How Sharpe and Information Ratios Differ by Benchmark
Summary
The document compares the Sharpe ratio and information ratio through their return-over-risk definitions. It explains that the traditional Sharpe ratio measures excess return relative to a risk-free rate, while the information ratio measures active return relative to a chosen benchmark. Their calculations share the same structure, so the benchmark return used in the numerator and denominator is the central distinction in the traditional interpretation.
It gives examples of evaluating a strategy against a risk-free or zero-return baseline and assessing a fund manager against an index. It also notes that later definitions of the Sharpe ratio use a general reference benchmark, making the two measures effectively equivalent under those definitions. A second answer describes an alternative use of “information ratio” as annualized compound return divided by return volatility, with annualization adjusted to the market calendar. The document presents differing conventions, so the metric’s definition should be checked in context.
Key ideas
- The traditional Sharpe ratio compares an asset’s excess return with a risk-free rate.
- The information ratio compares portfolio returns with a selected benchmark and scales by active-return variability.
- Both ratios share a return-to-risk structure, and broader Sharpe definitions can make them equivalent.
- Some sources use “information ratio” for annualized compound return divided by volatility, so terminology varies.
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# Difference between Sharpe Ratio and Information Ratio
# Difference between Sharpe Ratio and Information Ratio
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
## Answer by RA334 (score 6, accepted)
https://quant.stackexchange.com/a/20622
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."
## Answer by Ismail Mohamed (score 0)
https://quant.stackexchange.com/a/42374
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.