Int. J. Business Performance Management, Vol. X, No. Y, XXXX

Risk insolvency predictive model maximum expected utility Daria Marassi* Eu-Ra Europe Rating S.p.A., L.go Don Francesco Bonifacio 1, Trieste 34125, Italy E-mail: daria.marassi@eu-ra.com *Corresponding author

Valetino Pediroda Dipartimento di Ingegneria Meccanica, Università degli Studi di Trieste, via Valerio 10, Trieste 34127, Italy E-mail: mciprian@units.it E-mail: pediroda@units.it Abstract: This paper presents a new approach to develop the probability of default for private firms. This work provides a global perspective regarding the credit risk prediction, starting from the work of the Basel Committee on Banking Supervision, with a deep study of the more predictive variables for default prediction and, finally, building a new mathematical model based on machine learning. The used method is called Maximum Expected Utility (MEU) and represents the most promising methodology for the default prediction. The main idea is to use the interaction between variables to improve the final model efficiency. The development model has been tested on complex analytical function (in which the classical models fault) and finally has been developed to assess the distress of industrial companies according to Basel II guidelines. The evidence was related to Italian industrial enterprises and took into consideration, the situation of the Italian economy both from a micro and macro perspective. Keywords: rating; probability of default; credit risk prediction; Basel II; methodology for the default prediction; risk insolvency model; default model; predictive model; interaction between variables; predicted default variables; self organizing maps; statistical analysis; credit rating; risk model development; cumulative accuracy profile. Reference to this paper should be made as follows: Marassi, D. and Pediroda, V. (XXXX) ‘Risk insolvency predictive model maximum expected utility’, Int. J. Business Performance Management, Vol. X, No. Y, pp.XXX–XXX. Biographical notes: Daria Marassi is a Senior Database Analyst and Statistical Analyst at Eu-Ra Europe Rating Spa, an Italian credit rating agency that makes use of the most modern analysis and techniques in the field of Company Finance and Financial Engineering. She is engaged, in particular, into research of innovative methodologies for credit rating analysis and for risk management. Copyright © XXXX Inderscience Enterprises Ltd.

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D. Marassi and V. Pediroda Valentino Pediroda is a Research Assistant in the Department of Mechanical Engineering at the University of Trieste. His research field is mainly related to the development of numerical algorithm for the design, as Multi Objective Optimisation Algorithm (based on Game Theory), numerical model to approximate complex physics phenomena and Multi Criteria Decision-Making. Recently, his algorithms have been used with success in Finance problems, as default prediction. He is the Author of more than 50 international papers.

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Introduction: default risk â€“ problem presentation

The evaluation problem of credit rating, within the economic sciences, concerns different aspects. The key problem, that is, analysed in this work, defines the concept of credit risk like the risk of insolvency (or default risk), within the investigation of the company risk. The enterprise risk is canonically divided into three kinds of risk: operating risk, financial risk and default risk. The risk cannot be consider separately. In fact, it is obvious that the operating risk and the financial risk are in the medium period aggravated each other, and they carry together to default, or to insolvency for a company. In economic science, the definition of the insolvency risk is not univocal. In general, it is concerned with the risk that the debtor will not be able to or, in some cases, does not want to comply with the obligations assumed in the adverse parties comparisons, in terms of repayment of both interests and capital share (e.g. the obligations). This work, it has reached the consideration of the risk of insolvency like that risk of a company not to succeed to make forehead to the own obligations within a prefixed technical time. For this reason, the definition of default risk taken into consideration is wide and prudential and the same covers a temporal period of an accounting exercise. In fact, the constructed model, able to study the health state of companies, is based on the annual balance sheet data that is issued once a year. This meaning of the insolvency risk will carry, in continuation of this work, to consider all those enterprises that, not only are in failed state of liquidation, but also that they introduce it marks them of elevated vulnerability, that is introduces to the inside of the own accounts considerable pathological situations for which, the solvency financial institution is thought insufficient or null. Four factors are the reason for a strong credit risk management processes evolution: 1

the theoretical and applied development of the credit risk management theories

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the change in action of the vigilance norm in terms of financial market

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the change of the credit risk management theories from the profile of the portfolio management to the processes of selection and pricing characterises them

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transformation of the characters of the banks loan activity.

