Financial Ratios: A New Geometric Transformation

International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 20 (2008)
© EuroJournals Publishing, Inc. 2008
http://www.eurojournals.com/finance.htm

Financial Ratios: A New Geometric Transformation


Alireza Bahiraie
Institute for Mathematical Research, University Putra Malaysia
43400 Serdang, Selangor, Malaysia
E-mail: alirezab@math.upm.edu.my, bahiraie_upm@yahoo.com
Tel: (+60)-19-2203709

Noor Akma Ibrahim
Institute for Mathematical Research, University Putra Malaysia
43400 Serdang, Selangor, Malaysia

Ismail Bin Mohd
Department of mathematics, Faculty of Science, University Malaysia Terengganu
21030, Terengganu, Malaysia

A.K.M. Azhar
Graduate School of Management, University Putra Malaysia
43400 Serdang, Selangor, Malaysia


Abstract

This paper presents a complementary technique for the empirical analysis of risk
and bankruptcy using financial ratios. Within this framework, we propose the use of a new
measure of risk, which is Share Risk measure, and provide evidence of the extent to which
changes in values of this index are associated with changes in each axis values and how this
may alter our economic interpretation of changes in the patterns and direction of risk.
Solving some methodological problems concerned using financial ratios such as ratio
outliers, non-proportionality, non-asymetricity, non-scalicity and non-normal distribution
are illustrated. Then results of Multiple Discriminat Analysis (MDA) and Genetic
Programming (GP) are compared for common and modified ratios and higher accuracy
achieved.


Keywords: Altman Z-Score model, risk box, risk isoclines, bankruptcy prediction, ratio
analysis

1. Introduction
This paper presents a complementary perspective on the study of risk and bankruptcy. Since global
increases in bankruptcies, accurate predictions of companies' distress and bankruptcies have been
extensive. One of the most well known anomalies of the risk factors is the effect of some ratios on
bankruptcy risk and firm returns. One possible explanation for this effect that is consistent with the
“efficient market hypothesis” that ratio is a proxy for risk. Also in banking, the ratios are taken to be a
proxy for the charter value of banks (Landskroner, et. al. 2006).

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International Research Journal of Finance and Economics - Issue 20 (2008)
Since Altman (1968), the literature on predicting bankruptcy has witnessed numerous
extensions and modifications and various techniques been developed to measure risk and predict
bankruptcy. However, none of them had a perfect predictor functional form and all procedures utilised
the use of common ratios without any theoretical basis (Grice et. al. 2001). Many applications and
studies over ratios have been discussed and numerous models have been used but there are still some
limitations on using financial ratios derived from financial statement information1. These limitations
motivated researchers to explore and employ methods like sample trimming, negative observation
elimination, logarithmic, square root, logit and using rank transformation to achieve more predictive
independent variables (Kane et. al. 1998). Previous researchers all tested their models empirically
using data sets and the main results shows that, financial ratios have significant effect on bankruptcy
risk, return, credit risk, commercial risk, market and economic conditions.
Limitations and problems with these approaches are the subjective aspect of the prediction,
which makes it difficult to make consistent estimates and they tend to be reactive rather than predictive
(Kane et. al. 1998). In recent decades, more quantitative and objective systems for bankruptcy
prediction been developed. However, most of these attempts have utilised the use of common ratios
which may exceeded cost of errors in analysis and mis-specification and in general, no equally
convenient or superior alternative transformed ratio has been developed and applied2.
According to literature, variables that are used in previous studies, generally exhibit non-normal
distribution (Barnes, 1982; Ooghe & Verbaere, 1985 and McLeay & Omar, 2000) and non-
proportionality of ratios remove the influence of sector and firm size effect from corporate financial
indicators (Trigueiros, 1997). Some researchers made correction for univariate non-normality and tried
to approximate univariate normality by transforming the variables prior to estimation of their model.
Deakin (1976) used log transformation, then square root and log-normal transformation of financial
ratios were used by Foster (1986) and So (1987).
Other researchers approximate univariate normality by 'trimming' or 'outlier deletion', which
involves segregating outliers by reference to normal distribution (Ezzamel, Molinero and Beecher
(1987). Furthermore rank transformation been used by Perry et. al. (1986) and Kane et. al. (1996).
Recently Ooghe, et. al. (2005) used Logit transformation to achieve better accuracy. However, in the
literature, there are no general guidelines concerning the appropriate transformation in order to
approximate normality.
Our objective in this paper is to propose a complementary approach, which involves data
transformation and we illustrate the use of this methodology for measuring credit-risk and prediction of
bankruptcies. For illustration of this new methodology, book and market ratio values are used as
nominator and denominator of common ratio values and represented as Cartesian coordinates in our
constructed modification box in which we derive the isoclines of associated components of bankruptcy
risk.
The remainder of the paper proceeds as follows. Section 2 defines summary of existing
methodology of the dimensions and the general framework of the risk box. Section 3 briefly derives
the locus of each risk component analytically. Subsequent changes in each risk components associated
with changes in the risk coordinates can be then viewed geometrically. Section 4 illustrates an
empirical application using Multiple Discreminant Analysis (MDA) and Genetic Programming (GP) as
classification methods and we summarise and conclude in Section 5.


