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The Distribution of Cross Sectional Momentum Returns

Although there is a vast empirical literature on cross sectional momentum (CSM) returns, there are no known analytical results on their distributional properties due, in part, to the mathematical complexity associated with their determination. In this paper, we derive the density of CSM returns in analytic form, along with moments of all orders, under the assumption that underlying asset returns are multivariate normal. The resulting expressions are highly non-trivial in general and involve truncated normal distributions. The distribution of CSM returns can be formally described as a mixture of the unified skew-normal family of distributions. However, if the asset returns are independent, then the density of the CSM returns is shown to be a mixture of univariate normals. In order to shed light on the general case, we present a detailed analysis of the case of two underlying assets, which is shown to explain many of the key features of CSM returns reported in the empirical literature.