A Proof that Portfolios of Assets with Equal Pairwise Correlations Can Never Have Normally Distributed Returns

And a Demonstration of that via Monte-Carlo Simulation

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In my prior article, here, I demonstrated how the effect of correlation on portfolio members returns is sufficient to kill any “normalization” that occurs in large portfolios due to the Central Limit Theorem.

No, the Central Limit Theorem Doesn’t Make Portfolio Returns Normally Distributed. Here’s Why…

In that article I dodged the issue of how to construct multivariate correlated returns, instead using a formula for the number of “effective degrees of freedom” in a portfolio, N*(ρ), when all pairwise correlations are equal.

To recap, this is the “Grinold-Kahn” structure

A Simple Covariance Model for Stocks

and we can, from it, simply compute the “effective degrees of freedom” by computing the variance of an equal weighted portfolio, which is the variance of the mean cross-sectional return of such…