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The Risk Statistics formulas on this page are listed in alphabetical order.

 

Absolute Downside Risk

The annualized standard deviation of the negative account returns.

This measure is derived from at least 12 negative observations and may require a longer period (e.g., 2-3 years) to calculate.

Formula:

Absolute Downside Risk- Account

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
rm < 0
n = number of months in time period
n > or = 12
_
r = average monthly rate of return
_
r < 0

 
Formula:

Absolute Downside Risk- Index

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
rm < 0
n = number of months in time period
n > or = 12
_
r = average monthly rate of return
_
r < 0

Annualized Standard Deviation - Account

The difference between the account total fund rate of return and the benchmark rate of return.

Formula:

Annualized Standard Deviation (Portfolio) Based on a Population

This is a measure of the variability of returns around the average return of the portfolio. This measure is available after twelve months of performance have passed; shorter periods do not provide a good indication of a fund's variability.

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
_
r = average monthly rate of return
n = number of months in time period

The standard deviation used to measure investment portfolios is the squared deviations from the average of monthly returns, divided by the number of observations (12 for one year, etc) with that product then multiplied by the square of the frequency of returns used.

Monthly return frequency is the square root of 12, with 12 being months in a year. The calculation uses the actual number of observations in the denominator of the calculation, as opposed to the standard deviation of a sample which uses n-1. It is an investment tenet that if you have two portfolios with the same return, the one with a lower standard deviation is more desirable, because the return was achieved with less risk.

Formula:

Annualized Standard Deviation (Portfolio) Based on a Sample

This is a measure of the variability of returns around the average return of the portfolio. This measure is available after twelve months of performance have passed; shorter periods do not provide a good indication of a fund's variability.

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
_
r = average monthly rate of return
n = number of months in time period

 
Note:

This calculation estimates Standard Deviation based on a sample. It is a measure of how widely values are dispersed from the mean. The Standard Deviation calculation that is based on a sample assumes that its arguments are a sample of the population.

If available data represents the entire population, then standard deviation should be computed using the calculation based on a population.

Annualized Standard Deviation - Index

Formula:

Annualized Standard Deviation (Index) Based on a Population

This is a measure of the variability of returns around the average return of the index. This measure is available after twelve months of performance have passed; shorter periods do not provide a good indication of a fund's variability.

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
_
r = average monthly rate of return
n = number of months in time period

 
Formula:

Annualized Standard Deviation (Index) Based on a Sample

This is a measure of the variability of returns around the average return of the index. This measure is available after twelve months of performance have passed; shorter periods do not provide a good indication of a fund's variability.

Where:

V = annualized variability of monthly returns
rm = unannualized time weighted rate of return in month m
_
r = average monthly rate of return
n = number of months in time period

 
Note:

This calculation estimates Standard Deviation based on a sample. It is a measure of how widely values are dispersed from the mean. The Standard Deviation calculation that is based on a sample assumes that its arguments are a sample of the population.

If available data represents the entire population, then standard deviation should be computed using the calculation based on a population.

Capture Ratio

Formula:

Down Market Capture Ratio Equation

((1+AR0 ) * (1+AR1 ) * … * (1+ ARn )) - 1/ ((1+ IR0 ) * (1+IR1 ) *…* (1+IRn ))-1

Where:

AR = Account ROR
IR = Index ROR
IR < 0
n = number of periods

 
Formula:

Up Market Capture Ratio Equation

((1+AR0 ) * (1+AR1 ) * … * (1+ ARn )) - 1/ ((1+ IR0 ) * (1+IR1 ) *…* (1+IRn ))-1

Where:

AR = Account ROR
IR = Index ROR
IR > 0
n = number of periods

Methodology for Producing Up Market and Down Market Capture Ratios:

For any chosen time period there are X pairs or returns (account/consolidation and index). For example, there will be 12 pairs of data for a one-year time period.

Identify the up and down market pairs and separate them into two time series. This is done by examining the index return for each pair result. If the index return is greater than zero it is defined as an up market result and conversely, if the return is less than zero it is put into the down market time series.

For the up market time series, link the account results and then the index results. (You don't need to annualize when there are more than 12 observations).

