Review of Business and Economics Studies, 2018, том 6, № 3

Review of  
Business and 
Economics  
Studies

EDITOR-IN-CHIEF
Prof. Alexander Ilyinsky
Dean, International Finance 
Faculty, Financial University, Moscow, 
Russia
ailyinsky@fa.ru 

EXECUTIVE EDITOR
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Adam Smith Business School, 
The Business School, University 
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Research Director, IEFE Centre for 
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Economics and Policy, Università 
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Dept.of Mathematical Sciences, Brunel 
University, UK

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Department of Economics, Birmingham 
Business School, University 
of Birmingham, UK

Prof. Chien-Te Fan
Institute of Law for Science and 
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Director, Asia Business Studies, College 
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Faculty of Economics, Novosibirsk State 
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Development Economics, Cambridge 
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School of Economics, Moscow State 
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Department of Mathematical and 
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Prof. Thomas Renstrom
Durham University Business School,
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Saïd Business School, Fellow in 
Management, University of Oxford; 
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ISSN 2308-944X

Вестник 
исследований 
бизнеса  
и экономики

ГЛАВНЫЙ РЕДАКТОР
А.И. Ильинский, профессор, декан 
Международного финансо вого факультета Финансового университета 

ВЫПУСКАЮЩИЙ РЕДАКТОР
Збигнев Межва, д-р экон. наук

РЕДАКЦИОННЫЙ СОВЕТ

М.М. Алексанян, профессор Бизнесшколы им. Адама Смита, Университет 
Глазго (Великобритания)

К. Вонг, профессор, директор Института азиатско-тихоокеанского бизнеса 
Университета штата Калифорния, 
Лос-Анджелес (США)

К.П. Глущенко, профессор экономического факультета Новосибирского 
госуниверситета

С. Джеимангал, профессор Департамента статистики и математических финансов Университета Торонто 
(Канада)

Д. Дикинсон, профессор Департамента экономики Бирмингемской бизнесшколы, Бирмингемский университет 
(Великобритания)

В.Л. Квинт, заведующий кафедрой 
финансовой стратегии Московской 
школы экономики МГУ, профессор 
Школы бизнеса Лассальского университета (США)

Г. Б. Клейнер, профессор, член-корреспондент РАН, заместитель директора Центрального экономико-математического института РАН

Э. Крочи, профессор, директор по 
научной работе Центра исследований 
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директор Центра политики 
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Депар та мента математических 
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университета

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Ворчестерского политехнического 
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народнохозяйственного прогнозирования РАН

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Бизнес-кол леджа Университета 
Стони Брук (США) 

Т. Ренстром, профессор,
Школа Бизнеса Даремского 
университета,
Департамент Экономики и Финансов

Б.Б. Рубцов, профессор,  
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и банков по НИР Финансового 
университета

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Сассекский университет 
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Review of  
Business and  
Economics  
Studies

Volume 6, Number 3, 2018

Вестник 
исследований 
бизнеса  
и экономики

№ 3, 2018

Анализ инвестиционной деятельности  

на основе количественных мер, настроенных на риск

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Review of Business and Economics Studies  
doi: 10.26794/2308–944X‑2018–6–2‑5‑18
2018, Vol. 6, No. 3, 5‑18
Performance Analysis Based on Adequate 
Risk-Adjusted Measures

Alexander Melnikov*, daria Vyachkileva**

* Professor, doctor of Science in Physics and Mathematics, melnikov@ualberta.ca
** MSc., student, Mathematical Finance Program, vyachkil@ualberta.ca
department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Canada, T6G 2G1
Phone: +1‑780‑492‑0568; Fax: +1‑780‑492‑6826

Abstract
There are many potential investment options for investors and they should be able to compare them on a 
risk‑adjusted basis. if investors rely only on pure return they can be exposed to a high risk. Therefore, many 
investors rely on adequate performance measures to evaluate potential investment opportunities. in this paper, 
we describe widely used risk‑adjusted performance measures and add correlation through the M3 measure. 
We apply described measures to real financial data in order to rank managers and compare rankings between 
measures. We also look at the following year measures to compare the results with predictions.
Keywords: performance measures; correlation; manager ranking
JEL classification G11, G17

