# Research Paper: Methods Used for the Prediction of Bankruptcy

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This section of the paper discusses the various models that are used for bankruptcy prediction.

1. Discriminant analysis

Discriminant analysis operates by trying to develop a linear combination that consists of two or more variables and has the ability to discriminate between two priori defined grouped in an effective and efficient manner. This discrimination is achieved through the deployment of a statistical rule that maximizes the variance that exists between the groups relative to the variance that exists within the group. (Karamzadeh, 2013)

The above mentioned relationship is expressed in the form of ratio of between the group and within the group variance. (Back & Laitinen et al., 1996) The linear combination is derived by the discriminate analysis on the basis of an equation that takes the following form:

Z = W1X1+ W2X2+...+WnXn

(Back & Laitinen et al., 1996)

Where,

Z represents the discriminant score

Wi (i=1, 2, ..., n) represents the discriminant weights

Xi (i=1, 2, ..., n ) represents independent variables or the financial ratios (Back & Laitinen et al., 1996)

On the basis of the above mentioned equation each firm gets an individual discriminate scores. This score is then compared to a cut off value. This comparison determines to which group a particular firm belongs. (Back & Laitinen et al., 1996)

If the variables that are included in various groups follow a multivariate normal distribution and all of the groups have equal covariance matrices, then in such conditions a discriminate analysis performs in a very effective and accurate manner. (Back & Laitinen et al., 1996)

It has, however, been determined through empirical testing that generally the failing firms violate the condition of normal distribution. In addition to that, the condition of equal covariance matrices of the group is also often violated. (Back & Laitinen et al., 1996)

In addition to that, if the stepwise procedure is employed then the multicollinearity among the independent variables also acts as one of the major and serious problems. It has, however, been identified through empirical testing that the assumptions in relation to normal distribution of variables were not weakening the classification capability of the model, instead they were weakening the prediction ability of the model. (Back & Laitinen et al., 1996)

The two basic methods that have been deployed for the determination of the discriminate models are:

Simultaneous (Direct) Method: This method is based on construction of models on theoretical grounds. It ensures that the model is ex-ante defined and then deployed in discriminant analysis. (Back & Laitinen et al., 1996)

The Stepwise Method: The stepwise method goes for the construction of the models through the selection of a subset of variables, which ca be used to produce a good discrimination model. These variables are selected through forward selection, backward elimination, or stepwise selection. (Back & Laitinen et al., 1996)

A number of other modelling strategies are also suggested by various professionals, an example of which could be the one suggested by Hosmer and Lemeshow (1989). It has been identified by a number of studies that the major weakness of the stepwise method is that it overlooks the economic importance of the variables and put increased emphasis of the statistical grounds of the variables. (Back & Laitinen et al., 1996)

The modelling strategy that is suggested by Hosmer and Lemeshow, however, is less mechanical in nature. This is because it allows the opinions of the analysts, in relation to the variables, to be included in the model. (Back & Laitinen et al., 1996)

2. Logit analysis

Logistic regression analysis has also been used to determine the relationship that exists between binary or ordinal response probability variables and explanatory variables. (Klobucnik & Sievers, 2013)

By the method of maximum likelihood, the Logit analysis fits the method for logistic regression between binary or ordinal response probability variables and explanatory variables. Ohlson (1980) was among one of the first users of Logit analysis in financial situations. (Klobucnik & Sievers, 2013)

Similar to the discriminate analysis, this method also weights various independent variables and assign a Z score, which represents the probability of failure, to each of the organization that is included in a sample. (Klobucnik & Sievers, 2013)

The basic advantage of this method is that, unlike discriminate analysis, this method does not assume that all the normality of distribution and equality of covariance matrices among the groups. (Klobucnik & Sievers, 2013) In order to predict predicting a bankruptcy nonlinear effects are incorporates and cumulative logistic functions are used by this method, i.e.,

(Klobucnik & Sievers, 2013)

Logistic regression is restricted to the prediction of discrete sets, unlike linear regression. Therefore, the dependent variables that are included in logistic regression are restricted. These variables are restricted to a discrete number set. (Klobucnik & Sievers, 2013)

Another major disadvantage of this method is the underlying assumption of linearity between the variables that is deployed by the method. Furthermore, this method restricted to only between subject designs and does not address the within subject designs. (Klobucnik & Sievers, 2013)

3. Neural networks

A large number of processing elements, which are known as neurons, and connections between them are two important components of an artificial neural network. A function is implemented by this method. Through this function a set of input values, represented as x, are mapped to a set of output values, represented as y. The function, therefore, is represented as y = f (x). (Makeeva & Neretina et al., 2012)

The neural network aims at find out the best approximation of the function. Through the deployment of the weights, which are associated with each of the neurons, the above mention approximation is coded into the neurons of the network. (Makeeva & Neretina et al., 2012)

A genetic algorithm is another method to select the variables that would be a part of the network. The Darwinian evolution is simulated by a genetic algorithm. A population of chromosomes is maintained by a genetic algorithm. Here, chromosome can be defined as the solution of the problem that we wish to solve. (Makeeva & Neretina et al., 2012)

In a genetic algorithm context, chromosomes are often referred to as strings. These strings consist of a number of genes. The genes can take some value of numbers, which is referred to as alleles. The genetic algorithm terms that are used to represent genes and alleles are features and values. (Makeeva & Neretina et al., 2012)

In order to determine the goodness of a string, a fitness value is associated with each of the strings. The fitness value of a string is determined through a fitness function, which can be defined as measure of the goodness or profit that is required to be maximized. The three basic operates that play an important getting goods results in a genetic algorithm, are reproduction, crossover, and mutation. (Makeeva & Neretina et al., 2012) These operators are discussed in the following section

Reproduction: This process operates by copying the strings of one generation to the next generation. Strings that have a higher fitness value have a greater probability of getting copied into the next generation. A number of different methods are used to determine which strings would move to the next generation. (Makeeva & Neretina et al., 2012)

Crossover: In this method parts of two strings are combined with each other. The method aims at combining good parts of two strings and producing an even better string. (Makeeva & Neretina et al., 2012)

Mutation: Under this method an existing element of the string takes a new value. By adopting this procedure, the operator aims at introducing a new value in the string. In addition to that, it also prevents the loss of an existing value of the string. (Makeeva & Neretina et al., 2012)

4. Distance to Default

The Distance to Default score model of Morningstar is a slightly modified structural version of the model that was developed by Black, Scholes and Merton and. This model was commercialized by KMV, which is now referred to as Moody's KMV. (Miller, 2009)

This method has an underlying assumption that the equities of the organization can be converted into an option. This option has a strike price that is equal to the book value of the liabilities of the organization and the market price of the option is equal to the market value of the assets of the organization. (Miller, 2009)

This implies that the worth of company is nothing. That is when the market value of the assets of the organization fall below the book value of the liabilities of the organization then the organization defaults. (Miller, 2009)

The cash accounting values that are specifically examined in a bankruptcy or default scenario are not looked into by this method, therefore, this method is less intuitive in nature as compared to the Z score method. (Miller, 2009)

In addition, the financial covenants that are the true determinants of the fact that whether a distressed organization would default on its obligations or not are not examined…
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