Journal of Banking and Finance 18 (1994) 505-529.

North-Holland

Corporate distress diagnosis:

Comparisons using linear discriminant
analysis and neural networks (the Italian
experience)*
Edward I. Altman
New York University, Stern School of Business, New York, NY 10012-1126, USA and Centrale
dei Bilanci, Turin, Italy

Giancarlo Marco and Franc0 Varetto

Centrale dei Bilanci, Turin, Italy

Received November 1993, final version received January 1994

This study analyzes the comparison between traditional statistical methodologies for distress
classification and prediction, i.e., linear discriminant (LDA) or logit analyses, with an artificial
intelligence algorithm known as neural networks (NN). Analyzing well over 1,000 healthy,
vulnerable and unsound industrial Italian firms from 1982-1992, this study was carried out at
the Centrale dei Bilanci in Turin, Italy and is now being tested in actual diagnostic situations.
The results are part of a larger effort involving separate models for industrial, retailing/trading
and construction firms.
The results indicate a balanced degree of accuracy and other beneficial characteristics between
LDA and NN. We are particularly careful to point out the problems of the ‘black-box’ NN
systems, including illogical weightings of the indicators and overlitting in the training stage both
of which negatively impacts predictive accuracy. Both types of diagnostic techniques displayed
acceptable, over 90x, classification and holdout sample accuracy and the study concludes that
there certainly should be further studies and tests using the two techniques and suggests a
combined approach for predictive reinforcement.

Key words: Distress diagnosis; Discriminant models; Neural networks; Corporate bankruptcy
risk; Financial ratio analysis
JEL classiJication: G33; C49; C88

Correspondence to: Franc0 Varetto, Centrale dei Bilanci s.r.l., Societa per gli Studi Finanziari,
Corso Vittorio Emanuele II, 93, 10128 Torino, Italy. Tel. 39-l l-562 7366, Fax 39-1 l-562 7490.
*We would like to thank Professor P. Coats of Florida State University for the documents
she kindly supplied us, and Professor L. Saitta of the University of Turin for the numerous and
profitable discussions about neural networks and the applications of artificial intelligence. We
have protited by the comments of Professor Piero Terna and Professor Francesco Borazzo of
the University of Turin. An earlier version of this paper was presented at the ‘International
Seminar on European Financial Statement Data Bases: Methods and Perspectives’, Bressanone,

0378-4266/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved

SSDI 0378-4266(94)00006-O
506 E.I. Altman et al., Corporate distress diagnosis

1. Introduction
The Centrale dei Bilanci (CB) is an organization established in 1983 by the
Banca d’ltalia, the Associazione Bancaria Italiana and over forty leading
banks and special credit institutions in Italy. In 1993, the ‘Sistema Informa-
tivo Economico e Finanziario’ (the Economic and Financial Information
System of the CB which monitors Italian businesses) included approximately
seventy members.’
One of the ‘products’ of the CB is a system designed to provide banks
with a tool to quickly identify companies that are in financial trouble. The
development of this System commenced in 1988 with the creation of an
initial version based on a pair of linear discriminant functions, working
parallel to one another and adapted to the industrial sector. The functions
were estimated from a sample of 213 unsound (distressed) companies
compared to a sample group of the same number of healthy companies; the
estimation was made on the second year prior to the time that the state of
distress was recognized.’ This system correctly classified, in the year
immediately prior to distress, 87.6% of healthy companies and 92.6% cases of
unsound companies. For a description of the features of this initial System,
see Varetto (1990). In 1989, the System was distributed to half of the banks
belonging to CB for actual application in credit analysis at their head offices.
The result of the experiment confirmed the System’s soundness. In practical
terms, automatic diagnosis systems can be used to preselect businesses to
examine more thoroughly, quickly and inexpensively, thereby managing the
financial analyst’s time efficiently. These systems can also be used to check
and monitor the uniformity of the judgements made about businesses by the
various branches of the bank, without replacing credit analyst personnel.
On the basis of the experiments performed and making use of an extended
data base, the CB created a second version of the Diagnostic System that
was completed and distributed to the banks belonging to Centrale’s infor-
mation system during 1991. In the same year, initial tests were conducted
into the use of neural networks for the identification of businesses showing
economic and financial distress.
The aim of this paper is to illustrate the results achieved with neural
networks, comparing them with discriminant analysis results and its appli-

Italy, September l&17, 1993.

‘In addition to the management of data bases with information on the financial statements of
over 37,040 companies collected every year, the Centrale dei Bilanci is actively engaged in
several lines of operation: the development of financial analysis methodology, production of user
software, management education and financial and industrial economic research.
‘At the time that the distressed state is recognized, there is a break in the historical series of
the annual reports in the data base. For companies subject to the law governing bankruptcy and
failure, a period of time passes between the suspension of the availability of balance sheets and
the moment of the final declaration of bankruptcy or composition.
 E.I. Altman et al., Corporate distress diagnosis 507

