Value Added Monthly Index (VAMI) - This index reflects the growth of a hypothetical $1,000 in a given investment over time. The index is equal to $1,000 at inception. Subsequent month-end values are calculated by multiplying the previous month’s VAMI index by 1 plus the current month rate of return.

Where Vami 0 = 1000 and
Where R _N = Return for period N

Vami _N = ( 1 + R _N ) ´ Vami _N-1

Average Return (Mean) - This is a simple average return (arithmetic mean) which is calculated by summing the returns for each period and dividing the total by the number of periods. The simple average does not take the compounding effect of investment returns into account.

^{Where N = Number of periods}

Where R _I = Return for period I

_N

Average Return = ( S R _I ) ¸ N

^I=1

Average Gain (Gain Mean) - This is a simple average (arithmetic mean) of the periods with a gain. It is calculated by summing the returns for gain periods (return ³ 0) and then dividing the total by the number of gain periods.

^{Where N = Number of periods}

Where R _I = Return for period I

Where G _I = R _I ( IF R _I ³ 0 ) or 0 ( IF R _I < 0 )

N _G = Number of periods that R _I ³ 0

_N

Average Gain = ( S G _I ) ¸ N _G

^I=1

Average Loss (Loss Mean) - This is a simple average (arithmetic mean) of the periods with a loss. It is calculated by summing the returns for loss periods (return < 0) and then dividing the total by the number of loss periods.

Where N = Number of periods

Where R _I = Return for period I

Where L _I = 0 ( IF R _I ³ 0 ) or R _I ( IF R _I < 0 )

N _L = Number of periods that R _I < 0

_N

Average Loss = ( S L _I ) ¸ N _L

^I=1

Compound (Geometric) Average Return - The geometric mean is the monthly average return that assumes the same rate of return every period to arrive at the equivalent compound growth rate reflected in the actual return data. In other words, the geometric mean is the monthly average return that, if applied each period, would give you a final Vami (growth) index that is equivalent to the actual final Vami index for the return stream you are considering. In PerTrac 2000, compound quarterly and annualized returns are calculated using the compound monthly return as a base.

Where N = Number of periods

Where Vami (0) = 1000

Compound Monthly ROR = ( Vami _N ¸ Vami ₀ ) ^{1/ N} - 1

Compound Quarterly ROR = ( 1 + Compound Monthly ROR ) ³ - 1

Compound Annualized ROR = ( 1 + Compound Monthly ROR ) ¹² - 1

Standard Deviation - Standard Deviation measures the dispersal or uncertainty in a random variable (in this case, investment returns). It measures the degree of variation of returns around the mean (average) return. The higher the volatility of the investment returns, the higher the standard deviation will be. For this reason, standard deviation is often used as a measure of investment risk.

Where R _I = Return for period I

Where M _R = Mean of return set R

Where N = Number of Periods

_N

M _R = ( S R _I ) ¸ ^N

^I=1

N

Standard Deviation = ( S ( R _I - M _R ) ² ¸ (N - 1) ) ^½

^{I = 1}
Annualized Standard Deviation

Annualized Standard Deviation = Monthly Standard Deviation ´ ( 12 ) ^½

Annualized Standard Deviation * = Quarterly Standard Deviation ´ ( 4 ) ^½

* Quarterly Data

Gain Standard Deviation - Similar to standard deviation, except this statistic calculates an average (mean) return for only the periods with a gain and then measures the variation of only the gain periods around this gain mean. This statistic measures the volatility of upside performance.

Where N = Number of Periods

Where R _I = Return for period I

Where M _G = Gain Mean

Where G _I = R _I ( IF R _I ³ 0 ) or 0 ( IF R _I < 0 )

Where GG _I = R _I - M _G ( IF R _I ³ 0 ) or 0 ( IF R _I < 0 )

N _G = Number of periods that R _I ³ 0

_N

M _G = ( S G _I ) ¸ N _G

^I=1

N

Gain Deviation = ( S (GG _I ) ² ¸ (N _G - 1) ) ^½

^I=1

Loss Standard Deviation - Similar to standard deviation, except this statistic calculates an average (mean) return for only the periods with a loss and then measures the variation of only the losing periods around this loss mean. This statistic measures the volatility of downside performance.