2. Overview of Existing Methodology & Measurement Literature
2.1. The Risk Box
The framework is a two-dimensional box in which pair values of risk ratios, which are nominator and
denominator, are represented as Cartesian coordinates (Bahiraie et. al. 2008). Assume a hypothetical

1 For more details about financial ratios properties, see Watson (1990) and Tippett (1990)
2 Some exposition on some of the weaknesses in the use of common ratios such as scaling, proportionality and symmetric effects are provided in Bahiraie
et al. (2008) and, Azhar and Elliott (2006).

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International Research Journal of Finance and Economics - Issue 20 (2008) 166
study of risk covering n years for sector j3. For expositional purposes suppose our proxy for risk chosen
is the widely employed Book to Market (B, M) values. All risk components measure indices such as,
Total Risk (TR), Net Risk (NR), overlap Risk (OR), and lastly the proposed Share Measure of Risk
(SR) as we define below, are linear functions of B and M. Following Bahiraie et. al. (2008), we can
construct a two dimensional box that encapsulates all of these variables for n years.
Suppose that there are n years i.e. t = 1, 2, 3,..., n
and t
? =1, 2,3,..., n , we have: B , M ,TR , NR ,OR , SR ? 0
t
t
t
t
t
t
By definition;
Book value + Market value = Total Risk = Net Risk + Overlap Risk
(1)
where:
NR = B ? M (2)
OR = (B + M ) ? B ? M (3)
and
OR
(B + M )? | B ? M |
2 min(B, M )
SR =
=
=
(4)
TR
(B + M )
(B + M )
The dimensions of the risk box are generated by the maximum value of either the book value or
market value during the period of study. For example, each respective risk box will have sides equal to
max(B ) if for i ? t max(B ) > max(M ) or max(M ) if otherwise. Our exposition of the dimensions
i
i
i
i
of the box is as follows:
From (1), max(TR) = max(B) + max(M ) (5)
From (2), max(NR) = max ( B ? M )
i.e. max ( B ? M ) = ?max ( B) ? min(M )?
?
? or ?max (M ) ? min(B)?
?
? (6)
From (3), max(OR) = max ((2 min(B, M )) (7)
Suppose that we have max(B ) > max(M ) for i ? t
i
i
Let (5) be TR, (6) be NR, (7) be OR and max(M ) be M, and max(B ) be B
i
i
If
2
2
B > M ? BB = B > MM = M
2
? B > MB ; 2
B > NR ; 2
B > OR
2
? (5),(6),(7) ? B = max(B ) max(M )
i
i

3 For expositional simplification, sectoral subscripts are dropped from the rest of the paper.

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167
International Research Journal of Finance and Economics - Issue 20 (2008)
Figure 1: Share Risk Isoclines

?
Bmax

TR*
B

NR*
A
D
OR*
C
?1 ?2
45°
O
Mmax


A
o
45 line from the origin bisects the box into two equal triangles (see Figure 1). This positive
slope diagonal (as demonstrated in section 3) is the locus of balanced risk where B=M, TR equals OR,
SR equals unity, and NR equals zero. This is the risk components' axis of symmetry.4 The two
triangular planes in the box consists of an upper triangle containing coordinate points (B , M ) where
i
i
B > M in the net book value (NB) plane and points M > B in the lower triangle, defined as the net
i
i
i
i
market value (NM) plane.


