## Treynor Ratio

The Treynor Ratio differs from the Sharpe Ratio insofar as the beta to the Market Benchmark is used as a measure of risk rather than the standard deviation of the manager series.

Treynor Ratio = (AnnRtn(r1, ..., rn) - AnnRtn(c1, ..., cn)) / (beta of manager to market)

where:

r1, ..., rn = manager return series

## Style Drift

Since the inception of returns-based style analysis and the ensuing development of sophisticated software that brought returns-based style analysis to the masses, investors have used rolling asset allocation graphs and style maps to gain a visual feeling for the style consistency of managers and mutual funds. The opposite of style consistency is style inconsistency or style drift.

## Style Analysis

StyleADVISOR implements returns based style analysis as set forth by Stanford professor and Nobel Prize winner William F. Sharpe. Returns based style analysis calculates a Style Benchmark for a manager from the manager’s return series and the return series of the indices that are to be used in the Style Benchmark. Thus, we are given:

## Standard Deviation of Excess Return

The variance and standard deviation of excess return are simply variance and standard deviation applied to the excess return series e1, ... , en.

Var(e1, ..., en) =

StdDev(e1, ..., en) =

AnnStdDev(e1, ..., en) = StdDev(e1, ..., en) *

## Standard Deviation

Standard deviation of return measures the average deviations of a return series from its mean, and is often used as a measure of risk. A large standard deviation implies that there have been large swings in the return series of the manager.

Standard deviation can be calculated in two ways:

## Sortino Ratio

The Sortino Ratio is an analog to the Sharpe Ratio, with the standard deviation replaced by the downside deviation. Accordingly, there are two versions: one uses the downside deviation with constant MAR, the other uses the downside deviation with cash as the MAR.

SortinoRatioConstantMAR = (AnnRtn(r1, …, rn) – c)/DownsideDeviationConstantMAR(r1, …, rn)

## Skewness

Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values. Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values.

Skewness (r1, ..., rn) =

where r1, ..., rn is a return series, i.e., a sequence of returns for n time periods.

## Single Window vs. Rolling Window Style Analysis

For single window style analysis, the program calculates one set of style weights per manager, using the entire selected date range. In this case, the style map graph will show one point per manager, and the asset allocation graph will show one set of weights per manager.

## Significance Level

The significance level of a manager series vs. a benchmark series indicates the level of confidence with which the statement "the manager's annualized excess return over the benchmark is positive" or "the manager's annualized excess return over the benchmark is negative," as the case may be, holds true. The significance level is calculated from the T-Statistic using a numerical approximation known as the incomplete beta function. For further details, we refer the reader to Press et al. , Chapter 14.2.

## Sharpe Ratio

The Sharpe Ratio of a manager series is the quotient of the annualized excess return of the manager over the cash equivalent and the annualized standard deviation of the manager return.

Sharpe Ratio = (AnnRtn(r1, ..., rn) - AnnRtn(c1, ..., cn)) / AnnStdDev(r1, ..., rn)

where:

r1, ..., rn = manager return series

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