cations. The next section gives a brief description of the existing version of
the Diagnostic System obtained using what is now recognized as traditional
statistical discriminant analysis methodology. The third section examines the
essential aspects of the neural network approach. The main conclusions that
can be drawn from the experiments in the use of the neural networks (NN)
may be summed up as follows:
a) Neural networks are able to approximate the numeric values of the
scores generated by the discriminant functions even with a different set
of business indicators from the set used by the discriminant functions;
b) Neural networks are able to accurately classify groups of businesses as
to their financial and operating health, with results that are very close
to or, in some cases, even better than those of the discriminant analysis;
c) The use of integrated families of simple networks and networks with a
‘memory’ has shown considerable power and flexibility. Their perfor-
mance has almost always been superior to the performance of single
networks with complex architecture;
d) The long processing time for completing the NN training phase, the
need to carry out a large number of tests to identify the NN structure,
as well as the trap of ‘overlitting’ can considerably limit the use of NNs.
The resulting weights inherent in the system are not transparent and
are sensitive to structural changes;
e) The possibility of deriving an illogical network behavior, in response to
different variations of the input values, constitutes an important
problem from a financial analysis point of view;
f) In the comparison with neural networks, discriminant analysis proves
to be a very effective tool that has the significant advantage for the
financial analyst of making the underlying economic and financial
model transparent and easy to interpret;
g) We recommend that the two systems be used in tandem.
Perhaps the main conclusion of this study is that neural networks are not
a clearly dominant mathematical technique compared to traditional statisti-
cal techniques, such as discriminant analysis. The tendency for recently
published articles on the use of NN approaches in financial distress
classification (a number of references to these studies follows shortly) is that
this ‘new’ technique is clearly superior. We find that a more balanced
conclusion is appropriate, indicating advantages and disadvantages of the
‘black-box’ NN technique.
In addition, our study is one that is being applied and tested within an
operation that has the potential for being implemented in an actual business
and financial context by concerned practitioners. Finally, our samples,
consisting of over 1,000 Italian firms, is by far the largest of any distressed
prediction study to date - including those using discriminant analysis or NN
approaches.
508 E.I. Altman et al., Corporate distress diagnosis

2. Centrale dei Bilanci’s system of diagnostic risk of distress

Distressed firm risk analysis is one of the CB’s permanent projects aimed
at developing analytical methodologies concerning business credit. This
project allows for the periodic updating of the discriminant functions to
maintain or enhance their diagnostic capabilities. The integral parts of the
project are the construction and maintenance of a specific data base of
unsound companies and the development of research on the companies’
dynamics of economic decline leading to distress and bankruptcy.
The System is based on the application of the traditional linear discrimi-
nant analysis methodology on the basis of two samples of businesses
representative of healthy and unsound companies.3 A numerical score is
obtained from the discriminant function that expresses the ‘risk profile’ of the
business.
Unlike the first version of the System, the new release includes special
models each for trading and construction companies as well as the industrial
model developed earlier.4 Work discussed in this study only refers to the
existing model for industrial companies.
The essential points are as follows:
a) The Diagnostic System has been designed and set up to be applied to
the medium and small sized businesses in Italy. For this reason,
companies with sales of more than 100 billion liras (i.e. 60 million U.S.
dollars) have been excluded from the sample. Our tests involve data
covering the period 1985-1992.
b) We have utilized a balanced sample of healthy and unsound companies,
rather than to consider all the collected companies in the tiles of the
CB (around 37,000 companies a year) since our sample is quite large in
and of itself. This methodological line is common to other models of
discriminant analysis.
c) The discriminant models had only modest ex-post accuracies while
using large samples of ‘healthy’ (non-bankrupt) businesses due to the
fact that these companies are broken down into at least three large sub-
sets: ‘outstanding’, ‘normal’ and ‘vulnerable’ companies. And, the
breadth of these categories, just as their features, varies over time. The
discriminant analysis model seems limited in its ability to differentiate
between unsound companies and companies that are ‘live’ but belong
to the vulnerable subset. Certainly, it is far more difficult to discrimi-
nate between two ‘sick’ firm samples (unsound and vulnerable) than
between the clearly healthy vs. unsound firms. Consequently, with the

3For a description of the methodological aspects of discriminant analysis and the main
models available in different countries, see Altman (1993).
4The trading and building sector models are still being tested and will be reported on in a
subsequent publication.
 E.I. Altman et al., Corporate distress diagnosis 509

Fig. 1. Diagnostic system flow. This chart indicates the basic progression of discriminant analysis
models performed within the corporate monitoring system at the Centrale dei Bilanci, Torino,
Italy.

increase in the size and industrial scope of the sample, rates of

recognition decrease because of the increase in the variability of
possible situations. The accuracy does not improve even if use is made
of more sophisticated bayesian-type statistical methodologies.
4 To tackle this problem, it was decided to take another path in the
revised version of the System. The Diagnostic System was broken down
into two sub-models working in sequence. The first model (Fl) was
estimated on samples of 404 unsound companies and 404 healthy
companies: the former were identified from the entire population of
companies collected in the tiles of Centrale dei Bilanci which underwent
(1) some form of bankruptcy proceeding, (2) were wound up in
temporary receivership or (3) had stated they were in dire straits with
regard to their payments to the banks. The sound-firm sample was
obtained from the ‘live’ company tile excluding ‘vulnerable’ businesses
identified through the use of tests on a restricted number of business
ratios over the span of a few years. These ratios are not part of those
included in the discriminant functions. The sample of businesses
‘running normally’ was obtained by matching with similar distressed
companies by size (in terms of net assets), industry and location. The
first model consisted of a nine ratio linear function (Fl) that dis-
tinguished between ‘healthy’ businesses and ‘unsound’ or ‘vulnerable’
businesses (Fig. 1).
4 The second phase of the model (F2) comes into play after Fl has
diagnosed the business to be ‘unsound’. The second discriminant
function was estimated from two balanced samples, again each of 404
businesses, of unsound (the same ones used for the Fl estimation) and
‘vulnerable’ companies. The latter were extracted from among the ‘live’
sample but found to be diagnostically ‘unsound’ by Fl. Both functions
510 E.I. Altman et al., Corporate distress diagnosis

Table 1
Rate of successful recognition (Fl discriminant function)

Healthy Unsound
firms firms
Estimation sample (404 companies
in each group)
Estimation period T-3 90.3% 86.4%
Control period T-l 92.8% 96.5%
Holdout sample (150 companies
in each group) T-l 90.3% 95.1%

Table 2
Rate of successful recognition (F2 discriminant function)

