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Monitoring Models of Firms’ Growth: A Discriminant Analysis of the
DJIA
Dr. Tarek Ibrahim Eldomiaty*
Associate Professor of Finance
Misr International University
Faculty of Business Administration
Finance Department
PO Box 1 - Heliopolis 11341
Cairo
EGYPT
(Tel: +02 (2) 4477 1560)
(Fax: +02 (2) 4477 1566)
E-mail: tarek_eldomiaty@hotmail.com
Sahar Charara
University of Dubai
Finance & Banking Department
College of Business Administration
PO Box: 14143 Dubai
United Arab Emirates
(Tel: +9714-2072718)
(Fax: +9714-2242670)
E-mail: sahar_charara@hotmail.com
May 2008
*
Corresponding Author
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Monitoring Models of Firms’ Growth: A Discriminant Analysis of the
DJIA
Abstract
This paper examines three possible explanations of firm’s growth: (1) a firm grows
according to its sales growth, (2) a firm grows according to saving costs, and (3) a firm
grows according to the two factors being used simultaneously; that is increasing sales by
saving costs.
The paper uses corporate quarterly data from the DJIA30 covering the period 1990-2006.
The paper introduces a new measure of firm growth based on sales-weighted fixed assets
growth. The methodology utilizes the properties of the discriminant analysis to build
three Z-Score models each of which discriminates low-growth firms from high-growth
firms. The first model discriminates between low-growth and high-growth firms based
on sales ratios. The second model discriminates between low-growth and high-growth
firms based on cost ratios. The third model discriminates between low-growth and highgrowth firms based on both sales and cost ratios.
According to the properties of the discriminant models, the discriminant power of the
model is a determinant of the power of the model’s information content. The results
show that the three discriminant models have relatively the same discriminant power
(60%). This means that the revenue and cost factors are used simultaneously as drivers of
firm growth. Nevertheless, the information content in the third model (which examines
the simultaneous use of sales and cost factors) reveals high relative dependence on sales
ratios than cost ratios. This result provides support to the practice of the growth-size
theory of firm growth.
The results have practical implications to corporate managers regarding the sales and
cost factors that are to be considered to promote firm growth.
JEL classification: M21
Key words: Theory of Firm Growth, Discriminant Analysis, Z-Score Models, DJIA index
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I. Introduction
The theory of firm growth has always been considered a critical factor in the
evolution of the business literature (mainly economics, finance, marketing and
accounting). The literature has been characterized by a diversity of the factors affecting
firm’s growth and the lack of consensus on these factors. This diversity has resulted in
two competing theories trying to explain the factors that determine firm’s growth; the
growth-size theory and growth-learning theory. The former focuses on the trend of the
revenue curve as a measure of firm’s growth, while the latter focuses on the trend of the
cost curve as a proxy for the progress of the learning process and firm’s growth. Since
the two theories go in opposite directions, there has not been a consensus on what theory
explains the practice of firm’s growth. The former is concerned with the relationship
between firm’s growth and its size (using sales revenue, employees and assets as
proxies). The latter is concerned with the behavior of the cost function as a result of
firm’s learning process. The main concern of this paper is that it is possible that the two
relationships contradict each other which may not provide a complete picture on how
firms grow. That is, firms may try to increase sales revenues, number of employees or
the assets’ size which result in increases in costs. In this case, the growth-learning
relationship may not provide a possible explanation on how firms grow. If the firm is
trying to minimize costs, it is also possible that sales revenue, number of employees and
assets would be affected negatively. That is why the authors are trying to explore a new
approach to examine firm’s growth which is to examine the proportionate weight of the
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sales revenue and costs to firm’s growth. In this case the discriminant analysis provides
much help.
The approach of this paper is to examine the drivers of firms’ growth based on the
use of sales information and cost information. The former takes into account the
relationship between sales and the other items in the firm’s income statement and balance
sheet. The latter takes into account the relationship between cost of goods sold and the
other items in the firm’s income statement and balance sheet. The sales ratios are used as
a proxy for examining the growth-size relationship since the authors introduce a new
measure of firms’ growth based on ‘sales-weighted fixed assets growth.’ The cost ratios
are used as a proxy for examining the growth-learning relationship.
