## Statistic

## Benchmark Turnover

Asset turnover is a statistic that applies to any fund (i.e., a combination of assets) over a period of time. Asset turnover is a measurement of how much rebalancing was necessary over time to achieve the actual weightings of the assets in the fund.

## Batting Average

The batting average of the manager is the ratio between the number of periods where the manager outperforms a benchmark and the total number of periods. In other words,

Batting Average = numOutperform / numTotal

where

## Average Up and Down Returns

To calculate this statistic for a given return series r_{1}, ..., r_{n}, the program partitions the series in two parts, one made up of the positive returns, the other of the zero and negative returns. The average up and down returns are the respective averages of these two series.

Average Up Return =

Average Down Return =

where

## Annualized Standard Deviation

The Annualized Standard Deviation is the standard deviation multiplied by the square root of the number of periods in one year.

AnnStdDev(r_{1}, ..., r_{n}) = StdDev(r_{1}, ..., r_{n}) *

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

## Annualized Return

The annualized return is the geometric mean of the returns with respect to one year. If we denote by NumYears the number of years covered by the returns, the formula becomes:

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

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

**All returns greater than one year are annualized in StyleADVISOR.**

## Annualized 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:

AnnExRtn = AnnRtn(r_{1}, ..., r_{n}) - AnnRtn(s_{1}, ..., s_{n})

## Annualize Results: Less Than One Year

The Advanced Parameters tab in the Edit Analysis Parameters dialog allows the user to set a number of parameters that modify the way the calculations are made.

If this option is unselected, which is the default, the program will not annualize any returns, standard deviations (including downside risk) or alphas if the underlying time period is less than one year. If the option is selected, all these statistics will be annualized regardless of the underlying time period.

## Alpha

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.

Alpha is the mean of the excess return of the manager over beta times benchmark:

mean(i = 1, ... , m)( mi - Beta * bi )

Alpha is a measure of risk (beta)-adjusted return.

## Adjusted R-Squared

This applies only to a Style Benchmark; for Market Benchmarks, it is the same as the standard R2. The Adjusted R2 is based on the Standard R2, but it imposes a penalty for each additional index that is used to build the Style Benchmark.

Adjusted R2 = 1 – ((m - 1) / (m - n)) * (var(*e*) / var(*M*))

where: