Workstation Calculation Guide - Morningstar

Morningstar ® Workstation
Calculation Guide
July 2011
©2010 Morningstar, Inc. All rights reserved.
Page 1 of 21

Table of Contents
Table of Contents....................................................................................................................... 2
Introduction................................................................................................................................ 4
Parameters Required for Calculations .................................................................................... 4
Standard Statistics & Series’ .................................................................................................... 6
1 Alpha ................................................................................................................................. 6
2 Alpha (Ann) ...................................................................................................................... 6
3 Alpha (Jensen’s) ............................................................................................................... 6
4 Annual Average / Maximum Loss......................................Error! Bookmark not defined.
5 Appraisal Ratio.................................................................................................................7
6 Average Return................................................................................................................. 7
7 Average Return (Ann)...................................................................................................... 7
8 Average Return (Rel) ....................................................................................................... 7
9 Average Return (Ann Rel)............................................................................................... 7
10 Average Return (XS)........................................................................................................ 7
11 Average Return (Ann XS) ............................................................................................... 7
12 Beta .................................................................................................................................... 7
13 Beta (Bear) ........................................................................................................................ 8
14 Beta (Bull) ......................................................................................................................... 9
15 Coefficient of Variation.................................................................................................... 9
16 Correlation...................................................................................................................... 10
17 Correlation (Bull) ........................................................................................................... 10
18 Correlation (Bear) .......................................................................................................... 10
19 Covariance ......................................................................................................................10
20 Down-Down Periods....................................................................................................... 10
21 Down-Up Periods............................................................................................................ 11
22 Downside Capture Ratio................................................................................................ 11
23 Downside Risk................................................................................................................. 11
24 Efficiency......................................................................................................................... 12
25 Gain/Loss Ratio .............................................................................................................. 12
26 Gain (Longest) ................................................................................................................ 12
Page 2 of 21

27 Gain (Max) ......................................................................................................................12
28 Information Ratio........................................................................................................... 13
29 Investor Return............................................................................................................... 13
30 Loss (Longest)................................................................................................................. 13
31 Loss (Max)....................................................................................................................... 13
32 Months (%-ve) / Months (%+ve) .................................................................................. 14
33 Quarter (Best) / Quarter (Worst).................................................................................. 14
34 Return (Max) .................................................................................................................. 14
35 Return (min) ................................................................................................................... 15
36 Return (Range) ............................................................................................................... 15
37 R-Squared .......................................................................................................................15
38 Sharpe Ratio ................................................................................................................... 16
39 Sortino Ratio...................................................................................................................16
40 Standard Error............................................................................................................... 17
41 Sandard Error (beta) ..................................................................................................... 17
42 Tracking Error ............................................................................................................... 18
43 Treynor Ratio ................................................................................................................. 18
44 Up / Down Periods (% of total) ..................................................................................... 19
45 Up / Up Periods (% of total) .......................................................................................... 19
46 Upside Capture Ratio..................................................................................................... 19
47 Upside Potential Ratio.................................................................................................... 19
48 Upside Risk .....................................................................................................................20
49 Variance .......................................................................................................................... 20
50 Volatility.......................................................................................................................... 20
Page 3 of 21

Introduction
All descriptions, formulae and examples utilize a Reinvested Income Price Series (RIPs) rather
than the fund or indices pricing series. This is because RIPs are an aggregation of prices,
income, splits and other corporate actions thus allowing fair and easy comparison between funds
and indices.
Parameters Required for Calculations
Calculation
Annualized Option
Relative Option
Excess Option
Benchmark Option
In Simple Terms:
Alpha Y The measure of a funds theoretical
total return if the benchmark return
were zero.
Alpha (Ann) Y The measure of a funds theoretical
annualised return if the benchmark
return were zero.
Alpha (Jensen’s)
The funds alpha, adjusted by the
relative risk compared to the
benchmark.
Annual Average / Maximum Loss
The ratio of annualised return over
worst possible performance.
Appraisal Ratio Y
Bear Beta Y
Bear Correlation Y
Beta Y
Best / Worst Quarter
Bull Beta Y
Bull Correlation Y
Coefficient of Variation
Correlation Coefficient Y
Covariance Y
Down/Down Periods (% of Total)
Down/Up Periods (% of Total)
Efficiency
Gains : Losses
Benchmark Required
Manual Entry Option
Manual Entry Required
Page 4 of 21

Information Ratio Y Y Y Y The Information Ratio is based on
Excess or Relative returns and
benchmark is mandatory
Jensen's Alpha Y Y Requires a user to enter the Risk
Free Rate
Longest Gain Period
Longest Loss Period
Maximum (Return over Period) Y Y Y A Benchmark is mandatory when
calculating the Annualised or Excess
variants of this statistic
Maximum Gain Y Y Y A Benchmark is mandatory when
calculating the Annualised or Excess
variants of this statistic
Maximum Loss Y Y Y A Benchmark is mandatory when
calculating the Annualised or Excess
variants of this statistic
Minimum (Return over Period) Y Y Y A Benchmark is mandatory when
calculating the Annualised or Excess
variants of this statistic
Negative Periods (%)
Positive Periods (%)
R Squared Y
Sharpe Ratio Y Y Requires a user to enter the Risk
Free Rate
Sortino Ratio Y Y If Below-Benchmark downside risk is
selected the user is required to
select a benchmark.
If Below-Target downside risk is
selected. The user is required to
enter a Minimum Acceptable Return
Tracking Error Y Y Y Y
Treynor Ratio Y Y Requires a user to enter the Risk
Free Rate
Up/Down Periods (% of total)
Up/Up Periods (% of total)
Upside/Downside Ratio
Upside/Downside Capture Ratio Y
Upside Risk Y Y If Above-Benchmark downside risk
is selected the user is required to
select a benchmark.
Variance
Volatility Y
If Above-Target downside risk is
selected. The user is required to
enter a Minimum Acceptable Return
Page 5 of 21

Standard Statistics & Series’
Fund Monthly RIPs Fund Monthly Returns Index Monthly RIPs Index Monthly Returns
p 1
q 1
p 2
x 1 p2
/ p1
1
q 2
y 1 q2
/ q1
1
… … … …
p n1
x n pn1 / pn
1
q n1
y n qn1 / qn
1
1 Alpha
The Alpha is the y-intercept of the regression line of fund’s returns (y’s) against index returns
(x’s). It’s the fund’s theoretical return when its benchmark return is zero.
n
i1
n
x
i
n
i1
n
y
i
2 Alpha (Ann)
Annualised Alpha is simply:
12
annual
3 Alpha (Jensen’s)
Jensen’s Alpha is the difference between a fund’s actual returns and those that could have been
earned on a benchmark portfolio with the same amount of market risk (the same beta). It
measures a fund’s performance relative to a zero risk investment (Risk Free Rate) and the
performance of the fund’s benchmark.
1/ n
1/ n
p
n1
Jensen’s Alpha =
1/ m
qn1
1/
m
( 1 R f )
(1 R f ) ,
p1
q1
12 for monthly data
m
52 for weekly data
R is user-supplied and quoted on annual basis.
f
Page 6 of 21

4 Appraisal Ratio
Alpha divided by Standard Error, or non systematic risk.
Appraisal Ratio =
SE
1
n
n 2 i1
y i
x i
2
5 Average Return
The geometric average of the fund’s returns between the start and end date.
1/ n
p
n1 Average Return
1
p1
6 Average Return (Ann)
The Annualised geometric average of the fund’s returns between the start and end date.
m/
n
p
n1 12
Annualised Average Return
1,
m
p1
52
for monthly data
for weekly data
7 Average Return (Rel)
The geometric average of the fund’s returns between the start and end date divided by the
geometric average of the benchmark returns.
8 Average Return (Ann Rel)
The annualised geometric average of the fund’s returns between the start and end date divided
by the annualised geometric average of the benchmark returns.
9 Average Return (XS)
The geometric average of the fund’s returns between the start and end date minus the geometric
average of the benchmark returns.
10 Average Return (Ann XS)
The annualised geometric average of the fund’s returns between the start and end date minus the
annualised geometric average of the benchmark returns.
11 Batting Average
BattingAverage
where:
n
b
100
b
n
= number of months/weeks over which data was taken
= number of months/weeks in which the fund beats or matches the index
Page 7 of 21