Estimation sample (404 cos.) Vulnerable firms Unsound firms

Estimation period T-3 99.0% 60.1%
Control period T-l 97.8 82.7
Holdout sample
(150 cos.) T-l 96.8 81.0

have been estimated based on ratio values from the annual report of
the third year prior to the distress date.
f) All variables of the Fl and F2 models which contained coefficients with
counter-intuitive signs were eliminated (even if they were statistically
significant). Also, variables with unstable behavior were eliminated and
only those that increased the capacity to classify the unsound compa-
nies as the time prior to distress approached and maintained (or
increased) the capacity to classify healthy businesses were retained.
d Estimations were made using logit as well as discriminant analysis but
no significant progress was made on ex-post classification. Therefore,
we retained the discriminant functions.
h) The discriminatory capacity of the principal function (Fl), on which
most of the experiments with neural networks are compared, is shown
in Table 1.
The percentage of correct ex-post classification improves as distress
approaches; for the unsound companies it goes from 86.4% in T-3 (estima-
tion period) to 96.5% in period T-l. The accuracy of the classification was
checked with a holdout sample of 150 unsound businesses and 150 healthy
ones, obtaining results that were similar to the estimation sample (90% and
95% in period T-l).
The second function, as expected, has a lower discriminant capacity,
especially for the unsound firms. Table 2 lists the F2 function results showing
82.7% correct classification of the unsound firms in the control period (T-l)
and 81.0% in the holdout sample for that group (vs. about 95% in the Fl
Function).
 E.I. Altman et al., Corporate distress diagnosis 511

?!

I compuly
I 65 66 67 88 El

Fig. 2. Top: Function Fl - example of unsound company. Bottom: Function F2 - Example of

unsound company.

To make it easier to interpret the results, the scores of the functions are
represented on graphs where the business under examination is positioned on
the two different reference systems: Fig. 2a is an example of an unsound
business monitored over the last five years of its life (1985-1989). From Fl,
the firm is identified as a distressed company in the fifth year prior to failure.
At this stage, the system does not yet distinguish if the unsound business is
simply vulnerable (with a greater or lesser degree of vulnerability) or if it
belongs to the set of unsound companies. Fig. 2b shows the diagnosis of the
same firm made by F2 and places the business in the uncertain area between
vulnerability and risk of bankruptcy in the first two years of the series and
then signals a rapid decline into the higher risk bankruptcy zone. As can be
seen, the diagnosis of the company is carried out on the basis of a joint
analysis of the two functions with additional reference points supplied by
quartile comparisons with the entire CB data base of comparable companies.
The classificatory space described by Fl has been divided into five zones
on the basis of the distribution of healthy, vulnerable and unsound compa-
nies. These include: (al) high security; (bl) security; (cl) uncertainty between
security and vulnerability; (dl) vulnerability; and (el) intense vulnerability.
Function F2 is calculated as soon as Fl’s score falls in one of the zones (cl),
512 E.I. Altman et al., Corporate distress diagnosis

(dl) or (el): F2 has been split into zones of: (a2) high vulnerability; (b2)
vulnerability; (~2) uncertainty between vulnerability and risk; (d2) risk; and
(e2) high risk of bankruptcy.
Score values separating the different zones constitute the ordinates of the
fixed classification system shown on the graphs5

3. Neural networks

For many years, neural network models have been analyzed both by
academics and practitioners, including those efforts outside the circle of
artificial intelligence experts. 6 It is too early to say whether the use of
experimental N.N. is simply a fad or it will result into something more
permanent. Some aspects of the neural networks, however, do seem promis-
ing in the area of business and finance applications7
The application of the NN approach to company distress prediction,
although relatively new, has seen a number of researchers attempt to
improve upon the traditional discriminant analysis technique. An interesting
procedure by Coats and Fant (1993), used a limited number of financial
ratios to duplicate the ‘going-concern’ determination by accounting auditors.
They utilize the cascade-correlation NN approach (Fahlman and Lebiere,
1992) to duplicate the auditor-expert conclusion on a sample of 94 manufac-
turing and non-manufacturing failed firms and conclude that it clearly
dominates the LDA method in this application.* In addition, studies by

‘The coetlicients of all the functions are protected by secrecy for the purpose of safeguarding
the investments of the CB’s owners made in research, testing and data base creation. This latest
version of the two-function system has been inserted in a procedure on the PC and distributed
to around thirty of the member banks. Actual application in the field is underway and has
already given significant, favorable signs.
6For an introduction to the theory of neural networks and the operating mechanisms, see
Rumelhart and McClelland (1989); Cammarata (1990); Khana (1991); Freeman and Skapura
(1991); and Hertz et al. (1991).
‘In the area of finance, there have been a number of recent attempts to apply NN. Cadden
(1991) has applied neural networks to insolvency analysis by adopting a Boolean transformation
of the financial ratios divided into quartiles; Chung and Tam (1993) have compared the
performance of the neural networks with that of other inductive learning algorithms for
bankruptcy forecasting in the banking industry; Bell et al. (1990) have compared neural
networks with logistic regression for the prediction of bank failures. The networks have also
been assigned to the rating of bonds (Dutta and Shelber, 1992) to the prediction of the progress
of historical series of company data to the selection of investments and to operations on the
financial market (Swales and Yoon, 1992), (Wong et al., 1993), and (Trippi and de Sieno, 1992),
and the recognition of accounting data patterns (Liang et al., 1993). Kryzanowski, Galler and
Wright (1993) applied NN for positive vs. negative common stock return prediction and
Kryzanowski and Galler (1994) have analyzed the financial statements of small businesses using
neural nets, For a partial list of applications in the financial field, see Pau and Gianotti (1990)
and Trippi and Turban (1993).
‘While the Coats and Fant (1993) analysis is of relevance, we must point out that the
auditors’ qualification is itself an inexact and subjective process and as we have shown in an
earlier study (Altman and McGough, 1974), that the discriminant analysis Z-score approach was
far more accurate in predicting the actual bankruptcy of a sample of failed firms than was the
 E.I. Altman et al., Corporate distress diagnosis 513

Fig. 3. General scheme of neural unit.

