  Using detailed statistical analyses of the size distribution of a universe of
equity exchange-traded funds (ETFs), we discover a discrete hierarchy of sizes,
which imprints a log-periodic structure on the probability distribution of ETF
sizes that dominates the details of the asymptotic tail. This allows us to
propose a classification of the studied universe of ETFs into seven size layers
approximately organized according to a multiplicative ratio of 3.5 in their
total market capitalization. Introducing a similarity metric generalising the
Herfindhal index, we find that the largest ETFs exhibit a significantly
stronger intra-layer and inter-layer similarity compared with the smaller ETFs.
Comparing the performance across the seven discerned ETF size layers, we find
an inverse size effect, namely large ETFs perform significantly better than the
small ones both in 2014 and 2015.