This produces the following ratio: linked account ror/linked index ror. The ratio is then expressed as a percentage The same is done with the down market times series.

Note:

If the result for Up Market Capture Ratio is < or = 0, a zero will be displayed. If the result for the Down Market Capture Ratio is > or = 0, a zero will be displayed.

Correlation

This is a standard correlation calculation measure which should only be performed on periods greater than one year. The dependant variable could be fixed on the returns of the S&P 500 or we could allow clients to select the index.

Formula:

Correlation to Index

Where X is the independent Variable, (portfolio return) and Y is the dependent variable (index return).

Excess Return

The arithmetic difference between the account total fund rate of return and the benchmark rate of return.

Formula:

Excess Return Equation

excess return = (monthly account total fund return) - (monthly benchmark return) linked and annualized

Kurtosis

Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

Formula:

The following is the calculation of kurtosis:


The standard deviation used in this calculation is not annualized; we cannot use the existing as a component of this calculation.

Note: Kurtosis on the normal distribution = 3

Maximum Drawdown

Maximum Drawdown is the greatest loss from peak monthly return to lowest monthly return. This is a classic risk measure for hedge funds, often cited by hedge fund managers.

Formula:

Maximum Drawdown

MD = Min(X1, X1+X2,…, X1+X2+…XN,  X2+X3+…XN,   …,   ,XN)

Maximum Recovery

Maximum Recovery is the greatest gain from the lowest monthly return to the peak return.

Formula:

Maximum Recovery

MD = Max(X1, X1+X2,…, X1+X2+…XN,  X2+X3+…XN,   …,   ,XN)

Relative Return

A measure of the risk of deviating from the benchmark's return for the period. A low tracking error indicates that there is a good match between the benchmark and the account, calculated geometrically.

Formula:

Relative Return Equation

relative return = [((1 + monthly account total fund return / 100) / (1 + monthly benchmark return / 100)) -1] * 100

Sterling Ratio

The Sterling Ratio is a risk-reward measure which depicts the highest returns with the least volatility.  This measure is conceptually similar to the Calmar ratio, but user the average drawdown of each annual period.  This measure is valid for three year periods and greater.  A higher value is desirable since it depicts higher return relative to risk.

Formula:

Sterling Ratio

Sterling Ratio for three years = (D1+D2+D3)/3

Where: D1 is the max drawdown for the 1st 12 months, D2 is the max drawdown for the 2nd 12 months, and D3 is the max drawdown for the 3rd 12months.

Tracking Error

A measure of the risk of deviating from the benchmark's return for the period. A low tracking error indicates that there is a good match between the benchmark and the account.

Formula:

Tracking Error

tracking error = annualized standard deviation of ((account total fund return) - (benchmark total fund return))

Historical Tracking Error is essentially the standard deviation of monthly excess return the manager has generated over or under the index. The standard deviation of the excess returns is annualized and the calculation uses the actual number of observations as opposed to a sample calculation which uses n-1.

Note:

Fundamentals displays two tracking error measures.

One is the standard historical tracking error calculation that uses the standard deviation based on a population; the alternate calculation uses standard deviation based on a sample.

For more information, see Annualized Standard Deviation.

Relative Tracking Error

Formula:

Relative Tracking Error Equation

relative tracking error = annualized standard deviation of ((1 + account total fund return / 100) / (1 + benchmark total fund return / 100))*100

Information Ratio

A measure of the incremental return earned by an account versus its benchmark relative to the incremental risk assumed versus its benchmark.

Formula:

Information Ratio Equation

information ratio = (excess return) / (tracking error)

 
Note:

Fundamentals displays two Information Ratio measures.

One is the standard calculation that uses a tracking error that is calculated using the standard deviation based on a population. The alternate calculation uses a tracking error that is calculated using the standard deviation based on a sample.

For more information, see Annualized Standard Deviation.

Jensen's Alpha

A measure that involves an ordinary least squares regression return to calculate risk-premium for an account's return against its benchmark. A positive alpha means the account performed better than its benchmark in risk-adjusted terms. A negative means the opposite.