The paper is organized as follows. Section II reviews the relevant literature on the
models of firm growth. Section III outlines the research objectives. Section IV outlines
the research hypothesis. Section V describes the data and research variables examined in
the study. Section VI describes the structure of the discriminant analysis employed in this
paper. Section VII shows the results. Section VIII discusses the results. Section IX
concludes.
II. Models of Firm Growth: A Review of the Relevant Literature
The literature on firm’s growth is very rich and has been subject to intensive debate
for decades. That debate has resulted in the evolution of the literature of industrial
organization. The underlying concern in the literature is how firms grow. Many
significant efforts have presented possible explanations (or theories). This paper does not
attempt to provide a review to the literature since it is voluminous; rather the paper
focuses on two basic relationships upon which firms’ growth has been examined
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independently, namely; the growth-size relationships and growth-learning relationships.
What follows is a review of the relevant studies that examined each relationship
independently.
1. Growth-Size Relationships
These relationships are considered the oldest to explain firm’s growth. Robert Gibrat
(1931) presented the first explanation of growth with applications to inequality of
income distribution, concentration of enterprises and populations of cities. Since then,
the literature evolved upon extended applications to the economics of firm’s growth
based on Gibrat’s proportional growth law. It assumes that firm’s growth rate is
independent of both its current size and its past growth history. Kalecki, (1945), Hart
& Prais (1956) Hart (1962) and Clarke (1979) presented a validation to Gibrat’s law.
Simon and Bonini (1958) found that Gibrat’s law holds for firms associated with
above the minimum efficient size level. Mansfield (1962) describes the growth-size
independence as “the probability of a given proportionate change in size during a
specified period being the same for all firms in a given industry regardless of their
size at the beginning of the period. Although Gibrat’s growth-size independence was
examined in many studies using static specifications, Ijiri and Simon (1964) reached
the same conclusion using dynamic specifications. Lucas (1967) and Lucas and
Prescott (1971) found that firm’s capital adjustment, employment and output follow
Gibrat’s law. In an extended distinguished effort, Lucas (1978) used Gibrat’s law to
prove the existence of equilibrium in both developed and developing economies. It is
worth noting that the previous theories apply to the complete size distribution.
Nevertheless, Scherer (1980) considered the small-size firms and reported a negative
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relationship between firm’s size and growth. Nelson and Winter (1982) report a
concave relationship which implies that firm growth increases with firm size, then
decreases with firm size. The contradiction to Gibrat’s law has also been presented by
Geroski (1995), Sutton (1997) and Caves (1998). An extended strand to the literature
reported a negative relationship between firm growth and size (Evans, 1987; Hall
1987; Dunne et al., 1989; Mata, 1994; Dunne and Hughes, 1994; Audretsch, 1995;
Hart and Oulton, 1996; Weiss, 1998; Audretsch et al., 1999; Almus and Nerlinger,
2000; Bechetti and Trovato, 2002; Goddard et al., 2002). Chen and Lu (2003) add
another element to the literature by examining the Taiwanese service sector in
addition to the usual manufacturing sector. They found that Gibrat’s law does not
hold for the latter sector but does for the former.
It is quite obvious that taking firm’s size into consideration results in some
contradicting results. That is, a positive growth-size relationship exists in the large
firms while a negative relationship does exist in the small firms. Sutton (1997)
supports this conclusion pointing out that these contradictory results lies in systematic
differences in the samples selected. The earlier studies included only large firms,
while more recent studies included small firms. In the present study, the authors
examine the determinants of growth apart from size. The reason is two fold. First, the
authors believe that the firms’ growth matters relatively more than size because any
firm considers drivers of growth as first priority rather than changing its size. The
decision to change size occurs after the growth rate has been determined. Second, the
conclusions regarding firm’s growth-size are not deterministic since firm’s size
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changes over time. This is the main reason that the authors examine firm’s growth
from a different angle which is to classify firms according to growth rates.
2. Growth-Learning Relationships
The cost elements have also been examined in the literature on firm’s growth. The
early beginning was in Jacob Viner’s paper (1932) about the size distribution of
firms. Viner introduced the theory that a unique size of a firm exists within an
industry under the assumption that firms have a U-shaped long run average cost
functions which assume that each firm produces at the minimum point of this curve.
