# Information ratio

The Information ratio is a measure of the risk-adjusted return of a financial security (or asset or portfolio). It is also known as Appraisal ratio and is defined as expected active return divided by tracking error, where active return is the difference between the return of the security and the return of a selected benchmark index, and tracking error is the standard deviation of the active return; i.e., the information ratio $IR$ is:

$IR = \frac{E[R_p-R_b]}{\sigma} = \frac{\alpha}{\omega} = \frac{E[R_p-R_b]}{\sqrt{\mathrm{var}[R_p-R_b]}}$,

where $R_p$ is the portfolio return, $R_b$ is the benchmark return, $\alpha = E[R_p-R_b]$ is the expected value of the active return, and $\omega = \sigma$ is the standard deviation of the active return, which is an alternate definition of the aforementioned tracking error.

Note in this case, $\alpha$ is defined as excess return, not the risk-adjusted excess return or Jensen's alpha calculated using regression analysis. Some analysts, however, do use Jensen's alpha for the numerator and a regression-adjusted tracking error for the denominator (this version of the information ratio is often described as the appraisal ratio to differentiate it from the more common definition).[1]

The information ratio is often used to gauge the skill of managers of mutual funds, hedge funds, etc. In this case, it measures the active return of the manager's portfolio divided by the amount of risk that the manager takes relative to the benchmark. The higher the information ratio, the higher the active return of the portfolio, given the amount of risk taken, and the better the manager. Top-quartile investment managers typically achieve annualized information ratios of about one-half.[2] There are both ex ante expected and ex post observed information ratios.

Generally, the information ratio compares the returns of the manager's portfolio with those of a benchmark such as the yield on three-month Treasury bills or an equity index such as the S&P 500.

The information ratio is often annualized. While it is then common for the numerator to be calculated as the arithmetic difference between the annualized portfolio return and the annualized benchmark return, this is an approximation because the annualization of an arithmetic difference between terms is not the arithmetic difference of the annualized terms.[3] Since the denominator is here taken to be the annualized standard deviation of the arithmetic difference of these series, which is a standard measure of annualized risk, and since the ratio of annualized terms is the annualization of their ratio, the annualized information ratio provides the annualized risk-adjusted active return of the portfolio relative to the benchmark.

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.

Some hedge funds use Information ratio as a metric for calculating a performance fee.

One of the main criticisms of the Information Ratio is that it considers arithmetic returns and ignores leverage. This can lead to the Information Ratio calculated for a manager being negative when the manager produces alpha to the benchmark and vice versa. A better measure of the alpha produced by the manager is the Geometric Information Ratio.[4]

## Characteristics of the information ratio

The main characteristics of the information ratio are as follows:

• The information ratio estimates ex-post value added and relates this to ex-ante opportunity available in the future.
• The residual frontier that describes the opportunities accessible to the active manager is identified by the information ratio.
• The level of aggressiveness for each manager is decided by his/her information ratio.
• Sometimes intuition can give a good clue about the information ratio and residual risk aversion.
• Value added depends on the managers’ prospects and aggressiveness.