Journal of Naval Science and Engineering
2003, Vol. 1, No.2, pp. 123-140
Clustering of Financial Ratios of the Quoted Companies
Through Fuzzy Logic Method


Ekrem Tufan
Assistant Professor, Open Education Faculty
Anadolu University anakkale 17100
email:etufan@yahoo.com

Bahattin Hamarat
(Lecturer) College of Tourism and Hotel Management
ǃanakkale Onsekiz Mart University,


Abstract
Financial rates, in terms of investors, are of great importance in the
expansion of the companies to whose share they are going to invest.
Financial rates have been benefited from especially in determining the
risk while a portfolio is being formed, in changing the existing risk or in
determining the alternative shares. In literature has been researhed
whether more than average income of the market is possible to obtain
considering financial rates. In this respect, using fuzzy logic method the
companies that are the subjects in the study are clustered and it is
presented to the investors as a risk scale. The investor, according to the
portfolio he/she is going to form, can not only determine which shares
he/she should buy but he/she will be able to learn about the share which
has the same profit level and the same risk as well if the share he/she
wants to buy is not available. What is aimed to be achieved in this study
is helping the investor form a portfolio and the companies at question be
clustered with the help of financial rates.

Key words: Financial rates, Fuzzy logic, Factor Analysis

I. Introduction

Financial ratios have a great importance for investors in respect of
company evaluation or in the selection of shares. Particularly, the ratios
often used in the preferences of portfolios deal with many issues in literature

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Clustering of Financial Ratios of the Quoted Companies Through Fuzzy Logic
such as the characteristics of determining, changing of portfolio structure or
determining the shares substituting each other, estimating the returns of
shares to be obtained in the future. For example, Lǃders [1], in his
discussion of how the risk choices of investor would affect the price shares,
claims that the return will be able to be estimated even markets have an
informational efficiency. The author also claims that financial that are
commonly used will be able to be used in the estimation of returns to be
obtained in the future.
Pahor and Mramor [2] tested the hypotheses of specific industry
independent nonlinear relationships between financial ratios and excess rate
of return on equity for U.S. and Japan. The authors put forward that they
reached the findings previously in the direction of supporting the results of
similar investigation which was carried out for Slovakia and the relation
between the financial ratios and the return ratio of over the market was
independent from the sector.
Lewellen [3] provides a new test of the predictive ability of
aggregate financial ratios. The author has investigated dividend yield
predicts aggregate market returns from 1946 ܢ 2000, for New York Stock
Exchange as well as in various subperiods and reported that book-to-market
and the earnings-price ratio predict returns during the shorter 1963 – 2000
sample. Author also claims the evidence remains strong despite the unusual
price run-up in recent years.
Küçükkiremitçi [4] tested that whether or not the financial ratios
taken into consideration in forming portfolio in the sectoral base of 104
industrial companies quoted in Istanbul Stock Exchange (ISE) by using the
date of 1995 year. The author pointed out that the financial ratios showed
differences in the sectoral base and claimed that the most popular ratios in
the selection of shares were low market value/book value and total debt/total
assets value with high assets development turn over ratio, gross income
margin and stock turn over ratio.
Demir, Küçükkiremitçi, Pekkaya and Üreten [5] investigated the
ratio of price/earning, the size of company and negative earning effects by
taking advantage of the data of the companies quoted in ISE in the periods
of 1990-1996. The authors stated that they found out the results of the
returns of shares decreased as proceeding from the lowest price/earning