Karels and Prakash (1987), Odom and Sharda (1990), Ragupathi et al. (1991)
and Rahimian et al. (1992) have all assessed NN for bankruptcy prediction.
Interestingly, at least three of the above studies utilized the -ame five
financial variables found in Altman’s (1968) study.
This paper will explore the basic theory of neural networks but we do not
plan to discuss in detail the reasons that inspired the connectionist approach.
Connectionist processing models (neural networks) consist of a potentially
large number of elementary processing units; every unit is interconnected
with other units and each is able to perform relatively simple calculations.
The network’s processing result derives from their collective behavior rather
than from the specific behavior of a single unit. The links are not rigid but
can be modified through learning processes generated by the network’s
interaction with the outside world or with a set of symbolic signals.
The individual units and the connections linking them can be shown as in
Fig. 3: each unit (i) receives an input (xi) from the outside, or from other
neurons with which it is linked, with an intensity (weight) equal to Wji. The
overall input that the ith neuron receives equals an assumed potential (Pi)
equal to:

Pi=-p4$*Xi-Si
j
where Si represents an excitation threshold value that limits the neuron’s
degree of response to the stimuli received: for example the neurons give a
response signal in the ‘jump-type’ response function only if the total input
arriving from outside and/or other neurons is greater than Si. It is possible to
eliminate the Si threshold and replace it with a dummy input (k) of a value

so-called accountant-expert. Still, the auditing disclaimer report is an unambiguous, although

possibly incorrect, indicator of distress. And, the Coats and Fant sample of distressed firms were
those that discontinued operations after receiving a going-concern qualification.
514 E.I. Altman et al., Corporate distress diagnosis

equal to 1 (X,= 1) and by setting Wkl= -Si, obtaining the general

expression.

Pi=C Wji.Xj,where k is included in Xj

j

The neuron’s response (YJ depends on the transfer of potential (Pi) to the
output function. One of the most widely used functions in the literature and
used in our tests is the logistic function, according to which:

x=----- 1
1 +empi

Generally, the response function determines values between a minimum

and a maximum; in our case x is included between 0 and 1. Output (IJ of
the neuron can be either a total response value of the network (if it is the
final output value) or an input for further neuron units. The network, made
up of many elementary units of the ith type, can have different degrees of
complexity. The simpler networks consist of a single neuron layer (in extreme
cases by a single neuron) each of which is in direct contact with the outside
stimuli i and generates output from the network directly. A slightly more
complicated network has two layers: an intermediate, hidden layer, that
receives stimuli from outside the network, and an output layer that generates
the network’s responses. Networks can be constructed with circuits for feed-
back between neurons from one layer to those of previous levels, just like
self-connecting links.
Considerable limitations of a single-layer network have been shown.
Networks with one layer, in addition to the input layer, can only perform
linear separations of the input space. Two-layer networks can generate
convex geometrical shapes, while networks with at least three layers enable
the input space to be separated into shapes of any configuration (the
complexity of the regions is determined by the number of neurons). There are
no general rules to establish the optimal degree of network complexity.
The crucial aspect of neural networks lies in the fact that the weightings of
the connections are not fixed but can be modified on the basis of a learning
procedure derived from the comparison of the network responses with those
required by actual results. The network, in other words, behaves as an
adaptive dynamic system that reacts to response differences.
The network is given a set of inputs generating a response that is
compared with the response required; the weightings are not changed if the
response obtained corresponds with the response required. If the difference
exceeds a certain tolerance level, revisions are introduced into the weightings
and learning starts again; then a new case is input. The analysis of all the
cases supplied constitutes the maximum extension learning cycle. After the
 E.I. Altman et al., Corporate distress diagnosis 515

interaction of a large number of cycles, the error is reduced to acceptable

levels and, once the holdout set accuracy has been exceeded, the learning
ends and the weightings are locked. The network has achieved a stable
equilibrium configuration that represents ‘its capacity to solve a problem’.
The learning mechanism involves a number of problems; however:
a) The learning stage can be very long (slow learning);
b) The system might not achieve a stable absolute minimum configuration
(optimal error reduction) but might lock on local minimums without
being able to move to the optimum;
c) The system might give rise to oscillating behavior in the learning phase,
i.e., when the minimum point is reached and then exceeded. Hence, it
then returns to the previous point;
d) When the actual situation is significantly different, or changes, com-
pared to the situation implicit in the training examples; it is then
necessary to repeat the learning phase. The same applies when the set
of examples is not representative of the reality of the problem or
concept to be learned;
e) The analysis of the weightings is complex and difficult to interpret.
There is, in other words, little network transparency as far as the
examination of the system’s logic is concerned. This makes it difficult to
identify the causes of the errors or defective responses.
The algorithm determining the network’s learning is of fundamental
importance for the final performance of the network itself.’
Neural networks do not require the pre-specification of a functional form,
nor the adoption of restrictive assumptions about the characteristics of
statistical distributions of the variables and errors of the model. Moreover,
by their nature, NN make it possible to work with imprecise variables and
with changes of the models over time, thus being able to adapt gradually to
the appearance of new cases representing changes in the situation. As noted
earlier, the price to be paid using networks of neurons is their lack of
transparency in the use of the variables within the network connections.
While one is able to identify the explanatory importance of each variable
with the usual estimation techniques, signs of influence on the endogenous
variables and the degree of their mutual correlation with each neuron
remains unclear.
We know many things about how companies can fall into economic
distress, about crisis processes and company decline, but we do not have a
complete theory. One of the ways of tackling this problem in operative terms
is that of using company classification techniques making use of the tools
that statistical methodology supplies. Multiple discriminant analysis is one of