Locus of Equi Risk Components
2.2. Locus of Equi TR
From (1), we have:
B = ?M + TR (8)
Comparing (8) with y = mx + c , we have the gradient m equals minus unity. Hence, locus of
equi TR is perpendicular to the axis of symmetry.

2.3. Locus of Equi NR
From (2), we have the following:
B
? = M then NR = 0
B
? > M then NR = B ? M = B ? M i.e. B = M + NR (9)
B
? < M then NR = B ? M = M ? B i.e. B = M ? NR (10)
Comparing (9) with y = mx + c , we have, for a net book value, m = 1 with a vertical intercept
c = NR . Since the line of balanced trade is the axis of symmetry for NR, we have similar values for
(10) i.e. m = 1 and c = NR (see Figure 1). Consequently, locus of equi NR values is perpendicular to
1
?
lines of equi TR (i.e. m
=
). Following (6), maximum NR occurs at the maximum value of both
TR
mNR
axes.

2.4. Locus of Equi OR
From (3), we have the following;
B
? = M
? OR = (B + M ) = TR (11)

4 See Schattsneider (1978) for a broad discussion on plane symmetries.

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International Research Journal of Finance and Economics - Issue 20 (2008) 168
B
? > M ? OR = (B + M ) ? (B ? M ) = 2M (12)
B
? < M ? OR = (B + M ) ? (M ? B) = 2B (13)
? (11), (12), (13), imply that as min (B, M) increases, the corresponding OR values, which
equal 2min (B, M), also increases.

2.5. Proposed Share Measure of Risk and Locus of Equi SR
Consider our proposed index of risk measure, the share measure of risk denoted as SR index. The
followings are obtained:
(i) From (4), we have:
? B or M = 0 ? SR = 0 , and B
? = M ? SR = 1.
(ii) From figure 1, consider rays OB and OC subtending equi angles ? ,? from the symmetry
1
2
axis. Let A, B, C, and D represent points on the risk plane with A, B and C sharing equi total risk
values, TR*. In addition, B, C, and D share equi OR values, OR*.
defn
From (8), OA = TR *, and from (1), TR * ?OR * =
*
NR = AB
defn AB
TR * OR
?
*
OR *
Hence; tan? =
=
= 1?
= 1? SR *
1
OA
TR *
TR *
?
*
SR = 1 ? tan?1 (14)
SR values are constant along any ray from the origin.
We now provide demonstration of the convergence of equi SR rays to unity as they approach
the axis of symmetry from either the net book value or net market value plane.
From figure 1, consider ? BOA:
Since 0 ? ?
tan?
1 ?
o
45 ? 0 ?
1 ? 1
(15)
From (15), as ? decreases (i.e. approaches OA) ? tan? approaches zero.
1
1
Moreover, from (14), this implies
*
SR ? 1 . Thus, the SR index approaches unity as the ray
sweeps from the vertical axis (line of zero Market value) towards the axis of symmetry.
Similar reasoning follows for ? A0C. SR * tends towards unity as ? decreases i.e. as the equi
2
SR ray sweeps from the horizontal axis (line of zero Book value) towards the symmetry axis.

2.6. Weighting and Aggregation Procedures
Suppose we want to present weighted versions of the risk measures in accordance to the sectoral
significance of each respective sector. We can formulate the use of weights (w) for the proposed risk
B ? M
measure SR
= 1?
, which can be as follows:
BM
B + M
B ? M
B ? M
We have 0 < SR
= 1?
< 2 ? ?1< SR
=
< 1 (17)
BM
BM
B + M
B + M
Define:
t
B ? M
BM
t
BM
W =
=
i
SR
= SR
i
n
n
?
BMW
BM
n
(B + M )
?
so we have
1
(BM )t
?
(16)
(BM )t
i
i
i
This shows that weighted version of ratio values can be used instead of aggregated version.


3. An Illustrative Empirical Application
The data set used in our illustrative empirical study consists of 144 Iranian companies from Tehran
Stock Exchange (TSE) which 72 companies went bankrupt under paragraph 141 of Iran trade law5 and

5 Under paragraph 141 of Iran Trade Law, a firm is bankrupt when its total value of retained earning is equal or greater than 50% of its listed capital.