Where N = Number of Periods

Where R _I = Return for period I

Where M _L = Loss Mean

Where L _I = R _I ( IF R _I < 0 )or 0 ( IF R _I ³ 0 )

Where LL _I = R _I - M _L ( IF R _I < 0 ) or 0 ( IF R _I ³ 0 )

N _L = Number of periods that R _I < 0

_N

M _L = ( S L _I ) ¸ N _L

^I=1

N

Loss Deviation = ( S ( LL _I) ² ¸ (N _L - 1) ) ^½

^I=1

Downside Deviation - Similar to the loss standard deviation except the downside deviation considers only returns that fall below a defined Minimum Acceptable Return (MAR) rather then the arithmetic mean. For example, if the MAR is assumed to be 10%, the downside deviation would measure the variation of each period that falls below 10%. (The loss standard deviation, on the other hand, would take only losing periods, calculate an average return for the losing periods, and then measure the variation between each losing return and the losing return average). In PerTrac 2000, there are 3 downside deviation calculations, each using a different value for the MAR: 1)Uses a MAR which is defined by the user on the Preferences screen, 2) Uses the Sharpe risk free rate (which can also be defined in Preferences) as the MAR, and 3) uses zero as the MAR.

Where R _I = Return for period I

Where N = Number of Periods

Where R _MAR = Period Minimum Acceptable Return

Where L _I = R _I - R _MAR ( IF R _I - R _MAR < 0 )or 0 ( IF R _I - R _MAR ³ 0 )

_N

Downside Deviation = ( (S ( L _I ) ² ) ¸ N ) ^½

^I=1

Semi Deviation

Where R _I = Return for period I

Where N = Number of Periods

Where M = Period Arithmetic Mean

Where L _I = R _I - M ( IF R _I - M < 0 )or 0 ( IF R _I - M ³ 0 )

Where N_L = Number of Periods where R _I - M < 0

^N

Semi Deviation = ( (S ( L _I ) ² ) ¸ (N_L -1)) ^½

^I=1

Sharpe Ratio - A return/risk measure developed by William Sharpe. Return (numerator) is defined as the incremental average return of an investment over the risk free rate. Risk (denominator) is defined as the standard deviation of the investment returns. In PerTrac 2000, the user enters the value for the risk free rate.

Where R _I = Return for period I

Where M _R = Mean of return set R

Where N = Number of Periods

Where SD = Period Standard Deviation

Where R _RF = Period Risk Free Return

_N

M _R = ( S R _I ) ¸ ^N

^I=1

N

SD = ( S ( R _I - M _R ) ² ¸ (N - 1) ) ^½

^{I = 1}

Sharpe Ratio = ( M _R - R _RF ) ¸ SD

Annualized Sharpe Ratio

Annualized Sharpe = Monthly Sharpe ´ ( 12 ) ^½

Annualized Sharpe * = Quarterly Sharpe ´ ( 4 ) ^½ * Quarterly Data

Sortino Ratio - This is another return/risk ratio developed by Frank Sortino. Return (numerator) is defined as the incremental compound average period return over a Minimum Acceptable Return (MAR). Risk (denominator) is defined as the Downside Deviation below a Minimum Acceptable Return (MAR). Just as with the Downside Deviation calculation, PerTrac 2000 calculates the Sortino using 3 different values for the MAR: 1) a MAR defined by the user under Preferences, 2) the Sharpe ratio risk free rate (also set under Preferences), and 3) zero.

Where R _I = Return for period I

Where N = Number of Periods

Where R _MAR = Period Minimum Acceptable Return

Where DD _MAR = Downside Deviation

Where L _I = R _I - R _MAR ( IF R _I - R _MAR < 0 )or 0 ( IF R _I - R _MAR ³ 0 )

_N

DD _MAR = ( (S ( L _I ) ² ) ¸ N ) ^½

^I=1

Sortino Ratio = ( Compound Period Return - R _MAR ) ¸ DD _MAR

Annualized Sortino Ratio

Annualized Sortino = Monthly Sortino ´ ( 12 ) ^½

Annualized Sortino* = Quarterly Sortino ´ ( 4 ) ^½

* Quarterly Data

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.

