Mean-variance efficient portfolios with many assets: 50% short

Moshe Levy*, Ya'acov Ritov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


Any given set of asset parameters yields a specific mean-variance optimal tangency portfolio. Yet, when the number of assets is large, there are some general characteristics of optimal portfolios that hold 'almost surely'. This paper investigates these characteristics. We analytically show that the proportion of assets held short converges to 50% as the number of assets grows. This is a fundamental and robust property of mean-variance optimal portfolios, and it does not depend on the parameter estimation method, the investment horizon, or on a special covariance structure. While it is known that optimal portfolios may all have positive weights in some special situations (e.g. uncorrelated assets), the analysis shows that these cases occupy a zero measure in the parameter space, and therefore should not be expected to be observed empirically. Thus, our analysis offers a general explanation for the empirical finding of many short positions in optimal portfolios.

Original languageAmerican English
Pages (from-to)1461-1471
Number of pages11
JournalQuantitative Finance
Issue number10
StatePublished - Oct 2011

Bibliographical note

Funding Information:
We are grateful to the editors and two anonymous referees for their helpful comments and suggestions. Financial support from the Zagagi Fund is thankfully acknowledged.


  • Mean-variance analysis
  • Portfolio analysis
  • Portfolio optimization
  • Stochastic matrix analysis


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