LAPPEENRANTA UNIVERSITY OF TECHNOLOGY
Faculty of Technology
M.Sc (Computational Engineering)
Oundo Herbert Masinde
SIMILARITY BASED TOPSIS APPLIED TO EQUITY PORTFOLIO
Examiners: Assoc Prof Pasi Luukka.
Prof. Mikael Collan.
i
ABSTRACT
Lappeenranta University of Technology
Faculty of Technology
M.Sc(Computational Engineering)
Oundo Herbert Masinde
Similarity based TOPSIS applied to equity portfolio
Master's thesis for the degree of Master of Science in Technology
2016
57 pages, 17 gures, 23 tables, 1 appendix
Examiners: Assoc Prof Pasi Luukka
Prof. Mikael Collan.
Keywords:
Similarity, TOPSIS, Ranking, Equity, Portfolio, Returns.
The aim of this thesis is to study applicability of similarity based TOPSIS to equity portfolios.
Average annual returns are a basis to analyse portfolios.
They
are annually computed at same time for all nancial ratios by applying similarity
based TOPSIS. To form the ratios, nancial values P,B,Ev, Ebit, E,S and Ebitda
are selected from data set quoted on main list of Helsinki stock exchange (HEX) for
period 1996-2012. They are chosen because from those values one can form widely
used nancial ratios. The following nancial ratios;
EV /Ebitda are formed out of the values.
EV /Ebit, P/B, P/S, P/E
and
Any two of these ratios are joined to form a
combination. Since similarity based TOPSIS is multiple criteria decision making, by
combining two of the ratios, we examine whether combinations of these ratios bring
added value compared to using a single nancial ratio. Ranking is done according
to closeness coecient computed from similarity based TOPSIS and ve equal size
portfolios simultaneously. Portfolios are formed by dividing ranked companies into
about equal size sets.
Results obtained generally indicate that 1st portfolios have highest average annual
returns while 5th portfolios have the lowest.
Similarly best performing portfolios
p-values are less than 1, implying that reducing p-values greatly improved
performance of portfolios. Specically combination (EV /Ebit, P/E) has highest
average annual returns of 15.71 and corresponding p−value of 0.75. This average
annual return is higher than 15.29 for (EV /Ebit) which was the best single ratio
occur when
result.
ii
(EV /Ebit, EV /Ebitda) has
5th portfolio returns of 14.49
Another interesting comparison is that combination
highest dierence between 1st portfolio returns and
which is higher than
13.73
for
EV /Ebit
the highest single ratio dierence. Hence
we can note that there is added value to use similarity based TOPSIS.
The above results are in conformity with critical objectives of the study that; using
two criteria instead of one brings added value since higher average returns have
been gained this way and also that dierence between 1st portfolio returns and 5th
portfolio returns is higher when we use combinations of ratios as compared to single
nancial ratios.
Therefore, similarity based TOPSIS approach is practically robust and ecient in
analysing portfolios.
iii
Acknowledgements
I would like to thank department of Mathematics at Lappeenranta University of
Technology for the scholarship given to me during my studies.
My sincere gratitude goes to my supervisor Associate professor Pasi Luukka for
sparing your valuable time to guide, encourage and support me. You truely mentored
me through your constructive ideas.
Special thanks goes to Prof. Mikael Collan for examining my thesis and all sta at
Lappeenranta University of Technology who taught me various courses.
I would also like to thank all my family members; my parents Mr. and Mrs. Masinde
for bringing me to this world and gaving all the necessary support i needed.
My
Sisters Lonah and Lovisa, brother Fred, i am so grateful for all the family ideas we
have shared. Special thanks Lovisa and your husband John for taking care of my
education at a critical time. My wife Pamelah and children Larry and Helsa, you
have been patient and understanding during my absence from home, i thank you for
keeping the home going.
I also thank my friends; Constance, Margaret and Simon, we have shared alot academically and not forgeting Dr. Isambi and Idrissa for their assistance. Thanks to
all my friends both in Uganda and Finland.
Lappeenranta, November 30, 2016.
Oundo Herbert Masinde
CONTENTS
iv
Contents
Acknowledgements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Symbols and Abbreviations
iii
. . . . . . . . . . . . . . . . . . . . . .
vi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of Tables
1 INTRODUCTION
1
2 RESEARCH PROBLEM
5
3 MATHEMATICAL CONCEPTS
9
3.1
Fuzzy sets and crisp sets . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2
Properties of fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3
Operations on fuzzy sets
. . . . . . . . . . . . . . . . . . . . . . . . .
13
3.4
Fuzzy numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.5
Fuzzy relations and equivalence
3.6
. . . . . . . . . . . . . . . . . . . . .
16
. . . . . . . . . . . . . . . . . . . .
16
3.5.1
Crisp and Fuzzy relations
3.5.2
Binary relations on a single set
. . . . . . . . . . . . . . . . .
16
3.5.3
Fuzzy equivalence relations . . . . . . . . . . . . . . . . . . . .
17
Fuzzy Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.6.1
Individual decision making . . . . . . . . . . . . . . . . . . . .
17
3.6.2
Multiperson decision making . . . . . . . . . . . . . . . . . . .
18
3.6.3
Multiple criteria decision making
. . . . . . . . . . . . . . . .
19
3.6.4
Multistage decision making
. . . . . . . . . . . . . . . . . . .
20
CONTENTS
3.7
v
Fuzzy ranking methods . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.7.1
First type ranking methods . . . . . . . . . . . . . . . . . . .
21
3.7.2
Second type ranking methods . . . . . . . . . . . . . . . . . .
25
4 SIMILARITY BASED TOPSIS
27
5 RESULTS AND DISCUSSION
34
5.1
Results from individual value ratios with ve portfolios
. . . . . . . .
5.2
Results for value ratio combinations and TOPSIS with parameter p=1 38
5.3
Rankings with TOPSIS for varying parameters . . . . . . . . . . . . .
6 CONCLUSIONS AND FUTURE WORK
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
40
48
49
7 Appendix I: Analysis of Returns w.r.t p-parameters and respective
combinations
54
CONTENTS
List of Abbreviations
ME
Market Value of Equity
EV
Enterprise Value
E
Earnings
BE
Book Value of Equity
S
Sales
EBITDA
Earnings Before Interest, Taxes, Depreciation and Amortization
EBIT
Earnings Before Interest and Taxes.
vi
LIST OF TABLES
vii
List of Tables
1
Ten combinations formed from ve nancial ratios . . . . . . . . . . .
6
2
Valid records from data source
. . . . . . . . . . . . . . . . . . . . .
7
3
p-values with corresponding similarities . . . . . . . . . . . . . . . . .
29
4
Sample data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5
Decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
6
Positive and negative ideal solutions . . . . . . . . . . . . . . . . . . .
32
7
Relative closeness to ideal solutions . . . . . . . . . . . . . . . . . . .
32
8
Returns for single rankings . . . . . . . . . . . . . . . . . . . . . . . .
36
9
Ten applied combinations for the ve variables using two value ratios
38
10
Average annual returns using similarity based TOPSIS with
11
Highest returns for similarity based TOPSIS with
with highest individual returns
p = 1
p=1
. .
38
compared
. . . . . . . . . . . . . . . . . . . . .
12
Best returns, Volatility and Sharpe with respective p-values
13
Highest deviations
40
. . . . .
42
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
14
Returns w.r.t p-parameter with combination (EV/Ebit,P/E) . . . . .
43
15
Returns w.r.t p-parameter with comb. (P/B, P/E)
44
16
Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda)
17
Returns w.r.t p-parameter with comb.
18
Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda)
. . . . . . . . . .
. . . . . .
(EV /Ebit, EV /Ebitda)
45
. . . .
46
. . . . . .
47
19
Returns w.r.t p-parameter with combination (EV/Ebit, P/B) . . . . .
54
20
Portfolio returns w.r.t p-parameter with comb. (P/E, P/S)
55
. . . . .
LIST OF TABLES
viii
21
Returns w.r.t p-parameter with combination(P/S, EV/Ebitda)
. . .
56
22
Returns w.r.t p-parameter with combination (EV/Ebit,P/S)
. . . . .
57
23
Portfolio returns w.r.t p-parameter with comb. (P/B, P/S) . . . . . .
58
LIST OF FIGURES
ix
List of Figures
A = [a1 , a2 , a3 ]
1
Fuzzy number
2
α−cut
3
Similarity between x and y
4
Flow chart
5
Returns for individual rankings
6
Portfolio returns with TOPSIS (p=1)
7
structure of combinations
8
Returns w.r.t p-parameter with combination (EV/Ebit, P/E)
9
Returns w.r.t p-parameter with comb. (P/B, P/E)
10
Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda)
11
Returns w.r.t p-parameter with comb. (EV/Ebit, EV/Ebitda)
12
Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda)
13
Returns w.r.t p-parameter with combination (EV/Ebit, P/B)
14
Returns w.r.t p-parameter with comb. (P/E, P/S)
15
Returns w.r.t p-parameter with comb. (P/S, EV/Ebitda)
16
Returns w.r.t p-parameter with combination (EV/Ebit, P/S)
17
Returns w.r.t p-parameter with comb. (P/B, P/S)
of fuzzy number
. . . . . . . . . . . . . . . . . . . . . .
14
. . . . . . . . . . . . . . . . . . . . . . . . .
15
. . . . . . . . . . . . . . . . . . . . . . .
29
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35
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37
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39
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41
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43
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44
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45
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46
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47
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54
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55
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56
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57
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58
1
1
INTRODUCTION
1
INTRODUCTION
The TOPSIS method presented by Hwang and Yoon in 1981 [17] is one of the Multiple Criteria Decision Making (MCDM) methods and has the basic principle that
chosen alternatives should have the shortest distance from the positive ideal solution
(PIS) and the farthest distance from the negative ideal solution (NIS). Ideally, positive ideal solution aims to maximize the benets and minimize the costs whereas
the negative ideal solution aims to maximize the costs and minimize the benets.
Decision making involves nding feasible alternatives (see i.e. Jahanshahloo
et al.,
[21]). The criteria used in selection of feasible alternatives usually conict with each
other (see i.e.
Ling
et al.,
[25]).
For example in design of a car, the criteria of
higher fuel economy might mean a reduced confort rating due to the smaller passenger space. So there may be no solution satisfying all criteria simultaneously . Since
its discovery, TOPSIS has been applied in wide range of elds with a great deal of interesting results such as decision making and support systems, negotiation systems,
logistics management, wireless networks, project management, ecology, building and
construction and feature selection.
Examination of decision approaches for portfolio selection by value ratios can be derived from some researchers. Nguyen
et al.,
[46], initiated a new risk measure; the so
called fuzzy sharpe ratio in the modeling context for assessing portfolio performance.
Research done by Yue
et al.,
[52] using mean variance eciency and diversication
on Chinese stocks joint construct portfolio constraints of upper bounds market values, P/E ratios, turn over ratios and industries found that upper bounds are eective
in alleviating the contradiction, while market values, P/E ratios, turn over ratios
and industries have much dampened inuence when applied separately or joints. In
the same way Wang
et al.,
[43], used TOPSIS method to measure the relative per-
formance index of each project to select for a portfolio the rms which demonstrates
the closeness of their overall nancial performance by listing companies in Vietnam
stock market using inventory Turn over, Net Income Ratio, Earnings per share and
current ratio, Return on total assets (ROA) and Return on common Equity (ROE)
as estimation standard. Yuzi
et al.,
[2], evaluated the returns performance of Is-
lamic mutual funds in Malaysia based on four asset portfolios i.e.
Equity, Debt,
Money market and asset allocation using Sharpe and adjusted Sharpe Ratio. Similarly Kadri
et al.,
[32], developed an improved equity valuation model that predicts
rm's market value using rm's balanced score card (BSC) metrics by associatiing
market value, book value and earnings.
1
INTRODUCTION
2
Studies on equity portfolios by MCDM methods particularly brought interesting
results. Panagiotis
et al.,
[48] presented a methodology for supporting decisions that
concern the selection of equities, on the basis of nancial analysis for Athens stock
exchange, in which ELECTRE Tri outranking classication method was employed
for selection of attractive equities.
Promethee V. Multiple criteria method in the second round of the participatory budgeting (PB) Fontana
et al.,
[31] was used in nding feasible alternative compatible
with the city's goal. Similarly the holistic approach for nding an eective allocation of available research and development (R & D) resources by Gackstatter [40]
involved puting all potential portfolios one of them selected using (MCDM) methods.
Studies by Mendoza
et al.,
[34], used four multiple criteria methods; ELECTRE,
PROMOTHEE, TOPSIS and also a new and simple method called FUCA to select the best alternative among three criterias; NPV, risk and makespan for a new
product Development (NPD) problem in the Pharmacentical Industry. The fuzzy
decision theory was employed by Pai
et al.,
[13], to tackle the uncertainty arising
