Hellinger’s distance to normal distribution as market invariant

Abstract

Main purpose of distance based portfolio constructions is for portfolio imitation. Here we used distance from normal distribution for other purpose. We attempted to find static market invariant within possible linear combinations of given random variables. We conjectured that “closeness” to normal distribution of possible portfolios in market may reliably represent market microstructure with possible correlations between assets. Taking the squared Hellinger’s distance, we sought for each level of desired mean return the portfolio whose return distribution is closest to Gaussian, with variance taken from efficient frontier found by initially solving mean-variance problem. We found that minimal distance differs significantly from market to market. The sensitivity check showed small average sensitivity for 5% change of a 5% portion of data, small sensitivity for adding new variables in simulated market, and extreme sensitivity to bin numbers. Though distance to normality differed among markets, its sensitivity being small enough in average sometimes showed extreme changes.