Where N = Number of Periods

Where R _I = Return for period I

Where M _R = Mean of return set R

Where SD = Period Standard Deviation

_N

M _R = ( S R _I ) ¸ ^N

^I=1

N

SD = ( S ( R _I - M _R ) ² ¸ (N - 1) ) ^½

^{I = 1}

N

Skewness = ( N ¸ ((N-1)(N-2)) ) ( S (R _I – M _R) ¸ SD) ) ³

^{I = 1}

If there are fewer than three data points, or the sample standard deviation is zero, Skewness returns the N/A error value.

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.

Where N = Number of Periods

Where R _I = Return for period I

Where M _R = Mean of return set R

Where SD = Period Standard Deviation

_N

M _R = ( S R _I ) ¸ ^N

^I=1

N

SD = ( S ( R _I - M _R ) ² ¸ (N - 1) ) ^½

^{I = 1}

N

Kurtosis = {(N(N+1) ¸ ((N-1)(N-2)(N-3))) (S (R _I – M _R) ¸ SD))⁴} - (3(N-1)² ¸ ((N-2)(N-3)))

^{I = 1}

If there are fewer than four data points, or if the standard deviation of the sample equals zero, Kurtosis returns the N/A error value.

Calmar Ratio - This is a return/risk ratio. Return (numerator) is defined as the Compound Annualized Rate of Return over the last 3 years. Risk (denominator) is defined as the Maximum Drawdown over the last 3 years. If three years of data are not available, the available data is used. ABS is the Absolute Value.

Calmar Ratio = Compound Annualized ROR ¸ ABS (Maximum Drawdown )

Sterling Ratio - This is a return/risk ratio. Return (numerator) is defined as the Compound Annualized Rate of Return over the last 3 years. Risk (denominator) is defined as the Average Yearly Maximum Drawdown over the last 3 years less an arbitrary 10%. To calculate this average yearly drawdown, the latest 3 years (36 months) is divided into 3 separate 12-month periods and the maximum drawdown is calculated for each. Then these 3 drawdowns are averaged to produce the Average Yearly Maximum Drawdown for the 3 year period. If three years of data are not available, the available data is used.

Where D1 = Maximum Drawdown for first 12 months

Where D2 = Maximum Drawdown for next 12 months

Where D3 = Maximum Drawdown for latest 12 months

Average Drawdown = ( D1 + D2 + D3 ) ¸ 3

Sterling Ratio = Compound Annualized ROR ¸ ABS ( (Average Drawdown - 10% ))

Drawdown - A Drawdown is any losing period during an investment record. It is defined as the percent retrenchment from an equity peak to an equity valley. A Drawdown is in effect from the time an equity retrenchment begins until a new equity high is reached. (i.e. In terms of time, a drawdown encompasses both the period from equity peak to equity valley (Length) and the time from the equity valley to a new equity high (Recovery).

Maximum Drawdown is simply the largest percentage drawdown that has occurred in any investment data record.

The Drawdown Table in PerTrac 2000 provides a comprehensive list of all drawdowns in the historical performance record ranked from largest to smallest.

Gain to Loss Ratio - This is a simple ratio of the average gain in a gain period divided by the average loss in a losing period. Periods can be monthly or quarterly depending on the data frequency.

Gain/Loss Ratio = ABS (Average Gain in Gain Period ¸ Average Loss in Loss Period)

$ Profit to Loss Ratio - This ratio combines the Gain to Loss Ratio with the ratio of the percentage of profitable periods to the percentage of losing periods. Since this ratio considers both the average size and the frequency of winning and losing periods, it tells you the historical ratio of dollars earned in the investment to dollars lost. For example, a $ Profit to Loss Ratio of 2.5 means that, historically, the investment earned $2.50 of profit for each $1.00 of risk taken.

$ Profit/Loss Ratio = (% Profitable Periods ¸ % Losing Periods) ´ Gain to Loss Ratio

Correlation Analysis - All of the following correlation related statistics use the following variables:

Where R _I = The return of the independent variable for period I

Where RD _I = The return of the dependent variable for period I

Where M _R = The mean return of the independent variable

Where M _RD = The mean return of the dependent variable

Where N = Number of Periods

Beta - Beta is the slope of the regression line. Beta measures the risk of a particular investment relative to the market as a whole (the “market” can be any index or investment you specify). It describes the sensitivity of the investment to broad market movements. For example, in equities, the stock market (the independent variable) is assigned a beta of 1.0. An investment which has a beta of .5 will tend to participate in broad market moves, but only half as much as the market overall.

