Isotropic Correlation Models for the Cross-Section of Equity Returns
This note discusses some of the aspects of a model for the covariance of equity returns based on a simple "isotropic" structure in which all pairwise correlations are taken to be the same value. The effect of the structure on feasible values for the common correlation of returns and on the "effective degrees of freedom" within the equity cross-section are discussed, as well as the impact of this constraint on the asymptotic Normality of portfolio returns. An eigendecomposition of the covariance matrix is presented and used to partition variance into that from a common "market" factor and "non-diversifiable" idiosyncratic risk. A empirical analysis of the recent history of the returns of S&P 500 Index members is presented and compared to the expectations from both this model and linear factor models. This analysis supports the isotropic covariance model and does not seem to provide evidence in support of linear factor models. Analysis of portfolio selection under isotropic correlation is presented using mean-variance optimization for both heteroskedastic and homoskedastic cases. Portfolio selection for negative exponential utility maximizers is also discussed for the general case of distributions of returns with elliptical symmetry. The fact that idiosyncratic risk may not be removed by diversification in a model that the data supports undermines the basic premises of structures such as the C.A.P.M. and A.P.T. If the cross-section of equity returns is more accurately described by this structure then an inevitable consequence is that picking stocks is not a "pointless" activity, as the returns to residual risk would be non-zero.