Overall, in equilibrium, the industry allocates production over firms so as to minimize
total costs. A criticism has been introduced to this theory on the ground that when
firms sells in a large product markets, costs would increase due to an increasing
demand. Nevertheless, Lucas (1967) presented a modified version of Viner’s theory
based on a ‘constant-returns-to-scale’ that assumes that firms produce at minimum
resources cost.1
The literature has considered another element of firm’s growth which is learning. It
has been observed that innovation helps firms grow fast. It is also well known that
innovation requires extra costs to incur. That is, firm’s growth can be explained by
observing the cost functions. Jovanovic (1982) and Lippman and Rumelt (1982)
conclude that firms’ efficiency can be observed through the behavior of the cost
functions. The latter have been studied based on the convexity assumption which
holds for cost functions such as linear-quadratic and Cobb-Douglas. In this regard,
1
Robert Lucas (1978) extended his analysis to the aggregate levels to support his claim that producing at
minimum resources cost does exist. His aggregate theory implies that policies to modernize backward
economies by consolidation and policies to resist conglomerates in advanced economies may both be
misguided.
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Jovanovic (1982) developed a model that assumes that output is a decreasing convex
function of managerial inefficiency. Actually, this is true since inefficiency implies
increasing costs to scale.
III. Research Objectives
The paper aims at examining the objectives that follow.
1. Examine the type and weights of the sales information that discriminates between
low-growth and high-growth levels.
2. Examine the type and weights of the cost information that discriminates between
low-growth and high-growth levels.
3. Examine the type and weights of the sales and cost information that discriminates
between low-growth and high-growth levels.
IV. Research Hypothesis
This paper aims at examining the hypotheses that follow.
H1: A negative relationship exists between firm’s growth and cost-based ratios.
H2: A positive relationship exists between firms’ growth and sales revenue-based ratios.
H3: A positive relationship exists between firms’ growth and both of cost-based and sales
revenue-based ratios.
The 3rd hypothesis is based on certain marketing considerations. That is, when firms are
expecting or actually facing high demand, the production volume increases which
requires increases in production costs.
V. Data and Research Variables
Data
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The data used in this study is obtained from Reuters® for the non-financial firms included
in the DJIA30 index. The data cover the years 1989-2006 on quarterly basis. The basic
forms of the data are the income statement and balance sheets. The sales ratios and cost
ratios are calculated by the authors.
Dependent Variables
The dependent variable is firm’s growth which can be measured as:

FA t 
Sales t
 
Firm Growth   Ln
 FA t -1  Max Sales
The advantage of this measure is that it takes into account the growth of sales and fixed
assets simultaneously. The methodology examines two levels of firms’ growth: lowgrowth (1st quartile) and high-growth (3rd quartile).
Independent Variables
The independent variables are the cost ratios and sales ratios. The methodology examines
also the combined effect of the both types of ratios at a time.
Discriminant, Content and Construct Validity
The effectiveness of the discriminant analysis and the resulted discriminant
models requires a test for discriminant validity, content and construct validity (Podsakoff,
and Organ, 1986). In this case, the classifications and use of sales ratios and cost ratios
provide quite distinctive dimensionality. This means that the issue of discriminant
validity is well settled. Regarding the issues of content and construct validity, the sales
ratios and cost ratios are drawn from relevant literature that adequately provides a multidimensional perspectives. In addition, these ratios are considered an adequate coverage of
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the important content, therefore, provide a good basis for content validity (Nunnally,
1978). Since the variables have been empirically examined in many related studies in the
literature of firm growth, they provide an adequate evidence of construct validity.
VI. Method
The discriminant analysis is the most common technique used to develop a Zscore model. The problem that is addressed with discriminant analysis is how well it is
possible to separate two or more groups of observations (i.e., individuals, companies
…etc.), given measurements for these observations on several variables (Manly, 1998;
Hair, et al., 1995). Therefore, the discriminant analysis is a statistical technique used to
classify an observation into one of several a periori groupings dependent upon the
observation’s individual characteristics. It is used primarily to classify and/or make
predictions in problems where the dependent variable appears in qualitative form, e.g.,
high or low risk stocks, bankrupt or non-bankrupt …etc. In general, the qualitative form
is to be classified into two different groups.
