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

Abstract

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.