Introduction
According to the Investment Company Institute (2018) in 2017 total net assets of worldwide 
regulated open-end funds was more than $ 49 
trillion and have more than doubled in the past 
decade. Therefore, investors need instruments 
to analyse and choose the best funds.
Investors rely on risk-adjusted measures. However, there are a lot of measures that can be used, 
and it is not clear which ones are better or worse 
since none of the investors uses the same measures. 
In addition, such variety can be explained by investors not being able to define risk in such a way that 
it would incorporate all necessary parameters. We 
present several performance measures discussing 
the advantages and disadvantages of each.
In addition, to giving different risk definitions we will incorporate a correlation using 
the M3 measure (see Muralidhar (2000)). Correlation is an important parameter when one 
wants to create a portfolio rather than investing 
in a single stock. If we define risk as a standard 
deviation of stock returns (as most investors 
do) or some sort of standard deviation, then if 
we have a portfolio of two or more stocks then 
the combined standard deviation is not a sum 
of single standard deviations. Instead, we have 

a correlation term and a portfolio standard 
deviation looks like this

2
2
2
2
2
,
2
,
Portfolio
A
A
B
N
A B
A
B
A
B
σ
= σ ω + σ ω
+ ρ
ω ω σ σ

where 
A
σ  and 
B
σ  are standard deviations of 
stocks A and B respectively, 
A
ω  and 
B
ω  are 
weights invested om stocks A and B respectively, 

,A B
ρ
 is a correlation between stocks A and B.
It is clear from this equation that investors should 
seek negative or small correlation to decrease the 
portfolio risk. That is why it is important to be able to 
select new investments not only by their returns but 
also making an adjustment for correlation between 
new stock and an existing portfolio.
Another way to incorporate correlation is to 
use the approach that was proposed by Dowd 
(2000). His method shows how to adjust for correlation using the most popular and widely used 
risk-adjusted measure —  Sharpe ratio. Dowd’s 
basic idea is the following: calculate the Sharpe 
ratio before accepting new stock and calculate 
on the stock was accepted a while ago, then we 
can compare these two ratios and if it increased 
then we would proceed with the deal. In this case, 
we account for correlation when we calculate the 
new Sharpe Ratio.

In addition, in a recent research done by Ornelas, Silva Júnior, and Fernandes (2010) shows 
that performance ratios matter. Previously, Eling 
(2008) showed that some measures have a very 
high correlation in ranking with the Sharpe ratio 
and it might be enough just to look at Sharpe Ratio. 
However, Ornelas, Silva Júnior, and Fernandes 
(2010) exploited other measures in their research 
and agreed with Eling to some degree but not all 
measures produced a high correlation. Therefore, 
we should look at and compare measures and we 
cannot use only the Sharpe Ratio.

Risk-Adjusted Performance Measures
As was mentioned previously there are many 
risk-adjusted performance measures and their 
variations. In this paper, we describe some of 
the most commonly used measures and discuss 
their advantages and disadvantages. More detailed information can be found in Bacon (2013), 
Dowd (2000), Goodwin (1998), Grinold and Khan 
(1999), Harlow (1991), Lo (2002), Madgon-Ismail 
and Atiya (2004), Modigliani and Modigliani 
(1997), Muralidhar (2000, 2001, 2005), Papageorgiou (2005), Rollinger and Hoffman (2015), 
Prokopczuk, Rachev, and Truck (2004), Sharpe 
(1966, 1994), Sortino and van der Meer (1991) 
and Young (1991). Table 1 shows some widely 
used measure.
As we can see from Table 1 there are multiple 
definitions of risk and it is almost impossible to 
choose one. However, investors can choose the 
one that suits their vision of the market the best. 
We will apply all these measures to real-life financial data and later we will describe how investors 
can incorporate correlation using Dowd’s (2000) 
approach.