In the field of business, the discriminant analysis has been initially utilized by
Altman, (1968, 1971), Altman and Sametz, (1977) and Altman and Fleur, (1981) by
building a Z-Score model to discriminate between bankrupt and non-bankrupt firms
using public accounting information. There are a lot of Z models that are used in the
discriminant analysis. Most of these models are derived for the evaluation of company
solvency. In the literature of finance, interests in this model, and the methodology itself,
have continued in the work of Wilcox (1971), Edmister (1972), Deakin (1972), Blum
(1974), Sinkey (1975), Taffler (1978, 1982, 1983, 1984) and Sudarsanam (1981).
Discriminant Function Analysis
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It is sometimes useful to be able to determine functions of the variables X 1 , X 2 ,
…,X p that in some sense separate the m groups as well as is possible. The simplest
approach involves taking a linear combination of the X variables
Z = a 1 X 1 + a 2 X 2 +…+a p X p
For this purpose, in such a way that Z reflects group differences as much as
possible. Groups can be well separated using Z if the mean value changes considerably
from group to group, with the values within a group being fairly constant. One way to
choose the discriminant coefficients a 1 , a 2 ,…, a p in the index is therefore so as to
maximize the F ratio for a one-way analysis of variance. Hence a suitable function for
separating the groups can be defined as the linear combination for which the F ratio is as
large as possible. When this approach is used, it turns out that it may be possible to
determine several linear combinations for separating groups. In general, the number
available is the smaller of
p
and m-1. This is one of the advantages of the linear
discriminant analysis. That is, the reduction of the analysis space dimensionality, i.e.,
from the number of different independent variables X to m-1 dimension (s). As this paper
is concerned with only two groups, consisting of below-median and above-median risk
(systematic and unsystematic), the resulted Z function is only one function (i.e., onedimension analysis).
When the discriminant coefficients are applied to the actual ratio, a basis for
classification into one of the mutually exclusive groupings exists. In this regard, the
discriminant analysis technique has the advantage of considering an entire profile of
characteristics common to the relevant observations (i.e., companies) as well as the
interaction of these characteristics. The linear discriminant analysis has also the
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advantage of yielding a model with a relatively small number of selected measurements,
which has the potential of conveying a great deal of information (Altman, 1968, 1971;
Altman and Sametz, 1977).
VII. Results and Discussion
The Z-Score Models
According to the methodology, three linear discriminating functions with their Z
index (Z model) are derived. This procedure is designed to develop a set of
discriminating functions which can help predict grouping companies in the DJIA30 index
based on the values of sales and cost ratios. The two groups considered are the lowgrowth firms and high-growth firms.
Using a stepwise selection algorithm, certain variables are determined significant
predictors of grouping. The algorithm was carried out three times: for cost ratios, sales
ratios and both of cost and sales ratios all together at a time. The discriminating functions
with p-value  0.05 are statistically significant at the 95% confidence level. Table (1)
shows the discriminating functions with its standardized coefficients for cost, sales and
cost & sales ratios.
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Table 1: The Components of the Discriminant Models for Firms’ Low-Growth and HighGrowth Levels
Components of the Z models
Equation Coefficients2
Cost Ratios
COGS/Preference Dividends
0.564
COGS / Current Assets
-0.488
COGS / Total Liability
-1.334
COGS / Current Liability
1.310
3
Eigenvalue
0.063
% of Variance
100%
Canonical Correlation
0.243
Wilks-Lambda
0.941
2
55.43*
x
n
917
Sales Ratios
Preference Dividend Turnover
-0.51
Current Assets Turnover
0.573
Total Liability Turnover
1.356
Current Liability Turnover
-1.269
Eigenvalue
0.071
% of Variance
100%
Canonical Correlation
0.258
Wilks-Lambda
0.934
2
62.81*
x
n
917
Cost & Sales Ratios
COGS / Fixed Assets
0.68
COGS / Current Liabilities
1.331
Preference Dividend Turnover
0.43
Current Assets Turnover
-0.33
Total Assets Turnover
-1.149
Total Liability Turnover
-0.951
Eigenvalue
0.089
% of Variance
100%
Canonical Correlation
0.286
Wilks-Lambda
0.918
2
78.04*
x
n
917
* Significant at 1% significance level.
2
3
Standardized Canonical Discriminant Function Coefficients.
The variance in a set of variables explained by a factor or component, and denoted by lambda. An
eigenvalue is the sum of squared values in the column of a factor matrix, or
factor loading for variable i on factor k, and m is the number of variables.