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Ekrem Tufan and Bahattin Hamarat
portfolio to the highest price/earning portfolio that were formed when the
companies declaring their deficits were excluded. Besides this, they
explained that the market value of the shares of the companies declaring
deficits had high returns in the following period and found out the results
that they would be able to beat the market in case of being formed portfolio
according to the size of a company.
Karan [6] formed four portfolios by taking advantage of the ratios of
price/earning and market value/book value of the companies quoted in ISE
for the period of 1988-1995 and tested them whether or not the returns
would be able to be obtained excess rate of return in the long term. The
author stated that returns would be able to be obtained excess rate of return
in the long term by taking advantage of the ratios of price/sales and market
value/book value, that is, emphasized that ISE has not be weak form
efficiency.
Also Demir, Küçükkiremitçi, Pekkaya and Üreten [7] investigated
the relationships between the financial ratios and the returns of shares of the
companies quoted in ISE by taking advantage of the data of the years of
1993 – 1994 and ranged the companies. The authors stated that there was
not a meaningful relationship between the returns of shares and the ratios of
price\earning in the relevant to that term as a result that they found out. This
result was contradictory with the study which they previously carried out
similarly (1996). Besides this, the authors stated that the highest relationship
between the returns of shares and the financial ratios was seen in 1994 (with
83%) and 1999 (with 22%).
In this study, it is aimed at targeting clustering the companies
involved in the investigation through their financial ratios and helping
investors in forming portfolios. Study was ranged as follows besides the
introduction, data structure and methodology in the section II, where
empirical results in the section III and conclusions and evaluation in the
section IV of the study.

II. Data and Methodology

In this study, the companies being excluded from financial sectors,
which quoted in 2002 out of the companies, which were involved in ISE

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Clustering of Financial Ratios of the Quoted Companies Through Fuzzy Logic
100 Index were chosen. It was seen that some of the companies affected the
results negatively because they had both incomplete data and extremity
values. Therefore, these companies were determined through Factor
Analysis, were extracted from data set and the size was eliminated. The
companies involved in the study were presented at appendix 2. The data,
size of 47X43, were taken from ISE web page address. In the study, 43
financial ratios were used and presented at appendix 1.
In the study, the method of fuzzy cluster was used as a method
within the framework of fuzzy logic. Some samples can be ranked as
follows concerning the usage of the method in social sciences. Tay and Linn
[8] explained the reasons affecting the choices of investors in buying and
selling shares through fuzzy logic. Toraman [9] presented the method of
fuzzy logic to auditors as an alternative way in removing the indefiniteness
they met in the audience of company. Şahin and Hamarat [10] clustered the
countries forming the organizations towards international integration and
cooperation with the method of fuzzy cluster by using 30 variations showing
socio–economic structures.
In this study the fuzzy cluster method and oclid distance were used.
The fuzzy cluster method appears a suitable method if the clusters cannot be
separated from each other clearly or if some units in cluster membership are
undecided. Fuzzy clusters are the functions determining each unit between 0
and 1 being defined as a membership of unit in cluster. The units quietly
resembling to each other take place according to the relation of high
membership in the same cluster. That’s why the fuzzy cluster method
calculates the coefficients of the units belonging to the cluster or clusters.
The total of membership coefficients is always equal to 1. So, the
unit is assigned to the cluster that has the highest membership coefficient.
The membership functions are the functions that characterize the fuzziness
in a fuzzy cluster whether or not the elements in cluster are continual or
transitory.
Fuzzy cluster has two basic methods. From these, c average cluster
method depends on c divisions. The other method depending on fuzzy
equality relation is called as a graded cluster method depending on fuzzy
equality relation (11). The similarity structures of companies were found as
a basis of Fanny Algorithm that depends on fuzzy equality relation. Fuzzy

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Ekrem Tufan and Bahattin Hamarat
cluster technique used in this algorithm aims at minimization of target
function below the membership functions in this target function have these
limitations:
u
≥ 0 if so i=1,......., n and v=1,............, k

(1)
i v
k
∑u =1= %100 if so i=1,.......,n



(2)
i v
v 1
=
Here, each unit i and each cluster v will be a member of uiv. uiv
shows how much unit I belongs to cluster v. Under these circumstances, the
target function is as follows;
n
2
2
k ∑
u u d(ij)
C ∑ i,j=
=
1
i v
j v






(3)
n
2
V=1
2∑ u
j=1
j v
Here d(ij), i and j mean the resemblance among the units, whereas
uiv defines the unknown membership of unit i to cluster v. Total of
membership coefficients of each unit in comparison with all the clusters is
always positive as a figure 1 in fuzzy cluster. Fuzzy cluster is evaluated how
far it is from certain cluster through Dunn decomposition coefficient. This
coefficient gives an idea about how fuzzy the cluster gained is.
Decomposition coefficient of Dunn takes place between (0 and 1). Dunn
coefficient shows the situation of 0 as a complete fuzzy and the situation of
1 as a certain cluster. This coefficient can be normalized from the number of
clusters independently from 1 (certain cluster) to 0 (complete fuzzy) out of
cluster number. Normalized Dunn coefficient takes place at interval of (0,1)
and is called as Non-fuzziness index. It is determined how gradually well
the units cluster through silhouette coefficient in fuzzy cluster method. s(i),
shows how gradually well the unit clusters and takes place between -
1≤s(i)≤s(i) is an average silhouette image coefficient for all units in a
cluster, s(i) is a coefficient showing how gradually well the units cluster
according to the number of clusters for all units and equals to the average of