Fund case-studies
In this section, we describe the procedure that 
was used to calculate all ratios and the way 
funds were selected.
Article by Bill Harris “‘The 10 Biggest Mutual 
Funds: Are They Really Worth Your Money?” in 
Forbs brought our attention and 8 out of 10 funds 
presented were chosen for the illustration. Two 
funds from this article were eliminated because 
they are fixed income funds. To be able to compare 
apples to apples they were not selected because 
comparison would not be fair when we have to 
select a benchmark.

Monthly data was taken for the 11 years from 
1/1/2006 up to 1/1/2017 for all 8 funds. First, ten 
years were used to analyse the risk-adjusted performance of all funds when the last year was used 
to compare the results for the previous 10 years 
and the following year. The data were obtained 
for all funds and for the benchmark which was 
chosen to be S&P500 because funds are different 
by their nature and we need a common benchmark. 
Also, one should note that the financial crisis of 
2008–2010 was included in calculations. Therefore, 
some returns were small however we decided they 
are not outliers because it is a part of the risk of 
investing in a market.

Sharpe Ratio
First, let’s show how the Sharpe Ratio can be applied to the 8 selected funds. As we know from 
the definition of the Sharpe ratio we need an appropriate risk-free rate. In this example, US 10 
years T-Bond rate as of 12/31/2005 was chosen 
and equal to 4.39%. 10 years T-Bonds were chosen because we want to make sure we would 
make more on our investments rather than investing in a risk-free rate and leaving money 
there for 10 years. In Table 2 we can see the 
Sharpe ratio and all the information needed for 
all 8 funds:
As we can see from the Table 2 all funds produce positive returns and greater than the risk-free 
rate. Therefore, it would be more beneficial for 
investors in a long run to invest in any of these 
funds rather than risk-free rate even though the 
financial crisis of 2008–2010 are included in this 
dataset.
Table 2 allows us to make the following conclusions:
If we compared pure return without adjusting 
for risk, then the fund 3 would be the most attractive. Fund 3 was the best even after adjusting for 
the risk (standard deviation) because its return 
to risk had the best ratio.
Fund 6 produced a quite small annual return 
over the last 10 years comparing to the risk they 
took. They produced only 5.13% return per year, 
but they took 22.05% of the risk, which was the 
highest risk among all 8 funds.
In addition, maybe fund 5 did not produce the 
highest return but its risk was relatively small 
keeping in mind that the Financial Crisis period 
was included and fund 5 got the 5th rank.

Performance Analysis Based on Adequate Risk-Adjusted Measures

Performance Analysis Based on Adequate Risk-Adjusted Measures

Table 1
Risk-adjusted performance measures

Name
Definition
Advantages
Disadvantages

Sharpe 
Ratio
 

f
R
SR
µ −
=
σ

– Allows to compare and 
rank fund /managers
– Most of its advantages 
and disadvantages are 
known

– σ  is not always an 

appropriate risk measure
– σ  punishes companies for 

upward momentum
– no interpretation of the 
number

information 
Ratio
 

ER
IR =

σ

– Useful measure when 
the benchmark is carefully 
chosen

– Not a complete statistic
– only maximizing iR can 
lead to wrong decisions

M3
 

(
)
(
)
1
B
f
r CAP
a
bR
a
b R
= µ +
+
− −

– Adjust for correlation
– Provide guidance on how 
to build a portfolio

– Correlation is not stable 
over time
– Hard to compare funds on 
the after‑fee basis

Sortino 
Ratio
 
T
R
S
TDD
µ −
=

– Accounts only for the 
downside deviation
– Accounts for risk better 
if the distribution is not 
symmetric

– does not account for 
correlation
– No guidance on how to 
build a portfolio

Calmar 
Ratio
 

f
R
Calmar
MDD

µ −
=

– Shows a long‑term 
perspective
– Shows the cumulative loss 
investors can have
– Not sensitive to 
momentum changes

– No easy way to change 
frequencies
– Needs a lot of time to 
reflect momentum changes

RARoC
 RAROC
VaR
µ
=

– Allows to compare 
businesses with different 
sources of risk
– A powerful tool in asset 
allocation and risk control

– Hard to determine Cost of 
Capital Rate
– More accounting‑based 
ratio
–Hard to calculate VaR if a 
small number of returns are 
present

Source: the authors.