12 Beta
Beta is the slope of a regression line. A fund’s Beta is an estimate of how much a fund’s return
will move if its benchmark moves by 1. It’s a measure of the sensitivity of a fund’s return to the
changes in its benchmark’s return.
n
n
x
y
i i i i
i1
i1
i1
n
n
2
2
n
yi
yi
i1 i1
n
x
n
y
13 Beta (Ann)
annual
n
n
i1
n
x
n
m
i
i1
2
n n
n
m
yi
i1 i1
y
n
m
i
n
x
n
m
i
n
i1
y
n
m
i
y
2
n
m
i
where:
n = number of months/weeks over which data was taken
m = 12 for monthly data, or 52 for weekly data
14 Beta (Bear)
Bear Beta is a measure of the sensitivity of a fund’s return to negative changes in its benchmark’s
return.
Bear
m
x
m
yneg
y
y
yneg
2
yneg
where m = number of negative index (y) returns.
x
yneg
y
yneg
y
yneg
2
15 Beta (Bear Ann)
BearAnnual
where:
n
m
g
g
i1
g
g
g
x
i1
yneg
y
n
m
yneg
y
yneg
n
m
2
n
m
g
i1
g
i1
x
n g
m
yneg
i1
y
yneg
y
n
m
yneg
= number of months/weeks over which data was taken
= 12 for monthly data, or 52 for weekly data
= number of negative index (y) returns.
n
m
2
Page 8 of 21

16 Beta (Bull)
Bull Beta is a measure of the sensitivity of a fund’s return to positive changes in its benchmark’s
return.
Bull
m
x
m
ypos
where m = number of positive index (y) returns.
17 Beta (Bull Ann)
BullAnnual
g
g
i1
g
g
x
i1
ypos
y
n
m
ypos
y
n
m
ypos
2
n
m
g
i1
g
i1
x
y
y
n g
m
ypos
i1
y
ypos
ypos
2
ypos
y
n
m
ypos
n
m
2
x
ypos
y
ypos
2
y
ypos
where:
n
m
g
= number of months/weeks over which data was taken
= 12 for monthly data, or 52 for weekly data
= number of negative index (y) returns.
18 Calmar Ratio (Ann Avg / Max Loss)
This is the fund’s Geometric Annualized Average return divided by the fund’s Maximum
Drawdown (greatest ever loss, including temporary up periods). The higher the ratio, the better
the performance of the fund.
Annual Average / Max Loss =
m / n
pn1
1
p1
( 1)
,
Maximum Loss
12
m
52
for monthly data
for weekly data
See item 24 for Maximum Loss.
19 Coefficient of Variation
Coefficient of Variation is the inverse of the efficiency ratio, i.e., standard deviation (risk) divided
by mean return.
COV =
p
p
n1
1
1/ n
1
Page 9 of 21

20 Correlation
Correlation coefficient measures the extent to which the returns of a fund and a benchmark are
related. Values vary between 1 and –1. A correlation of 1 would indicate a perfect positive
correlation, i.e., a fund’s returns move exactly in line with its benchmark. A correlation of -1 would
indicate a perfect negative correlation, i.e., a fund’s returns move in the exact inverse of its
benchmark.
where
r xy
xy
n
n
2
n
xi
xi
i1
i1
x
,
n(
n 1)
x
2
y
y
n
n
i1
n
2
y
i
i
n(
n 1)
1
2
y
i
21 Correlation (Bull)
Bull correlation measures the relationship between a fund and positive market movements.
xy
r xy
, where y i is positive.
See Item 11 for and , and Item 12 for .
x
y
x
y
xy
22 Correlation (Bear)
Measures the relationship between a fund and negative market movements.
xy
r xy , where y i is negative.
See Item 11 for and , and Item 12 for .
x
y
x
y
xy
23 Covariance
A measure of how much the fund and its benchmark move together. A positive covariance shows
that the returns of a fund and its benchmark move in the same direction. A negative covariance
shows that the returns of a fund and its benchmark move in opposite direction.
xy
1
n 1
n
i1
x
xy
y
i
i
24 Down-Down Periods
It’s the percentage of the total number of observations where a negative return was followed by
another negative return. Note: The divisor is the number of observations minus 1.
Page 10 of 21