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where
aik is the
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Table (1) reports the significant coefficients of cost ratios and sales ratios that
discriminate between low-growth and high-growth firms. With regard to cost
information, the ratios of cost of goods sold (COGS) to current assets and total liability
have negative coefficient. This indicates a negative relationship between firms’ costs and
growth. Nevertheless, the coefficient of COGS/current liability is positive which
indicates that long-term debt affects firms’ growth negatively. This is to imply that
capital structure matters when planning for firm’s growth.
With regard to sales information, firms’ liabilities present an implication
analogous to cost ratios. That is, the negative coefficient of current liabilities turnover
and the positive coefficient of total liabilities turnover imply that sales are affected by
long term debt positively. This also indicates that capital structure matters when planning
for sales growth. The positive coefficient of current assets turnover indicates that sales
growth is affected by current rather than fixed assets. The negative coefficient of
preference dividend turnover is expected since preference dividends are constant, thus not
associated with the increases in sales revenues.
When both of cost ratios and sales ratios are examined all together at a time,
extended implications can be reached. The negative coefficients of current assets and
fixed assets turnover indicate that firms’ growth is negatively associated with assets. This
implies that Gibrat’s law does not hold for large size firms as much as it does not for
small firms as mentioned above in the literature review. The positive coefficient of
COGS/fixed assets indicates that firms’ growth is positively affected by both types of
costs which carries the same implication as the results of the cost ratios do. The positive
coefficient of preference dividend turnover contradicts the negative coefficient reported
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in the sales ratios which could be due to data discrepancies. Nevertheless, the authors
believe that the negative coefficient is more realistic according to the nature of preference
dividends.
Since the two groups are in equal size, the model can be used operationally. 4 The
estimated prior probability ratios were used to calculate the cut-off points on the Z-Scale
as shown in tables 2 and 3. The cut-off points are calculated as Ln(P1/P2), where P1= the
prior probability of low-growth level and P2= the prior probability of high-growth level.
Table 2: The Cut-Off points for the Cost, Sales and Cost-Sales Discriminant Models
Prior Probability
Cut-Off Points
Low
High
Growth
Growth
Cost Ratios, Sales ratios, and Cost &
-0.004
5
Sales ratios
49.9%
50.1%
Relative Contribution of the Model’s Discriminatory Power
The usefulness of the discriminant analysis requires that the final variables profile
is to show the relative contribution of each variable to the total discriminating power of
the Z-Score model and the interaction between them as well. The common approach used
to assess the relative contribution is based on measuring the proportion of the
Mahalanobis D 2 -distance between the centriods of the two constituent groups accounted
for by each variable (Mosteller and Wallace, 1963; Taffler, 1981, 1983).6
4
The prior probability ratio is an estimate of the proportion of companies with a ratios’ profile more
similar to that of group 1 and group 2.
5
Since the two groups are in equal size, the cut-off point is the same for the three types of ratios.


 r jf  r js 

cj 


6
P
j
=
2
 c  r
i 1


r
4
if
and

i

 r is 


if
Where P j = The proportion of the D -distance accounted for by ratio j
r is = The means of the below-median and above-median groups for ratio i respectively.
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Table 3: Relative Contribution of the Models’ Discriminatory Power
Components of the Z model
Cost Ratios
COGS/Preference Dividends
COGS / Current Assets
COGS / Total Liability
COGS / Current Liability
Sales Ratios
Preference Dividend Turnover
Current Assets Turnover
Total Liability Turnover
Current Liability Turnover
Cost & Sales Ratios
COGS / Fixed Assets
Relative Contribution *
15.26%
13.20%
36.09%
35.44%
13.75%
15.45%
36.57%
34.22%
13.96%
27.32%
COGS / Current Liabilities
Preference Dividend Turnover
8.83%
6.77%
23.59%
19.52%
Current Assets Turnover
Total Assets Turnover
Total Liability Turnover
* Mosteller-Wallace measure.
Table 3 shows that:
1.
With regard to the cost ratios, the ratios of Cost of Goods Sold (COGS) to firms’
total and current liabilities contribute the most to model discriminatory power
(36.09% and 35.44% respectively). This outcome shows that when firms are to
adjust costs, firms’ liabilities are to be considered.
2.
