prediction analysis of bankruptcy risk

Constanta Maritime University Annals Year XI, Vol.14

PREDICTION ANALYSIS OF BANKRUPTCY RISK USING BAYESIAN NETWORKS

1CRACIUN MIHAELA-DACIANA, 2BUCERZAN DOMINIC, 3RATIU CRINA

1,2“Aurel Vlaicu” University of Arad, 3Daramec srl Arad, Romania

ABSTRACT

The Bayesian probability, is widely misunderstood by the general public, as well as some economists. On the other hand, bankruptcy risk can be estimated in the static and dynamic analysis of the financial balance that outlines the former performance of the enterprise. A global evaluation of the enterprises future becomes interesting for the management of the enterprise and especially for its business partners: banks, clients, capital investors. Therefore, in this paper we mould the Anghel Prediction Model for bankruptcy risk using the Bayesian probability. To this purpose, we use Bayesian Networks (BN) and the AgenaRisk Tool. The result of this mould is a solution of bankruptcy risk prediction using BN. Keywords: Bayesian probability, Bayesian Network (BN), bankruptcy risk prediction, AgenaRisk Tool, Anghel Prediction Model 1. INTRODUCTION

A Bayesian Network (BN) is a way of describing

the relationships between causes and effects, and is made up of nodes and arcs. The collection of nodes and arcs is referred to as the graph or topology of the BN. In addition, in a BN each node has an associated probability table, called the Node Probability Table (NPT). The nodes represent variables. The arcs in a BN represent causal or influential relationships between variables. The key feature of BN is that they enable us to model and reason about uncertainty. The NPT for any node gives the conditional probability of each possible outcome given each combination of outcomes for its parent nodes. Usually, there are several ways of determining the probabilities in any of the tables. Alternatively, if no such statistical data is available we may have to rely on subjective probabilities entered by experts. A key feature of BN is that we are able to accommodate both subjective probabilities and probabilities based on objective data, as specified in [1].

Having entered the probabilities we can now use Bayesian probability to do various types of analysis. Bayesian probability is all about revising probabilities in the light of actual observations of events. When we enter evidence and use it to update the probabilities in this way, we call it propagation. In theory we can enter any number of observations anywhere in the BN and use propagation to update the marginal probabilities of all the unobserved variables. This can yield some exceptionally powerful analyses that are simply not possible using other types of reasoning and classical statistical analysis methods, as you see in [5]. BN offer the following benefits, subject founded in [2]: - Explicitly model causal factors: this key benefit is in

stark contrast to classical statistics whereby prediction models are normally developed by purely data-driven approaches.

- Reason from effect to cause and vice versa: A BN will update the probability distributions for every unknown variable whenever an observation is

entered into any node. So entering an observation in an “effect” node will result in back propagation, i.e. revised probability distributions for the “cause” nodes and vice versa. Such backward reasoning of uncertainty is not possible in other approaches.

- Overturn previous beliefs in the light of new evidence: The notion of explaining away evidence is one example of this.

- Make predictions with incomplete data: There is no need to enter observations about all the “inputs”, as is expected in most traditional modelling techniques. The model produces revised probability distributions for all the unknown variables when any new observations (as few or as many as you have) are entered. If no observation is entered then the model simply assumes the prior distribution.

- Combine diverse types of evidence including both subjective beliefs and objective data. A BN is “agnostic” about the type of data in any variable and about the way the NPTs are defined.

- Arrive at decisions based on visible auditable reasoning: Unlike blackbox modelling techniques (including classical regression models and neural networks) there are no “hidden” variables and the inference mechanism is based on a long-established theorem (Bayes). This range of benefits, together with the explicit

quantification of uncertainty and ability to communicate arguments easily and effectively, makes BN a powerful solution for all types of risk assessment.

The first working applications of BN (during the period 1988-1995) tended to focus on classical diagnostic problems, primarily in medicine and fault diagnosis. Intelligence Group at Aalborg University produces the MUNIN system. Companies such as Microsoft and Hewlett-Packard have used the early BN for fault diagnosis, and in particular printer fault diagnosis. There have also been numerous uses of BN in military applications, for example the TRACS system for predicting reliability of land vehicles. Another high-stakes application domain where BN have been used

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Constanta Maritime University Annals Year XI, Vol.14 extensively by commercial organizations is fault prediction; subject is founded in [4].

Because of historical limitation even Bayesian statisticians have shunned BN for problems that involve continuous variables and complex stochastic models. Instead they have used tools like the WinBUGS software package, which are based on intensive sampling algorithms collectively known as Markov Chain Monte Carlo (MCMC) methods. Fortunately, there have been some recent breakthroughs in algorithms for hybrid BN. Building on the work of Koslov and Koller, Neil have developed and implemented a dynamic discretisation algorithm which works efficiently for a large class of continuous distributions.

