Antonio Blanco*, Ana Irimia** y María Dolores Oliver***

Transcription

1 32 ANÁLISIS FINANCIERO Antonio Blanco*, Ana Irimia** y María Dolores Oliver*** The Prediction of Bankruptcy of Small Firms in the UK using Logistic Regression Predicción de la quiebra de las pequeñas empresas en el Reino Unido utilizando regresión logística ABSTRACT Since the Basel Accord regulation (Basel II) was implemented, considerable studies have been undertaken in an effort to predict the bankruptcy of SMEs and to analyze the effects of this accord upon these kinds of firms. However, no research has focused on devising bankruptcy models specifically designed for small firms despite the relevance of this kind of firm in economic development. This paper proposes a bankruptcy model specifically designed for small firms, by employing a sample with financial information of almost 40,000 small unlisted British firms. We find that bankruptcy models work well in the case of small firms, and when the original financial variables are transformed through the hyperbolic tangent function, then the accuracy performance of bankruptcy models increases significantly. Key Words: Small-Business Failure; Bankruptcy Model; Financial Ratios; Logistic Regression. RESUMEN Tras la implementación de los Acuerdos de Basilea II, numerosos estudios han intentado predecir la insolvencia empresarial de las PYMEs y analizar los efectos que dicho Acuerdo tiene sobre la financiación de este tipo de empresas. No obstante, ninguna de estas investigaciones se ha focalizado en desarrollar un modelo de predicción de la insolvencia empresarial específicamente para las pequeñas empresas a pesar de la relevancia de este tamaño empresarial en el desarrollo económico. En este artículo elaboramos un modelo de insolvencia empresarial específicamente diseñado para este segmento empresarial empleando una base de datos con información financiera de casi pequeñas empresas británicas. Encontramos que los modelos de insolvencia empresarial funcionan bien al ser aplicados a las pequeñas empresas, y que al transformar las variables financieras originales mediante la función tangente hiperbólica, la capacidad predictiva del modelo desarrollado aumenta significativamente. Palabras Claves: Quiebra de Pequeñas Empresas; Modelos de Insolvencia; Ratios Financieros; Regresión Logística. Recibido: 8 de febrero de 2012 Aceptado: 10 de febrero de 2012 * Assistant Professor. Department of Financial Economics and Operations Management. University of Seville. ** Professor. Department of Financial Economics and Operations Management. University of Seville. *** Professor. Department of Financial Economics and Operations Management. University of Seville.

2 THE PREDICTION OF BANKRUPTCY OF SMALL FIRMS IN THE UK USING LOGISTIC REGRESSION INTRODUCTION Small and Medium-sized Enterprises (SMEs), particularly small firms, have historically faced significant difficulties in accessing funding due to the lack of credible information available to potential providers of funds (Ang, 1991). These companies are typically much more informationally opaque than large corporations since they often have no certified audited financial statements to provide credible financial information on a regular basis. Difficulties in accessing funding are especially relevant for small firmssince this kind of firm has low assets with which to secure funding. One way to encourage lenders to increase lending towards small firms is through theuse of bankruptcy models, specifically designed for this kind of firm.nevertheless, the problem of developing a bankruptcy model specifically for small firms has either not been addressed or only briefly considered since in recent years, research on bankruptcy for SMEs has dealt with the problem of the possible effects of Basel II on bank capital requirements, but not with the problem of modelling financial distress specifically for small firms (see Berger and Udell, 1998). Moreover, the 2007 financial crisis and the New Basel Capital Accord (Basel III) have provided renewed impetus for lenders to research and develop adequate failure prediction models. Therefore, the main goal of this paper is to develop a bankruptcy model specially designed for small firms. In order to attain this objective, we apply the logistic regression method. Furthermore, since previous literature (see for example, Altman and Sabato, 2007) suggests that transformations in financial predictors increase the accuracy performance of the model, we explore the effect caused by these transformations by developing another model and comparing its performance with that whose predictor variables were not transformed. The remainder of our paper proceeds as follows. In Section 2, an overview of the extensive literature on failure prediction and methodology is included. In Section 3, we provide an analysis of the UK sample used in this study. In Section 4, a bankruptcy model for small firms is presented. A second model with hyperbolic tangent transformations on the original financial ratios is also created in this section, and the two models are tested and compared in a validation sample. Finally, Section 5 provides a concluding discussion and proposes future research lines. 2. LITERATURE REVIEW 2.1. Small and medium enterprise failure After theimplementation of the Basel Accord regulation (Basel II), considerable studies have been undertaken in an effort to predict the failure of SMEs and to analyze the effects of this accord upon these kinds of firms. As early as the 1960 s, however, the first default prediction model was developed by Beaver (1966). Using univariate analysis and a matched sample consisting of 158 firms (79 failed and 79 non-failed), he analyzed 14 financial ratios. Altman (1968) was the first author who applied the multiple discriminant analysis (MDA) in bankruptcy prediction. On a matched sample of 66 manufacturing firms (33 failed and 33 non-failed) during the period , and by examining 22 potentially predictive financial ratios classified into five categories (activity, solvency, leverage, liquidity, and profitability), he found that working capital/total assets, retained earnings/total assets, earnings before interest and taxes/total assets, market value of equity/book value of total liabilities and sales/total assets are the most predictive ratios 1. Ohlson (1980) developed the first corporate failure study by employing the logistic regression on a sample of 2,000 non-failed and 105 failed industrial firms during the period Focusing on SMEs, Altman and Sabato (2007) developed one of the most relevant models specifically made for this size of firm. Their study compares the traditional Z-score model with two new models which consider other financial variables and use traditional logistic regression. On a panel of data of over 2,000 US SMEs in the period , these authors find that the new models outperform the traditional Z-score model by almost 30 per cent, in terms of prediction power. Based on the above research, Altman et al. (2010) explore the effect of the introduction of non-financial information as predictor variables into the models developed by Altman and Sabato (2007). Employing a large sample (5.8 million) of sets of accounts of unlisted firms from the UK in the period , they find that non-financial information makes a large contribution (by approximately 13% in terms of the area under the receiver operating characteristics curve 2 ) towards increasing the default prediction power of risk models.

