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I am finding it difficult to understand the difference between the sharpe ratio and the information ratio and the relationship between the two, and cannot find a decent reference that breaks it down in terms of the actual mathematical definitions of the two. The Wiki page for the two is quite confusing as well, since it defines sharpe ratios almost identically to information ratios, but SR is in terms of a single asset, IR is in term of a portfolio of assets.

update: for clarity, the definiton of sharpe ratio (Wiki): $$ S_a = \frac{E(R_a-R_b)}{\sqrt{Var(R_a - R_b)}} $$ and definition of information ratio: $$ IR_p = \frac{E(R_p-R_b)}{\sqrt{Var(R_p - R_b)}} $$ where a is the reference asset, b is the benchmark and p is the reference portfolio

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Sharpe's 1966 equation had $R_b$ defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent.

If we refer to the original definitions, then that is the primary difference - Sharpe's ratio looks at reward/risk of the excess return for an asset over the risk-free rate while the information ratio looks at the reward/risk of the excess return for an asset over some reference benchmark.

Example: Sharpe Ratio could be used by someone developing a trading strategy who wants to study the average risk/reward profile over time (signal-to-noise) such that $R_b$ is set to the risk free rate, or even 0, while the Information Ratio could be used by a mutual fund manager whose job is to beat the S&P 500, thus $R_b$ could be the expected index return.

The mechanics of the two formulas are the same, e.g. there really isn't a difference especially since Sharpe has updated his formula.

From Wiki: https://en.wikipedia.org/wiki/Information_ratio

"The information ratio is similar to the Sharpe ratio but, whereas the Sharpe ratio is the 'excess' return of an asset over the return of a risk free asset divided by the variability or standard deviation of returns, the information ratio is the 'active' return to the most relevant benchmark index divided by the standard deviation of the 'active' return or tracking error."

https://en.wikipedia.org/wiki/Sharpe_ratio

"This is often confused with the information ratio, in part because the newer definition of the Sharpe ratio matches the definition of information ratio within the field of finance. Outside of this field, information ratio is simply mean over the standard deviation of a series of measurements."

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I have also seen a definition of Information Ratio that doesn't compare returns to a benchmark. According to Kaufman (Trading Systems and Methods, 2013), Chapter 2 and Chapter 21, the Information Ratio is defined as the compound Annualized Rate of Returns divided by the volatility of said returns. The $\mathrm{AROR_{compound}}$ can be calculated as:

$$\mathrm{AROR}_\mathrm{compound} = \left[ \left( \frac{\mathrm{Final Balance}}{\mathrm{Initial Balance}} \right)^ {\frac{252}{\mathrm{length-of-testing-period}}} \right]- 1$$

252 represents the number of trading days in a typical American calendar, so this would work on assets traded in American markets such as NYSE or Nasdaq. When dealing with other markets, you might need to adjust that number accordingly.

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