Information efficiency in investment

We answer the question, what should we say about V when we want to gamble on X, and what is it worth? If V=X, we show that every bit of description at rate R is worth a bit of increase Δ(R) in the doubling rate. Thus the efficiency Δ(R)/R is equal to 1. For general V, we provide a single letter characterization for Δ(R). When applied specifically to jointly normal (V,X) with correlation ρ, we find the initial efficiency Δ´(0) is ρ². If V and X are Bernoulli random variables connected by a binary symmetric channel with parameter ρ, the initial efficiency is (1-2p) ². We finally show how much increase in doubling rate is possible when the sender can provide R bits of information about V and side information S is available only to the investor