Diversification and Gerneralized Tracking Errors For Correlated Non-Normal Returns

TL;DR

This paper explores how diversification affects the distribution of portfolio returns when asset returns are correlated and non-normal, showing that increasing the number of assets reduces tracking errors under certain symmetry assumptions.

Contribution

It generalizes previous results on tracking errors to non-normal, symmetric distributions, demonstrating convergence properties as the subset size increases.

Findings

Generalized tracking errors decrease with larger subset size n.

Distribution of relative returns becomes symmetric and peaked as n grows.

Results extend prior work to broader classes of asset return distributions.

Abstract

The probability distribution for the relative return of a portfolio constructed from a subset n of the assets from a benchmark, consisting of N assets whose returns are multivariate normal, is completely characterized by its tracking error. However, if the benchmark asset returns are not multivariate normal then higher moments of the probability distribution for the portfolio's relative return are not related to its tracking error. We discuss the convergence of generalized tracking error measures as the size of the subset of benchmark assets increases. Assuming that the joint probability distribution for the returns of the assets is symmetric under their permutations we show that increasing n makes these generalized tracking errors small (even though n<<N). For n>>1 the probability distribution for the portfolio's relative return is approximately symmetric and strongly peaked about the origin. The results of this paper generalize the conclusions of Dynkin et. al. (2002) to more general underlying asset distributions.