Note:

(
)

1

1
T

t
t

t
E R
R
T
=
µ =
= ∑
 —  mean return

(
)
2
2

1

1
T

t
t
R
T
=
σ =
−µ
∑
 —  variance of returns

(
)
1

1

t
t

T

P
B

t
ER
R
R
T
=
=
−
∑
 —  mean excess return over the benchmark, where 

tP
R
 —  return of the portfolio, 

t
B
R
 —  return of 

the benchmark


(
)

2

1

1
1

T

t
t
ER
ER
T
=
σ =
−
− ∑
 —  standard deviation of excess return, where 

t
t
t
P
B
ER
R
R
=
−

As we can see Sharpe ratio gives different ranking rather than a pure return. In addition, it allows 
easy calculations and comparison between the 
fund’s return and risk.
Now let’s compare the results for the following year. Risk-free rate was chosen as 1-year US 
rates of 0.89%.
As we can see from the Table 3 that all Sharpe 
ratios increased because in the previous examples 
Financial Crisis was included. Table 3 allows us 
to make the following conclusions:
If we compare just pure annual returns, then 
fund 2 would have the first place but fund 2 has 

one of the highest risks among all 8 funds which 
brings fund 2 to the 4th place.
Previous Sharpe Ratio ranked fund 6 as the 
least attractive fund. However, as we can see from 
its performance in the following year fund 6 got 
one of the highest returns and one of the lowest 
risks. That is why Sharpe ratio ranked fund 6 as 
the first one.
Another big change was for fund 5. Even with 
Financial Crisis fund 5 had the average annual return of 5.13%. However, in 2017 it returns dropped 
to 2.4% which brought it to the last place even 
though it has the lowest risk among all 8 funds.

Table 2
Sharpe ratio case-studies for 2006–2016

Fund
Return
Standard Deviation
Sharpe Ratio
Rank

F
4.39%
0

1
6.56%
14.15%
0.15313
Vi

2
6.33%
21.33%
0.09116
Vii

3
10.00%
18.77%
0.29881
i

4
9.29%
19.83%
0.24690
iV

5
5.13%
4.54%
0.16248
V

6
5.51%
22.05%
0.05064
Viii

7
9.13%
18.38%
0.25801
iii

8
9.60%
18.53%
0.28103
ii

Source: the authors.

iσ  —  standard deviation of stock i  or portfolio i

B
σ
 —  standard deviation of the benchmark

(
)

(
)

2
2

,2
2

1
1,1
1

B
T B

B
a
σ
−ρ
=
σ
−ρ

 —  portion invested in a fund

1
,1,T B
B
B

b
a σ
= ρ
−
ρ
σ

 —  portion invested in the benchmark

T
R
 —  target return

(
)
(
)

2

1

1
0,T

t
T
t
TDD
Min
R
R
T
=
=
−
∑
 —  target downside deviation

ThroughvaluePeakvalue
Peakvalue
MDD =
 —  maximum drawdown

(
)
( )
:
orVaR
VaR P X
VaR
f x dx

−

−∞
< −
= α
= α
∫
 —  where X  is a random variable that the represents the profit and loss of 

the business.