out of possible market scenarios in the fund manager's view point. The performance
eciencies of the optimal fuzzy portfolios were measured using Sharpe and Treynor
ratios and compared with those of the crisp counterparts.
Wachowicz
et al.,
[45] designed a TOPSIS based approach to scoring negotiating
oers in Negotiation support systems (NSS), in which a Simple Additive Weighing (SAW) model was used in negotiations preference analysis. Raia
et al.,
[22],
had similarly used SAW by applying formal models, which allow for analyzing negotiators preferences and determining a scoring system for negotiation oers. This
system was indeed real in building negotiators own proposals and analyzing partner's
counter oers. Raia
et al.,
[37] later modied this into a new scientic displine
called negotiation analysis which was implemented as a software solution in form of
negotiation support system (NSS). Hordijk L [18] developed a system using RAINS
model which supports real world negotiation problems for instance to resolve the
dispute between the European countries negotiating air pollution limits. Recently,
the basic supportive ideas derived from SmartSettle system (see i.e Thiessen
et al.,
[44]) have been used for supporting First Nations Negotiation in Canada.
Other
models are also applied into NSSs, based on dierent analytical approach,like the
AHP (see i.e. Mustajoki
et al.,
[35]) or ELECTRE (see i.e. Wachowicz
[47]). On
the other hand, the Internet expansion and e-commerce development cause that the
vast majority of the business processes, including the negotiations, are conducted by
means of computers and the web, using both simple communication software such
as electronic mail clients and instant messaging systems, and more sophisticated ne-
1
INTRODUCTION
3
gotiation support systems (NSS) or electronic negotiation systems [45] . The Deep
Ocean Mining Model used in the United Nations UNCLOS III negotiations (see i.e.
Sebenius
[38]) on the rights to exploit the natural resources from beneath of the
sea bed and sharing the prots yielded from the exploitation was also developed.
Several studies have still emerged bringing newer techniques involving TOPSIS, for
instance Chamodrakas
et al.,
[6], developed a model for aggregating function of
TOPSIS based on Fuzzy set representation of the closeness to ideal and negative
ideal solution.
Further Chamodrakas
et al.,
[20], presented a method that takes
into account user preferences, network conditions, QoS and energy consumption requirements in order to select the optimal network which achieves the best balance
between performance and energy consumption.
The proposed network selection
method incorporates the use of parameterized utility functions in order to model
diverse QoS elasticities of dierent applications, and adopts dierent energy consumption metrics for real time and non- real- time applications.
Maryam
et al.,
[29], used graph theory and matrix methods as decision analysis tools for contractor
selection as decision support system for identifying eligible contractor to be awarded
a contract. Zavadskas
et al.,
[53], used grey theory technique for performing pre-
dictive, relation analysis and decision making for assesing contractors competitive
ability. Krohling
et al.,
[23], presented fuzzy TOPSIS to handle uncertain data and
proposed a fuzzy TOPSIS for group decision making which was applied to evaluate
the ratings of response alternatives to a simulated oil spill. Hence combat responses
in case of accidents with oil spill in the sea. Chen [7], extended TOPSIS to fuzzy
environment in which the ratings of each alternative and weight of each criterion are
described by linguistic terms which can be expressed in triangular fuzzy numbers.
He then proposed a vertex method to calculate the distance between two triangular
fuzzy numbers. Milani
et al.,
[30], employed entropy method and TOPSIS to weigh
selected failure criteria and to rank the selected material IDs, respectively. This was
applied specically to the gear material for selection of power transmission.
There are also lots of theoretical works on TOPSIS extensions showing how the
method may be modied to solve problems of a particular formal structure with
additional assumptions (see i.e. Jahanshahloo
et al.,
[21] and Shih
et al.,
[41]).
In this study, similarity based TOPSIS [26] is applied to equity portfolios for Finnish
non-nancial stocks.
This will involve computation of average annual returns for
each combination of nancial ratios.
Results are obtained simultaneously for ve
portfolios on annual basis so that we can examine performance of similarity based
TOPSIS.
1
INTRODUCTION
4
This thesis is organized as follows. The rst chapter is an introductory part which
highlights the background, In chapter two, research problem is introduced and we
also look at objectives of the study, data and methodology used.
Chapter three
contains some mathematical concepts . In chapter four, the Similarity based TOPSIS
is introduced. Chapter ve shows the results from our computation and discussions
about the results.
future studies.
In chapter six, we conclude the study and give prospects for
2
RESEARCH PROBLEM
2
5
RESEARCH PROBLEM
Similarity based TOPSIS has been introduced (see i.e Luukka
et al. [26]), in which
histogram ranking is as well introduced to relax parameter dependency for problems
where suitable parameter values are exactly unknown.
In this study, similarity
based TOPSIS will be applied to criteria which are gained from nancial ratios.
To examine performance of similarity based TOPSIS, average annual returns for
ve portfolios are computed. The ve portfolios are formed by ranking companies
based on their closeness coecient value and forming ve sets based on these values.
The portfolios employed in testing the applicability of adjusted valuation measures
as a basis of stock selection criterion are composed of Finnish non-nancial stocks
quoted on main list of the Helsinki stock market (HEX) for the period 1996- 2012.
Financial values, P,B,Ev, Ebit, E,S and Ebitda are selected because they are easily
measurable. They are explained below;
•
Ebit- Earnings Before Interest and Taxes
•
EV(Enterprise value) Market value of Equity (ME) plus Short term Debt
plus Long term Debt Plus preferred stock value Minus Cash and Short term
investments.
ME is stock price multiplied by shares outstanding from the
CRSP monthly le & obtained as end of April of year
•
t throughout the paper.
B- Book value of equity is the stock holders equity plus deferred taxes minus
preferred stock.
•
P- Price
•
E-(Earnings) is income before extra ordinary items minus preferred Dividends
plus income statement deferred taxes.
•
S(Sales)
•
Ebitda- Earnings Before Interest, Taxes, Depreciation and Amortization.
From the above values, we derive ve most widely used nancial ratios;
P/B , P/S , P/E
and
EV /Ebitda
EV /Ebit,
through division. Then we form ten combinations
out of ve given nancial ratios by joining one ratio with atleast each of the other
four remaining ratios.
table 1 below.
Basically this should lead to unique combinations seen in
2
RESEARCH PROBLEM
6
(EV/Ebit,P/B)
(EV/Ebit,P/E)
(P/B, P/E)
(EV/Ebit,P/S)
(P/B,P/S)
(P/E,P/S)
(EV/Ebit, EV/Ebitda)
(P/B, EV/Ebitda)
(P/E,EV/Ebitda)
(P/S,EV/Ebitda)
Table 1: Ten combinations formed from ve nancial ratios
Ranking is done according to closeness coecient computed from similarity based
TOPSIS and ve equal size portfolios. The returns of portfolios are examined with
respect to parameter changes in similarity measure and with respect to dierent
value ratio combinations. The results for dierent combinations and respective
p-
values are obtained to determine best performing and also the eect of changing
p−values.
The objectives of this study are to;
•
Study the Similarity based TOPSIS
•
Apply similarity based TOPSIS to equity portfolios and compute returns from
the available data basing on ve valuation ratios;
•
Perform ranking to determine a portfolio which gives better percentage returns
and the corresponding
•
p-
values.
Analyse eect of changing
p-
values with similarity based TOPSIS on equity
portfolios.
•
Compare results for single criteria and two criteria to nd if we can ascertain
which gives higher average returns and hence see if there is added value in
using combinations of ratios.
•
Compare dierence between 1st portfolio average annual returns and 5th portfolio average annual returns.
To achieve the above objectives, data was collected, consisting of Finnish nonnancial stocks quoted on the main list of the Helsinki Stock exchange (HEX)
during the period 1996-2012.
This sample comprehensively includes all Finnish
non-nancial companies that have been quoted on the main list of the OMX HEX
and have met all the criteria for inclusion.
The stocks in sample are rst ranked
based on conventional individual valuation ratios.
2
RESEARCH PROBLEM
7
Normalization has been done to near minimum by reversing the valuation ratios.
The valuation ratios and respective parameter(p)-values are the inputs while the
%
returns are the outputs. However, two important considerations are made on the
ratios;
•
Missing values:
This kind of scenario could lead to
0
or undened, hence
non-representable.
•
Very small numbers:
This was leading to penny stocks i.e.
stocks with
price less than 1 euro. It is a common practice not to include such and hence
they were also removed.
Therefore, particular companies were removed if for one nancial ratio one of these
conditions was valid. All together, in the 17 years, total amount of companies was
160 records but as a result of above eects, data set is now less than original 160
records in each excel worksheet (each year).
New sample of
1279
valid records is
displayed in table 2 below:
Years
No. of valid records
1996
49
1997
50
1998
55
1999
73
2000
83
2001
81
2002
80
2003
75
2004
68
2005
75
2006
75
2007
80
2008
95
2009
84
2010
85
2011
87
2012
84
Total No. of valid records
1279
Table 2: Valid records from data source
Ranking is done according to closeness coecient computed from similarity based
TOPSIS and ve equal size portfolios on yearly basis. Portfolios are examined with
2
RESEARCH PROBLEM
respect to parameter
8
(p) changes in similarity measure and with respect to dierent
value ratio combinations. Specically we;
•
Form portfolios based on ranking companies with respect to value ratios;
EV /Ebit, P/B, P/E, P/S ,
•
and
EV /Ebitda
Apply similarity based TOPSIS to compute average annual returns for the ten
combinations of valuation ratios with varying parameter which are formed out
of ve given nancial ratios by joining any two of the ratios as seen in table 1.
•
Determine best returns with respective
p−values and also corresponding volatil-
ity and sharpe.
•
Compare returns from single criteria and two criteria to see which one gives
highest average annual returns.
•
Find the dierence between rst portfolio average annual returns and 5th
portfolio average annual returns.
3
3
MATHEMATICAL CONCEPTS
9
MATHEMATICAL CONCEPTS
Professor L.A Zadeh [28] introduced the concept of a fuzzy set which has played a
major part in many science models. We examine three main aspects of fuzzy sets
discussed by Dubois [11] to understand the set concept.
Uncertainty:
it is the ability to judge whether a proposition is true or false for
example we can describe the weather today as sunny if we dene any cloud cover of
20% or less sunny, that implies that cloud cover of 20.5% is not sunny (see Klir [14]).
Impreciseness:It
is a characteristic of language and pertains to measurable con-
cepts and particularly metric properties.
In traditional theories, world represen-
tations are forced to comply with extremely precise models, avoiding and rejecting
imprecise as a perturbation fact [5]. However, impreciseness plays an important role
in information representations where increase in precision would otherwise become
unmanagable.
Vagueness:
A notion is said to be vague when its meaning is not xed by sharp
boundaries. Example of vague information; data quality is 'good' or transparency of
optical element is acceptable. Dubois [11] generally observe that impreciseness and
vagueness refer to the contents of a piece of information expressed in some language,
while uncertainty refers to ability of an agent to claim whether a proposition holds
or not.
3.1 Fuzzy sets and crisp sets
Let us consider that we have elements of a set
range
0 ≤ x ≤ 1.
A
with membership values in the
For a crisp set, an element is either a member of the set
A or not,
while for fuzzy sets, elements can be partially in a set with a degree of membership
such that for value 0,
x∈
/A
and for value
extreme membership values of
0
and
1
1, x ∈ A.
On the other hand if only the
are allowed, then it is a crisp set.
Crisp sets
A crisp set is dened in such a way as to classify the individuals in some given
universe of discourse X into two groups: members and nonmembers (see i.e.
[10]
3
MATHEMATICAL CONCEPTS
and
[14]).
10
Klir [14], further outlines three basic methods by which sets can be
dened within a given universal set X:
The list method:
A set is dened by naming all its members. This method can
only be used for nite sets.
written as
Set A, whose members are
{a1 , a2 , . . . , an }
is usually
A = {a1 , a2 , . . . , an }.
The rule method:
A set is dened by a property satised by its members.
A
common notation expressing this method is
A = {x|P (x)} ,
where
0 0
|
denotes the phrase "such that" and
P (x)
destinates a proposition of the
form "x has the property P". That is, A is dened by this notation as the set of all
elements of
x for which the proposition P (x) is true.
P be such that for any given
Characteristic function:
x ∈ X,
the proposition
It is required that the property
P (x)
is either true or false.
A set A is dened by its characteristic function that
declares which elements of X are members of the set and which are not. Set A is
dened by its characteristic function as follows;