_{N N}

Beta = (S (R _I - M _R ) (RD _I - M _RD) ) ¸ (S (R _I - M _R ) ² )

^{I=1 I=1}

Alpha - Alpha is a measure of value added. It is the Y intercept of the regression line.

Alpha = M _RD - Beta ´ M _R

Annualized Alpha – Annualized Alpha is the annualized value of Alpha.

Annualized Alpha = ((1 + Alpha)¹² - 1 (Monthly Data)

Annualized Alpha = ((1 + Alpha)⁴ - 1 (Quarterly Data)

Correlation and Correlation Coefficient - Correlation measures the extent of linear association of two variables. The Coefficient of Determination ( R² ) is a measure of how well the regression line fits the data (variation explained by the regression line). Unexplained variation is simply 1- R² .

Correlation Coefficient ( r )

Where S _XY = Sample Covariance

Where S _X = Standard deviation of independent variable

Where S _Y = Standard deviation of dependent variable

_N

S _XY = (S (R _I - M _R ) (RD _I - M _RD) ) ¸ (N - 1 ) )

^I=1

N

S _X = ( S ( R _I - M _R ) ² ¸ (N - 1) ) ^½

^{I = 1}

N

S _Y = ( S ( RD _I - M _RD ) ² ¸ (N - 1) ) ^½

^{I = 1}

Correlation Coefficient = S _XY ¸ (S _X ´ S _Y )

Coefficient of Determination ( r ² )

Where Y _I = Alpha + Beta ´ R _I

_{N N}

Coefficient of Determination = ( S ( Y _I - M _RD ) ²) ¸ ( S ( RD _I - M _RD ) ²)

^{I = 1 I=1}

Tracking Error (Annualized) - Tracking Error is a measure of the unexplained portion of an investments performance relative to a benchmark. Annualized Tracking Error is measured by taking the square root of the average of the squared deviations between the investment’s returns and the benchmark’s returns, then multiplying the result by the square root of 12.

N

Tracking Error = ( ( S ( R _I - RD _I) ² ¸ (N-1) ) ^½) ´ 12^½

^{I = 1}

Treynor Ratio – The Treynor Ratio, developed by Jack Treynor, is similar to the Sharpe Ratio, except that it uses Beta as the volatility measurement. Return (numerator) is defined as the incremental average return of an investment over the risk free rate. Risk (denominator) is defined as the Beta of the investment returns relative to a benchmark. In PerTrac 2000, the user enters the value for the risk free rate.

Where M _R = Annualized Return of Investment

Where R _RF = Annualized Risk Free Return

Treynor Ratio = ( M _R - R _RF ) ¸ Beta

Jensen Alpha - The Jensen Alpha, developed by Michael Jensen, quantifies the extent to which an investment has added value relative to a benchmark. The Jensen Alpha is equal to the Investment’s average return in excess of the risk free rate minus the Beta times the Benchmark’s average return in excess of the risk free rate. In PerTrac 2000, the user enters the value for the risk free rate.

Where R _I = Benchmark Return for period I

Where RD _I = Return for period I

Where M _R = Mean of return set R (Benchmark)

Where M _RD = Mean of return set RD

Where N = Number of Periods

Where R _RF = Period Risk Free Return

_N

M _R = ( S R _I ) ¸ ^N

^I=1

_N

M _RD = ( S RD _I ) ¸ ^N

^I=1

Jensen Alpha = ( M _RD - R _RF ) - Beta ´ ( M _R - R _RF )

Active Premium - A measure of the Investment’s annualized return minus the Benchmark’s annualized return.

Active Premium = Investment’s annualized return - Benchmark’s annualized return.

Information Ratio - The Information Ratio is the Active Premium divided by the Tracking Error. This measure explicitly relates the degree by which an Investment has beaten the Benchmark to the consistency by which the Investment has beaten the Benchmark.

Information Ratio = Active Premium ¸ Tracking Error

Up Capture - The Up Capture Ratio is a measure of the Investment’s compound return when the Benchmark was up divided by the Benchmark’s compound return when the Benchmark was up. The greater the value, the better.

