Averaging financial ratios?

Ratios represent a special kind of relation between two magnitudes, and computing the average of ratios is fairly common among academics and Finance practitioners. How should price to-earnings (P/E) ratios be aggregated (averaged) at the portfolio level to provide a unified number? The arithmetic mean is the natural alternative. However, in case of financial ratios, it is generally accepted that the much less familiar harmonic mean may be more valuable, because it solves the upward bias encountered when using arithmetic mean. However, and to the best of our knowledge, there is no statistical evidence to show the superiority of the harmonic mean when computing the average of ratios. In this paper, by bootstrapping P/E ratios and earnings yield of companies listed in eight common stock indices, we compare the traditional averages and it is shown that geometric, not the harmonic average, as it is commonly accepted, is more suitable to average the ratios.