Let’s represent a strategy’s return stream using a random variable X with mean μ, variance σ2, and risk-free rate r. The growth rate over n time steps (approximated by the 2nd order taylor expansion) is:
Allow n to go to infinity:
This function is maximized at the optimal kelly-betting fraction / leverage level:
Let r=0 and apply f*:
Refactor:
Now recall the definition of the Information Ratio:
Substitute in the definition of the information ratio:
Refactor:
Indeed, an optimally levered strategy’s growth rate is proportional to it’s Information Ratio (equivalent to a Sharpe Ratio with a zero risk-free-rate). Put differently, a high information ratio strategy not only has a high return per unit of risk, but can also be safely operated at a higher risk-level using leverage. High risk-return strategies deserve to be run hot.