9The method considered here is the well-known Error Back Propagation Algorithm by
Rumelhart et al. (1986).
516 E.I. Altman et al., Corporate distress diagnosis

Fig. 4. Discriminant analysis and multilayer neural network (NN).

the tools most often used and was described earlier in our Fl and F2
functions.
Results obtained appear to be very promising. Linear discriminant analysis
can be considered equivalent to a network made up of a single neuron that
receives signals from the set of indicators and generates an output with a
linear transfer function without transformation, x= Pi. To exploit the
advantages offered by the network of neurons, we have used a three layer
network based on a combination of simple (two-layer) elementary networks
in a ‘cascade’ fashion. Fig. 4 illustrates the differences between Discriminant
Analysis and a multi-layer Neural Network System.
The experimental program is subdivided into four parts:
Part 1: Check the capacity of a neural network to reproduce the numeric
values of the scores obtained using linear discriminant analysis, receiving, as
input, the signals of ratios different from those employed in discriminant
analysis. Note that in this first experiment, the multi-layer network has been
forced to behave linearly, not exploiting its wealth of descriptive potential.
Nonetheless, within this constraint, we can verify the network’s capacity to
approximate the discriminant analysis’ linear functions using a different set of
ratios.
Part 2: Check the capacity of the neural network to separate the samples
between bankrupt and healthy companies. The network’s output unit is not
the value of a score as in the previous section, but simply the binary values 0
(= healthy) and 1 (= unsound). The network’s training stage was carried out
in period T-3 while the test of its correct recognition was done in either
period T-l of the training sample or on an independent sample.
 E.I. Altman et al., Corporate distress diagnosis 517

t-1
t-1

t-1
t-z
t-2

t-2
t-3
t-3

Xi t-n

L:
x2 t-n

XN

Fig. 5. Networks with memory of input.

Part 3: This section considers the change in company performance over time.
One of the problems involved in identifying distressed companies is that of
making the classificatory functions sensitive to the passing of time, and the
changes of the companies’ business patterns. See the work of Theodossiou
(1993) for an analysis of the time series properties of distressed prediction.
An attempt was made to capture these aspects by constructing complex
networks divided into three segments.
The output of the first sub-network summarizes the conclusions about the
economic and financial profile observed in period T-3; these are linked to the
conclusions relating to period T-2. If, during this period, the profile follows a
trend that is consistent (inconsistent) with the trend in the prior period, then
the conclusions come out reinforced (weakened). The same applies to the
pattern of period T-l. An alternative way of tackling the problem of time
pattern analysis lies in using networks with ‘memories’; the simplest network
of this type is that of including among the input data the change in value of
the variables. Fig. 5 illustrates an example of such a network ‘with memory’.
From an economic-logic point of view, it is as if there has been an attempt
to reproduce the reasoning of the financial analyst when he examines a
historical series of business data. The analyst forms an opinion on the state
of business by observing how it has evolved over the entire time span
available.
Part 4: The aim of this section is to check the capacity of networks to
518 E.I. Altman et al., Corporate distress diagnosis

separate the three categories of company: healthy, vulnerable and unsound.

The networks used for this section have an output level comprising two
output neurons: the first distinguishes healthy businesses from the unsound
ones while the second separates the unsound businesses from the vulnerable
and unsound businesses. The two neurons have the same role as the two
functions Fl and F2.
Two types of experiment were carried out for this purpose: in the first case,
network training was carried out over period T-3 while the check of the
capacity for generalization was conducted on period T-l and on an
independent sample. In the second case, networks with memories were used
over the whole three-year monitoring period, while the control sample was
limited to healthy and vulnerable companies extracted from the continuing
company data base.
All the experiments were carried out on the same samples used to line tune
the discriminant functions: an initial sample involved 1212 businesses: 404
each of healthy, unsound and vulnerable firms. A second independent sample
of 453 companies was used, 151 of each type, with data limited to the last
year prior to bankruptcy. A final sample, independent of the other two, was
analyzed comprising 900 healthy and 900 vulnerable companies for three
years of historical series. These were taken from the files of ‘live’ companies.

4. Results

4.1. Healthy vs. unsound firms

The first tests were conducted to estimate the accuracy of the numeric
values of the linear discriminant function. We limited the analysis to
approximating the function that separates the healthy from unsound compa-
nies (Fl) for period T-3. If these approximations can be obtained with a
smaller set of indicators (input signals) than what was used for the estimation
of the discriminant function, it will be a direct check of the neural network’s
capacity for adaptation and simplification. The experiments were conducted
using networks of varying complexity in terms of the number of input
indicators, number of layers and the number of connections.
The best results were obtained with a three-layer network: one initial
hidden layer of ten neurons, a second hidden layer with four neurons and an
output layer consisting of a single neuron. The input comprised ten financial
ratios: four relative to the firms’ financial structure and indebtedness, two to
liquidity, and four representative of company profitability and
internal-financing.
The network neurons are totally interconnected. This means that each
neuron on a layer is connected to all the others on the next level, including
the input signals which are connected to all of the neurons on the first layer.
Training was interrupted after 1000 learning cycles; each of which examined
 E.I. Altman et al., Corporate distress diagnosis 519

Table 3
Distribution of companies by score intervals

Score Score
required calculated
High security 15.2 10.2
Security 34.5 31.2
Uncertainty 11.3 12.0
Vulnerability 23.8 25.4
High level of vulnerability 15.2 15.2
Total 100.0X lOO.Oo/,

808 companies, adjusting the weighting after each cycle. The resulting profile
was extremely close to the desired level.
Another measure of the network results is summarized in Table 3. This
shows the distribution of the categorization of company credit worthiness by
score intervals. The classification differences based on the scores and the
actual categories seem small and concentrated mainly towards positive values
near 1 (best credits).
Results obtained after 1000 learning cycles are quite encouraging and lead
one to believe that if the learning phase lasted longer the error could be
reduced still further. It should be noted that the network built to replicate
the discriminant function is comprised of completely different indicators from
those included in the functions. The latter’s selection required a significant
number of man-hours. In the case of the neural network, machine-hours were
used more, while the selection of indicators, albeit careful and well thought
out, required a tiny fraction of the total time. This is a clear indication of the
network’s capacity for adaptation.