Users of AgenaRisk Tool, which implements this algorithm, can simply define continuous nodes by their range and distribution. Without any of the complexities associated with the MCMC approach, they can achieve results of matching or greater accuracy for many classes of models, especially for models that include discrete variables, as specified in [6]. On a wider scale, there is considerable research into how to model extremely large problems involving hundreds of data points, with many variables, over long periods of time, or involving complex sequences of variables and data. A number of extensions to BN beyond the classical inference algorithms are being used for this purpose, including: Relational BN, Statistical parameter learning, Sensitivity analysis, Safety and reliability modelling, Operational risk in finance, Recommendation engines and information retrieval.

2. THE I. ANGHEL MODEL IN BANKCRUPTCY RISK PREDICTION

Anghel has developed a model based on

discriminatory analysis, starting from a sample of 276 enterprises, grouped into non-bankrupt (60%) and bankrupt (40%), and belonging to a number of 12

industries of the national economy. The analysis covered the period 1994 -1998 and has initially used a number of 20 economic -financial indices.

After the selection stage, four financial rates have been established for the development of the score function: - X1 - earning after taxes / incomes; - X2 - Cash Flow / total assets; - X3 - liability / total assets; - X4 - liability/ sales * 360 All the above rates have been aggregated in the following score function: A = 5.667 + 6.3718 * X1 + 5.3932 * X2 – 5.1427 * X3 – 0.0105 * X4, subject is founded in [3].

Varying within the values established for this function, enterprises are included in one of the following three situations: - When A < 0, bankruptcy/failure situation; - When 0 ≤ A ≤ 2.05, uncertainty situation demanding prudence; - When A > 2.05, a good financial situation.

The analysis of the previously presented models has revealed a certain facility in detecting bankruptcy in time.

Subject to bankruptcy risk prediction was been treated with interest over the years. The use of BN for BP was study, as you see in [7], by Lili Sun and Prakash P. Shenoy.

3. THE ANGHEL PREDICTION MODEL (PM) FOR BANKRUPTCY RISK (BR) EXPLAINED USING A BAYESIAN NETWORK

Accepting the Bayes’ Theorem and the accuracy of

the AgenaRisk software it is possible to explain the Anghel PM for BR without exposing the mathematical details. The vehicle for doing this is a visual model called Bayesian Network (BN) as shown in the Figure 1.

Figure 1 – BN showing causal structure

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In this structure we have four types of nodes: sample, probability, result and assumption nodes. The sample nodes represent the probability that Xi is faulty (i = 1, 2, 3, 4). The probability nodes are Xi faults in number of trials, where number=20, 10, 15, 25. The result nodes are the following: Mediate Node1, Mediate Node2 and A Z score. The Hypothesis node is the assumption node. As mentioned in the previous section, the Anghel PM for BR is based on the function score A = 5.667+6.3718*X1+5.3932*X2-5.1427*X3-0.0105*X4. In this case we are handling nodes with multiple parents. The initially model we built, was that all four sample nodes were parents for the result node. In this case the calculation was very slowly and difficult. So we introduce the two Mediate Nodes and so we reduce the number of parents’ node and of the calculation time, too.

Next we explain how we built the nodes. The sample nodes are simulation nodes, with

continuous interval type. The lower bound is 0.0 and the upper bound is 1.0. The NPT is a Uniform Expression with lower bound 0 and upper bound 1. The graph types associated to this node are Histogram.

The probability nodes are simulation nodes, with integer interval type. The lower bound is 1 and the upper

bound is 9. The NPT is a Binomial Expression with 20, 10, 15 and 25 trials and the probability of success given by the parents’ node probability p_Xi_faulty. The graph types associated to this node are Histogram.

The result nodes are simulation nodes, too. They divide in two categories. The Mediate Nodes and the A Z score node. The types of Mediate Nodes are continuous interval with values between -10 and 50. The NPT is an arithmetic expression 6.3718*p_ X1_faulty +5.3932* p_ X2_faulty. The graph types associated to this node are Histogram. The type of A Z score node is continuous interval with values between -20 and 100. The NPT is an arithmetical expression MN1+MN2. The graph type associated to this node is Histogram.

The assumption node Hypothesis is a simulation node, with Boolean type. The state options are customised, with positive Outcome “Good financial situation” and the Negative Outcome “Bankruptcy / failure situation”. The NPT is a comparison expression: if(zscore<2.05,"Bankruptcy / failure situation", "Good financial situation"). The graph type associated to this node is Histogram.