 1,
λA (x) =
 0,
for
x∈A
for
x∈
/A
That is, the characteristic function maps elements of X to the elements of the set
{0, 1},
which is formally expressed by
λA : X → {0, 1}
For each
x
(1)
x ∈ X, when λA (x) = 1, x is declared to be a member of A; when λA (x) = 0,
is declared as a nonmember of A.
Fuzzy sets
According to Klir [14], a fuzzy set can be dened mathematically by assigning to
each possible individual in the universe of discourse a value representing its grade
of membership in the fuzzy set. Dubois [10] presents a discussion for concept of a
fuzzy set as follows;
3
MATHEMATICAL CONCEPTS
11
Let X be a classical set of objects called the universe, whose generic elements are
denoted
x, membership in a classical subset A of X is often viewed as a characteristic
function;
µA
from X to valuation set
{0, 1}
such that pairs

 1,
µA (x) =
 0,
i
x∈A
i
x∈
/A
[0, 1],
If the valuation set is allowed to be real interval
µA (x)
i.e Zadeh [28]).
value of
A is called a fuzzy set (see
denotes the grade of membership of x in A. The closer the
µA (x) is to 1,the higher the certainty is that x belongs to A. A is completely
characterized by the set of pair
A = {(x, µA (x), x ∈ X)}
Let
α, β, γ
and
δ
(2)
be real numbers, some commonly used fuzzy sets [27] are dened
below:
• Γ-Shaped fuzzy set:
[0, 1]
A function with one variable and two parameters
is dened by
Γ(x; α, β) =
•



0,


if
x−α
β−α



 1,
if
if
x<α
α≤x≤β
x>β
S-shaped fuzzy set is dened by
S(x; α, β, γ) =








0,
2
if
(x−α)
(β−α)
x−γ)
γ−α


1−2





1,
•
Γ:x→
if
x<α
α≤x≤β
where
if
β=
β≤x≤γ
if
x>γ
L-Shaped fuzzy set: is decreasing piecewise continuous function
dened by
L(x; α, β) =
• Λ-shaped
fuzzy set: is dened as;



1,


β−x
β−α



 0,
α+γ
2
if
x<α
if
α≤x≤β
if
x>β
L : x → [0, 1]
3
MATHEMATICAL CONCEPTS
12
Λ(x; α, β, γ) =
•



0,




 x−α
β−α
γ−x



γ−β



 0,
if
x<α
if
α≤x≤β
if
β≤x≤γ
if
x>γ
Bell-shaped fuzzy set: is dened by;

 S(x; γ − β, γ−β ),
2,γ
π(x; β, γ) =
 1 − S(x; γ, γ+B , γ + β),
2
• Π-shaped
if
x≤γ
if
x>γ
(Trapezoidal fuzzy set):



0,




x−α



 β−α
Π(x; α, β, γ, δ) =
1,



δ−x



δ−γ



 0,
if
x<α
if
α≤x≤β
if
β <≤ γ
if
γ≤x≤δ
if
x>δ
3.2 Properties of fuzzy sets
Consider the universe of discourse X; as crisp sets, Klir [14] denes the following
main properties of fuzzy sets;
•
Given two fuzzy sets A and B, if
A⊆B
and also
the same members and are called equal sets.
write
•
B ⊆ A, then A and B contain
If A and B are not equal, we
A 6= B
The support of a fuzzy set A within a universal set X is the crisp set that
contains all the elements of
x
that have nonzero membership grades in A.
Supp(A) = {x ∈ X|A(x) > 0}.
•
(3)
The core of a fuzzy set A is a crisp set
Core(A) = {x ∈ X|A(x) = 1} .
(4)
3
MATHEMATICAL CONCEPTS
•
13
The height, h(A) of a fuzzy set A is the largest membership grade obtained
by any element in that set.
hgt(A) = Supx∈X A(x)
(5)
A fuzzy set A is called normal when h(A)=1; it is called subnormal when
h(A) < 1.
•
Given a fuzzy set A dened on x and any number
the strong
α-cut
,
α+
A,
α ∈ [0, 1],
the
α-cut α A and
are the crisp sets
α
A = {x ∈ X|A(x) ≥ α} .
and
α+
That is, the
α-cut
(or the crisp set
α+
{x ∈ X|A(x) > α}
A =
(or the strong
α − cut)
of a fuzzy set A is the crisp set
(6)
α
A
) that contains all the elements of the universal set X whose
membership grades in A are greater than or equal to (or only greater than )
the specied value of
α.
3.3 Operations on fuzzy sets
We present three special operations of fuzzy sets often called standard fuzzy operations discussed by Klir [14]. Consider two fuzzy subsets A and B of the universe X,
and
A(x), B(x)
1. The
their respective membership values for all
x ∈ X.
intersection of two fuzzy sets A and B, A(x) ∧ B(x), is dened as:
(A ∩ B)(x) = A(x) ∧ B(x) = min{A(x), B(x)}.
2. The
union of two fuzzy sets A and B, A(x) ∧ B(x), is dened as:
(A ∪ B)(x) = A(x) ∨ B(x) = max{A(x), B(x)}.
3. The
(7)
(8)
complement of a fuzzy set A, is Ā is dened as:
Ā(x) = A(x) = 1 − A(x).
(9)
3
MATHEMATICAL CONCEPTS
14
3.4 Fuzzy numbers
Shang
et al., [39] dened a fuzzy number as an ordinary number whose precise value
is somewhat uncertain. Klir [14], further explains fuzzy numbers as special types of
fuzzy sets that are dened by the set of real numbers
of the form
A : R → [0, 1], i.e.
R, with membership functions
they are close to a given real number or numbers that
are around a given interval of real numbers. Therefore, a fuzzy set A on
R
must
possess atleast the following properties for a fuzzy number.
(i) A must be a normal fuzzy set;
(ii)
α
A
must be a closed interval for every
(iii) the support of A,
0+
A,
α ∈ (0, 1];
must be bounded .
Operations of fuzzy numbers
If a fuzzy set is convex and normalized, and its membership function is dened in
R and piecewise continuous, it is called a fuzzy number.
Hence a fuzzy number /set
represents a real number interval whose boundary is fuzzy.
Figure 1: Fuzzy number
A = [a1 , a2 , a3 ]
3
MATHEMATICAL CONCEPTS
15
Further, a fuzzy number can be expressed as a fuzzy set dening a fuzzy interval in
the real number
R. Since the boundary of this interval is ambiguous, the interval is
also a fuzzy set. Generally a fuzzy interval can be represented by two end points and
a1
a peak point. Let use consider end points
and
a3
with a peak point
a2
as shown
in Figure 1. We also consider that the fuzzy number is normalized and convex, i.e.
∃x0 ∈ R, µĀ (x0 ) = 1.
The
α-cut operations can be also applied to the fuzzy number.
α-cut
h intervali for fuzzy number A as Aα , the obtained interval Aα is
(α) (α)
Aα = a1 , a3 . We can also know that it is an ordinary crisp interval
If we denote
dened as
as shown in Figure 2
Figure 2:
α−cut
of fuzzy number
α-cut is continuous and
(α) (α)
[a1 , a3 ]
The convex condition is that the line by
satises the following relation
Aα =
α-cut
interval
3
MATHEMATICAL CONCEPTS
16
3.5 Fuzzy relations and equivalence
The basic ideas of fuzzy relations and concepts of fuzzy equivalence, compatibility
and fuzzy orderings were rst introduced by Zadeh(1965) [28].
3.5.1 Crisp and Fuzzy relations
A crisp relation represents the presence or absence of association, interaction or
interconnectedness between the elements of two or more sets (see i.e.
[14]). A relation among crisp sets
Xi∈Nn Xi .It
X1 , X2 , . . . , Xn
can be denoted by either
R(Xi |i ∈ Nn ).
Klir
et al.,
is subset of the cartesian product
R(X1 , X2 , . . . , Xn )
or by the abbreviated form
Thus
R(X1 , X2 , . . . , Xn ) ⊂ X1 × X2 × . . . × Xn ,
(10)
Each crisp relation R can be dened by a characteristic function which assigns a
value of
1
to every n tuple if the universal set belongs to the relation and
0
to every
tuple not belonging to it.

