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Is Minimum-variance Investing Really Worth the While? An Analysis with Robust Performance Inference
Authors: Patrick Behr, André Güttler, Felix Miebs
Source: Working Paper
Date: November 2008

There are two interesting portfolios on the efficient frontier: the tangency portfolio and the minimum-variance portfolio. The minimum-variance portfolio is interesting because it does not require computation of expected asset returns, but only of the covariance matrix, which is more stable. Many researchers have estimated the performance of this portfolio and compared it to other portfolios and identified an advantage in terms of performance for this portfolio. Some authors may have found that this portfolio outperforms a value-weighted portfolio (Baker and Haugen 1991; Clarke et al. 2006) or, under short sale constraints, the tangency portfolio (Chan et al. 1999; Jagannathan and Ma 2003; DeMiguel et al. 2007), but it is harder to identify any outperformance of an equally weighted portfolio.

In this article, Behr, Güttler and Miebs examine the performance of the minimum-variance portfolio using robust performance measures. Their major contribution is to use non-parametric performance tests to compare the minimum-variance portfolio, a naively diversified portfolio and a value-weighted benchmark portfolio. This technique had not previously been used in the related literature; its advantage is that it does not require the assumptions of normally and independent and identically distributed returns.

Their work is based on the whole US equity universe represented by the CRSP database over the period from April 1968 to December 2007. They built a constrained myopic minimum-variance portfolio based on Markowitz portfolio selection and using the constraints proposed by Jagannathan and Ma (2003) to reduce estimation error. The portfolio is derived with upper bound constraints to ensure diversification, and with no short sales constraints. Varying the upper bound constraints and using rolling periods of three years, they obtain a set of minimum-variance portfolios. They then evaluate the out-of-sample performance of this portfolio and compare it to the performance of a value-weighted portfolio, which serves as market proxy and to the performance of an equally weighted portfolio. They use various performance measures including the certainty equivalent, the Sharpe ratio, the Sortino ratio, alpha measures based on the one-factor model, the Fama-French (1992) three-factor model, and the Carhart (1997) four-factor model.

The minimum-variance portfolios seem to outperform the value-weighted portfolio for all performance measures that include a measure of total risk. On the contrary, they underperform the equally weighted portfolio for all these measures. Alternatively, if performance measures based on alpha are considered, the minimum-variance portfolios outperform both value-weighted and equally weighted portfolios. The authors used a non-parametric bootstrap test to confirm the significance of their results. The superiority of the constrained minimum-variance portfolio to the value-weighted portfolio holds statistically for different sub-periods, including recessions and periods of low and high volatility. The results also hold for different portfolio revision frequencies, with the most favourable results obtained for the lowest revision frequency. On the contrary, that the constrained minimum-variance portfolio performs better than the equally weighted portfolio is not confirmed statistically.

In their conclusion, the authors suggest further exploration of two questions relating to the construction of the constrained minimum-variance portfolio that, in their view, call for additional investigation: the sensitivity of the portfolio to the revision frequency and the optimal choice of upper bound constraints.


Baker, N., and R. Haugen, “The Efficient Market Inefficiency of Capitalization-Weighted Stock Portfolios”, Journal of Portfolio Management, vol. 17, n°1, 1991, p. 35-40.

Carhart, M., “On the Persistence in Mutual Fund Performance”, Journal of Finance, vol. 52, 1997, p. 57-82.

Chan, L. K. C., J. Karceski, and J. Lakonishok, “On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model”, The Review of Financial Studies, vol. 12, 1999, p. 937-974.

Clarke, R., H. d. S., and S. Thorley, “Minimum-variance Portfolios in the U.S. Equity Market”, Journal of Portfolio Management, vol. 33, 2006, p. 10-24.

DeMiguel, V., L. Garlappi, and R. Uppal, “Optimal versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy”, The Review of Financial Studies, Forthcoming, 2007.

Fama, E. F., and K. R. French, “The Cross-Section of Expected Stock Returns”, Journal of Finance, vol. 47, 1992, p. 427-465.

Jagannathan, R., and T. Ma, “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps”, Journal of Finance, vol. 58, 2003, p. 1651-1684

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