## Statistic

## Information Ratio

The Information Ratio of a manager series vs. a benchmark series is the quotient of the annualized excess return and the annualized standard deviation of excess return.

Information Ratio = (AnnRtn(r_{1}, ..., r_{n}) - AnnRtn(s_{1}, ..., s_{n})) / AnnStdDev(e_{1}, ..., e_{n})

where:

r_{1}, ..., r_{n} = manager return series

## Exponential Weighting

When a style analysis is performed with exponential weighting, all input series to the analysis (i.e., manager and index series) are multiplied by an exponential function. Thus, if the original series was r_{1}, ..., r_{n}, the analysis will now use the series

## Drawdown

A manager’s drawdown at time t is defined as follows:

- Find the time s where the maximum of the manager’s cumulative return from the beginning of the analysis period to time t occurs.
- The drawdown at time t is the manager’s cumulative return from time s to time t. Note that this cumulative return must be zero or a loss, because the manager’s high water mark from the beginning of the analysis period to time t occurs at s.

## Downside Standard Deviation

The downside standard deviation, also referred to as downside risk, differs from the ordinary standard deviation insofar as the sum is restricted to those returns that are less than the mean:

DownsideStdDev(r_{1}, ..., r_{n}) =

## Downside Deviation

**(MAR = const)**

Here, MAR stands for “minimal acceptable return.” To calculate this, we first determine the sum of the squared distances between the returns and the MAR constant, where the sum is restricted to those returns that are less than the MAR. This sum is then divided by n, and the square root of the result is taken:

## Cumulative Return

The cumulative return is simply the compound return of the series.

CumRtn(r_{1}, …, r_{n}) =

where r_{1}, …, r_{n} is a return series, i.e., a sequence of returns for n time periods.

## Cumulative Excess Return

It is very important to realize that annualized and cumulative excess return are not calculated in the naive way, by taking the annualized or cumulative return of the excess return series. Instead, one must take the annualized and cumulative return of the two original series and then form the difference between the two:

CumExRtn = CumRtn(r_{1}, ..., r_{n}) - CumRtn(s_{1}, ..., s_{n})

## Correlation Squared

This is the classical statistical method for measuring how closely related the variances of two series are. The correlation is defined as:

## Beta

The alpha and beta of a manager vs. a benchmark are obtained by fitting a straight line to the points in a scatter plot of the market returns vs. the manager’s returns. Alpha is the intercept of this straight line, while beta is the slope. Hence, if the market returns change by some amount x, then the manager returns can be expected to change by Beta * x.

Beta is defined as:

## Best and Worse Statistics

Best and Worst Period Returns

The best period return for a time window is simply the maximum of the returns inside this window. Similarly, the worst period return for a time window is the minimum of the returns inside this window. Thus, if the return series is r_{1}, ..., r_{n}, we have

Best Period Return = max(r_{1}, ..., r_{n})

Worst Period Return = min(r_{1}, ..., r_{n})