Performance Analysis Based on Adequate Risk-Adjusted Measures

Information Ratio
First, let’s discuss how the benchmark was selected and the details of these calculations.
Since funds that were selected have different 
nature then it would be beneficial for all of them 
to select a benchmark which is a whole market or 
S&P500 since some of these funds are stock market indexes, some are growth funds, etc. Therefore, to be consistent, S&P500 was selected as a 
benchmark.
As we know from the definition of the Information ratio we need to have an average annual 
excess return and standard deviation of the excess 
return. Therefore, to obtain these values annual 
returns for each fund were used then S&P500 
annual returns were subtracted from the fund’s 
returns. Further, the average was taken and the 

standard deviation for each fund. Hence, we can 
see the result of the calculations in Table 4.
As we can see from Table 4 not many funds 
managed to produce a positive excess return over 
the 10 years if the market (S&P500) was selected 
as a benchmark.
Table 4 allows us to make the following conclusions:
As in the Sharpe ratio fund, 3 managed to produce the highest excess return. However, in the 
relationship to a benchmark, this fund was exposed to one of the highest risks among all 8 funds.
Fund 8 produced almost the same excess return as fund 3. However, fund 8 did not take as 
much “extra” risk as fund 3. Therefore, now fund 
8 has the highest Information ratio and the lowest tracking error among all funds. It means that 

Table 3
Sharpe ratio case-studies for 2017

Fund
Return
Standard 
Deviation
Sharpe
Previous 
Sharpe
Ranking
Previous 
Ranking
Increase/
Decrease

F
0.89%
0.00%

1
8.99%
3.83%
2.12
0.1531
Vi
Vi
increase

2
24.63%
7.17%
3.31
0.0912
iV
Vii
increase

3
24.39%
9.15%
2.57
0.2988
V
i
increase

4
17.84%
8.94%
1.90
0.2469
Vii
iV
increase

5
2.39%
1.79%
0.84
0.1625
Viii
V
increase

6
23.83%
4.42%
5.20
0.0506
i
Viii
increase

7
19.48%
4.33%
4.30
0.2580
iii
iii
increase

8
18.98%
4.18%
4.33
0.2810
ii
ii
increase

Source: the authors.

Table 4
Information ratio case-studies for 2006–2016

Fund
Return/Excess Return
Standard 
Deviation
Information Ratio
Rank

B
7.25%
18.74%

1
–0.69%
8.85%
–0.0781
Vi

2
–0.91%
12.17%
–0.0750
V

3
2.75%
10.18%
0.2702
iii

4
2.04%
8.50%
0.2399
iV

5
–2.12%
18.30%
–0.1158
Vi

6
–1.74%
11.95%
–0.1456
Viii

7
1.88%
5.73%
0.3290
ii

8
2.35%
5.53%
0.4251
i

Source: the authors.

Performance Analysis Based on Adequate Risk-Adjusted Measures

funds 8 is more attractive for the investor rather 
than funds 3 if we compare it to the Sharpe ratio.
As we compare Information Ratio ranking with 
the Sharpe ratio overall there is a difference but 
most of the funds are changed places by one ranking. However, Information ratio allows us to compare returns not only with a risk-free rate but it 
can be interpreted as how much “additional” risk 
each fund brings to the market risk.
Finally, if we use Grinold and Khan (1999) approach and compare Information ratio with 0.5, 0.75, 
and 1.0 we can see that none of the funds produced 
even “good” Information ratio over the 10 years period.
Now let’s compare the results for the following year.

There are few things could be noted from the 
Table 5.
Almost all funds except for fund 2 had a negative excess return which means that all of them 
did not manage to beat the benchmark for the 
following year.
Fund 8 that was previously ranked the worst 
fund now got the third rank and it is one of two 
funds which information ratio increased even 
though it is still negative.
Since almost all funds have negative information ratio then based on the information ratio 
investor shouldn’t invest in any of the funds. Even 
fund 2 which have a positive information ratio 
have a ratio of 0.01.

Table 5
Information ratio case-studies for 2017

Fund
Return/Excess 
Return
Standard 
Deviation
Information 
Ratio
Previous 
IR
Ranking
Previous 
Ranking
Increase/
Decrease

B
17.32%
5.68%

1
–12.64%
8.82%
–1.4333
–0.0781
Vii
Vi
decrease

2
0.15%
10.65%
0.0137
–0.0750
i
V
increase

3
–0.21%
13.23%
–0.0160
0.2702
ii
iii
decrease

4
–5.54%
12.90%
–0.4294
0.2399
iV
iV
decrease

5
–17.90%
6.04%
–2.9618
–0.1158
Viii
Vi
decrease

6
–0.39%
7.65%
–0.0515
–0.1456
iii
Viii
increase

7
–3.98%
8.09%
–0.4923
0.3290
V
ii
decrease

8
–4.39%
7.88%
–0.5569
0.4251
Vi
i
decrease

Source: the authors.