Where R _I = Return for period I

Where RD _I = Benchmark Return for period I

Where N = Number of Periods

Where L _I = R _I (IF RD_I ³ 0) or 0 (IF RD_I < 0)

Where LD_I = R_DI (IF RD_I ³ 0) or 0 (IF RD_I < 0)

T = ((1+L₀) ´ (1+L₁) ´ … ´ (1+L_N)) -1

TD = ((1+LD₀) ´ (1+LD₁) ´ … ´ (1+LD_N)) -1

Up Capture = T ¸ TD

Down Capture - The Down Capture Ratio is a measure of the Investment’s compound return when the Benchmark was down divided by the Benchmark’s compound return when the Benchmark was down. The smaller the value, the better.

Where R _I = Return for period I

Where RD _I = Benchmark Return for period I

Where N = Number of Periods

Where L _I = R_I (IF RD_I < 0) or 0 (IF RD_I ³ 0)

Where LD _I = RD_I (IF RD_I < 0) or 0 (IF RD_I ³ 0)

T = ((1+L₀) ´ (1+L₁) ´ … ´ (1+L_N)) -1

TD = ((1+LD₀) ´ (1+LD₁) ´ … ´ (1+LD_N)) -1

Down Capture = T ¸ TD

Up # - The Up Number Ratio is a measure of the number of periods that the Investment was up, when the Benchmark was up, divided by the number of periods that the Benchmark was up. The larger the ratio, the better.

Where R_I = Return for period I

Where RD_I = Benchmark Return for period I

Where N = Number of Periods

Where L_I = 1 (IF R_I ³ 0 AND RD_I ³ 0) ELSE 0

Where LD_I = 1 (IF RD_I ³ 0) ELSE 0

_N

T = (S L _I)

^I=1

_N

TD = (S LD _I)

^I=1

Up Number Ratio = T ¸ TD

Down # - The Down Number Ratio is a measure of the number of periods that the Investment was down when the Benchmark was down, divided by the number of periods that the Benchmark was down. The smaller the ratio, the better.

Where R _I = Return for period I

Where RD _I = Benchmark Return for period I

Where N = Number of Periods

Where L _I = 1 (IF R_I < 0 AND RD_I < 0) ELSE 0

Where LD _I = 1 (IF RD_I < 0) ELSE 0

_N

T = (S L _I)

^I=1

_N

TD = (S LD _I)

^I=1

Down Number Ratio = T ¸ TD

Up % - The Up Percentage Ratio is a measure of the number of periods that the Investment outperformed the Benchmark when the Benchmark was up, divided by the number of periods that the benchmark was up. The larger the ratio, the better.

Where R_I = Return for period I

Where RD_I = Benchmark Return for period I

Where N = Number of Periods

Where L_I = 1 (IF R_I ³ RD_I AND RD_I ³ 0) ELSE 0

Where LD_I = 1 (IF RD_I ³ 0) ELSE 0

_N

T = (S L _I)

^I=1

_N

TD = (S LD _I)

^I=1

Up Percentage Ratio = T ¸ TD

Down % - The Down Percentage Ratio is a measure of the number of periods that the Investment outperformed the Benchmark when the Benchmark was down, divided by the number of periods that the benchmark was down. The larger the ratio, the better.

Where R_I = Return for period I

Where RD_I = Benchmark Return for period I

Where N = Number of Periods

Where L_I = 1 (IF R_I ³ RD_I AND RD_I < 0) ELSE 0

Where LD_I = 1 (IF RD_I < 0) ELSE 0

_N

T = (S L _I)

^I=1

_N

TD = (S LD _I)

^I=1

Down Percentage Ratio = T ¸ TD

% Gain - The Percent Gain Ratio is a measure of the number of periods that the Investment was up divided by the number of periods that the Benchmark was up. The larger the ratio, the better.

Where R_I = Return for period I

Where RD_I = Benchmark Return for period I

Where N = Number of Periods

Where L_I = 1 (IF R_I ³ 0) ELSE 0

Where LD_I = 1 (IF RD_I ³ 0) ELSE 0

_N

T = (S L _I)

^I=1

_N

TD = (S LD _I)

^I=1

Percent Gain Ratio = T ¸ TD