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BUSINESS INSOLVENCY PREDICTION MODELS: A COMPARATIVE ANALYSIS
OF MULTIVARIATE DISCRIMINANT ANALYSIS AND LOGISTIC REGRESSION
Olimjonova Munisa Nuriddin kizi
Tashkent state university of economics
Accounting faculty
Abstract:
This study looks at predictive approaches that use logistic regression and multivariate
discriminant analysis (MDA) to evaluate a company's insolvency. The Z-Score model developed
by Edward Altman and William Beaver is the main focus of this historical review of bankruptcy
prediction techniques. The use of a binary logistic regression model to assess the importance of
individual financial ratios in predicting a company's bankruptcy risk is also covered in the paper.
According to the results, MDA and logistic regression provide strong frameworks for
comprehending the financial health of businesses, with applications in risk management and
decision-making. To preserve accuracy and relevance in a changing economic environment, the
study does stress the necessity of constant validation and adaptation.
Key word:
Insolvency prediction models, multivariate discriminant analysis, logistic regression,
risk mitigation, univariate analysis, bankruptcy, ratios, Altman.
Introduction.
Over the years, an interest arose among various finance professionals to analyze the risk of
business (corporate) insolvency. Therefore, many experts developed diverse methods to predict
bankruptcy of businesses.
The predictive capacity was carried out at two levels (Jiménez Cardoso et al., 2002, pp. 257-260):
a)
Univariate analysis, based on the use of only one financial ratio for the elaboration of
business bankruptcy prediction. The pioneering author resulting from this study was William
Beaver (1966), who considered that through the use of
cash-flow/total debts
and
net profit/total
assets ratios
, the bankruptcy situation of the company could be identified, noting that the former
of these provides greater predictive value. A company would be in “bankruptcy” when it
presents negative cash balances or the non-payment of obligations or dividends.
b)
Multivariate analysis, corresponding to the integration of more than one independent
variable (several financial ratios) with the objective of classifying companies according to their
situation.
These comprehensive models aim to provide a more robust assessment of a company's financial
stability by considering various financial dimensions simultaneously, thus offering a more
holistic view of potential risks. They move beyond single indicators to capture the complex
interplay of factors contributing to insolvency.
Literature review.
In this case, the precursor was Edward Altman (1968), who based his linear model on a
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multivariate approach, that is, he considered the simultaneous influence of two or more variables
through a Multiple Discriminant Analysis (MDA) on a series of financial variables.
According to Jiménez Cardoso et al. (2002), "the classification of a company's situation and
future is usually not done based on the value presented by a single financial ratio, which is why,
in practice, it makes no sense to speak of a strictly univariate analysis" (p. 257). For this reason,
this work will focus primarily on the multivariate analysis used in Edward Altman's model.
The choice has also been due to the fact that, as will be seen below, these ratios are calculated
through any company's accounting data, which allows for simple calculation because cases will
not be disregarded due to lack of financial information. It has also been one of the most applied
prediction models both in finance and in accounting research, which is why it has been
considered very useful for the study.
Likewise, the choice has also been made for the logistic regression or Logit model, which allows
obtaining the significance of the ratios, used with the aim of identifying which of them, is the
most relevant and requires more attention compared to the others. In addition, it will be observed
if it has the same prediction capacity as the previous model.
In 1968, Edward I. Altman, a professor at New York University, wanted to develop a predictive
method based on the Multiple Discriminant Analysis (MDA) statistical technique, given that it
had previously been applied favorably in financial problems, including consumer credit
evaluation and investment classification.
According to Marchese et al. (2006), "multivariate analysis is a set of statistical and
mathematical methods designed to describe and interpret data that come from the observation of
various statistical variables studied jointly" (p. 3).
Logistic regression analysis is a nonlinear econometric model used to predict the outcome of
categorical variables (variables that can contain a limited number of classes) based on independent
variables. Generally speaking, the probabilities obtained in the function should be between 0 and 1.