The introduction to the credit rating logic represents the key moment of such evolution. We remember that the credit rating normally is presented as the classification of a buyer or a specific financial operation in predefined risk classes, normally expressed in letters or number.

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The quantification of the insolvency risk

According to the Basel Committee on Banking Supervision the ratings represent a risk assessment of loss consequent to the counterparty insolvency, based on quantitative and qualitative information. Classification of every class are drawn like characterised from specific and measurable losses. A rating system comprises the elements that play a role in the process, includes the concepts of loss and the relative measures, the methodology in order to estimate the risk, the responsibilities of the operating ones and the production functions of rating use. In synthesis, the rating is a discreet measure of the default probability or loss probability. Due to the adopted rating definition, of the methodologies of the rating determination and of the risk level expression, it must be asserted that the methodological possible alternatives in order to assign the rating are extremely variegate. The goal of this work is to reach the construction of a model that comes from the class of the family of non-linear regression models, with which we define the level of credit risk. That choice is done according to the thought that a possible way for the solution of the risk insolvency assessment problem in a company falls under the statistical-economic sciences.

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The model for risk study already exists in literature

The use of numerical methodologies to evaluate the default risk of a company is a well-known problem in Finance and Statistical sciences. The modern era of commercial default prediction really begins with the work of Beaver and Altman in the late 1960s. Beaver (1967), with univariate analysis, found that a number of indicators could discriminate between matched samples of failed and non-failed firms for as long as five years prior to failure. The most interesting example for the definition of a numerical model for the default prediction is the Zscores method of Altman (1968) which, starting from the financial ratios of the companies, calculates a multivariate discriminant model to divide the companies into two categories: bankrupt or non-bankrupt. The discriminant function, of the formula Z = V1X1 + V2X2 + " + VnXn, transforms the individual variable values to a single discriminant score or z value, which is then used to classify the object where:

X1 , X 2 ,â€Ś, X n = discriminant coefficients V1 , V2 ,â€Ś, Vn = independent variables The advantage of this methodology is the simplicity of the analytical form, with direct correlations between the input parameters (financial ratios) and the discriminant score. A common criticism of this method is that it depends on financial environment (Country or economic sector). One of the advantages of the Zscores methodology is the demonstration of the necessity to have a numerical method to derive the probability of default. Starting from the Altman ideas, new statistical theories have been developed, based on the conditional

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probability or machine learning. From a mathematical point of view, the problem of determining a model for the default probability becomes: Let x denote our vector of explanatory variables (financial ratios), let the random variable Y ∈ {0, 1} indicate default (Y = 1) or survival (Y = 0) over d some fixed time interval from the observation of x ∈ R . We seek the conditional probability measure p (y|x) = Prob(Y = 1|x).

To solve this problem, many different numerical methodologies have been developed: Neural Networks (Rojas, 1996), Support Vector Machines (Cortez and Vapnik, 1995) and the classic Logit and Probit models (Hastie et al., 2002). These methods tend to optimise a likelihood function, in order to find the best matching between the numerical model and the known database. The advantage is that the discriminant coefficients of the model are not user defined, but they are automatically determined by the optimisation algorithm. One new important method, based on machine learning, is the Maximum Expected Utility (MEU) Principle developed at the Standard & Poor’s Risk Solutions Group (Friedman and Huang, 2003; Friedman and Sandow, 2003). This model has a clear economic interpretation and it measures the performance in economic terms. The main idea is to seek a probability measure that maximises the out-of-sample expected utility of an investor who chooses his investment strategy so as to maximise his expected utility under the model he believes to be efficient. The Group of authors demonstrate how this new numerical method outperforms the Logit and Probit methodology, since it takes into consideration the interactions between the financial ratios in order to obtain a better approximation of the real probability of default. In this paper, the authors present an iterative model to determine the most default predictive financial ratios for avoiding the arbitrariness in the input variables choice. Another default risk predictor is RiskCalcTM (RiskCalcTM for private, 2000) used by Moody’s. This model is based on the Merton model (Merton, 1973) in which the firm’s future asset value has a probability distribution characterised by its expected value and standard deviation. The method defines a ‘distance to default’, which is the distance in standard deviation between the expected value of assets and the value of liabilities at time T. The greater the value of the firm and the smaller its volatility, then the lower is the probability of default. The authors demonstrate how the Merton methodology outperforms the Zscores model using the well-known CAP index of accuracy, reaching a value of 0.54 when Zscores reaches 0.45. Interesting new emerging techniques are the methodologies connected to the Multi Criteria Decision-Making (MCDM) algorithm; as observed in Spronk et al. (2005), Zopounidis and Doumpos (2002) and Bernè et al. (2006) this new rating approach could outperform the existing methodologies based on machine learning, in order to decrease the importance of the database and to improve the use of the financial analyst knowledge. In this paper the MEU principle has been developed and linked with new interesting statistical methods to improve the efficiency in rating evaluation. The new model has been tested on an Italian economical sector: wholesale trade.