 1,
R(x1 , x2 , . . . , xn ) =
 0,
⇒
if(x1 , x2 , . . . , xn )
∈R
(11)
otherwise
3.5.2 Binary relations on a single set
Types of relations
R(X, X)
can be distinguished basing on three dierent charac-
teristic properties [14]:
1.
Reexivity:
A crisp relation
R(X, X)
is reexive i
(x, x) ∈ R
for each
x ∈ X , that is, if every element of x is related to itself otherwise it is irreexive.
If
2.
(x, x) 6= R
Symmetric:
and
for every
A crisp relation
(y, x) ∈ R
an element
y
x ∈ X,
where
the relation is called antireexive.
R(X, X)
x, y, ∈ X .
Thus, whenever an element
through a symmetric relation,
it is asymmetric. If both
< x, y >∈ R
(x, y) ∈ R
is symmetric i for every
and
y
x
is also related to
< y, x >∈ R
is related to
x.
implies
Otherwise
x=y
then
3
MATHEMATICAL CONCEPTS
17
the relation is called antisymmetric. If either
whenever
3.
x 6= y ,
Transitive:
relation of
x
and
to
< y, x >∈ R,
y
R(X, X) to be transitive (x, z) ∈ R whenever
< y, z >∈ R
and
y
to
z
for at least one
implies the relation
y ∈ X.
x
to
not satisfy this property is called non transitive.
whenever both
or
then the relation is called strictly antisymmetric.
For a crisp relation
< x, y >∈ R
< y, x >∈ R
< x, y >∈ R
and
< y, z >∈ R,
z.
In other words the
A relation that does
However, if
< x, z >∈
/ R
then the relation is called
antitransitive.
3.5.3 Fuzzy equivalence relations
A crisp binary relation
R(x, x)
that is reexive, symmetric, and transitive is called
an equivalence relation. We can dene a crisp set
that are related to
x
Ax
containing all elements of
x
by the equivalence relation.
Ax = {y| < x, y >∈ R(x, x)} .
(12)
A fuzzy binary relation that is reexive, symmetric, and transitive is known as a
fuzzy equivalence relation or similarity relation. (see i.e. Klir
et al., (1995) [14]).
3.6 Fuzzy Decision Making
The concept of decision making has been applied in various elds such as logistics
management, wireless networks, project management, building and construction
and ecology.
Klir
[14] denes decision making as nding the best option among
the available alternatives. Decision problems are further categorized into four main
classes; individual decision making, multiperson decision making, multiple criteria
decision making and multistage decision making (see i.e.
[14] and [3]).
3.6.1 Individual decision making
This is a model of decision making in which one decision maker is involved in nding
the best alternatives. Relevant goals and constraints are expressed in terms of fuzzy
sets and a decision is determined by an appropriate aggregation of these fuzzy sets.
It is made up of the following components.
3
MATHEMATICAL CONCEPTS
•
a set X of possible actions.
•
a set of goals Pi (i
•
a set of constraints
∈ Nn ),
18
each expressed in terms of a fuzzy set dened on X;
Qj (j ∈ Nn ),
each of which is also expressed by a fuzzy set
dened on X.
0
0
If we let Pi and Qi to be fuzzy sets dened on sets Ai and Bi , respectively, where
i ∈ Nn
and
j ∈ Nm
and assume that these fuzzy sets represent goals and constraints
expressed by the decision maker. Then, for each
i ∈ Nn
and
j ∈ Nm , we can describe
the meanings of actions in set X in terms of sets Ai and Bj by functions
pi : X → A i ,
(13)
qj : X → Bj ,
(14)
0
If we express goals Pi and constraints Qj by the compositions of pi with Pi and the
0
compositions of qj and Qj ; then,
for each
Pi (i
a ∈ X.
∈ Nn ),
Pi (a) = Pi0 (pi (a)),
(15)
Qj (a) = Q0i (qi (a)),
(16)
Now given a decision situation characterized by fuzzy sets X,
and Qj (j
∈ N m ),
a fuzzy decision, D, is represented in form of a fuzzy
set on X. That is,
D(a) = min inf Pi (a), inf Qj (a)
i∈Nn
j∈Nm
for
all
a∈X
This simultaneously satises the given goals Pi and constraints Qj .
(17)
We can now
choose the best single crisp alternative from this fuzzy set by selecting an alternative that attains maximum membership grade in D.
3.6.2 Multiperson decision making
This arises when decisions made by more than one person are modeled. There are
two dierences to consider from the case of single decision making;
3
MATHEMATICAL CONCEPTS
•
19
the goals of the individual decision makers may dier such that each places a
dierent ordering on the alternatives.
•
the individual decision makers may have access to dierent information upon
which to base their decision.
Each member of a group of
n-individual
decision makers is assumed to have a re-
exive, antisymmetric, and transitive preference ordering Pi , i
∈ Nn
which totally or
partially orders a set X of alternatives. A social choice function must then be found
which produces the most acceptable overall group preference ordering from the individual preference orderings.
The model allows an individual decision maker to
have dierent aims and values while assuming that the overall purpose is to reach a
common, acceptable decision. Let the social preference S be represented by a binary
relation with membership grade function to deal with the multiplicity of opinions.
S : T × T → [0, 1]
S(ti , tj )
which assigns the membership grade
alternatives
over
tj
ti
over
tj .
(18)
to show the degree of reference of
We then use the method of popularity of alternatives
which involves dividing the number of persons prefering
N(ti , tj ), by the total number of decision makers,
ti
to
tj ,
denoted by
n
N (ti , tj )
n
S(ti , tj ) =
ti
(19)
S is then converted into its resolution form to determine the trial non fuzzy group
preference.
S = Uα∈[0,1] αα S
which is the union of the crisp relation
S, each scaled by
α. α
α
(20)
S comprising the α−cuts of the fuzzy relation
represents the level of agreement between the individual
concerning the particular crisp ordering
unique compatible ordering on
T ×T
α
S.
The largest value of
α
for which the
is found represents the maximum level of
agreement of the group while the crisp ordering represents the group decision.
3.6.3 Multiple criteria decision making
Each object is assigned several numerical evaluations which refer to dierent criteria
of the objects. (see i.e Dubois
et al., [10]).
Hence relevant alternatives are evaluated
3
MATHEMATICAL CONCEPTS
20
according to a number of criteria, each inducing a particular ordering of alternatives.
We therefore need a procedure by which to construct one overall preference ordering.
The number of criteria and alternatives are assumed to be nite.
Let
X = {x1 , x2 , . . . , xn } be a set of alternatives to be evaluated and C = {c1 , c2 , . . . , cm }
be a set of criteria to be followed for a decision problem. We can represent this as
a matrix.
X1
X2 , . . . ,
Xn
r11
r12
...
r1n

C2  r21
R = .. 
 ..
.  .
Cn rm1
r22
...
.
.
.
.
.
.
rm2
...