There are two modes of logistic regression analysis (López-Roldán, P., and Fachelli, S., 2015):
Methodology.
This study employs a quantitative methodology to predict business insolvency, primarily utilizing
two distinct statistical approaches: Multivariate Discriminant Analysis (MDA) and Logistic
Regression. MDA is used to classify companies into solvent and insolvent groups by finding a
linear combination of financial ratios that best discriminates between them. Complementing this, a
binary Logistic Regression model is applied to estimate the probability of insolvency, allowing for
the assessment of individual financial ratios' significance and their impact on a company's
likelihood of bankruptcy, providing a comprehensive framework for risk assessment.
Results and discussion.
1)
Altman's Z-Score Model
As mentioned, Marchese et al. define multivariate analysis as a collection of mathematical and
statistical techniques intended to characterize and interpret data derived from the observation of
many statistical variables under joint study. Therefore, this study allows examining the
combination of one or two predefined groups. By virtue of this, "discriminant analysis is a
multivariate analysis technique whose objective is to find the linear combination of independent
variables that best differentiates the groups".
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Therefore, Altman first selected two groups, which are made up of companies that are in
bankruptcy (insolvent) and those that are not in bankruptcy (solvent). The prediction of each of
them is established in the dependent variable of function (1). Once the grouping has been carried
out, the financial data belonging to each of the companies that are part of the study are collected.
Subsequently, following Marchese et al. (2006), thanks to the MDA model, it allows determining
a set of coefficients that best discriminate the set of groups, which are weighted and a linear
equation is constructed.
Finally, he made use of a one-dimensional discriminatory function, which is given by the
following equation:
Z=V
1
X
1
+V
2
X
2
+...+V
n
X
n
(1)
where, as stated in the book by Edward I. Altman (1968), Z is the discriminant variable used to
classify the company (global indicator); V
1
,...,V
p
are the weighting coefficients of the
discriminatory variables; and X
1
,...,X
p
represent the independent variables corresponding to
the company's financial ratios.
Once the MDA had been studied, the first questions that arose for the development of its model
were the following (Altman, E. I., 1968):
1.
What ratios are the most important for predicting such insolvency?
2.
What weights should be assigned to each assigned proportion?
3.
How should the obtained result be objectively established, once the previous questions
have been resolved?
For his study, he based it on the classification of a total of 66 US manufacturing companies listed
on the stock exchange, of which 33 belonged to insolvent companies, and the rest were healthy
companies.
From the obtained sample, Altman selected a total of 22 financial ratios, but finally, he relied on
the popularity of the literature and the potential for the study, choosing five of them, of which
four variables correspond to the balance sheet and one of them to the income statement. These
variables could be classified into five distinct categories: liquidity, profitability, leverage,
solvency, and activity (Altman, E. I., 1968). And, for each of them, he established five different
coefficients, which he combined using a linear discriminant function.
After the companies had been evaluated with their respective coefficients and ratios, the results
obtained in said discriminant equations (denominated "Z") would be expressed through a series
of areas, which are shown in Figure 6.
Table 1. Classification of Z-Score Areas.
Danger zone
“Grey” zone
Secure zone
Insolvent
Uncertain
Solvent
Z < A
A < Z < B
Z > B
In a broader sense, each of these areas is defined in the following way:
Danger Zone:
The probability of presenting insolvency problems in the future is very
high, so it can be considered that there will be an imminent bankruptcy.
1
Author’s elaboration, based on Altman, E. I. (1968)
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"Grey" Zone:
There is a high probability that companies may enter into an insolvency
situation in the next two years.
Secure Zone:
The company does not present insolvency problems in the future.
As represented in Table 2, Altman established five indicators in 1968 for those companies listed
on the stock exchange, thus obtaining the equation called Z-Score. Subsequently, in 1983,
Altman re-estimated the original model by changing both the parameters and the ratios, to adapt
it to those companies not listed on the stock exchange, as the previously used financial ratio was
not compatible with these societies. He called this new discriminant equation Z
1
-Score.
