Distribution of the Information Ratio // Mean and Variance Product

We are investigating the distribtuion of the information ratio. However, instead of using the original information ratio defined as $$\begin{equation} IR=\frac{E(r_1)-E(r_2)}{\sqrt{Var(r_1-r_2)}}, \end{equation}$$ where $$r_1$$ and $$r_2$$ are random returns, we are interested in the statistical properties of $$\begin{equation} \text{reversed }IR=\left(E(r_1)-E(r_2)\right)\times\sqrt{Var(r_1-r_2)}. \end{equation}$$ Note that $$r_1$$ and $$r_2$$ exibit the usual stylized facts of financial time series (so non-iid). It might also be that $$r_2$$ is not random to some extent.

To make it a little easier for a first step, lets assume independence and normality of $$r_1$$ and $$r_2$$ and the variance follows a Gamma distribution.

• I assert the starting point that this distribution is, in general, dependent upon the distributions of $r_1$ and $r_2$. If not can you supply the reasoning, otherwise can you supply your assumption of the underlying $r_1,r_2$ distributions. – Attack68 May 21 at 10:46
• Results for iid normality would be great, I just try to get an intuition for the thing, if it is well behaved etc... – dsforecast May 21 at 12:24