The efficiency of investment information

We investigate how the description of a correlated information V improves the investment in the stock market X. The objective is to maximize the growth rate of wealth in repeated investments, We find a single-letter characterization of the incremental growth rate Δ(R), the maximum increase in growth rate when V is described to the investor at rate R. The incremental growth rate specialized to the horse race market is related to source coding with side information of Wyner and Ahlswede-Kormer. We provide two horse race examples: jointly binary and jointly Gaussian. The initial efficiency Δ´(0) is the maximum possible increase in the growth rate per bit of description, We show that the initial efficiency is related to the dependency between V and the market. In particular, for the horse race market, the initial efficiency is the square of the Hirschfeld-Gebelein-Renyi maximal correlation between V and X. This provides a connection with the hypercontraction of the Markov operator of Ahlswede and Gacs. For the general market the initial efficiency is 1 when the side information V is equal to the stock market outcome X