It is worth mentioning that Altman also considered the so-called Z
2
-Score based on the Z
1
-Score,
but excluding the ratio X
5
since it could distort the analysis, being the variable of sales a variable
of easy manipulation by companies.
Table 2. Ratios employed by Altman in the Z-score model
X
n
Z-Score (1968)
Z
1
-Score (1983)
X
1
Working Capital / Total Assets
Working Capital / Total Assets
X
2
Retained Earnings / Total Assets
Retained Earnings / Total Assets
X
3
EBIT / Total Assets
EBIT / Total Assets
X
4
Market Value of Equity / Total
Liabilities
Book Value of Equity / Total Liabilities
X
5
Sales / Total Assets
Sales / Total Assets
It is worth mentioning that Altman also considered the so-called Z
2
-Score based on the Z
1
-Score,
but excluding the X
5
ratio since it could distort the analysis, with the sales item being a variable that
is easily manipulated by companies.
Next, we explain each of these ratios and the reasons for their use by Edward I. Altman (1968):
X
1
: This ratio measures a company's liquidity through the relationship between working capital and
total assets. The numerator, working capital, is defined as the difference between current assets and
current liabilities. The denominator contains total assets and allows for comparison with other
elements of the balance sheet or income statement (Marchese et al., 2016). Therefore, it can be said
that the greater the working capital relative to total assets, the higher the level of liquidity. Altman,
of the three different ratios that measure liquidity, considered X
1
to be one of the most essential
ratios for its calculation.
X
2
: This ratio shows the company's cumulative profitability, since its definition uses the year-end
profit, thereby determining whether the company is profitable. As Altman noted in his book, he
considered young companies to be more discriminatory compared to others, because they have
recently entered the market and have not had enough time to accumulate profits. Therefore, these
companies have a high probability of facing bankruptcy. However, older companies may also incur
losses for several consecutive years, which will result in a negative result in this ratio.
X
3
: This is a measure of a company's profitability. Altman considered it an appropriate ratio for
studying corporate failure. Therefore, companies with negative earnings before interest and taxes
will have a lower ratio and a greater likelihood of bankruptcy than those with profits.
2
Author’s elaboration. Source: Altman, E. I., & Hotchkiss, E. (2006)
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X
4
: This is the variable that relates equity to total debt. This ratio refers to the company's financial
structure; in other words, it measures solvency. In this case, the methodology applied to listed
companies uses the market value of capital, which means it takes into account the number of shares
the company holds and the market price. However, for unlisted companies, the book value of capital
is chosen, which gives the total value of equity.
X
5
: Represents the company's ability to generate sales from its assets; in other words, it measures
the company's efficiency. In these cases, Altman estimated the following models with their
corresponding classifications according to the area to which the results obtained in the model
pertain, which are reflected in Table 3 below.
Table 3. Altman, EI, and Hotchkiss, E. (2006)
Manufacturing Companies Listed on the Stock Exchange
Area
Z=1.2X
1
+1.4X
2
+3.3X
3
+0.6X
4
+9.99X
5
(2)
Danger Zone: Z<1.81
Gray Zone: 1.81<Z<2.99
Safe Zone: Z>2.99
Manufacturing Companies NOT Listed on the Stock
Exchange
Area
Z
1
=0.717X
1
+0.847X
2
+3.107X
3
+0.42X
4
+0.998X
5
(3)
Danger Zone: Z<1.23
Gray Zone: 1.23<Z<2.9
Safe Zone: Z>2.9
Z
2
=6.56X
1
+3.26X
2
+6.72X
3
+1.05X
4
(4)
Danger Zone: Z<1.1
Gray Zone: 1.1<Z<2.6
Safe Zone: Z>2.6
Finally, it is worth mentioning that, according to the studies conducted, reliability was demonstrated
between 80 and 90 percent. Edward I. Altman (1968) added that "this method was not probabilistic,
but rather descriptive-comparative." In other words, this model should be used as a warning tool for
a company's potential insolvency, as it identifies its orientation toward one group or another.