4.2. Multi-layer networks

Networks with varying degrees of complexity were trained using ratios

from period T-3 followed by testing in period T-l from the same sample and
also an independent sample. The most satisfactory results were obtained with
a three-layer network, comprising fifteen neurons in the first hidden layer, six
neurons in the second hidden layer and one neuron in the output layer. The
fully interconnected network is fed with the numeric values of fifteen business
ratios; these are a broader set than the one in the ten-ratio network
described in the previous section. Not having observed substantial benefits
with using a random selection of the cases, we used a sequential ordering of
the observations.
Although the network training used slightly greater than 2000 cycles, the
analysis of the error sequence indicated a typical oscillating phenomenon.”
“Training was conducted with a 0.75 learning rate and null momentum: these values were
obtained on the basis of the results of experiments with alternative learning and momentum
rates.
520 E.I. Altman et al., Corporate distress diagnosis

At the end of the training period, the network was able to recognize correctly
97.7% of healthy and 97.0% of unsound companies. All the other networks
which used a lower degree of complexity, even if trained with a higher
number of cycles, did not achieve the same recognition capability as obtained
using the 15, 4, 1 network. r1 This compares favorably with the recognition
rates obtained by the linear discriminant function Fl in period T-3: 90.3% of
healthy companies and 86.4% of unsound ones.
The network’s identification capability is clearly greater than the discrimi-
nant function’s although it is obtained with a higher number of indicators:
fifteen as opposed to nine. This aspect is important since the network is more
complicated and uses a large number of learning cycles. The results of the
learning, however, behave erratically. There are great and rapid improve-
ments in the capacity to identify the two groups with the first cycles;
nevertheless, as the cycle procedure continues the convergence becomes
slower with frequent oscillations and jumps backward and with deterioration
in the recognition rates that are sometimes significant. As can be seen, the
network had already achieved recognition levels that were not far off the
final results, especially in the healthy group, in the earlier cycles. The
unsound firm errors were reduced considerably as the number of cycles
increased until the last 560 cycles, when the classification accuracy became
erratic.
This network, trained in period T-3, showed a lower recognition capacity
than the one in the training period using period T-l; period T-l’s identifica-
tion error was 10.6% for healthy businesses and 5.2% for unsound firms.
Compare these rates of error with those obtained with the discriminant
functions: 7.2% for healthy and 3.5% for unsound companies (Table 1).
This neural network shows a lower capacity for generalization than the
traditional discriminant function’s This conclusion is reinforced by the
results obtained on the independent samples of 302 companies for period
T-l: rates of error are 15.9% for the healthy and 9.5% for the unsound
companies as opposed to the 9.7% and 4.9%, respectively, obtained with the
discriminant functions. l2
The simpler network’s results are more modest than the ones obtained
from traditional discriminant analysis, but show a greater capacity for

“Experiments were also conducted, among others, using Cascade-Correlation but we did not
obtain superior results; for the methodological aspects relating to Cascade-Correlation see
Fahlman and Lebiere (1992).
‘*Increased generalization was achieved with the simpler, 10, 4, 1 type networks, fed with the
ten ratios used in the first section of experiments. After two thousand learning cycles, this
network showed a recognition capacity of 93.3% for healthy companies and 84.7% for unsound
companies in period T-3, far lower than results obtained with the more complex 15, 6, 1
network. Nonetheless, the simpler network was able to limit the errors on the T-l sample to
8.2% and 3.7x, respectively, and, on the independent sample, to 14.6% and 6.8% for the two
samples of firms.
 E.1. Altman et al., Corporate distress diagnosis 521

Table 4
Comparison of recognition rates: NN vs. LDA

Linear
discriminant
Neural network function (Fl)
Sample size = 404
in each group Healthy Unsound Healthy Unsound
Estimation period T-3 89.4% 86.2% 90.3% 86.4%
Control period T-l 91.8% 95.3% 92.8% 96.5%

generalization than the more complex networks. This confirms what others
have shown, i.e., the network judged to be most effective at the end of the
learning cycle might not be as suitable with other sets of independent cases.
The network is the victim of a phenomenon known as ‘overfitting’. We
encountered a similar phenomenon when we observed the holdout sample
accuracy of quadratic discriminant functions vs. the less complex linear
function, see Altman et al. (1977).

4.3. Multi-layer networks with discriminant function ratios

The results obtained in the previous section use networks fed with ratios
different from the ones used in discriminant functions. The reason for this
choice is the need to estimate the classification capacity of the networks
using a standard information base (ratios) such as are normally available in
financial analysis reports published by the CB. In a related test, nine of the
11 Fl discriminant function’s indicators are utilized with networks of
differing complexity. The intention was to check the networks’ capacity to
reproduce the ‘knowledge’ built into the discriminant functions and convert
it into knowledge distributed over the neural connections.
The best result was obtained with a 9, 5, 1 network after 4030 learning
cycles with a 0.75 learning rate and 0.30 momentum.i3 Table 4 shows the
rates of recognition of businesses in the T-3 period (network estimation) and
period T-l (control period). The results are not dissimilar, although slightly
lower, from those obtained using the discriminant function. It is not,
however, certain that the formalization of the knowledge built into the
network is totally equivalent to the knowledge of the linear function since
companies that the network recognized incorrectly were, in part, different
from those incorrectly recognized by the discriminant function. Moreover,
while the discriminant function always behaves in the same way when the
values of the exogenous variables vary, with the use of the network we have
seen behavior that is not always consistent when the input changes. We will

13These values were obtained from the results using alternative parameter experiments.
522 E.I. Altman et al., Corporate distress diagnosis

postpone this discussion until the next section where it can be treated more
thoroughly.