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The theoretical bases of the risk insolvency model

The statistical problem to determine the default risk of an enterprise could be defined in mathematical terms as the search of the conditional probability p(y|x) = Prob(Y = 1|x),

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where x is the free problem variable (in our case the balance sheet ratios that are predictive with the default, chosen by the statistical methodologies as Default Frequency and Self Organising Maps). Y is the random variable, which represents the default probability, Y âˆˆ {0, 1} indicates default (Y = 1) or survival (Y = 0) over some fixed time interval from the observation. As above described, that problem could be solved using the numerical methodologies obtained by the theory of Statistical Learning. The more used methodologies are the logistic regression (Fitting Logistic Regression Model) and more recently Neural Networks and Support Vector Machines. These numerical methodologies, even if they represent optimal models to solve the exanimate problem, suffer from not having a theoretical financial base. To avoid that gap, a new numerical model has been recently presented: MEU. This methodology, developed inside the research group of Standard & Poorâ€™s, takes origin from an economical interpretation and measures the quality of the model in financial terms, obtained therefore, a clear improvement in comparison to the methodologies purely statistical. A interesting point of the methodology described below is that the maximisation of the numerical model quality is not simply mono-objective, but multiobjective: at the same time (according to Pareto) will be attempted the consistency with the data (known dataset, normally called training set) and with the probability measure to which the investor believes before knowing the data. The relative weight between the two objects became parameterised by a parameter Îą defined by the customer.

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Developed risk insolvency predictive model MEU

The predictive model for the insolvency risk study called MEU analyses the Italian company to determine the risk degree in terms of default probability. The output that it generates is the probability for a company to be classified in an insolvent company category, based on the evaluation of its balance sheet data. The definition of default risk, for a company, used in this work is the risk for an enterprise not to succeed to make forehead to the own obligations within a prefixed technical time, in our case an accounting exercise. The rating analysed, has the goal to investigate the debtor credit merit, in rigorous and transparent way, through appraisal of one factor series concerning risk activity and characteristics of the same debtor. The analysis of the default probability (like defined above) leads to determine the rating class for every analysed company. The fundamental objective in a default model study is to reach the definition of the insolvency risk.

5.1 The development process of MEU model The diagram in Figure 1 introduces the fundamental steps followed for the development of model MEU described below. The data on which has been lead the development and the validation (testing) of MEU model are the annual financial statements of the enterprises, operating in Italian industrial sectors, filed in Chambers of Commerce (according to Italian low D.Lgs.

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127/91). All the data have been placed on file and organised in the public database DBComplex. Figure 1

Process flow for the rating definition

The financial statement data are processed by applying the Fund Accounting methodology (Flow and Fund Analysis, Fanni, 2000), scheduled in the database DBComplex manager, which underlines company strength and weakness factors and its possible pathologies. In particular, it offers drivers to evaluate financial and economics equilibrium of a company and drivers for cash flow analysis. The same methodology is able to give warning to ratios of financial statements, which do not meet system request standards.

