Studying a Mean-Variance Product Related to the Information Ratio

Summary

The document proposes studying a quantity formed by multiplying the difference in expected returns of two return series by the square root of the variance of their return difference. This reverses the variance term’s placement in the conventional information ratio, which divides the expected return difference by the tracking-error scale. The author is interested in the distribution and statistical properties of this alternative quantity, with financial returns that may exhibit dependence and other familiar time-series features.

As a simplifying first step, the document suggests assuming the two returns are independent and normally distributed, while allowing variance to follow a Gamma distribution. It also raises the possibility that the second return series may not be entirely random. No derivation, resulting distribution, empirical analysis, or conclusions are provided. The proposed simplifying assumptions therefore define an open research setup rather than a validated performance measure, and the relevance of the setup to dependent, non-normal market returns remains unresolved.

Key ideas

• The proposed statistic multiplies the expected return difference by the square root of the variance of the return difference.
• This construction differs from the conventional information ratio, which divides by a tracking-error scale.
• The initial analysis assumes independent, normally distributed returns and a Gamma-distributed variance.
• Dependence, non-normality, and a partly non-random benchmark remain open complications.

Tags

# Distribution of the Information Ratio // Mean and Variance Product


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.