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169
International Research Journal of Finance and Economics - Issue 20 (2008)
72 companies are "matched" companies from the same period of listing. For testing the accuracy of
common and new transformation, first Multiple Descriminant Analysis (MDA) technique is applied for
traditional Altman Z-score (1968) and Etemadi et al. (2008) with five ratios as independent variables
model for each two sections6.
The common ratios are first applied followed by the transformed data representation, using
SPSS software version 16. Then implementing Genetic Programming (GP) process and testing the
accuracy in which GeneXpro Tools software version 4.0 was used.

Table 1:
Accuracy levels for previous researches in two classification methods


Multiple
Discriminant
Genetic
Analysis (MDA)
Programming (GP)
Altman's variables (1968)
Original Ratios
80.6 %
88.28 %

New approach (Share Measure)
83.1 %
88.97 %
Etemadi's variables (2008)
Original Ratios
81.6 %
91.03 %

New approach (Share Measure)
84.5 %
93.1 %

Table 1, exhibits the accuracy of the Altman’s Z-score and Etemadi's procedure. The results
have been improved under data transformation procedure from 80.6 and 81.6 percent to 83.1 and 84.5
percent on MDA classification and GP respectively. Due to better performance testing of this new
transformation, data set is not collected form particular industry type and similar firm size or apply any
outlier deletion to overcome any potential explanatory effect errors that will be caused by independent
variables distribution.7
The accuracy of the model is also analyzed using the classification accuracy on both
transformed and common ratios sample by Receiver Operating Characteristic (ROC) curve. In
addition, the significance level and discriminanting power of both discriminant functions can be
assessed through Wilks' Lamba measure testing, which is used in this research8.

Table 2:
Wilk's Lamba test for traditional ratios and Share Risk Measure model

Functions test
Wilks'-Lamba
Chi-Square
df
Sig
Altman's variables Original Ratios
0.610 48.626 5 0.000
(1968)
New approach (Share Measure)
0.559 69.485 5 0.000
Etemadi's
Original Ratios
0.617 47.510 5 0.000
variables (2008)
New approach (Share Measure)
0.583 53.078 5 0.000

Table 2 shows how significant each discriminant function can classify cases. It is equal to the
proportion of the total variance in the discriminant scores not explained by differences among the
groups.


4. Summary and Conclusions
Since Altman (1968), the growth in the literature on predicting risk and bankruptcy without any
theoretical background is considerable. In this paper, a new dimension to measurement of risk,
bankruptcy, and ratio transformation with the advent of the risk box was proposed. We briefly derived
the respective isoclines of each risk component in the Risk Box as illustration of risk proxies of
Cartesian coordinates. Our simple methodology, called SR index, provides a geometric illustration of
our new proposed risk measure and transformation behaviour.

6 The basic assumption of MDA is the multivariate normality of the variables (Kleinbaum, et .al. (1988), but it can be applied when the observations are
not less than 50 (Gu, 2002). Our test case of 200 firms applies hence there is no need for normality test.
7 MDA is a prevalent technique in bankruptcy prediction (Aziz and Dar, 2006). In term of classification and ability among traditional models, while some
techniques like Probit reported the same results.

Deakin (1976) found that financial ratios might be more normally distributed within a specific industry groups.
8 Level of significance of a discreminant function can be proved through different measures like Walks' Lamba, Hotelling's Trace and Pillai's criterion.

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International Research Journal of Finance and Economics - Issue 20 (2008) 170
Alternatively, the risk box can be used as a tool of analysis in providing a crucial first stage for
analysing studies associated with changes in risk patterns, in particular those assumed to be correlated
with potential bankruptcies. The versatility of our proposed methodology is emphasised by its
applicability for any number of years on sectoral or cross-country studies on risk and bankruptcy
studies. More testing is remained for further studies and research.