Table 6
M2 ratio case-studies for 2006–2016

Fund
Return
Standard Deviation
 d
RAP
Rank

F
4.39%
0.00%

B
7.25%
18.74%

1
6.56%
14.15%
0.3247
7.26%
Vi

2
6.33%
21.33%
–0.1214
6.10%
Vii

3
10.00%
18.77%
–0.0015
9.99%
i

4
9.29%
19.83%
–0.0551
9.02%
iV

5
5.13%
4.54%
3.1264
7.43%
V

6
5.51%
22.05%
–0.1501
5.34%
Viii

7
9.13%
18.38%
0.0196
9.22%
iii

8
9.60%
18.53%
0.0111
9.66%
ii

Source: the authors.

Performance Analysis Based on Adequate Risk-Adjusted Measures

M2 Ratio
As we know from the definition of M2 we need 
a benchmark and a risk-free rate. Risk-free rate 
and the benchmark were chosen the same way 
and the same values as in Sharpe and Information ratios. Results of the calculations can be 
found in Table 6.
As we know from the nature of M2 measure it 
produces the same ranking as a Sharpe Ratio but 
instead of having a number which can be hard 
or impossible to interpret (Sharpe ratio), RAP 
gives investors a risk-adjusted return that was 
calculated based on the leverage/deleverage of 
the portfolio.
Table 6 allows us to make the following conclusions:
Funds 1, 5, 7, and 8 produced higher risk-adjusted return rather than a pure return. However, 
funds 2, 3, 4, 6 produce a lower risk-adjusted return.
On a pure return fund, 5 did not look very 
attractive to the investors. However, it was not 
exposed to a lot of risks (just 4.54%) and after adjusting for risk fund 5 produced a 7.43% 
return.
Fund 6 was exposed to the highest risk among 
all funds which brought this fund to the 8th place.
Now let’s compare the results with the following year:
There are few things could be noted from the 
Table 7:
Fund 6 had the highest risk-adjusted return 
of 30%.

Fund 5 had the only RAP measure that decreased for the following year in comparison to 
the previous year. However, its risk-adjusted return 
was 5.6% when the pure return was only 2.4%.
Fund 3 moved from the first place to the fifth 
having a risk-adjusted return of 15.46% when the 
pure return was 24.4%.

M3 Ratio
As we established, in the beginning, it is important to adjust for the correlation between 
a benchmark and a fund’s return. One of the 
measures that adjust for the correlation is M3. It 
requires benchmark returns (S&P500), risk-free 
rate (US T-Bond) and a target tracking error. For 
the target tracking error was 7% selected. Which 
corresponds to 0.9302 of the target correlation
 

(
)

2

2
0.07
1
2
0.1874
−
×

. The choice of the tracking error 

was made based on the risk-free return and the return of a benchmark. Investors always should seek 
a target return higher than a risk-free therefore it 
is higher than 4.4% but it is lower than the market 
because we want to be conservative and prepare 
for a lower return of the market than in previous 
years. Investors can choose any target tracking error, but calculations will be exactly the same.
Table 8 allows us to make the following conclusions:
Correlation influences ranking funds/managers. For example, the Sharpe ratio suggested that 

Table 7
M2 ratio case-studies for 2017

Fund
Return
Standard 
Deviation
d
RAP
Previous 
RAP
Ranking
Previous 
Ranking
Increase/
Decrease

F
0.89%
0.00%

B
17.32%
5.68%

1
8.99%
3.83%
0.4831
12.90%
7.26%
Vi
Vi
increase

2
24.63%
7.17%
–0.2086
19.68%
6.10%
iV
Vii
increase

3
24.39%
9.15%
–0.3798
15.46%
9.99%
V
i
increase

4
17.84%
8.94%
–0.3651
11.65%
9.02%
Vii
iV
increase

5
2.39%
1.79%
2.1761
5.66%
7.43%
Viii
V
decrease

6
23.83%
4.42%
0.2854
30.38%
5.34%
i
Viii
increase

7
19.48%
4.33%
0.3113
25.27%
9.22%
iii
iii
increase

8
18.98%
4.18%
0.3591
25.48%
9.66%
ii
ii
increase

Source: the authors.