5.2 Data collection The second step has an interesting collection of the revised financial statement data (in the specific financial data, financial ratios or credit ratios) electronically organised. In line with the financial analysis standards, more than 80 financial ratios were calculated for each company and for a given year. In the development of the MEU model, the best possible use of the financial information available is required. For this reason, a high number of ratios are calculated. We remember that all the data represented in the MEU model input data are called explanatory data. In this phase, it is important to emphasise the fact that the database included not just the healthy company, but also a distressed group of companies that filed a petition in bankruptcy under Italian Bankruptcy Law. In order to understand the nature and in general, all characteristics of the available data, a careful, descriptive statistical analysis has been implemented.

5.3 Construction of the stratified sample In order to develop a correct insolvency risk model, a representative stratified sample of Italian companies, according to specific criteria (territorial distribution in Italy, economic sectors and company size more than Euro 1 million yearly revenues), is extracted from the database. The stratification of the sample allows instead of considering some variable characterising population object of study like, in the case of the problem that we are about to face, the field of economic activity or the territorial positioning, let alone the factors of legal nature and determine the proportions them. We remember that the data to which is reached are relating you to the enterprises of understood them that they deposit the budget of a sure dimension, that is database the DBComplex draft the economic truths that deposit the budget with equal greater annual turnover produced to 500 mila euro.

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5.4 Correlation analysis In order to construct a logistic regression statistical model developed by explanatory variable, it is necessary that they must be independent. So we proceeded to the study of the correlations between ratios (our explanatory variable). In order to avoid redundancy, these ratios were processed, excluding those that were highly statistically correlated, either directly or inversely. This means that, in the case of high correlation of two ratios, the one that is considered to have a lower relation with default status has been excluded. The cleaning process of the ratios with the correlation methodologies has allowed reducing their numerousness to approximately 42 ratios.

5.5 Explanatory variables selection process MEU model is shaped, like seen before, within the widest class of the non-linear multiple regression models. This class of models forecast/fit essentially the modelling (non-linear) for the expected value of an observable and predicted variable (in our case, risk of default) in the function of an unknown parameter vector and in the function of explanatory variables vector (in our case, the balance sheet ratios and drivers which can describe the state of enterprise solvency). To understand the importance of financial ratios in terms of distress predictive power, three different methods are used: t-Student parameter, Self Organising Maps and Default Frequency. By combining these three different methodologies, which show different aspects of the predictive power of each factor, it has been possible to identify the level of importance to be attached to each financial ratio.

5.5.1 t-Student parameter t-Student parameter is used to define whether the difference between two sets is genuine or not (Press et al., 1989). By applying the concept of standard error, the conventional statistic for measuring the significance of a difference of means is termed t-Student. The entire database of companies is divided into two sets: default and non-default, and whether an important financial ratio in the two set divisions has to be understood. In numerical terms: SD = t=

∑ (x one

− xone ) + ∑ two ( xi − xtwo ) ⎛ 1 1 ⎞ + ⎜ ⎟ N one + N two − 2 N N two ⎠ ⎝ one 2

i

xone + xtwo SD

where None and Ntwo are the number companies in the two sets, mean(xone) and mean(xtwo) are the means of the financial ratios for the two sets. If the t parameter is high, the means are different and the financial ratio is significant for default prediction. An example on the application of the t-student parameters could be observed from Figure 2; higher is the t-student parameter, more influence is the financial ratio for the default event.

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D. Marassi and V. Pediroda t-Student parameter distribution on different financial ratios

5.5.2 Default frequency It is important when considering, which ratios to include in a model to have prior expectation of, how they will be related to default. This relationship is shown clearly by Default Frequency, that is, the frequency, where a firm has defaulted over a given period. An example of Default Frequency is depicted in Figure 3. After dividing the sample into 10 buckets, which hold 10% of all observations (the x-axis shows the bucket in which a particular ratio value lies), the resulting default rates observed for firms with ratios in that bucket is shown on the y-axis. For example, it can be seen from Figure 3 that Return on Investment (ROI) has lower values that are associated with higher default rates. Figure 4 shows that Operating Income Interest Coverage Ratio has no relation to default. Figure 3

Ratio with a high predictive power

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It is clear how, for the exposed ratio, the distribution of the bankrupt companies (in green colour) is decreasing to growing ratio (in terms of value); analogous but opposite considerations can be made in analysing the distribution of the companies known health (in yellow colour); consequently the course of the default frequency is strongly monotone decreasing. Figure 4

Ratio with a low predictive power

It is clear how, for the exposed ratio, the distribution of the bankrupt companies (in green colour) is similar to the distribution of the companies known to be healthy (in yellow colour); in fact the default frequency value is around 0.5, it means that the ratio does not have a predictive power, because they do not discriminate between healthy and bankrupt companies.