References
[1]
Altman, E.I., (1968): "Financial ratios, Discriminant analysis and the Prediction of Corporate
Bankruptcy". The Journal of Finance, 23(4), 558 589–609.
[2]
Azhar, A.K.M. and Robert J.R. Elliott, (2006): "On the Measurement of Product Quality in
Intra industry Trade".Review of World Economics, 142(3), 476-495.
[3]
Aziz, M.A. and Dar, H.A., (2006): "Predicting Corporate Bankruptcy: Where we stand?".
Corporate Governance, 6(1), 18–33.
[4]
Bahiraie A.R., A.K.M Azhar, N.A Ibrahim, W.W. Abdullah and Ismail M., (2008): "On the
Measurement of Credit Risk: A New Geometric Approach".Presented in: Third International
Conference on Mathematics and Statistics, Bogor, Indonesia (icoms3).
[5]
Barnes P., (1982): "Methodological Implications of Non-Normality Distributed Financial
Ratios".Journal of business Finance & Accounting, 9, spring 1982, 51-62.
[6]
Deakin E., (1976): "On the Nature of Distribution of Financial Accounting Ratios: Some
Empirical Evidence". The Accounting Review, 51, January 1976, 90-97.
[7]
Etemadi H., Rostamy A.A., Farajzadeh H., (2008): "A genetic Programming Model for
Bankruptcy Prediction: Empirical Evidence from Iran", Expert Systems and Applications.
doi:10.1016/j.eswa.2008.01.12.
[8]
Ezzamel, M., Mar-Molinero C. and Beecher A., (1987), "On the Distributional Properties of
Financial Ratios", Journal of Business Finance & Accounting, 14, winter 1987, 463-481.
[9]
Foster, G. (1986): "Financial Statement Analysis", Second edition. Prentice-Hall. Englewood
Cliffs, NJ.
[10]
Grice J.S. and Dugan M.T., (2001): "The Limitation of Bankruptcy Prediction Models: Some
Cautions for the Researchers".Review of Quantitative Finance and Accounting, 17(2001): 151-
166.
[11]
Gu Z. (2002): "Analyzing Bankruptcy in Restaurant Industry: A Multiple Discriminant Model".
Hospitality Management, 21, 25-42.
[12]
Kane, G.D., Richardson F. and Meade N., (1996): "Rank Transformations and the Prediction of
Corporate Failure". Working paper University of Delaware, Newark: DE.
[13]
Kane, G.D. and Meade, N.L., (1998): "Ratio Analysis Using Rank Transformation”. Review of
Quantitative Finance and Accounting, 10(1998), 59-74.
[14]
Kleinbaum DG, Kupper LL, Muller KE. (1988): "Applied Regression Analysis and Other
Multivariate Methods". Second edition, Belmont, California: Duxbury Press, 1988.
[15]
Landskroner, Y. Ruthenberg, D. and Pearl, D., (2006): "Market to Book Value Ratio in
Banking: the Israeli Case". Bank of Israel banking, Supervision Department, Research Unit.
[16]
McLeay S. and Omar A., (2000): "The Senility of Prediction Models to the Non-Normality of
Bounded and Unbounded Financial Ratios". British Accounting Review, 32, 213-230.
[17]
Ooghe H., Speanjers CH., Vandermoere P., (2005): "Business Failure Prediction: Simple-
Intuitive Models versus Statistical Models". Working Paper Series. Vlerick Leuven Gent,
(22)2005.
[18]
Ooghe H., and Verbaere E., (1985): "Some Further Empirical Evidence on Predicting Private
Company Failure". Accounting and Business Research, 18, 57-66.
[19]
Perry L.G. and Cronan T.P., (1986): "A Note on Rank Transformation Discriminant Analysis".
Journal of Banking and Finance, 10, 605–610.

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171
International Research Journal of Finance and Economics - Issue 20 (2008)
[20]
Schattsneider D., (1978): "The Plane Symmetry Groups: Their Recognition and Notation".
American Mathematical Monthly, 85, 439-450.
[21]
So S.J., (1987): "Some Empirical Evidence on the Outlier and Non-Normal Distribution of
Financial Ratios". Journal of Business Finance & Accounting, 14, winter 1987, 483-496.
[22]
Tippet M., (1990): "An Introduced Theory of Financial Ratios". Accounting and Business
Research, 21(81), 77-85.
[23]
Trigueiros, D., (1997): "Non-Proportionality in Ratios: An Alternative Approach". British
Accounting Review, 29(1997), 213-230.
[24]
Watson C.J., (1990): "Multivariate Distribution Properties, Outliers and Transformation of
Financial Ratios". Accounting Review, 65(3), 682-695.

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