r2n 
. 
. 
. 
rmn
C1

It may happen that instead of matrix R with entries
0
R0 = [rij
],

[0, 1],
an alternative matrix
whose entries are arbitrary real numbers is initially given.
R0
can then
be converted to a desired matrix R by the formula
0
0
− min rij
rij
rij =
j∈Nn
0
0
max rij
− min rij
j∈Nn
∀i ∈ Nm
and
j ∈ Nn
(21)
j∈Nn
One approach is by converting to single criterion decision problems, whereby we nd
a global criterion,
gate of values
rj = h(rij , r2j , . . . rmj ),
r1j , r2j , rmj
i.e. for each
xj ∈ X
to which the individual criteria
is an adequate aggre-
c1 , c2 , . . . cn
are satised.
3.6.4 Multistage decision making
In this case, a required goal is achieved by solving a sequence of decision- making
problems. The decision making problems, which represent stages in overall multistage decision making are dependent on one another in the dynamic sense.
Gen-
erally, multistage decision making may be viewed as part of the theory of general
dynamic systems. The most important being that of dynamic programming, which
can be fuzzied (see i.e. Bellman
et al.,[3]).
A fuzzication of dynamic program-
ming extends its practical utility since it allows decision makers to express their
goals, constraints, decisions in appropriate fuzzy terms.
dynamic programing (see i.e. Bellman
The basic ideas of fuzzy
et al.,[3] ) are formulated as follows;
A decision problem concieved in terms of fuzzy dynamic programming is viewed as
a decision problem regarding a fuzzy nite- state automaton with two restrictions
3
MATHEMATICAL CONCEPTS
•
21
the state-transition relation is crisp and hence, characterized by the usual state
transition function of classical automata.
• n
special output is needed i.e.
next internal state is also utilized as output
and; consequently the two need to be distinguished.
From the above restrictions, we dene
A =< X, Z, f >,
(22)
where X and Z are respectively the sets of input states and output states of A, and
f :Z ×X →Z
(23)
is the state-transition function of A whose meaning is to dene, for each discrete
time
t (t ∈ N),
internal state,
the next internal state,
z
t
z t+1
of the automaton in terms of its present
, and its present input state
xt ,
i.e.
Z t+1 = f (z t , xt ).
(24)
3.7 Fuzzy ranking methods
The nal scores of alternatives can be represented in terms of fuzzy numbers, to
try to resolve the ambiquity of concepts that are associated with human beings
judgements. We need to construct crisp total ordering from fuzzy numbers in order
to express crisp preferences of alternatives.
Fuzzy ranking methods are common in establishing an ordering relation on
F.
In
comparing with previously studied methods, they are divided into two main types,
(see i.e., Matteo
3.7.1
et al., [33]).
First type ranking methods
These map fuzzy numbers directly into real line. The transformation is of the form
M:F
→ R.
Implying that they associate each fuzzy number with a real number and
then use the ordering
≥
on the real line. Hence a higher associated value indicates
a higher rank.
M (Ai ) ≥ M (Aj ) ⇒ Ai M Aj
(25)
3
MATHEMATICAL CONCEPTS
where
M
22
is the dominance relation induced by
M.
Several examples of rst type ranking methods have been proposed which include;
Hamming distance on the set R of all fuzzy numbers.
This is a method for ranking fuzzy numbers which is based on distance. For any
given fuzzy numbers A and B, the hamming distance
d(A, B)
is dened as;
Z
|A(x) − B(x)|dx
d(A, B) =
(26)
R
M AX(A, B) for the numbers A and B
Then calculate the Hamming distances d(M AX(A, B), A)
We therefore determine the least upper bound,
which we want to compare.
and
If
d(M AX(A, B), B) and dene A ≤ B if d(M AX(A, B), A) ≥ d(M AX(A, B), B).
A ≤ B,
then
M AX(A, B) = B
and hence
A ≤ B.
Other rst type ranking methods have been compared [33]. These are briey discussed below.
Adamo
When using this method, (see i.e. Adamo [1]), we simply evaluate the fuzzy numbers
based on the right most point of the
α-
cut for a given
ADα (A) = a+
α.
α.
(27)
Center of maxima
This is calculated, (see i.e. Klir [14]), as the average value of the end points of the
modal values interval by the formula
+
a−
1 + a1
CoM (A) =
.
2
(28)
3
MATHEMATICAL CONCEPTS
23
Center of gravity
The center of gravity of a fuzzy number is obtained [54] using
R∞
CoG(A) = R−∞
∞
xA(x)dx
−∞
A(x)dx
,
(29)
Median
The median value of a fuzzy number [4] and [9], generalizes the denition of median
to fuzzy numbers by minimizing the following expression.
Z
Z ∞
med(A)
A(x)dx −
A(x)dx
−∞
M ed(A)
(30)
Hence the median can be interpreted as the center of area (CoA) of a fuzzy number
A as it divides the area under the membership function into two equal parts.
Credibilistic mean
This is based on four axiomatic properties [24] and is proved that the original denition is equivalent to the following formulation
Cr(B) =
where
B⊂R
P os(B) + N ec(B)
,
2
(31)
i.e., the credibility measure is the arithmetic mean of the possibility
and necessity measures.
By this concept, the credibility expectation of a fuzzy
variable is dened as
Z
0
Z
∞
Cr(A ≥ x)dx −
CrM ean(A) =
−∞
Cr(A ≤ x)dx.
0
Chang's method
This ranking method is based on the index
Z
C(A) =
xA(x)dx.
x∈suppA
(32)
3
MATHEMATICAL CONCEPTS
24
From above [42], it can be observed that
C(A)
.
A(x)dx
−∞
CoG(A) = R ∞
(33)
Possibilistic mean
The possibilisitic mean value [19] of a fuzzy number
of the middle points of the
α-cuts
A∈F
is the weighted average
of a fuzzy number A;
1
Z
+
α(a−
∞ + aα )dα.
Ep (A) =
(34)
0
Yager's approaches
Four dierent ranking methods for fuzzy quantities in the unit interval are proposed
by Yager[49], [50], [51]. These methods are represented by equations (35), (36), (37)
and
(38) below.
-
R1
0
Y1 (A) =
where
g(x)
g(x)A(x)dx
R1
A(x)dx
0
measures the importance of
x,
(35)
can be seen as a generalization of
the ranking based on the center of gravity.
-
hgt(A)
Z
Y2 (A) =
M (Aα )dα,
(36)
0
where
hgt(A) = supx∈sup A A(x)
is the height of A and M is the mean value
operator. This can be used for ranking fuzzy numbers with arbitrary support.
In this case,
hgt(A) = 1
and
M (Aα ) =
-
Z
+
a−
α +aα
2
1
|x − A(x)|dx,
Y3 (A) =
(37)
0
-
Y4 (A) = sup min(x, A(x))
x∈[0,1]
(38)
3
MATHEMATICAL CONCEPTS
25
Chen's method
This is dened (see i.e.
Chen [8] ) using the concepts of fuzzy maximizing and
minimizing sets:
Amax (x) =
where
x − xmin
xmax − xmin
xmax = sup ∪ni=1 sup Ai
The left and right utility of a
k
, Amin (x) =
xmax − x
xmax − xmin
k
xmin = inf ∪ni=1 sup Ai and k > 0 is a real number.
fuzzy number Ai are dened as follows:
and
L(Ai ) = sup min(Amin (x), Ai (x)), R(Ai ) = sup min(Amax (x), Ai (x)),
x∈R
x∈R
Hence the nal ranking index is obtained as
1
CH k (Ai ) = (R(Ai ) + 1 − L(Ai ))
2
(39)
Kerre's method
The ranking index [16] is based on the Hamming-distance of fuzzy numbers by
determing the distance between
Ai
and
max(A1 , . . . , An ) :
Z
|Ai (x) − max(A1 , . . . An )|dx,
K(Ai ) =
(40)
x∈S
where
3.7.2
S = ∪ni=1 sup Ai .
Second type ranking methods
They generate fuzzy binary relations where by the methods are functions
F → [0, 1]
Ai
where the value of the relation
is greater than
Aj .
M (Ai , Aj ) ∈ [0, 1]
M :F×
is the degree to which
The fuzzy numbers for this type are ranked according to the
following rule;
M (Ai , Aj ) ≥ M (Aj , Ai ) ⇒ Ai M Aj
Examples of second type ranking methods compared by Matteo
discussed below.
(41)
et al.,
[33] are
3
MATHEMATICAL CONCEPTS
26
Baas and Kwakernaak's method
With this method, the value of the relation
which
Ai
is greater than
Aj
PBK (Ai , Aj )
quanties the degree to
as follows:
PBK (Ai , Aj ) = sup min(Ai (xi ), Aj (xj ))
xi ≥xj
which leads to the ranking of the fuzzy number as
BK(Ai ) = min PBK (Ai , Aj ).
(42)
j6=i
It is worth noting that
Dubois and Prade
PBK
coincides with the fuzzy relation PD introduced by
[12]. It is important to mention that the rankings produced by
the two methods can be dierent: Baas and Kwakernaak's approach is based on the
minimum value of
PBK
and Dubois and Prade's PD relation can be used according
to the ordering to the ordering procedure described in
[15],
Nakamura's method.
Here the parametric method, (see i.e. Nakamura [36]), is based on the fuzzy relation
PN λ(Ai , Aj ) =
with
λ ∈ [0, 1]
λdH (Ai , min(Ai , Aj )) + (1 − λ)(dH (Āi , min(Āi , Āj )))
λdH (Ai , Aj ) + (1 − λ)(dH (Āi , Āj ))
and where
dH (Ai , Aj ) =
R
|Ai (x) − Aj (x)|dx
is the Hamming dis-
Ai (x) = supy≤x Ai (y) and Āi (x) = supy≥x Ai (y).
λdH (Ai , Aj ) + (1 − λ)(dH (Āi , Aj )) = 0, the value of the relation is dened as
tance between two fuzzy numbers,
When
R
(43)
PN λ(Ai , Aj ) = 0.5.
4
SIMILARITY BASED TOPSIS
4
27
SIMILARITY BASED TOPSIS
The original TOPSIS is based on the concept that the chosen alternative should
have the shortest geometric distance from the positive ideal solution and the longest
geometric distance from the negative ideal solution [28].
The aim is to maximize
the benets and minimize the costs. Therefore TOPSIS is a muticriteria decision
making technique which aim to nd feasible alternatives. In this study we will apply
similarity based TOPSIS. The underlying idea in this method is that comparison
is done by computing similarity between alternatives and ideal solutions.
These
alternatives should have highest similarity to positive ideal solution and lowest to
negative ideal solutions.
Similarity for two elements
x1 ∈ [0, 1]
and
x2 ∈ [0, 1]
can be computed using the
formular:
For the case of two vectors,
S(x1 , x2 ) =
p
p
1 − |xp1 − xp2 |
(44)
x1 ∈ [0, 1]n
and
x2 ∈ [0, 1]n
similarity can be calculated
as
n
1X
S(x1 , x2 ) =
wi
n i=1
q
p
1 − | (x1 (i))p − (x2 (i))p |
(45)
The procedure of similarity based TOPSIS starts from the construction of an evaluation matrix
X = [xij ],
where
xij
denotes the score of the ith alternative, with
respect to the jth criterion, and can be summarized in the following steps;
Step I:
Calculation of normalized, decision matrix R
xij − | min(xij )|
rij =
i
(46)
max(xij ) − min(xij )
i
i
i = 1, . . . m, j = 1, . . . n
Step II:
Calculation of weighted normalized decision matrix V = [vij ]
vij = rij (.)wj
Step III:
j = 1, . . . , m,
i = 1, . . . , n.
(47)
Determine positive and negative ideal solutions A+ and A−
+
A+ = {v1+ , . . . , vm
} = {(max vij |j ∈ B), (min vij |j ∈ C)}
j
j
A− = {v1− , . . . , vn− } = {(min vij |j ∈ B), (max vij |j ∈ C)}
j
j
Where B is for benet criteria, and C is for cost/non-benet criteria.
(48)
4
SIMILARITY BASED TOPSIS
Step IV:
28
Calculation of the similarities of each alternative from positive ideal
solution and negative ideal solution.
m
Si+
1 Xq
p
=
1 − |(vij )p − (vj+ ) p | i = 1, . . . , n
n j=1
m
Si−
Step V:
1 Xq
p
1 − |(vij )p − (vj− ) p | i = 1, . . . , n
=
n j=1
(49)
Calculation of the relative closeness to Ideal solutions.
Si+
CCi = +
,
Si + Si−
i = 1, . . . , n
(50)
From the above steps, we are able to achieve important aspects of the similarity
based approach and they are discribed as follows;
•
The closer the CCi is to
1
implies the higher priority of the ith alternative.
Hence the alternative with highest value in interval
•
From
equation (49), the parameter
larity. The higher, the parameter
•
p
p
[0, 1]
will be selected.
is used to set the strength of the simi-
value, the higher the similarity degrees.
In step 1, normalization is done to ensure all elements
rij
are between
0 and 1
which is required in step 4.
•
We apply closeness coecient designed for similarity measure in step 5.
Example 4.1
x = [0.3, 0.8, 0.9, 0.3, 0.5] and y = [0.35, 0.9, 1, 0.5, 0.65].
S(x, y) using p values p = [1, 2, 3].
Let
Calculate similarity
Solution:
We calculate
S(x, y)
using the formular
n
S(x, y) =
1 Xp
p
1 − |(xi )p − (yi )p | i = 1, . . . , n
n i=1
The results are displayed in table
(51)
4
SIMILARITY BASED TOPSIS
29
P − values
S(x, y)
1
0.88
2
0.9276
3
0.9460
Table 3: p-values with corresponding similarities
From above results, we conclude that there is similarity between x and y since results
for all
p = 1, p = 2
and
p=3
are all very close to 1. Indeed vectors x and y are
highly similar.
If we apply higher values of of
p i.e. p = [1, 2, 3, . . . 10] to vectors x and y
to observe
similarity between them for the Example 4.1, we nd similar trend as seen in gure 3
below.
Figure 3: Similarity between x and y
Generally, we can see that values of
from
1
upto
10.
S(x, y)
Therefore the vectors
x
are closer to
and
y
1
as we increase
p−
values
are similar with high values of
p.
Example 4.2
Five companies A1, A2, A3, A4 and A5 are to be evaluated, using four criteria C1,
C2, C3 and C4. Let us consider data in table 4 below:
4
SIMILARITY BASED TOPSIS
30
Valuations
A1
A2
A3
A4
A5
Sales
1059
1109,7
215,7
218,3
83,5
EPS
19,1
12,1
3,76
2,57
0,13
B
122,3
118,08
24,05
29,37
15,62
P
154,00
240,00
45,58
46,50
13,20
EBIT
22,94
14,42
34,46
34,48
87,11
EV
189,59
200.28
174,92
192,57
551,33
Table 4: Sample data
The criteria are calculated as follows;
C1 =
C2 =
C3 =
C4 =
B
P
EP S
P
Sales
P
Ebit
EV
Assume C1, C2, C3 and C4 are all benets then form decision matrix as shown in
table 5 below.
We note that if we had calculated them as ;
C1 =
C2 =
C3 =
C4 =
P
B
P
EPS
P
Sales
EV
Ebit
Then they would be handled as cost criterias.
4
SIMILARITY BASED TOPSIS
31
C1
C2
C3
C4
A1
0.7942
0.1240
6.8766
0.1210
A2
0.4920
0.0504
4.6238
0.0720
A3
0.5344
0.0836
4.7933
0.1970
A4
0.6316
0.0551
4.6946
0.1790
A5
1.1833
0.0098
6.3258
0.1580
Table 5: Decision matrix
W = [1 1 1 1]
We now assume the weight vector for decision makers to be
Step I
Calculation of normalized, decision matrix R
xij − | min(xij )|
rij =
i
i = 1, . . . m, j = 1, . . . n
max(xij ) − min(xij )
i
i
From the formula, we obtain in matrix form the following results:

0.4371 1.0000 1.0000 0.3920


 0.0000 0.3555 0.0000


D =  0.0613 0.6462 0.0752


 0.2019 0.3967 0.0314

1.0000 0.0000 0.7555
Step II


0.0000 


1.0000 


0.8560 

0.6880
Calculation of weighted normalized decision matrix V = [vij ]
vij = rij (.)wj
and given that
Step III

W = [1, 1, 1, 1],
j = 1, . . . , m,
i = 1, . . . , n.
we obtain same decision matrix as in step I.
Determine positive and negative ideal solutions A+ and A−
+
} = {(max vij |j ∈ B), (min vij |j ∈ C)}
A+ = {v1+ , . . . , vm
j
−
A
=
{v1− , . . . , vn− }
j
= {(min vij |j ∈ B), (max vij |j ∈ C)}
j
j
Where B is for benet criteria, and C is for cost/non-benet criteria.
The positive and negative ideal solutions are in table 6 in which all criteria
are assumed to be benets.
4
SIMILARITY BASED TOPSIS
32
Criteria
C1
C2
C3
C4
A1
0.4371
1.0000
1.0000
0.3920
A2
0.0000
0.3555
0.0000
0.0000
A3
0.0613
0.6462
0.0752
1.0000
A4
0.2019
0.3967
0.0314
0.8560
A5
1.0000
0.0000
0.7555
0.6880
A+
1.0000
1.0000
1.0000
1.0000
A−
0.0000
0.0000
0.0000
0.0000
Table 6: Positive and negative ideal solutions
Step IV
Calculation of the similarities of each alternative from positive ideal
solution and negative ideal solution.
m
Si+
1X q
=
wj p 1 − |(vij )p − (vj+ ) p | i = 1, . . . , n
n j=1
m
Si− =
q
1X
Wj p 1 − |(vij )p − (vj− ) p | i = 1, . . . , n
n j=1
p = 1. The results from computations
S + = {0.7073, 0.0889, 0.4457, 0.3715, 0.6109}
We assume
are displayed below:
S − = {0.2928, 0.9113, 0.5543, 0.6285, 0.3891}
Step V
Calculation of the relative closeness to ideal solutions.
We carrry computations of relative closeness to ideal solutions using the formular
Si+
CCi = +
, i = 1, . . . , n.
Si + Si−
The results are shown in the table 7 below:
S+
S−
CCi
A1
0.7073
0.2928
0.7073
A2
0.0889
0.9113
0.0889
A3
0.4457
0.5543
0.4457
A4
0.3715
0.6285
0.3715
A5
0.6109
0.3891
0.6109
Table 7: Relative closeness to ideal solutions
We note that the closer the CCi is to
the results in Table 7,
A1
1
implies priority of the alternative. Using
has the highest with closeness coecient of
0.7073
and is
4
SIMILARITY BASED TOPSIS
preferred for selection while
A2
33
has the lowest and is least preferred. Over all, the
ranking order will be
A1 ≺ A5 ≺ A3 ≺ A4 ≺ A2
A3, A3
to
to
A4
and
A4
We have used parameter
i.e
A1
is preferred to
A5, A5
to
A2.
p = 1 for the calculation of similarities of alternatives from
positive ideal solution and also alternatives from negative ideal solution.
could use higher parameter values
But we
(p > 1) to increase the strength.The normalization
is performed to ensure the values of elements
rij
are between
0
and
1.
5
RESULTS AND DISCUSSION
5
34
RESULTS AND DISCUSSION
This chapter presents results and discussions about our experiment in which we
specically analyze the performance of portfolios basing on the average annual returns within a certain range of parameters.
This is done by applying similarity
based TOPSIS on equity portfolios to arrive at a portfolio which gives overall highest average annual returns.
To arrive at this, nancial values P, B, Ev, Ebit, E,
S and Ebitda are extracted from Finnish non nancial stocks. Division is carried
out on above nancial values to obtain value ratios; EV/Ebit, P/B, P/S, P/E and
EV/Ebitda. These reect cost criteria. From the above value ratios, we form portfolios by ranking companies. specically ve portfolios in this case. Similarity based
TOPSIS is then used to get ranking order of the companies. These ranking orders
are then used to form ve portfolios.
are computed.
By changing the
p-
For these portfolios average annual returns
value in similarity based TOPSIS we also get
abit dierent ranking order for the companies. This eects to portfolios in a sense
that somewhat dierent companies are selected together. This way we also notcied
a change in results for average annual returns.
Testing procedure
The experiment was carried out in three steps;
•
First we computed results for individual value ratios to get benchmark results.
In this, single ranking is performed on the value ratios directly.
•
Then we computed all two criteria combinations to see if these results can be
improved using multiple criteria decision making- method.
•
After this we examined also the eect of parameter value changes.
The initial inputs for this experiment are the value ratios and the nal outputs are
the average annual returns. The following steps are carried out before arriving at
nal outputs:
For each year
Step 1:
i=1
to
N
where
N
denotes the number of years do
Get nancial value for companies in year
i.
5
RESULTS AND DISCUSSION
Step 2:
35
Compute the value ratios for the companies and remove companies which:
(a) value ratio cannot be computed because of missing values.
(b) they belong to `penny stock' category.
Step 3:
Calculate ranking orders of companies based on these value ratios.
- for single criteria case simply by ordering directy.
- for two or more case using similarity based TOPSIS.
Step 4:
Divide ranked companies into ve dierent portfolios based on their ranking
order.
Step 5:
Compute yearly returns for each of the portfolios.
END for
Step 6:
Compute average annual returns for each portfolio.
The above steps are summarized in a ow chart below.
Figure 4: Flow chart
5
RESULTS AND DISCUSSION
36
5.1 Results from individual value ratios with ve portfolios
Individual value ratios;
EV /Ebit, P/B, P/E, P/S
with ve portfolios P1 - P5 .
and
EV /Ebitda
are examined
Ranking is performed directly without using simi-
larity based TOPSIS.
Results obtained indicate that best performing stocks are in 1st portfolio while 5th
portfolios clearly have worst. Specically, EV/Ebit performed well since its highest
return is 15,29 and clearly in rst portfolio and its lowest is 1.58, which is in 5th
portfolio. By observing deviations from 1st portfolio and 5th portfolio, we can clearly
see that EV/Ebit is also doing well since its dierence is 13.73 and the results are
linearly decreasing from 1st to 5th.
P/S is giving worst results because on contrary, its highest average annual returns
is 11.93 and in 2nd portfolio.
Its lowest is 6.82 and found in 3rd portfolio.
The
deviation between 1st and 5th portfolios for P/S is lowest i.e -2.27 and the average
annual returns are not linearly decreasing as we can see in table 8. This is not valid
for our measures of performance where by we expect highest average annual returns
in rst portfolio, lowest average annual returns in last portfolio, high dierence
between rst and last portio and also a linear decrease from rst portfolio to last
portfolio.
Value ratios
1
2
3
4
5
P1- P5
EV/Ebit
15,29
12,90
10,94
7,76
1,56
13.73
P/B
11,87
9,65
11,81
6,95
6,24
4.93
P/E
12,93
11,47
13,66
9,68
-0,036
12.96
P/S
8,33
11,93
6,82
10,27
10,59
-2.27
EV/Ebitda
13,64
13,02
10,54
6,50
4,05
9.59
Table 8: Returns for single rankings
Average annual returns obtained using single ranking is displayed in gure 5
5
RESULTS AND DISCUSSION
37
Figure 5: Returns for individual rankings
From the above table and graph, we obtain the following ranking orders.
•
Highest returns in 1st portfolio. Based on this result, we get
EV
EV
P
P
P
≺
≺
≺
≺
Ebit
Ebitda
E
B
S
•
Deviation between rst and last portfolio. From this analysis, we obtain the
following order;
EV
P
EV
P
P
≺
≺
≺
≺
Ebit
E
Ebitda
B
S
Based on these two ranking lists we can conclude that for individual value ratios,
EV
is giving best results and therefore best ranked i.e. rst portfolio returns of
Ebit
15.29 and a deviation of 13.73, while PS is giving worst results and is ranked last for
both conditons with rst portfolio returns of
8.33 and a deviation of −2.27.
1st portfolio average annual returns are highest in case of;
EV /Ebitda.
While for the case of
P/S ,
P/S
is worthy using.
EV /Ebit, P/B, P/E
and
the highest average annual returns are in
2nd portfolio. This therefore leads to conclusion that
while
Overall,
EV /Ebit
is the best to use
5
RESULTS AND DISCUSSION