25 Down-Up Periods
It’s the percentage of the total number of observations where a negative return was followed by a
positive return.
Note: The divisor is the number of observations minus 1.
26 Downside Capture Ratio
Downside Capture Ratio measures a manager's performance in up markets relative to the market
(benchmark) itself. It is calculated by taking the security’s downside capture return and dividing it
by the benchmark’s downside capture return.
DownsideCa ptureRatio
DnCapRtn
DnCapRtn
sec urity
benchmark
100
Downside Capture Return is a measure of the manager's performance in periods when the
market (benchmark) goes down
27 Downside Risk
It’s measured by semivariance: the average of the squared deviations of returns around the
average return of observations 1) < Mean Return, 2) < Benchmark Mean Return, 3) Minimal
Acceptable Return (MAR). The argument is that returns above the mean are desirable. The
only returns that disturb an investor are those below average.
Below MEAN semivariance:
SV
Below BENCHMARK semivariance:
Below TARGET semivariance:
SV
SV
t
m
b
1
n 1
n
max(0,
x x)
i
d 1
1
n 1
d
1
n 1
d
n
n
max0,
y xi
i1
max0,
MAR xi
i1
n d = count of data periods where return is less than the mean return x , benchmark mean return
y or target return MAR.
MAR= Minimum Acceptable Return, user supplied
Annualised Mean Semivariance:
12 for monthly data
Annualised SV m = SV m * √m , m
52 for weekly data
Annualised Benchmark Semivariance:
12 for monthly data
Annualised SV b = SV b * √m , m
52 for weekly data
2
2
2
Page 11 of 21

Annualised Target Semivariance:
12
Annualised SV t = SV t * √m , m
52
for monthly data
for weekly data
28 Efficiency
It’s the average return divided by the average risk over the selected time period and selected data
frequency.
Efficiency =
p
p n 1
1
1/ n
1
29 Gain/Loss Ratio
This is the sum of positive fund returns divided by the sum of negative fund returns over the
chosen calculation period. The higher the ratio, the greater the proportion of positive returns
versus negative returns, therefore the better the fund’s performance.
Gain / Loss = ( 1)
i
xi | x i 0
i
x | 0
i x i
30 Gain (Longest)
Largest number of returns where a positive return was followed by another positive return.
31 Gain (Max)
The Maximum Gain represents the best possible return within the performance period.
p j
Maximum Gain = Max 1
i
j pi
Relative:
p
j
1
pi
Max
1
i
j
q
j
1
qi
Excess:
p
Max
i
j p
j
i
q
q
j
i
Page 12 of 21

32 Information Ratio
The Information ratio shows the degree to which a fund outperforms or under performs its
benchmark, taking into account its sensitivity to its specified market (Tracking Error). The general
formula is mean(d)/std(d).
Inf Ratio
Inf Ratio (Ann)
Inf Ratio (Rel)
Inf Ratio (Ann Rel)
Informatio n Ratio
Informatio n Ratio
annual
Informatio n Ratio
Informatio n Ratio
annual
XS
XS
rel
rel
XS
XS
annual
annual
rel
rel
annual
annual
XS
XS
rel
rel
The higher the value the higher the extent to which the fund has consistently outperformed its
benchmark.
33 Investor Return
In order to calculate investor returns, Morningstar first calculates the monthly cash inflows or
outflows for each fund. The cash flow estimate for a month (C) is simply the difference in
beginning and ending total net assets (TNA) that cannot be explained by the monthly total return
(r).
C t = TNA t – TNA t+1 (1+r t )
Once monthly cash flows are available for the period in question, investor returns can be derived
with an iterative process. As with an internal rate of return calculation, investor return is the
constant monthly rate of return that makes the beginning assets equal to the ending assets with
all monthly cash flows accounted for. Morningstar runs a program that attempts to solve for this
constant rate of return, adjusting the estimate up and down until it converges on a solution. Then
the monthly investor return (r) is annualized ((1+ RIR)12-1).
34 Loss (Longest)
Largest number of returns where a negative return was followed by another negative return.
35 Loss (Max)
The Maximum Loss represents the worst possible return within the performance period.
p j
Maximum Loss = Min 1
, i, j 1,
2, , n 1
i
j pi
Page 13 of 21