4.4. Simple network connections

Neural networks can have difficulty when tackling particularly complex

problems. In our case, the complexity derives from the nature of the problem
and from the wide range of observations. While the wealth of data makes it
possible to construct a model that is general and robust, it also tends to
make the training of the network more difficult. Complex networks with
numerous inputs and neurons are perhaps better able to classify a more
heterogeneous sample of firms but has the disadvantage of making the time
required (and the expense) for training sometimes prohibitive. Moreover,
these networks tend to adopt oscillating or non-convergent behavior as well
as often being the victim of the overtitting trap.
One methodology for tackling these problems might be breaking down the
total network into simpler networks connected to each other. We carried out
experiments along these lines starting with the generation of elementary
networks which were then connected to each other in a second level network.
Fig. 7 shows how the networks used in this section were organized; we
illustrate this with three of the eight simple structure networks (with one
hidden and one output layer apiece). Every elementary network (e.g.,
leverage) is fed with a number of ratios that are representative of that
characteristic.
The second level network coordinates the results of the eight elementary
networks in order to generate the system’s final response. That is, it is
‘trained’ to combine the conclusions reached by the elementary networks.
This is equivalent to a multivariate discriminant or logit analysis with the
same potential benefits over a univariate structure.
The results obtained are shown in Table 5. As expected, the classification
accuracy of the elementary networks differs greatly from network to network
and generally it is not very high. Furthermore, it does not always increase
when passing from the estimation period (T-3) to the control period (T-l).
The second level network, however, which generates the system’s overall
responses, performs very well, with correct classification of healthy companies
of 98.3% and unsound companies of 92.5% in the estimation period (T-3).
This result is considerably better than that achieved through discriminant
analysis (cf. Table 1). In addition, the system of networks applied to the
control period (T-l) gives outstanding results in this case, too, with 92.8% of
healthy companies and 94.5% of unsound companies classified correctly.
On the whole, the test of the simple network system’s capacity for
generalization on the independent sample of 302 companies in period T-l is
also good. In the face of significant reduction in first level elementary rates of
 Table 5
Connection of simple neural networks with one output

Variable types analyzed Number % Correct classification

of
learning Sample of 808 companies Holdout sample
cycles Period T-3 Period T-l Of 302 Companies
Estimate Control Period T-l
Healthy Unsound Healthy Unsound Healthy Unsound
Specialized elementary NN
1) Leverage; asset/liab. strut. 4640 76.98 82.21 73.76 91.04 76.80 90.48
2) Ability to bear fin. debt 3800 74.50 79.20 69.06 92.04 35.20 97.28
3) Liquidity 4950 69.31 84.96 71.29 90.80 36.00 97.96
4) Profitability, internal tin. 4770 87.87 85.71 84.90 92.29 56.80 96.60
5) Profit accumulation 4050 81.44 75.94 88.61 71.14 59.20 85.03
6) Ability to bear cost of debt 5000 75.25 87.97 76.73 95.77 7.20 99.32
7) General efficiency 1260 62.38 60.40 65.10 69.65 63.20 57.82
8) Trade indebtedness 900 59.90 69.92 60.15 79.60 73.60 66.67
Second-level NN 2850 98.27 92.48 92.82 94.53 93.60 89.12
NN trained at T-3 with learning rate=0.75 and momentum =0.25. The revision of the weight is done for each company; in each cycle
there are 808 revisions.
524 E.I. Altman et al., Corporate distress diagnosis

identification, second level networks generate correct classification rates of

93.6% for the healthy companies and 89.1% for unsound companies.
Compared with results obtained with discriminant functions, there is a
significant drop in the number of unsound companies identified correctly
(- 6 percent) but not enough to cancel out the effectiveness of the system.
These results are encouraging for the use of neural networks. Note that the
system’s classification was obtained with a small effort on the part of the
analyst and by using annual report ratios that are not particularly complex.

4.5. Analysis of simple network systems - some concerns

As mentioned earlier, a problem that arises in the use of neural networks

concerns the low level of intelligibility of the knowledge base spread over the
network and built into the weighting of the connections. We carried out an
analysis of elementary and second level networks to try to better understand
how they work. Distributed knowledge mapping, see Hinton, et al., (1986),
implicit in the weighting values of the network, can be studied from various
points of view including the identification of the significance assumed by the
various neurons and the analysis of the network’s behavior when input
conditions change.
The significance of the neurons can be identified either by examining the
weightings matrix, in the simplest cases, or by studying the role that
individual neurons play in determining the output.r4 By modifying the initial
weightings and repeating the learning process, the weightings matrix is
modified which might also change the role of the various neurons.
Generally speaking, the behavior of a network can be studied on the basis
of the derivatives (or elasticities) of the output compared with the individual
inputs. For the analysis in the case where more than one input is changed at
the same time, it is possible to refer to the total differential of the output
compared with the inputs.
From a mathematical point of view, the neural network is a non-linear
system. Its input/output derivatives depend on the input value configuration
vector. Therefore, the network’s capacity to react to input changes is not
always the same but strictly depends on the starting position. This factor
makes the identification of the individual contributions of the inputs to the
formation of the output complex and uncertain.
Even in its simplicity, the second-level network shows peculiar behavior
patterns. We examined the values of the partial derivatives between the

l4The activation patterns that the hidden-layer neurons assume in response to different input
value configurations can be observed to try to understand how the network has formed
responses. An alternative method consists of causing voluntary ‘damages’ inside the network by
deactivating certain of its connections or removing entire groups or by altering the size of its
values.
 E.I. Altmnn et al., Corporate distress diagnosis 525

output and the individual inputs in the case of different starting contigu-
rations. The calculation of the partial derivatives showed significant depen-
dence on the base conditions, with sudden changes of sign. This feature of
the networks is particularly awkward for the financial analyst because the
behavior of the network may be unpredictable and contrary to business
logic. For example, for a business to be considered unsound by the network,
it only needs to have a very modest general efficiency, a relatively high
leverage and an uncertain ability to bear financial indebtedness while having
outstanding ratings in the other inputs. If the level of liquidity is worsened
under this profile, the network shows an improvement in the output and,
under some conditions, the company goes from being unsound to healthy,
thus altering the initial conclusions! Such behavior does not occur if the
profitability is worsened but it reappears in the case of increased commercial
indebtedness.

