Analysis of efficient markets

A simple trading model is presented in which Bayes’ rule is used to aggregate traders’ forecasts about risky assets’ future returns. In this financial market, Bayes’ rule operates like an omnipotent market-maker performing functions that in 1776 Adam Smith attributed to an “invisible hand.” We have analyzed two distinct cases: in the first scenario, the traders’ forecast errors are uncorrelated, and in the second scenario, the traders’ forecast errors are correlated. The contribution of our paper is fourfold: first, we prove that the “efficient market” mean-return can be expressed as a complex linear combination of the traders’ forecasts. The weights depend on the forecast variances, as well as on the correlations among the traders’ forecasts. Second we show that the “efficient” variance is equal to the inverse of the sum of the traders’ precision errors, and is also related to the correlations among the traders’ forecast errors. Third, we prove that the efficient market return is the best linear minimum variance estimator (BLMVE) of the security’s mean return (in the sense that it minimizes the sum of the traders’ mean squared forecast errors). Thus, an efficient market aggregates traders’ heterogeneous information in an optimal way. Fourth, we prove that an efficient market produces a mean return (price) as a Blackwell sufficient (most informative) experiment among all possible aggregated expected return (price) forecasts.