Subsequently, thanks to the development of the Altman model, a similar version was created, but
this time based primarily on the detection of insolvency in Spanish SMEs3, which was carried out
by Amat et al. in 2016. To do so, they again applied linear discriminant analysis to obtain a
combination of ratios that better discriminates, which is why it was necessary to readjust the
coefficients and variables used by Altman.
In his analysis, he studied a group of 2,000 companies that received short-term loans during 2008,
of which 144 presented repayment difficulties at the time in which the loans were granted (Amat et
al., 2016). Finally, according to the results obtained for the development of the new methodology,
through the following Table 4 we can observe equation (5) obtained by Amat et al., and the
interpretation given to the results.
Table 4. Equation of the Amat et al. model and its meaning
Equation
Area
Z
∗
=−3.9+1.28X
1
+6.1X
2
+6.5X
3
+4.8X
4
(5)
Financial Problems: Z<0
Financial Health: Z>0
where the ratios used in the equation are reflected in the following Table 5.
3
Author's elaboration. Source: Altman, EI, and Hotchkiss, E. (2006)
4
Source: Amat et al. (2016)
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Table 5. Ratios used by Amat et al.
X
n
Amat et al. (2016)
X
1
Current Assets / Current Liabilities
X
2
Own Funds / Total Liabilities
X
3
Net Profit / Total Assets
X
4
Net Profit / Own Funds
X
1
: Similarly to the numerator of the Altman X1 ratio, it corresponds to the liquidity ratio. This
refers to the working capital, so it analyzes the short-term financial stability of the company.
According to Oriol Amat (2008), the lower this ratio, the more likely the company is to default.
However, if it is too high, it would obtain idle current assets, which is the same as losing
profitability.
X
2
: This ratio is very similar to the debt ratio and divides equity by debt. It establishes the
company's financial independence vis-à-vis third parties. That is, it analyzes whether the company
is capable of meeting all its obligations and debts. The more autonomous the company, the more its
capital structure will be comprised of its own resources, so the company will be less dependent on
its creditors.
X
3
: This is a company profitability indicator that examines net profits relative to total assets. In
other words, it measures whether the company is using its assets efficiently.
X
4
: Determines the financial profitability, or Return on Equity (ROE), which indicates the net return
obtained from the company's equity after deducting interest and taxes. This ratio also indicates the
return earned by shareholders, based on the value created by the company. The higher this ratio, the
better the company's performance, as its equity will be more profitable.
2)
Logistic regression model
To examine the relationship between the dependent binary outcome and the selected explanatory
variables, a logistic regression model was employed. Logistic regression is a widely used statistical
method for modeling binary response variables, where the outcome takes on values of 0 or 1. This
model estimates the probability that a given event occurs as a function of one or more independent
variables, using the logit transformation of the probability.
First, there is binary logistic regression, where the dependent variable is dummy or dichotomous,
that is, it can only take two values (probability of an event occurring or not).
Second, multinomial logistic regression, which allows for qualitative explanations in a more general
case. Depending on the categorical variable selected, a distinction can be made between nominal
polytomous variables (e.g., the choice of a product brand) and ordinal polytomous variables (e.g.,
the level of satisfaction in a restaurant).
As will be seen later, this paper will focus solely on the study of a binary logistic regression, in
which the dichotomous dependent variable will represent the value 1 if the company is filing for
bankruptcy and 0 if it is currently in bankruptcy. Therefore, as previously discussed, the
dichotomous dependent variable can be explained by one or more explanatory or independent
variables, where the classification of each case can be predicted based on the group to which they
correspond. This result is therefore possible thanks to the weights or coefficients associated with
each of these variables in order to determine their significance in the study.