38
5.2 Results for value ratio combinations and TOPSIS with
parameter p=1
In this experiment, value ratio combinations are formed by joining any two of the
value ratios and by so doing, we can know which value ratio combination works best
compared to when used individually.
Ten combinations are formed out of ve value ratios as seen in table 9, by combining
one of the ratios with each of the other four ratios.
(EV/Ebit,P/B)
(EV/Ebit,P/E)
(P/B, P/E)
(EV/Ebit,P/S)
(P/B,P/S)
(P/E,P/S)
(EV/Ebit, EV/Ebitda)
(P/B, EV/Ebitda)
(P/E,EV/Ebitda)
(P/S,EV/Ebitda)
Table 9: Ten applied combinations for the ve variables using two value ratios
Combinations of value ratios are examined for ve portfolios. Ranking is performed
using similarity based TOPSIS with parameter value
(p = 1).
The results are
displayed in table 10 and gure 6
Combination
1
2
3
4
5
P1 − P5
(EV/Ebit,P/B)
12,70
10,15
12,44
9,05
2,45
10,25
(EV/Ebit,P/E)
15,47
12,15
11,35
6,64
2,58
12,89
(EV/Ebit,P/S)
10,94
11,67
8,86
11,12
5,30
5,64
(EV/Ebit,EV/Ebitda)
13,08
15,25
9,67
7,34
0,53
12,55
(P/B,P/E)
13,34
11,55
10,29
8,11
3,16
10,18
(P/B,P/S)
9,13
12,45
8,99
9,42
6,27
2,85
(P/B,EV/Ebitda)
13,75
10,25
10,00
9,59
1,75
12,00
(P/E,P/S)
11,87
13,60
10,08
8,02
3,92
7,95
(P/E,EV/Ebitda)
14,99
10,63
12,79
6,31
1,42
13,58
(P/S,Ev/Ebitda)
10,75
12,77
9,79
9,40
2,94
7,80
Table 10: Average annual returns using similarity based TOPSIS with
p=1
5
RESULTS AND DISCUSSION
39
Figure 6: Portfolio returns with TOPSIS (p=1)
From table 10, we obtain the best four perfoming ratios computed using similarity
based TOPSIS with parameter
p = 1 as (EV/Ebit, P/E), (P/B, EV/Ebitda), (P/B,
P/E) and (EV/Ebit, EV/Ebitda) with corresponding highest average annual returns
of
15.47, 13.75, 13.34
and
13.08
respectively.
We also compare results obtained using similarity based TOPSIS with parameter
p=1
and results for individual value ratios. In this analysis we nd that highest
combination returns is
15.47
which is higher than
15.29
obtained with single value
ratio (see table 11). Hence we can say that by using similarity based TOPSIS, there
is some added value since we have better performance.
5
RESULTS AND DISCUSSION
Combination
40
Highest
Highest
Highest
Combination
Individual
Individual
returns
returns
value ratio
(EV/Ebit,P/B)
12,7013
15,29
EV/Ebit
(EV/Ebit,P/E)
15,4689
15,29
EV/Ebit
(EV/Ebit,P/S)
11,6678
15,29
EV/Ebit
(EV/Ebit,EV/Ebitda)
15,2537
15,29
EV/Ebit
(P/B ,P/E)
13,3399
13,65
P/E
(P/B, P/S)
12,4485
11,93
P/S
(P/B,EV/Ebitda)
13,7507
13,64
EV/Ebitda
(P/E,P/S)
13,6037
13,65
P/E
(P/E,EV/Ebitda)
14,9929
13,64
EV/Ebitda
(P/S, EV/Ebitda)
12,7711
13,64
EV/Ebitda
Table 11: Highest returns for similarity based TOPSIS with
p=1
compared with
highest individual returns
5.3 Rankings with TOPSIS for varying parameters
We now extend our analysis from having one parameter value to varying parameters
p = [0.10, 0.25, 0.75, 1.00, 2.00, 3.00, 4.00, 5.00, 10.00]
with similarity based TOPSIS
for a combination of any two of the value ratios.
A total of 100 excel les are
generated (10 unique combinations and
10
parameters), each le having
which represent years (1996- 2012) as illustrated below.
17
sheets
5
RESULTS AND DISCUSSION
41
Structure of combinations
Figure 7: structure of combinations
Results obtained for varying parameters with similarity based TOPSIS indicate that
still most of the best perfoming stocks are in 1st portfolio and occur when the
value is less than
1
(EV /Ebit, P/E)
higher than 15.47
. Indeed combination
highest average returns of
15.71
similarity based TOPSIS with
. This is
p=1
p-
p = 0.75 has the
15.29 obtained for
with
and
and single ratio respectively. We observe that
by varying parameters, we have better results and therefore added value.
In the
same way, we can also conclude that better results are obtained with reduction in
parameter value.
Combination (P/B,P/S) has worst results i.e.
ratio and similarity based TOPSIS with
performing ratio is
10.86.
p = 1,
In fact as compared to single
we see that in single ratio, the best
EV /Ebit while by using similarity based TOPSIS, combinations
involving EV/Ebit are giving best results. Similarly, P/S is the worst performing
for single ratio while for similarity based TOPSIS, combinations involving P/S are
the worst performing.
Five of the best performing combinations with corresponding volatility, Sharpe and
5
RESULTS AND DISCUSSION
p−values
42
are given in table 13 and the remaining combinations are in appendix 1.
Combinations
Best returns
Volatility
Sharpe
P-values
(EV/Ebit,P/E)
15,71
0,18
0,19
0,75
(P/B,P/E)
15,44
0,18
0,19
0,25
(P/E,EV/Ebitda)
15,09
0,19
0,18
0,5
(EV/Ebit,EV/ Ebitda)
14,40
0,19
0,16
5,00
(P/B,EV/Ebitda)
14,27
0,19
0,16
0,5
Table 12: Best returns, Volatility and Sharpe with respective p-values
From the results, we now obtain the following ranking orders:
•
Highest returns in 1st portfolio.
Based on this result, we get
•
P P
P EV
EV
EV
P
EV
≺
,
≺
,
≺
,
≺
,
B E
E Ebitda
Ebit Ebitda
B Ebitda
EV P
P P
P EV
EV P
P P
≺
≺
≺
≺
≺
,
,
,
,
,
Ebit B
E S
S Ebitda
Ebit S
B S
EV P
,
Ebit E
Deviation between rst and last portfolio.
From this analysis, we obtain the following order;
EV
P EV
EV P
P P
P
EV
EV
,
≺
,
≺
,
≺
,
≺
,
Ebit Ebitda
E Ebitda
Ebit E
B E
B Ebitda
P P
EV P
P EV
EV P
P P
≺
,
≺
,
≺
,
≺
,
≺
,
E S
Ebit B
S Ebitda
Ebit S
B S
Combinations
Highest deviations
p-values
(EV/Ebit,EV/ Ebitda)
14,49
5
(P/E,EV/Ebitda)
14.48
0,5
(EV/Ebit,P/E)
14.31
0.25
(P/B,P/E)
13.97
2.00
(P/B,EV/Ebitda)
12.00
1.00
Table 13: Highest deviations
From the two conditions,
and
EV
, EV
Ebit Ebitda
EV P
,
Ebit E
is best ranked in terms of rst portfolio returns
is best ranked in terms of highest deviation.
P P
,
B S
is giving worst
5
RESULTS AND DISCUSSION
43
EV
is common to both best ranked
Ebit
P
combinations and also it was the best for the case of single ranking.
is common
S
EV
is the best to
to worst performing combinations for both conditions. Therefore
Ebit
P
use while
is the worst overall.
S
results for both conditions.
We notice that
Parameters
1
2
3
4
5
P1-P5
p=0.10
15,4564
12,4151
10,7272
7,8244
1,3447
14,1117
p=0.25
15,571
12,1623
10,9324
7,8234
1,2595
14,3114
p=0.50
15,6199
12,4035
10,9235
7,5506
1,3425
14,2775
p=0.75
15,7108
11,9718
11,0366
7,7956
1,4455
14,2651
p=1.00
15,4689
12,1474
11,3548
6,6388
2,5805
12,8884
p=2.00
10,8230
9,4676
13,8467
8,4769
5,1169
5,7061
p=3.00
14,7242
11,1180
12,2543
8,4805
1,07150
13,6527
p=4.00
9,8499
9,3664
12,7172
10,1007
5,1567
4,6932
p=5.00
13,8817
10,3204
13,5058
9,1772
0,5806
13,3011
p=10.00
9,6839
9,4277
14,3468
10,2889
3,9594
5,7245
Table 14: Returns w.r.t p-parameter with combination (EV/Ebit,P/E)
Figure 8: Returns w.r.t p-parameter with combination (EV/Ebit, P/E)
5
RESULTS AND DISCUSSION
44
Parameters
1
2
3
4
5
P1-P5
p=0.10
15,0772
11,7134
9,1890
7,1525
3,0230
12,0542
p=0.25
15,4358
9,6962
10,6592
7,1082
3,2408
12,1950
p=0.50
14,2688
9,8274
10,7268
7,5167
4,2314
10,0374
p=0.75
13,6873
10,7534
10,5507
7,6213
3,4421
10,2452
p=1.00
13,3399
11,5489
10,2868
8,1118
3,16478
10,1751
p=2.00
14,4125
12,0764
10,6299
9,0761
0,4383
13,9741
p=3.00
12,7596
11,7236
12,0245
9,6879
0,4880
12,2716
p=4.00
12,9280
10,8030
12,8519
9,5218
0,4344
12,4936
p=5.00
11,9658
11,8593
12,3893
10,3551
0,0907
11,8751
p=10.00
11,6368
11,3066
14,1406
8,0126
2,0001
9,6366
Table 15: Returns w.r.t p-parameter with comb. (P/B, P/E)
Figure 9: Returns w.r.t p-parameter with comb. (P/B, P/E)
5
RESULTS AND DISCUSSION
45
Parameters
1
2
3
4
5
P1-P5
p=0.10
14,6786
11,0558
11,8483
7,4118
0,7736
13,9049
p=0.25
14,8912
10,9407
11,3345
8,1391
0,6253
14,2659
p=0.50
15,0955
11,2637
11,2135
7,7747
0,6126
14,4828
p=0.75
14,9419
11,1825
11,3908
7,8844
0,5172
14,4247
p=1.00
14,9928
10,6332
12,7877
6,3131
1,4157
13,5771
p=2.00
13,3620
6,8379
11,4949
9,5299
4,9818
8,3808
p=3.00
13,8713
10,4115
12,1395
7,9477
1,9144
11,9568
p=4.00
12,1850
8,2202
11,2593
9,3212
5,2297
6,9553
p=5.00
13,8470
9,4910
12,0944