5.5.3 Self organising maps The visualisation of the interaction between financial ratios and default is one important point for improving the performance of credit risk estimation. The financial analyst can obtain considerable information about the direct control of the relation between the input and output parameter of the credit rating model. This is an important field in data mining and different methodologies have been developed: clustering, Principal Component Analysis and Self-Organising Maps (SOMs). The SOM (Kohonen, 2001) is an unsupervised neural network algorithm that projects high-dimensional data onto a two-dimensional map. The projection preserves the topology of the data so that similar data items will be mapped to nearby locations on the map. This allows the user to identify clusters, that is, large groupings of a certain type of input pattern. Further examination may then reveal what features; the members of a cluster have in common. Since its invention by Professor Teuvo Kohonen in the early 1980s (Kohonen, 1982), more than 4000 research papers have been published on the algorithm (Ojia et al., 2002), its visualisation and application. The maps comprehensively

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visualise natural groupings and relations in the data and have been successfully applied in a broad spectrum of research areas ranging from speech recognition to financial analysis. The SOM belongs to the class of unsupervised and competitive learning algorithms. It is a sheet-like neural network, with nodes arranged as a regular, usually two-dimensional grid. Each node is directly associated with a weight vector. The items in the input dataset are considered to be in a vector format. If n is the dimension of the input space, then every node on the map grid holds an n-dimensional vector of weights. The basic principle is to adjust these weight vectors until the map represents visualisation of the input dataset. If the number of map nodes is usually significantly smaller than the number of items in the dataset, then it is impossible to represent every input item from the data space on the map. Rather, the objective is to achieve a configuration in which the distribution of the data is reflected and the most important metric relations are preserved. In particular, the importance is in obtaining a correlation between the similarity of items in the dataset and the distance between their most similar representatives on the map. In other words, items that are similar in the input space should map to nearby nodes on the grid. By running the SOMs on the company’s database and by using the value 0 for non-default companies and value 1 for default companies, it is possible to visualise the correlation between financial ratios and distress (Figures 5 and 6). In order to understand SOMs, one should keep in mind that similar elements tend to fall into the same or neighbouring map nodes. Nodes may thus be viewed, in our case, as company categories. From the above maps, the direct correlation between default and ROI is shown. In fact, with a high value of the ROI ratio there are no companies in default; furthermore, by examining the map values, there are no nodes in the ‘non-default cluster’ with ROI lower than −10%. The correlation is different for the operating income interest coverage ratio. In the cluster characterised by default, there are companies with high and low values of this ratio, as well in the non-default cluster. The use of SOMs completes the information extracted by the t-Student parameter and the Default Frequency, thus helping the financial analyst to choose those financial ratios that have more predictive power over default. Furthermore, SOMs permit the financial analyst to measure critical values (dangerous, pathological, adequate, very good, etc.) for each ratio, which are used in the following steps described below. Figure 5

Ratio with a high predictive power

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Explanation of the diagrams: •

Diagram on the left introduces two clusters (red and blue) corresponding to the scale of the values 0 (blue – low values) and 100 (red – high values). The diagram represents, in red, the classification in the cluster of the failed enterprises, characterised exactly with a value 100, while in blue, the classification in the cluster of the companies not failed, characterised exactly with a value 0.

•

Diagram on the right introduces more cluster corresponding to the modification of several values of the ratio assumed from the same companies (bankrupt and healthy). The different scale of colours corresponds to a minimum (blue) until to a maximum (red) assumed from the ratio (e.g. min = −0.0235 and max = 0.213). Every neuron of the map on the right side corresponds to the neuron in the same position of the map on the left side. Therefore, the bankrupt companies, characterised with the red zone of the diagram on the left, correspond to the same zone of the map in the diagram on the right.

