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 A(B), 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-Körner. 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-Rényi maximal correlation between V and X. This provides a connection with the hypercontraction of the Markov operator of Ahlswede and Gács. For the general market the initial efficiency is 1 when the side information V is equal to the stock market outcome X.