Formula:

Jensen's Alpha Equation

Jensen's Alpha = [sum(square(X)) * sum(Y) - sum(X)sum(X * Y)] / [n * sum(square(x)) - square(sum(X))]]

Where:

n = number of periods and n > 0
X = (benchmark return) - (risk-free return)
Y = (account total fund return) - (risk-free return)

M Squared

M squared is the Risk-Adjusted Performance RAP(i) = risk-adjusted rate of return for portfolio i.

Formula:

M Squared Equation

(M/i)(ri - rf) + rf

Where:

M = standard deviation of monthly returns of the benchmark
i = standard deviation of monthly returns for portfolio i
ri = average monthly return of portfolio i
rf = risk-free rate of return (90 T-Bill rate for the month)

R Squared

A measure that varies from 0.0 (meaning no correlation between the benchmark and the account returns) to 1.0 for complete agreement between the two. This measure is sometimes referred to as the coefficient of determination.

Formula:

R-Squared Equation

R-squared = square[(n * sum(X * Y)) - (sum(X) * sum(Y) / (sqrt((n * sum(square(X))) - (square(sum(X)) * (sqrt((n * sum(square(Y))) - (square(sum(Y))]

Relative Downside Risk

The standard deviation of the negative results is the difference between account returns and index returns. This measure is derived from at least 12 negative observations and may require a longer period (e.g., 2-3 years) to calculate.

Formula:

Relative Downside Risk Equation

annualized standard deviation of ((account total fund return) - (benchmark total fund return))

Where:

((account total fund return) - (benchmark total fund return)) < 0

Relative Volatility (Beta)

A measure that uses an ordinary least squares regression return to compute the relative volatility of an account against its benchmark. A relative volatility measure of 1.0 means equal volatility as the benchmark, greater than 1.0 means more volatile, and less than 1.0 means less volatile.

Formula:

Relative Volatility Equation

Relative Volatility = [n * sum(X * Y) - (sum(X) sum(Y)] / [n * sum(square(X) - square(sum(X)]

Where:

n = number of periods and n > 0
X = (benchmark return) - (risk-free return)
Y = (account total fund return) - (risk-free return)

Risk Free Return

The rate of return from an investment that is considered risk-free or very near risk-free. Risk free investments will vary from country.

The risk-free investment used in calculations for accounts and groups with a US Dollar base currency is the 90 Day US T-Bill. Calculations for non-US dollar based accounts and groups use the JP Morgan Cash Index in the appropriate currency as the risk free investment.

Sharpe Ratio

A measure of the return earned per unit of risk.

This measure looks at the excess returns of an account relative to the volatility of the account's returns.

Formula:

Sharpe Ratio Equation - Portfolio

Sharpe Ratio = (portfolio total fund return) - (risk free return) / (annualized standard deviation of portfolio total fund return)

This measure looks at the excess returns of an index relative to the volatility of the index's returns.

Formula:

Sharpe Ratio Equation - Index

Index Sharpe Ratio = (index total return) - (risk-free return) / (annualized standard deviation of index).

 
Note:

Fundamentals displays two Sharpe Ratio measures.

One is the standard calculation using the standard deviation based on a population. The alternate calculation uses standard deviation based on a sample.

For more information, see Annualized Standard Deviation.

Skewness

Returns the skewness of a distribution. 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.

Formula:

Skewness

Where s is the standard deviation

The standard deviation used in this calculation is not annualized; we cannot use the existing as a component of this calculation.

Note: Skewness on the normal distribution = 0

Sortino Ratio

The Sortino ratio is the excess return over risk-free rate over the downside semi-variance. This ratio allows investors to go beyond simply looking at excess returns to total volatility, since such a measure does not consider how often the price of the security rises as opposed to how often it falls.

Formula:

Sortino Ratio Equation

(AR - RF) / (DSR)

Where:

AR = account rate of return
RF = risk free index rate of return
DSR = downside risk

Treynor Ratio

A risk-adjusted return measure that provides excess return per unit of risk where risk is defined as systematic risk (beta).

Formula:

Treynor Ratio Equation

Treynor ratio = (account total fund return - risk free return) / (systematic risk)