Like it could be read in overlapped diagrams, we can assert that the healthy companies assume all high values of the ratio that is, the discriminating values for the ratio are advanced to the 0.0947 (in green). In fact, some bankrupt companies have assumed the same value (yellow-green). Figure 6

Ratio with a low predictive power

Other considerations could be derived from the Figure 6. From the diagram (on the right side) it is possible to observe that there aren’t a clear division of the ratio values (colors); in the default companies group there is the entire scale of the assumed possible values of the ratio; it means that the financial ratio don’t have predictive power of the default. From the two opposite example, it is possible to derive how the Self Organizing Maps is a useful tool to investigate the relation between financial ratio and propability of default.

5.6 Maximum expected utility The basic idea of the MEU methodology is to find the probability measure which maximises the utility function of an investor on the unknown future data; the hypothesis

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is that, the investor will choose the investment strategy which maximises the own utility in respect of a model in which he/she believes. The MEU approach will find asymptotically that result, selecting the utility function on the unknown data (the data is not used for building the model, normally known as cross validation set). The interesting aspect of the methodology is that, the quality model maximisation is not simple mono objective, but it becomes multiobjective: at the same time (using Pareto approach) the model will seek the data consistency (with the used data, normally known as training set) and with the probability measures in which the investor believes before knowing the data. The relative weights between the two optimisation objectives are parameterised with a user defined parameter α. In our case, as there is the presence of two states for the random variable (0–1, no default-default), the problem becomes the maximisation of the difference between the probability measure p and the a priori probability p0:

(

)

D p p0 = ∑ p ( x ) ∑ p( y || x ) log y = 0,1

x

p( y | x ) p0 ( y | x )

(1)

Nc T Σ −1c ≤ α

(2)

c = E p [ f ] − E p [ f ]

(3)

E p [ f ] = ∑ p ( x ) ∑ p( y | x ) f ( y, x )

(4)

E p [ f ] = ∑ p ( x ) ∑ p ( y | x ) f ( y, x )

(5)

x

x

y = 0,1

y = 0,1

In the above relations, Equation (1) represents the improvement in the utility function (in this case logarithmic) by using the relative Kullback-Leibler entropy. The difference is between the probability p and the a priori model probability p0. The definition of the relative entropy between the two models q1 and q2 is:

(

)

(

DU ,O ( q1 q2 ) = ∑ qyU by∗ ( q1 ) Oy −∑ q yU by∗ ( q2 ) Oy y

y

)

(6)

where U is the utility function. Equation (6) defines the quality of a predictive model: the model 1 has to be preferred in comparison with the model 2 if DU ,O (q1 || q2 ) > 0 on the real data. In Equations (5) and (6), there is the presence of the terms f(y, x), which are defined ad kernel function. The use of that methodology is mainly used in the Machine Learning theory, with the Support Vector Machine approach. The utility of the kernel is to transform the dual couple (y,x) in a unique value in order to change the definition space of the function to reach an easier separation between default and no default enterprises. It is possible to use different kernel definition; we use three types of features: 1 Linear features

f ( y, x ) = ( x ) j 2

where (x)j denotes jth coordinate of the explanatory variable vector x Quadratic features f ( y, x ) = ( x )i ( x ) j

3

Cylindrical kernel features

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f ( y, x ) = g ( ( x ) j )

where g(( x ) j ) = e− ( x − a)

2

/σ 2

The logical algorithm implementation becomes: Find β ∗ = arg max h( β ) β

conh( β ) =

1 N

N

∑ log p β ( y ( )

k =1

p( β ) ( y | x ) =

k

xk ) −

α N

β T Σβ

(7)

T T 1 e β f ( y , x ) p0 ( y | x ) Z x ( β ) = ∑ p 0 ( y | x )e β f ( y , x ) Z x (β ) y = 0,1

As optimisation algorithm, we used a BFCG Quasi Newton Approach (Press, 1989), based on the function gradient. The algorithm choice is motivated from the high number of free parameter to be optimised. An interesting consideration could be observed from the second part of Equation (7). It is possible to note how that part takes to the minimisation of the absolute values of the free model parameters (weighted by the kernel): that minimises the over-fitting risk. The over-fitting is a well-known problem in Machines learning: it leads to a good minimisation error on the training set data, but with poor accuracy results on the validation set. Normally that behaviour could be avoided using a recursive methodology that however carries to higher computational time resources. The advantage in the MEU methodology is that the method to avoid the over-fitting is implemented directly in the model algorithm.