All necessary formulas of logistic regression model are given below in Table 5. The specification of
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the logit or binary logistic regression model is as follows
(formula 6)
, where Y is the dichotomous
dependent variable, indicating the occurrence of the event (Y=1) or its non-occurrence (Y=0); β
1
,
β
2
, …, β
k
are the population coefficients or parameters; and finally X
1
, X
2
, …, Xk are the
explanatory variables.
Table 5. Logistic Regression Model Formulas.
In the simple binary logistic regression model, as the dummy variable collects the probabilities
between 0 and 1, the probability of the event occurring (Y=1) is called “P
i
” and the probability of it
not occurring (Y=0) is 1 minus the previous probability
(formula 7)
.
This logistic function is represented by a logistic or sigmoidal curvature, which has an S shape and
is reflected as a growth curve since the Y axis increases monotonically with respect to the X axis
(formula 8)
.
As a consequence, as Pedro López-Roldán and Sandra Fachelli (2015) comment, “the interpretation
associated with each of the coefficients of logistic regression differs from that of linear regression,
because the coefficient is not the measure of how much “y” will vary in response to a variation in
one unit of “x”, but rather the change produced by a variation of one unit of “x” in the natural
logarithm “ln” of the quotient of probabilities of the two events, the so-called logit transformation”
(p. 18)
(formulas 9, 10)
The population parameters (βi) are determined through the relationship between the probability of
the event occurring with the probability of it not occurring, called odds ratio, and whose expression
is given as (
formula 11).
Therefore, the interpretation of the signs obtained in the coefficients is similar to that of linear
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models: if the beta is positive, the probability of the event occurring increases, and if it is negative,
it decreases. If the coefficient is equal to zero, there will be no change in the odds ratio.
When evaluating the marginal effect of each of the explanatory variables on the probability of the
event occurring, it will be necessary to do so for certain values of the explanatory variables in
(formula 11).
Following Maddala, G. (1983) and assuming that the observations are independent, the likelihood
function that captures the probability of obtaining the observed sample is the product of the
marginal probabilities of the binary variables (yi). Therefore, this function is as
(formula 12)
The population parameters (βi) are estimated using the maximum likelihood method, which consists
of obtaining those parameter values that, given the sample, maximize ln L
(formula 13)
. The
maximum likelihood estimator is obtained by solving the system of equations
(formula 14)
This system of equations
(formula 14)
for deriving the estimators is a nonlinear system that must
be solved by numerical optimization methods, i.e., an iterative procedure must be used to solve
them. The information matrix is the second derivative of the function
(formula 15).
The iterative procedure starts with an initial value of , called 0, and in each iteration the values
( ) and ( ) are calculated to obtain a new estimate of the parameters as follows
(formula 16).
In this equation ( ) is positive definite at each iteration step s. Therefore, the iterative procedure
will converge to a maximum of the function regardless of the starting value. If the final convergent
estimates are denoted by
̂ , then the asymptotic covariance matrix
( ̂
) is estimated by [
(
̂ )]−1 .
The maximum likelihood estimator is consistent and normally distributed in large samples.
Therefore, individual significance hypothesis tests are performed using the asymptotically
distributed t statistic N (0,1) under the true null hypothesis.
Conclusion.
This study has carefully examined models for predicting business insolvency, highlighting the
analytical strength of logistic regression and multivariate discriminant analysis (MDA). This work
demonstrates their significant effectiveness in identifying corporate financial health and predicting
distress by carefully examining their theoretical underpinnings and real-world applications,
including influential works like Edward Altman's framework and its modifications. Stakeholders
benefit greatly from the analytical insights obtained, which offer tools for risk reduction, well-
informed lending choices, and proactive strategic reactions. Although these models offer strong
analytical frameworks that have been shown to be reliable, their long-term usefulness requires
constant improvement through validation against new data and safe adjustment to changing
business complexity, regulatory changes, and economic landscapes. uch sustained scholarly
engagement is crucial for enhancing their predictive accuracy, analytical relevance, and long-term
efficacy in a dynamic global economy.
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