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Clustering of Financial Ratios of the Quoted Companies Through Fuzzy Logic
s(i), k the cluster number, being equivalent to the biggest s(i) ’s chosen as
the most suitable cluster number (12, 13).

III. Empirical Results

It was investigated that the companies quoted in ISE 100 Index by
making use of their financial ratios whether or not they clustered
homogeneously by using S-PLUS 2000 statistical packet program. A
suitable cluster number was determined by being changed the number of
cluster between 2 and 10. So, the inter-cluster variations of companies
started to be observed with the determination of the cluster numbers
reflecting the natural classification appearing by coming together of
variations. As a rule, the suitable cluster number was determined as 2 in
respects of all the separate factors.
Fuzzy clustering (FC), in terms of liquidity rates; when the
clustering of the companies are looked at in terms of liquidity rates,
according to not only the division coefficient indicating how well the
companies are clustered but also the silhouette coefficient, k=2 was found to
be the most convenient cluster number. The membership number of the
companies for k=2 was calculated and the findings obtained are shown in
Table 1. When the membership values of the companies to the clusters are
look at, the companies with 9, 14, 17, 23, 24, 26 and 43 numbers display a
consistent membership to the clusters. FC and silhouette width values s(i)
which signals the quality of clustering were calculated for k=2. These values
are big, which indicates a good clustering has been formed and s(i) averages
are big, which indicates a good and certain has been obtained. Now we can
claim that the companies in the first clustering are better clustered than
those in the second one. According to the cluster number k=2 the two
clusters obtained cannot be said to be clustered highly well. Especially 9th,
14th, 17th, 24th, 26th and 43rd companies in the second cluster are observed
not to be consistent in the cluster membership. Besides this, they are also
observed not to consistent in the certain cluster membership. The results are
shown in Table 1.

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When the results of fuzzy clustering are looked at in terms of
activity rates; k=2 was found to be the most convenient cluster number
according to not only division but also silhouette coefficient. Accordingly,
all the companies in the second cluster are not consistent in cluster
membership and can be said to be have fuzziness. The results are shown in
Table 2. When the certain clustering membership is looked at in terms of
activity rates of the companies the companies in the first cluster are highly
well clustered where as those in the second are, on the contrary, highly well
clustered. It was also found that some clusters have negative membership
coefficient.
When fuzzy clustering results are investigated in terms of financial
rates; k=2 was found to be the suitable cluster number. According to this,
the companies with 10, 20, 24, 27, 31, 34, 46 numbers in were found to be
close in cluster membership and although these companies can be said to be
fuzzy, only the companies with 1, 46 numbers are not consistent in their
cluster membership in the cluster that they belong to and can be said not to
clustered for certain. Besides this, when the averages values of the
companies s(i) in the first cluster are investigated, this cluster can be said to
be less inconsistent than the second one. In terms of financial rates, of the
companies, both clusters can be said to be well clustered. The results are
shown in Table 3. According to the rates which indicate the relationship
between the assets of the companies, when the fuzzy clustering results are
given, according to not only division but also silhouette coefficient, k=2 was
found to be the most convenient. According to this, when the similarities of
the companies are investigated, it can easily be seen that they have
membership at a high level. But, it was also found that the 18th company
was not clustered for certain and the second cluster had quite low average
silhouette coefficient. According to these rates it was found that s(i)
coefficient of the companies, which is the scale of clustering, was very high.
The results are shown in Table 4.
According to the rates which indicate the relationship between
income and sale of the companies, when fuzzy clustering is looked at, k=3
was found to be the convenient cluster number in terms of income rates.
According to this, the 11th company has a high membership to the second
cluster but it was low in terms of consistency in the certain cluster