With regard to the sales ratios, firms’ total and current liabilities turnover ratios
contribute the most to the model discriminatory power (36.57% and 34.22%
respectively). This outcome carries the same implications that the cost ratios do.
That is, when firms are to adjust sales, firms’ liabilities are to be considered. It is
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worth noting that firms’ liabilities are common factors in both cost and sales
ratios which carry the implication that firms’ growth is quite interrelated by the
capital structure.
3.
When the cost and sales ratios are examined all together at a time, a new
perspective is presented by two ratios which are associated with the highest
relative contribution to the model discriminatory power. The first is the ratio of
COGS/Current liabilities (27.32%) and the second is the total assets turnover
(23.59%). The first ratio carries the implication above mentioned that firms’
growth is affected by the capital structure. The second ratio carries the implication
that firm’s growth is also affected by its size (in terms of total assets). It is also
worth noting that in table 1, the coefficient of total asset turnover is negative and
statistically significant which does not provide support to Gibrat’s law in the case
of large size firms.
The Accuracy-Matrix of the Z model:
In the multi-group case, a measure of success of the discriminant analysis is to be
addressed. This measure results in a classification table or “Accuracy-Matrix,” as shown
in figure 1 (Altman, 1968).
Actual Group Membership
Predicted Group Membership
Below Median
Above Median
Below Median
H
M1
Above Median
H
M2
Figure (1) The Accuracy-Matrix of the discriminant analysis
The actual group membership is equivalent to the priori groupings and the model
attempts to classify correctly these companies. At this stage, the model is basically
explanatory. When new companies are classified, the nature of the model is predictive.
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The H’s (Hits) stand for correct classifications and the M’s (Misses) stand for
misclassification. M 1 represents a Type I error and M 2 represents a Type II error. The
jack-knife test, or Lachenbruch Holdout Test, (Lachenbruch, 1967) is the readily
statistical test for producing the classification table. The final results of the jack-knife test
are shown in Table IV. Therefore, Type I and Type II can be easily observed according
to the “Accuracy-Matrix” shown in figure 1. It is worth noting that tables (5) and (6)
show that Type I errors are all less than type II errors in both cases of systematic and
unsystematic risk. This is considered a support to the relatively high reliability of the
estimated discriminant models.
Table 4: Lachenbruch Holdout Test (jack-knife test), Low-High Growth Firms
Actual Group
No. of cases
Predicted Group Membership
Membership
Cost Ratios7
Low Growth
High Growth
Low Growth
458
263
195
57.4%
42.6%
High Growth
459
164
295
35.7%
64.3%
Sales Ratios8
Low Growth
High Growth
Low Growth
458
271
187
59.2%
40.8%
High Growth
459
180
279
39.2%
60.8%
Cost & Sales Ratios9
Low Growth
High Growth
Low Growth
458
275
183
60%
40%
High Growth
459
183
276
39.9%
60.1%
7
Percent of "grouped" cases correctly classified: 60.9%.
Percent of "grouped" cases correctly classified: 60%.
9
Percent of "grouped" cases correctly classified: 60.1%.
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Table 4 shows that the three discriminant models have relatively the same discriminant
power (60%). This means that the revenue and cost factors are used simultaneously as
drivers of firm growth. Nevertheless, table 3 shows very important implication (which is
also an advantage to the use of discriminant analysis for examining firm’s growth) that
sales ratios’ contribution (58.72%) is relatively higher than cost ratios’ contribution
(41.28%). This result shows that firms’ growth depends much relatively on the combined
effect of sales’ and assets’ growth.
VIII. Conclusion
This study examines the relationship between firms’ growth, sales information and cost
information. The study utilizes the benefits of the discriminant analysis to build up
quantitative models that discriminate between low-growth and high-growth firms. The
results indicate that firms’ growth is affected by cost elements and sales elements
although the latter is relatively more contributable to firm’s growth than the former. The
results also indicate that firm’s growth is negatively associated with assets growth which
implies that Gibrat’s law does not hold for large-size firms. This is true since the sample
include the firms in the DJIA30 index. The results also indicate that growth is also
affected by firm’s capital structure. The results carry implications to corporate managers
that firm’s growth can be achieved by initiating more sales and less cost.
October 18-19th, 2008
Florence, Italy
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8th Global Conference on Business & Economics
ISBN : 978-0-9742114-5-9
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