Consider a growth-optimizing investor in a fair game with mutually exclusive outcomes 
(e.g. a “horse race” in which the official odds add up to one).
The rate of return expected by such an investor is equal to the relative entropy 
between the investor's believed probabilities and the official odds.
This is a special case of a much more general connection between financial returns and divergence measures.
Financial risks are connected to  via information geometry. Investors views, the prevailing market view, and risky scenarios form triangles on the relevant manifold of probability distributions. The shape of the triangles determines key financial risks (both qualitatively and quantitatively). For instance, obtuse triangles in which investors views and risk scenarios appear on “opposite sides” relative to the market describe negative risks, acute triangles describe positive exposure, and the right-angled situation in the middle corresponds to zero risk.