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Clustering of Financial Ratios of the Quoted Companies Through Fuzzy Logic
membership. 44th company was determined to be similar to the first cluster
at 48.90% and to the second one 49.82%. Consequently, this company is
fuzzy and it is inconsistent between these two clusters. Its certain cluster
coefficients was calculated as –0.088 and was found to be inconsistent in
certain cluster membership. When s(i) average values which indicates FC
quality in term of income rates of the company are investigated, the
companies in the first cluster was found to be more homogenous than the
others clusters in terms of income indicators. The results are shown in
Table 5.
When the fuzzy clustering results are investigated in terms of the
rates which indicate the relationship between income and equity of the
companies, the companies, which belong to 1, 3, 4 and 5 clusters, are
observed to display consistency in similarity to these clusters. Whereas
those which belong to the other clusters are not so consistent in their
similarity. Especially, all the companies in the second cluster displays
similarity to the neighboring 8th cluster at the same level. Consequently, the
companies in the second cluster can be said to be inconsistent at fuzzy.
Similarly, the companies in the 8th cluster resemble the second cluster at the
same level. The 13th company in the 7th cluster is approximately at the same
similarity level to the neighboring cluster 2. Consequently, it’s inconsistent
between these two clusters. When the certain cluster memberships, the
companies with 13, 15, 30, 34, 42 numbers were found to have low certain
cluster membership when s(i) values, which signals the cluster quality, the
8th cluster can be said to be well-clustered but 6th cluster can be said not to
be well-clustered. The results are shown in Table 6.

IV. Conclusions

For an investor who wants to form a portfolio, it’s important that the
companies be classified according to the risks. In our study, the investor was
considered the one who does not avoid risk and if an portfolio is assumed to
be formed, we expect the investor to prefer any of the share with the
numbers 11, 15, 27, 31, 33, 34, and 42 according to the clustering formed by
the help of liquidity rates.

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In the same way, the investor may choose the companies in the first
cluster according to the clustering formed by the help of activity rates. All
the companies but 42nd resemble each other in terms of these rates. Thus,
they can substitute each other.
The investor may choose all the shares of all the companies
according to the clustering formed by the help of the rates used in the
analysis of the relation between the assets.
The investor may choose all the shares but that of the 34th company
whereas he/she may prefer second 4th 23rd, 27th, 31st, 47th. Companies in the
second cluster according to the clustering formed by the help of the rates
which indicate the relation between income and sales.
The investor may prefer shares of the companies with 43 and 44
numbers from the 8th cluster and the companies with 24, 26, 28, and 29
numbers from the 7th cluster and the companies with 7, 20, 36 from the 2nd
cluster according to the clustering formed by the help of the rates which
indicate the relationship between income and equity. If the investor is
someone who avoids risk she/he will be able to prefer the shares of the
companies which are negative and whose certain membership coefficient is
over %50 in all the clusters.

References

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Economic research, Discussion Paper, No:02-48, pp.1-30.
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Pahor, M., and Dusan M., (2001), Testing Nonlinear Relationships Between Excess
Rate of Return on Equity and Financial Ratios, www.ssrn.com, Working Paper
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Lewellen, J., (2002), Predicting Returns With Financial Ratios, MIT Sloan School
of Management, Working Paper No:4374-02, pp.1-35.
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Küçükkiremitçi, O., (1997), Sektörel Farklõlõklarõn Finansal Oranlara Etkisi (1995
Yõlõ için İMKB Üzerine Bir Varyans Analizi Denemesi), İktisat, İşletme ve Finans
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ve İMKB Üzerine Çalõşmalar, İşletme ve Finans Yayõnlarõ No:4, pp.41-69.

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Teknik: Bulanõk Mantõk, İktisat, İşletme ve Finans Dergisi, Vol:193, pp. 67-76.
[10] Şahin M., ve Hamarat, B., (2002), G10, Avrupa Birliği ve OECD Ülkelerinin
Sosyoekonomik Benzerliklerinin Fuzzy Kümeleme Analiziyle Belirlenmesi, VI.
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[13] S-PLUS
2000.

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Document Outline

• Ekrem Tufan
  • Abstract
    • Introduction
    • II. Data and Methodology
    • IV. Conclusions
      • References
• Financial Ratios
• Using Companies For Study

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