest tracking error with respect to the S&P 500 benchmark when three years of daily return data are used for estimating covariances. We find that our jackknife estimator provides more precise estimates and suffers less from in-sample optimism when compared to conventional estimators.

https://www.jstor.org/stable/info/40539275

Jackknife Estimator for Tracking Error Variance of Optimal Portfolios

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 co-variance 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 non-negativity constraint is imposed, minimum variance and minimum tracking error portfolios constructed using the sample covariance matrix perform as well as those constructed using covariance matrices estimated using factor models and shrinkage estimators

Tongshu Ma

Papers

The 52-week high momentum strategy in international stock markets

The economic growth of Central and Eastern Europe in comparative perspective, 1870–1989

Jackknife Estimator for Tracking Error Variance of Optimal Portfolios

Risk Reduction in Large Portfolios: A Role for Portfolio Weight Constraints

New Estimates of Income Levels in Central and Eastern Europe, 1870–1910