The statistic attached to the main risk graph is shown in Figure 2. In this case there is no observation.

Figure 2 - Complete Hypothesis Testing model

In case that observations appear we can attached to each probability node a certain number of trials. Let`s consider the following observation: X1 decrease the trials from 20 to 2, X2 decrease the

trials from 10 to 1, X3 decrease the trials from 15 to 1, X4 decrease the trials from 25 to 2. The result is shown in Figure 3.

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Figure 3 - Risk graph of Hypothesis after evidence has been entered

The risk map for this model has attached the following risk table, as shown in Figure 4.

Figure 4 – Risk table of Hypothesis generated after evidence has been entered

4. THE ANGHEL PREDICTION MODEL (PM) FOR BANKRUPTCY RISK (BR) USING HYPOTHESIS TESTING WITH EXPERT JUDGMENT

The structure of the risk map described at the previously section, will be changed. We add a new node at the top of the risk map named Prior Type. This node is a labelled type with Uniform and Beta label value. The NPT is a comparison expression with value 0. The graph type associated to this node is Histogram. (see Figure 5).

Figure 5 - Hypothesis Testing with Expert Judgment

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In this case the sample nodes are modified only by the NPT. The NPT Editing Mode changes into Partitioned Expression. We specify the distribution as follows: for nodes X1 and X4 the uniform distribution is given by the function Uniform(0,1) and the beta distribution is given by the function Beta(1,9,0.0,1.0). So

we obtain the chance of failure of 1 to 10. For nodes X2 and X3 the uniform distribution is given by the function Uniform(0,1) and the beta distribution is given by the function Beta(2,8,0.0,1.0). So we obtain the chance of failure of 1 to 5.

Figure 6 - Complete Hypothesis Testing model with Expert Judgment

Next, we define two scenarios one type Uniform and the second type Beta. The two scenarios will be correlated with the risk map entering the observation that

the Prior Type is Uniform in the scenario that we have named Uniform and Beta in the scenario that we have named Beta (see Figure 7).

Figure 7 - Results of hypothesis test with two different prior assumptions

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Comparing the last two figures we observe the difference of the results.

Combining Data and Prior Assumptions We change the probability nodes so that we

decrease 1/5 the trials: from 20 to 4, from 10 to 2, from 15 to 3 and from 25 to 5.

For the both scenarios, Uniform and Beta, we will introduce values as follow: for the probability nodes X1 and X4 Uniform and Beta receive the value 1, respectively the nodes X2 and X3 receive the value 0 for Uniform and Beta. The results are shown in Figure 8.

Figure 8 - Results of hypothesis test after entering sparse sample data

Figure 9 – Risk table of Hypothesis Testing with Expert

Judgment after entering sparse sample data

Changing the Simulation Settings We delete the Uniform scenario and we remove

from the probability nodes X1 and X4 the value 1 for Beta. After we calculate we obtain the mean 0.40 for X1 and 0.50 for X4 and the variance value 0.46 for X1 and 0.62 for X4.

Next, we modify the properties for the defined model. The maxim number of iteration defined in the simulation settings are 25. We work with 5 iterations. Running the calculation we will obtain different values for both probability nodes.

5. CONCLUSIONS We have shown that, using BN and AgenaRisk

Tool, it is possible to show all of the implication and results of a complex Bayesian argument without requiring and understanding of the underlying theory of mathematics. Economists can use the obtained analysis to predict the bankruptcy risk using Bayesian probability. 6. REFERENCES [1] JENSEN FINN V, GRAVEN-NIELSEN THOMAS - Bayesian Networks and Decision Graphs, Springer 2002 [2] POURRET OLIVIER, NAIMS PATRICK, MARCOT BRUCE – Bayesian Networks - A Practical Guide to Applications, John Wily & Sons Ltd, 2008 [3] ANGHEL ION – Falimentul – radiografie şi predicţie, Ed. Economică, Bucureşti, 2002 [4] NEAPOLITAN RICHARD E. – Learning Bayesian Networks, Prentice Hall Series in Artificial Intelligence [5] HECKERMANN DAVID – A Tutorial on Learning with Bayesian Network, March 1995 [6] Agena 2007, Press Release, http://www.agenarisk.com/agenarisk/case_13.shtml[7] SUN LILI, SHENOY PRAKASH P. – Using Bayesian Networks for Bankruptcy Prediction – Some Methodological Issues, European Journal of Operational Research, 2007

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prediction analysis of bankruptcy risk

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