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A strategy's growth rate is proportional to the square of it's Information Ratio

Posted on:July 30, 2023 at 12:00 PM

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:

Gn(f)=r+(μr)f(σf)22+O(n12)G_n(f) = r + (\mu - r)f - \frac{(\sigma f)^2}{2} + O(n^{-\frac{1}{2}})

Allow n to go to infinity:

G(f)=r+(μr)f(σf)22G_{\infty}(f) = r + (\mu - r)f - \frac{(\sigma f)^2}{2}

This function is maximized at the optimal kelly-betting fraction / leverage level:

f=μrσ2f^* = \frac{\mu - r}{\sigma^2}

Let r=0 and apply f*:

G(f)=μ2σ2μ2σ22G_{\infty}(f) = \frac{\mu^2}{\sigma^2} - \frac{\frac{\mu^2}{\sigma^2}}{2}


G(f)=(μσ)2(μσ)22G_{\infty}(f) = (\frac{\mu}{\sigma})^2 - \frac{(\frac{\mu}{\sigma})^2}{2}

Now recall the definition of the Information Ratio:


Substitute in the definition of the information ratio:

G(f)=S2S22G_{\infty}(f^*) = S^2 - \frac{S^2}{2}


G(f)=S22G_{\infty}(f^*) = \frac{S^2}{2}

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.