The Distribution of Cross Sectional Momentum Returns

Abstract

The cross sectional momentum (CSM) strategy, considered in Jegadeesh and Titman (1993), is a trading strategy in which assets with highest past returns are equally weighted and held long while assets with lowest returns are also equally weighted but held short. Although anomalous returns from such momentum strategies, indicating persistence or reversal in the relative performance of underlying assets, have been the subject of numerous empirical studies, very little appears to be known about the distributional properties of the CSM returns. Most of the known results in this regard, such as those obtained in Lo and MacKinlay (1990), Jegadeesh and Titman (1993), Lewellen (2002), and Moskowitz, et al. (2012), are limited to expected values and the first order autocorrelations in the returns of alternative momentum strategies in which the assets are weighted not equally, but instead in proportion to their returns over the ranking period. In this paper, we derive the density of CSM returns in analytic form, along with moments of all orders, under a reasonably general set of assumptions on the underlying asset returns. In particular, if the asset returns over the ranking and holding periods are independent then the density of the CSM returns is shown to be a mixture of univariate normals. We also obtain the density and the moments of equally weighted long only portfolios consisting of assets from an arbitrary return percentile band that we refer to as quantile portfolios.