5.7 Numerical test case To confirm the validity of the MEU method and our numerical implementation, the algorithm has been tested on a well-known mathematical test case (Friedman and Huang, 2003). The test is the reconstruction of a probably function model, defined by: p (1 x1 , x2 ) = 1 + 2 x1 x2 − x1 − x2 x1 , x2 ∈ [0,1]

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This probability density function is really interesting; in fact, the classical probability model reconstructs as logit regressions fail, building a uniform probability function with value of 0.5. The MEU result is presented in Figure 7. It is possible to observe how the original function has been almost perfectly reconstructed. A small difference is presented in the function behaviour near the boundary values variables: that because, there the original function is linear, unlike the MEU model. That behaviour is possible to be explained considering our kernel implementation with exponential terms; this carries to have a higher flexibility on complex data (as the real default model probability), but that could have small errors when a perfectly linear model is needed. Figure 7

Result of the MEU method on analytical function testcase

5.8 Results of the MEU method Observing the accurate result on the analytical probability density function reconstruction, the MEU model has been tested on real financial data: wholesale trade. The obtained result will be the default probability for an enterprise, derived by the ratios calculated from the balance sheet.

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The first phase is the selection of the no default enterprises (p = 0) and the default enterprises (p = 1) to build the training and validation set. The total number of enterprises is 60,532, with 371 enterprises known as in default. It could be observed that the number of default enterprises is low, but this is in accordance with the modern economical market data. The total number of ratios used is 11, selected by the statistical methodologies explained above. This is a fundamental phase to develop an accuracy model for the default prediction. We have to use the ratios (the input parameters for our model) which have the highest predictive power for the default in order to maximise the total information derived from the database; this in order to build in an accurate manner a statistical model, which divides the enterprises ‘in default’ and ‘no default’. The results could be summarised as shown in Figure 8 using the CAP curve. The cumulative accuracy profile (CAP) summarises the power curve for a model on a dataset, and compares this with that of the perfect and random model. The cumulative accuracy profile measures the area under the power curve for a model and compares this with the area under the perfect and random models, as shown in Figure 8. Thus, the perfect model would have an accuracy ratio of 100%, and the random model would have an accuracy ratio of 0%. When comparing the performance of two models on a dataset, the more powerful model on that dataset will have a higher accuracy ratio. Figure 8

Visualisation of the MEU rating model by CAP method

From Figure 9, it is possible to observe the accuracy of the MEU model for default prediction; a result in terms of CAP value of 57.2% for one-year prediction is reached. These results are comparable with the leading work in this field. From the model we could explain the relation between the financial ratios and the default probability (Figures 10 and 11); it is possible to note how the relation is highly non-linear, thanks to the kernel implementation with the exponential terms.

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Figure 9

Comparative representation of the quality of different rating methodologies

Figure 10 Capitalisation-ROS-Default probability

Figure 11 ROS-cash flow ratio-default probability (right)

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Conclusion

To sum up the model MEU, an important method based on machine learning, has a clear economic interpretation and it measures the performance in economic terms. The main idea is to seek a probability measure that maximises the out of sample expected utility of

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an investor who chooses his/her investment strategy so as to maximise his/her expected utility under the model he/she believes to be efficient. This method outperforms other numerical methods like Logit and Probit methodology, since it takes into consideration the interactions between the financial ratios in order to obtain a better approximation of the real probability of default. The disadvantage of this method, like other numerical methods based on statistical theory or machine learning is the necessity to have an accurate and complete database of the companies (universe of the companies), data that could be difficult to assemble, especially considering the data of the companies which experienced bankruptcy in the past.

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