8,8585
1,8599
11,9871
p=10.00
10,8197
10,5185
12,4816
9,4138
3,3915
7,4282
Table 16: Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda)
Figure 10: Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda)
5
RESULTS AND DISCUSSION
46
Parameters
1
2
3
4
5
P1-P5
p=0.10
12,5857
15,4610
9,6777
8,5914
-0,1394
12,7251
p=0.25
12,3419
15,6349
9,7943
7,1130
0,8110
11,5308
p=0.50
12,7999
15,2852
9,6764
7,2255
0,7899
12,009
p=0.75
12,8473
14,9566
10,2200
7,2854
0,5766
12,2707
p=1.00
13,0803
15,2537
9,6711
7,3197
0,5270
12,5533
p=2.00
9,8352
11,2003
7,8272
8,3415
8,1326
1,7025
p=3.00
13,9787
13,2372
10,4077
8,0880
-0,0353
14,0141
p=4.00
9,6825
11,0940
9,0436
8,0683
7,3282
2,3543
p=5.00
14,4038
12,2104
11,2501
8,1626
-0,0937
14,4976
p=10.00
12,0726
9,8676
9,3197
8,0120
7,0727
4,9999
Table 17: Returns w.r.t p-parameter with comb.
(EV /Ebit, EV /Ebitda)
Figure 11: Returns w.r.t p-parameter with comb. (EV/Ebit, EV/Ebitda)
5
RESULTS AND DISCUSSION
47
Parameters
1
2
3
4
5
P1-P5
p=0.10
11,3409
11,6619
11,5715
8,2017
2,4204
8,9205
p=0.25
12,2969
11,4025
11,0867
8,1948
2,1334
10,1635
p=0.50
14,2726
10,7651
8,3064
9,4270
2,3724
11,9002
p=0.75
14,2662
9,5984
10,7559
9,0964
1,4715
12,7947
p=1.00
13,7507
10,2508
10,0029
9,5889
1,7482
12,0024
p=2.00
12,6740
10,1142
11,0233
8,5386
2,7105
9,9635
p=3.00
11,9792
12,1653
10,1159
6,6952
4,0799
7,8992
p=4.00
11,8543
11,9811
9,5926
6,5756
5,3377
6,5166
p=5.00
11,9198
11,8974
9,2719
8,2774
3,6919
8,2278
p=10.00
11,2644
11,5565
10,6026
4,8181
7,1352
4,1292
Table 18: Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda)
Figure 12: Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda)
6
CONCLUSIONS AND FUTURE WORK
6
48
CONCLUSIONS AND FUTURE WORK
In this research, we have applied similarity based TOPSIS to equity portfolios. A
comparison has been carried out on average annual returns obtained using single
rankings and those with multiple criteria decision making-methods in which we
observe that using multiple criteria decision making methods brings added value
because of higher average annual returns obtained. Specically, the Finnish stock
market has been analyzed by computing average annual returns as a basis for nding
best performing portfolios.
Five value ratios EV/Ebit, P/B, P/E, P/S and EV/Ebitda where used for single
ranking in which EV/Ebit was the best perfoming, while for multiple criteria decision making methods, the above value ratios were joined to arrive at ten dierent
combinations. The results for this case indicate that
(EV /Ebit, P/E) has the highest
average annual returns and therefore the best perfoming. This combination infact
contains
(EV /Ebit)
which is already found as best performing for the case of single
ranking.
When comparing results with highest returns in 1st portfolio, combinations of two
criterias gave highest returns with
(EV /Ebit, P/E)
as
15.71
whereas compared to
EV /Ebit gave 15.29 as best possible option. Also by comparing results
with deviations, (EV /Ebit, EV /Ebitda) gave the highest as 14.49 whereas EV /Ebit
single criteria
gave the highest deviation of
13.73 for the case of single criteria.
The overall highest
results for combinations as compared to single criteria clearly indicates added value
in using similarity based TOPSIS.
Generally it was observed that best perfoming portfolios occur when
than
1
p-
value is less
and therefore, we can improve the performance of portfolios by reducing
p-
values. It is further noticed that best performing ratios are in rst portfolios and
worst are clearly in 5th portfolio for both single ranking and using multiple criteria
decision making methods with similarity based TOPSIS. It is also interesting to note
that the highest dierence between 1st portfolio returns and 5th portfolio returns
were high in the combinations which gave best results.
The work has widened the scope of application of TOPSIS and therefore provides a
good foundation for further studies which can be carried out. Other decision making
methods could be applied on a dierent selection of value ratios.
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7
APPENDIX I: ANALYSIS OF RETURNS W.R.T P-PARAMETERS AND RESPECTIVE C
Appendix I: Analysis of Returns w.r.t p-parameters
and respective combinations
Parameters
1
2
3
4
5
P1-P5
p=0.10
12,4665
10,7845
12,0256
9,5002
2,4692
9,9973
p=0.25
12,2593
11,8706
11,3761
8,5929
2,9855
9,2738
p=0.50
12,6369
11,3835
11,0165
7,6558
3,9149
8,7219
p=0.75
12,8412
10,9139
10,3184
8,7815
3,7877
9,0535
p=1.00
12,7013
10,1496
12,442
9,0489
2,4526
10,2488
p=2.00
13,4815
10,7553
11,7355
8,6849
2,5279
10,9530
p=3.00
12,4406
11,4009
12,5758
8,4007
2,1253
10,3154
p=4.00
12,4195
11,9177
11,3715
7,0363
4,6211
7,7983
p=5.00
12,4649
12,5177
12,864
6,5573
2,8674
9,5976
p=10.00
11,6947
12,4787
11,8558
6,1390
5,4890
6,2056
Table 19: Returns w.r.t p-parameter with combination (EV/Ebit, P/B)
Figure 13: Returns w.r.t p-parameter with combination (EV/Ebit, P/B)
7
APPENDIX I: ANALYSIS OF RETURNS W.R.T P-PARAMETERS AND RESPECTIVE C
Parameters
1
2
3
4
5
P1-P5
p=0.10
13,0857
14,7517
7,4791
10,6561
1,4136
11,6722
p=0.25
12,5130
12,6896
7,1835
8,6280
6,7544
5,7585
p=0.50
10,5016
15,9796
4,8820
10,6342
5,7377
4,7638
p=0.75
10,8351
15,1992
7,1027
9,0297
5,6260
5,2090
p=1.00
11,8652
13,6037
10,0786
8,0167
3,9156
7,9495
p=2.00
12,7848
11,0209
12,0615
10,3006
1,4596
11,3252
p=3.00
13,1393
10,8825
12,5598
8,5812
2,4407
10,6986
p=4.00
12,7307
11,4355
12,5622
9,5561
1,2295
11,5013
p=5.00
12,3156
11,9066
12,9162
9,6213
0,7785
11,5371
p=10.00
12,4830
10,4998
12,2813
9,2223
2,9691
9,5139
Table 20: Portfolio returns w.r.t p-parameter with comb. (P/E, P/S)
Figure 14: Returns w.r.t p-parameter with comb. (P/E, P/S)
7
APPENDIX I: ANALYSIS OF RETURNS W.R.T P-PARAMETERS AND RESPECTIVE C
Parameters
1
2
3
4
5
P1-P5
p=0.10
9,0432
14,5949
8,9235
8,5363
3,6293
5,4139
p=0.25
11,2408
10,6847
11,6637
6,3336
5,4137
5,8270
p=0.50
11,6625
9,6898
11,4054
7,9835
4,7022
6,9602
p=0.75
11,1254
11,1200
9,4056
9,7532
4,4728
6,6525
p=1.00
10,7465
12,7711
9,7927
9,4048
2,9417
7,8047
p=2.00
8,5633
10,8505
10,5702
8,8997
6,5276
2,0359
p=3.00
9,6536
13,9348
9,0935
8,4662
4,3114
5,3422
p=4.00
9,3759
11,5529
10,4487
8,5232
5,4806
3,8952
p=5.00
9,4993
15,4078
7,9731
7,7339
4,9392
4,5601
p=10.00
7,6338
11,5927
6,6709
9,9322
10,18695
-2,5532
Table 21: Returns w.r.t p-parameter with combination(P/S, EV/Ebitda)
Figure 15: Returns w.r.t p-parameter with comb. (P/S, EV/Ebitda)
7
APPENDIX I: ANALYSIS OF RETURNS W.R.T P-PARAMETERS AND RESPECTIVE C
Parameters
1
2
3
4
5
P1-P5
p=0.10
10,7110
9,8630
8,0491
14,0328
4,7033
6,0078
p=0.25
9,4393
12,4709
8,0386
10,2167
7,4782
1,9610
p=0.50
10,0759
12,8182
6,7996
10,3352
7,7216
2,3544
p=0.75
9,9639
13,3081
6,8311
13,7752
4,0819
5,8821
p=1.00
10,9391
11,6677
8,8612
11,1196
5,3039
5,6352
p=2.00
8,7347
10,8050
11,6177
10,5169
6,61338
2,1212
p=3.00
9,8055
14,4781
9,8502
9,2697
4,7782
5,0273
p=4.00
9,7900
12,3936
8,6394
10,0739
7,7542
2,0358
p=5.00
11,1445
14,2343
8,6842
8,4306
5,8766
5,2679
p=10.00
10,4038
11,2515
6,9199
12,0601
7,8327
2,5712
Table 22: Returns w.r.t p-parameter with combination (EV/Ebit,P/S)
Figure 16: Returns w.r.t p-parameter with combination (EV/Ebit, P/S)
7
APPENDIX I: ANALYSIS OF RETURNS W.R.T P-PARAMETERS AND RESPECTIVE C
Parameters
1
2
3
4
5
P1-P5
p=0.10
7,9761
14,8964
7,4898
9,8925
7,7078
0,2683
p=0.25
9,2138
12,3720
9,1670
10,5284
5,4622
3,7515
p=0.50
9,7983
12,2380
9,7828
8,7681
5,9237
3,8745
p=0.75
10,0099
12,2535
9,6056
8,3433
6,5771
3,4327
p=1.00
9,1262
12,4485
8,9956
9,4202
6,2719
2,8543
p=2.00
10,4884
12,8524
8,2124
9,1395
5,0127
5,4756
p=3.00
10,8575
11,4283
9,7658
8,3447
5,1950
5,6624
p=4.00
10,5226
12,2706
10,2755
8,0393
4,5237
5,9988
p=5.00
10,3263
11,6926
10,2673
7,9552
5,3633
4,9623
p=10.00
10,2898
12,3384
10,7784
7,8555
5,2076
5,0822
Table 23: Portfolio returns w.r.t p-parameter with comb. (P/B, P/S)
Figure 17: Returns w.r.t p-parameter with comb. (P/B, P/S)