Relative:
Excess:
p j
pi
Min
i
j
q
j
qi
p
Min
i
j p
j
i
1
1
1
q
q
j
i
36 Months (%-ve) / Months (%+ve)
This is the percentage of negative monthly returns over the performance period.
Count(
x 0)
Negative Months % = i
100
n
This is the percentage of positive monthly returns over the performance period.
Count(
x 0)
Positive Months % = i
100
n
This is also calculated as a count of months when using Months (-ve) or Months (+ve)
37 Quarter (Best) / Quarter (Worst)
The Best Quarter is the largest positive percentages change over any three months within the
performance period.
p
i 3
Best Quarter = Max
1
i
pi
The Worst Quarter is the largest negative percentages change over any three months within the
performance period.
pi3
Worst Quarter = Min 1
i
pi
Note: monthly data is used in this formula.
38 Return (Max)
The single highest observation for all data available.
Relative:
p
Max
p
i1,
,
n
i1
i
1
Page 14 of 21

Excess:
pi
pi
Max
i1,
,
n
qi
qi
pi
Max
1,
,
n
p
i
1
1
1
i
1
1
1
qi
q
1
i
39 Return (min)
The single lowest observation for all data available.
Relative:
Excess:
p
Min
p
i1,
,
n
pi
pi
Min
1,
,
n
qi
qi
i
pi
Min
1,
,
n
p
i
i1
1
1
1
i
i
1
1
1
1
qi
q
1
i
40 Return (Range)
The single highest observation for all data available minus the lowest observation for all data
available.
p i 1
p 1
Max 1
Min i
1
i1,
,
n
Range = [ pi
i1,
,
n
] – [ pi
]
41 R-Squared
R squared is the square of the correlation and
measures the strength of the association between
the returns of a fund and its benchmark. It’s a measurement of what portion of its performance
can be explained by the performance of the overall market or index. Ranges from 0 to 1 with 0
indicating no correlation and 1 indicating perfect correlation.
See Item Correlation for r xy .
2
R
2
r xy
Page 15 of 21

42 Sharpe Ratio
The Sharpe Ratio is an expression of a fund’s return relative to the risk incurred after subtracting
the returns of a zero risk investment (Risk Free Rate). It’s a measure of efficiency that removes
the risk-free rate of return from observation. The higher the ratio, the better the fund.
Sharpe Ratio =
1/ n
p n1
1/
m
( 1
R f )
p
1
,
12
m
52
for monthly data
for weekly data
Annualized:
Sharpe Ratio =
m / n
p
n 1
(1 R f )
p1
,
m
12
m
52
for monthly data
for weekly data
Where
R f = Risk Free rate, user supplied, quoted on annual basis.
is the standard deviation of p / , i 1, 2, ,
i 1 p i n
Sharpe Ratio can also be run with a Benchmark in place of the Risk Free Rate, these are the
Sharpe (Bmk) and Sharpe (Bmk Ann) functions.
43 Sortino Ratio
Sortino Ratio is calculated the same as Sharpe Ratio except you use the downside risk as the
divisor. The larger the Sortino Ratio, the less the likelihood of large losses occurring.
Sortino Ratio =
1/ n
p
n1
1/
m
(1 R
f
p )
1
SV
x
,
12
m
52
for monthly data
for weekly data
R f = Risk Free rate, user supplied, quoted on annual basis.
SVx
= Downs ide Risk, see item 15 for description
There are three calculation options within Workstation for the Sortino, the difference is described
below:
Sortino Mean – this uses the mean semi-variance of the fund for the Minimum
Acceptable Return (i.e. calculates the average return of the fund and uses that)
Sortino Target – this enables the manual input of your target MAR.
Sortino Benchmark – this one enables you to choose a benchmark as your MAR.
Page 16 of 21

Annualised Sortino Ratio:
Sortino Ratio = SR * √m ,
12
m
52
for monthly data
for weekly data
44 Standard Error
Standard Error provides an indication of the magnitude of the sampling error in the prediction of
the fund return for a given index return. The Standard Error shows the amount of error in the
position of the regression line.
Example:
If we use the regression line to estimate Y with a given value of X, the Standard Error gives us an
indication of the range that estimate may lie in.
Thus, if Y
X
and =5 and =1, and the estimated standard error = 2, then
X Y Range
Min Max.
10 10+5=15 13 17
12 12+5=
17 15 19
etc.
If the Standard Error is 2, then the range of Y is 2.
Formula:
er
1
n y y
nn 2 2
2
n
xy x y
n x
2 2
x
2
where:
=(Returns of Fund A-1) 100
x i
y i =(Returns of Fund B-1) 100
n=number of Monthly Returns
45 Standard Error (beta)
Stand ard Error provides an indication of the magnitude of the sampling error in the prediction of
the fund return for a given index return. The Standard Error shows the amount of error in the
position of the regression line.
Formula:
er
n xy x y
n y y
1
2 2
nn 2
n x
2 2
x
2
where:
=(Returns of Fund) 100
x i
y i =(Returns of Index) 100
n=number of Monthly Returns
Page 17 of 21