Performance Analysis Based on Adequate Risk-Adjusted Measures

Table 8
M3 ratio case-studies for 2006–2016

Fund
Return
Standard 
Deviation
 
1,B
ρ
d
TE
a
b
1-a-b
M3
Rank

F
4.39%
0.00%
0

B
7.25%
18.74%
1
100%

1
6.56%
14.15%
0.8921
132.47%
7.26%
1.0758
0.2057
–0.2815
7.31%
Vi

2
6.33%
21.33%
0.8231
87.86%
6.10%
0.5678
0.3983
0.0339
6.63%
Vii

3
10.00%
18.77%
0.8526
99.85%
9.99%
0.7014
0.3313
–0.0327
9.27%
iii

4
9.29%
19.83%
0.9043
94.49%
9.02%
0.8125
0.1526
0.0349
8.80%
iV

5
5.13%
4.54%
0.2155
412.64%
7.43%
1.5509
0.8492
–1.4001
7.96%
V

6
5.51%
22.05%
0.8404
84.99%
5.34%
0.5755
0.3611
0.0634
6.06%
Viii

7
9.13%
18.38%
0.9525
101.96%
9.22%
1.2292
–0.2182
–0.0110
9.60%
ii

8
9.60%
18.53%
0.9560
101.11%
9.66%
1.2653
–0.2661
0.0008
10.22%
i

Source: the authors.

Table 9
M3 ratio case-studies for 2017

Fund
Return
Standard 
Deviation
 
1,B
ρ
d
M3
Previous 
M3
Rank
Previous 
Rank
Increase/
Decrease

F
0.89%
0.00%
0

B
17.32%
5.68%
1
100%

1
8.99%
3.83%
–71.22%
148.31%
37.62%
7.31%
i
Vi
increase

2
24.63%
7.17%
–36.59%
79.14%
30.69%
6.63%
Vi
Vii
increase

3
24.39%
9.15%
–56.96%
62.02%
33.09%
9.27%
V
iii
increase

4
17.84%
8.94%
–53.50%
63.49%
27.30%
8.80%
Vii
iV
increase

5
2.39%
1.79%
–5.61%
317.61%
10.36%
7.96%
Viii
V
increase

6
23.83%
4.42%
–13.54%
128.54%
35.90%
6.06%
ii
Viii
increase

7
19.48%
4.33%
–29.64%
131.13%
34.56%
9.60%
iii
ii
increase

8
18.98%
4.18%
–26.33%
135.91%
33.92%
10.22%
iV
i
increase

Fund
a
b
1-a-b
Previous 
a
Previous 
b
Previous 
1-a-b
TE
Previous 
TE

1
2.051
1.224
–2.276
1.076
0.206
–0.282
12.90%
7.26%

2
0.826
0.621
–0.447
0.568
0.398
0.034
19.68%
6.10%

3
0.733
0.912
–0.645
0.701
0.331
–0.033
15.46%
9.99%

4
0.730
0.854
–0.584
0.813
0.153
0.035
11.65%
9.02%

5
3.089
0.294
–2.382
1.551
0.849
–1.400
5.66%
7.43%

6
1.260
0.372
–0.632
0.576
0.361
0.063
30.38%
5.34%

7
1.333
0.541
–0.874
1.229
–0.218
–0.011
25.27%
9.22%

8
1.368
0.504
–0.872
1.265
–0.266
0.001
25.48%
9.66%

Source: the authors.

Performance Analysis Based on Adequate Risk-Adjusted Measures