4.6. Interconnection of simple networks with memories

The next experiment made use of the logic of networks with memories on
the inputs (see Fig. 5 for a general description of such a structure). The
inputs of this type of network include the entire three-year historical series of
the indicators used. The network is trained to consider all the data available
about the company at the same time. This is like a financial analyst
examining the historical time series of financial statements.
The correct recognition rates of healthy and unsound companies is high,
even in several elementary networks, rising to over 99% in the second-level
network. The overall accuracy of the interconnected system of elementary
networks with memories commits errors of 4 healthy companies (out of 404)
and 1 unsound company (out of 404).”
We did analyze the overall functioning of the system on simple, intercon-
nected networks with memories. We found, in the second level network, the
same non-acceptable behavioral problems already identified above with a
frequent inversion of the output value when the inputs are uniformly
modified either individually or in limited subsets.

4.7. Multi-group analysis

The last set of experiments is aimed at the generation of a two-output

network for the simultaneous, not sequential, separation of healthy, vulner-
able and unsound companies. This test was very severe because of the

15The price paid for these performance levels has been the high number of elementary first-
level network learning cycles. Consider that with 5000 cycles there are over four million changes
made to the weightings via the backward propagation algorithm.
526 E.I. Altman et al., Corporate distress diagnosis

difficulty in identifying in a single solution the characteristics separating the

three groups of business. The best results were achieved with a network using
families of simple NNs with memory consisting of three layers: one hidden
layer with fifteen neurons, a second hidden layer with twelve neurons and
one output layer with two neurons. This is somewhat analogous to the three
group simultaneous distress s & L analysis of Altman (1977). Since these
results are still experimental and not germane to our overall conclusions, we
refer the reader to a lengthier working paper, Varetto and Marco (1993).

5. Conclusion

In the light of the experiments carried out, neural networks are a very
interesting tool and have great potential capacities that undoubtedly make
them attractive for application to the field of business classification. The
networks assessed on our samples have shown sign&ant capacities for
recognizing the health of companies, with results that are, in many cases,
near or superior to the results obtained through discriminant analysis. The
results of the two-output networks trained to simultaneously recognize the
three types of company performance: healthy, vulnerable and unsound, also
proved to be very interesting. Nonetheless, taking into account the results
obtained in the control periods and in the holdout samples, discriminant
analysis was deemed to be better, on the whole, than the networks trained in
our experiments.
The greatest problem concerns the existence of non-acceptable types of
behavior in the network. These are intrinsic to the non-linear nature of the
mathematical model underlying the network, combining a large number of
variables several times over in a complex fashion. These behavior patterns
are characteristic of networks of any complexity that have at least two
inputs.
The extent and frequency of illogical types of behavior (in the judgement
of the financial analyst) grow with the increase in the complexity of the
network architecture. Only extremely simple networks limit the probability of
meeting these unacceptable results. The construction of ultra simplified
networks cannot be a solution, however, because the problem is only
delayed. It does in fact crop again as a result of the need to coordinate
simple networks with others of higher level.
The problem of understanding these types of behavior and how to remedy
them is not an easy one to solve. As well as using real examples, it would be
possible, for example, to train the network with artificial cases constructed to
represent other possible combinations. Given the high number of artificial
cases required, the network’s capacity for analyzing real cases could be
totally distorted if errors are committed at this stage.
On the whole, linear discriminant analysis compares rather well when
 E.Z. Altman et al., Corporate distress diagnosis 521

compared to neural networks. The fine-tuning of the discriminant function

does take longer, but the greater estimation speed makes it possible to carry
out careful tuning at relatively low cost. Furthermore, the linear form, albeit
with the limitations of its ability to perform well, ensures consistent behavior
for any type of variable. This makes it possible to interpret the model’s
operating logic on the basis of the coefficients.
In the neural network, it is not possible to ascertain whether a particular
variable comes into the interpretative model with the wrong sign and to
eliminate or replace it, as can be accomplished with traditional econometric
and statistical tools. With discriminant analysis, it is possible to learn what
the most important variables are for explaining the differences between the
companies in the sample. In the network tine-tuning process, the length of
the training process, the rate of mean error decline, and the results on the
recognition rates are the keys for estimating the soundness of the variables,
ex-post.
Our conclusions on the use of neural networks are not straightforward
and they recognize the undoubted advantages such networks have. A path
we intend to adopt in the future is to integrate networks and discriminant
functions, applying the former to less clear and more complex problems of
classification in which the flexibility of networks and their capacity for
structuring into simple, integrated families could prove to be very useful. The
key determinant as to whether neural networks, in conjunction or not with
traditional classification procedures, will be integrated into practitioner
decisions is the accuracy, logic, and understandability of the process and its
components. It must be emphasized again, however, that we have found
illogical behavior patterns in all of the many NN systems tried in our
research. These results have not been emphasized in previous applications of
NN systems to business related problems.
Neural networks have shown enough promising features to provide an
incentive for more thorough and creative testing. Analysts’ fascination with
artificial intelligence models will, no doubt, motivate continued firm related
investigations.

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