46 Tracking Error
Relative Tracking Error measures the standard deviation of the relative returns.
Monthly/Weekly relative rel :
pi
1
1
pi
It’s the standard deviation of fund’s monthly/weekly relative returns, 1.
qi
1
1
qi
Annualized relative annual rel
:
12 for monthly data
annual rel
rel
m , m
52 for weekly data
Excess Tracking Error measures the standard deviation of the excess returns.
Monthly/Weekly excess
XS :
It’s the standard deviation of fund’s monthly excess returns,
Annualized excess
annual
XS
:
m ,
annual XS
XS
pi1 q i 1
.
p i q i
m
12 for monthly data
52 for weekly
data
The lower the tracking error the closer the relationship between the risk/return characteristic of
the fund and those of its benchmark or the market.
47 Treynor Ratio
Also known as the Reward-to-Volatility ratio. The Treynor Ratio is a measure of a fund’s
performance relative to its benchmark after subtracting the risk free rate.
Treynor Ratio:
where:
μ Mon = Monthly Average
R mon =Monthly Risk Free Rate
f
Treynor mon
mon
R f
mon
R %
n
m
f
1
100
where:
n=number of months you wish to express R in. f
Page 18 of 21

m=12 for monthly data
β =Beta
R f = Risk
Annualised Treynor Ratio:
Free rate, user supplied, quoted on annual basis.
where:
μ annual =Annualised Average
R =Annualised Risk Free Rate
f
Β =Beta
Treynor
annual
annual
R
f
48 Up / Down Periods (% of total)
It’s the percentage of the total number of observations where a positive return was followed by a
negative return.
Note: The divisor is the number of observations minus 1.
49 Up / Up Periods (% of total)
It’s the percentage of the total number of observations where a positive return was followed by
another positive return.
Note: The divisor is the number of observations minus 1.
50 Upside Capture Ratio
Upside Capture Ratio measures a manager's performance in up markets relative to the market
(benchmark) itself. It is calculated by taking the security’s upside capture return and dividing it by
the benchmark’s upside capture return.
UpCapRtn
UpsideCaptureRatio
UpCapRtn benc
sec urity
hmark
100
UpCapRtn is a measure of the manager' s performance in periods when the market (benchmark)
goes up
51 Upside Potential Ratio
The Upside Potential ratio is the expected return in excess of the MAR divided by the downside
risk.
Page 19 of 21

U-P ratio =
1
n n
1
n
1
MAR= Minimum Acceptable Return, user-supplied.
d
n
n
i1
max0,
MAR x
i
d 1
i 1
i
max 0, x MAR
2
52 Upside Risk
Volatility over range of returns that are greater than: 1) the Mean Return, 2) Benchmark Mean
Return, or 3) Minimal Acceptable Return (MAR).
Above-mean semivariance:
Above-benchmark semivariance:
Above-target semivariance:
SV
SV
SV
t
m
b
1
n 1
n
u i1
1
n 1
u
1
n 1
u
max0,
xi
x
n
max0,
xi
y
n
i1
max0,
xi
MAR
i1
2
2
2
nu
= count of data periods where return is greater than the mean return
return y or target return MAR.
x , benchmark mean
Note: MAR is user-supplied.
53 Variance
It measures the divergence of returns around the average return.
2
V
S ee Item 40 for .
54 Volatility
Volatility indicates the degree of variability present in a fund’s return history. The higher the
volatility, the higher the level of implicit risk associated with the fund.
Annualized
Volatility
n
n
i1
n
2
x
i
i
n( n 1)
1
2
x
i
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m ,
annual
m
12
52
for monthly data
for weekly data
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