3 34 ANÁLISIS FINANCIERO 2.2. Research Methodology Bankruptcy models assign firms to one of two groups: a good firm group that is likely to payanyfinancial obligation; or a bad firm group that has a high likelihood of defaulting on any financial obligation. Therefore, default prediction problems lie within the scope of the more general and widely discussed classification problems. Logistic regression is one of the most commonly used statistical tools for classification problems, although other parametric (such as linear discriminant analysis) and non-parametric (such as artificial neural networks) statistical models have been successfully applied to the default prediction problems. The non-parametric techniques often outperform the logistic regression approach (for example, see Piramuthu et al. 1994; West 2000), however their black-box nature hinders the interpretation of the resulting models. The linear discriminant analysis requires the independent variables included in the model to be normally distributed, and the variance-covariance matrices to be equivalent across the failing and the non-failing groups (Karels and Prakash 1987). Both these conditions are often violated when the linear discriminant analysis is employed in the development of default prediction models 3. Therefore, many studies favour logistic regression over linear discriminant analysis thereby implying that it is the best parametric technique (for example, see Ohlson 1980) 4. The logistic regression model assumes that the probability of a dichotomous outcome is related to a set of potential predictor variables in the form: where p is the probability of the outcome of interest, β 0 is the intercept term, and β i (i=1,..., n) represents the β coefficient associated with the corresponding explanatory variable x i (i=1,..., n) (for an example, see Pampel 2000). The dependent variable is the logarithm of the odds,, which is the logarithm of the ratio of the two probabilities of the outcome of interest. These variables are usually selected for inclusion by using some form of backward or forward stepwise regression technique (Pampel 2000) even though these selection techniques may be prone to problems. The maximization of the likelihood function is commonly applied as the convergent criterion to estimate the coefficients of corresponding parameters when the logistic regression models are utilized. 3. EMPIRICAL STUDY 3.1. The data set A dataset from the Credit Management Research Centre (CMRC) of the University of Leeds (UK) is used in this study. It contains 4,813,391 (98.32% non-failed and 1.68% failed 5 ) unlisted SMEs from the UK for the period In order to obtain an exclusive sample of the smallest microenterprises, all the firms which had an asset greater than 67,000 were eliminated. Following this elimination, around 85% of the microenterprises remaining in the sample were under ten years old. This percentage is usual in the real economy since the majority of start-ups are small microenterprises. After selecting the smallest microenterprises and eliminating missing cases 6, 2,089,140 cases remained. Among these, 20,228 (0.97%) were defaulted cases and 2,068,912 (99.03%) were not. Generally, financial ratios are contaminated by some degree of error, and if these items of data are not eliminated, then the established model may be unstable. Therefore, in order to build a more accurate model, the abnormal cases, which lie within the top 1% and the bottom 1% of each financial ratio, were also eliminated, and 2,020,492 cases remained (0.98% of which were defaulted cases and 99.01% were not). Similar to previous bankruptcy studies (for an example, see Fletcher and Goss 1993; Leshno and Spector 1996), this paper also adopts a matched-pair approach. Therefore, a random sampling was performed: 19,855 (50%) failure cases and 19,855 (50%) non-failure cases 7. Finally, in order to run all the models, our final dataset was split into two subsets; by 80% of the data is used for a training set and 20% for a validation set. Therefore, the models are first estimated using the training data; once the model is formed and the parameters are established, then these models are employed in the identification of default microenterprises from among all the firms available during the hold-out period (2008) Financial ratios description There are a large number of possible candidate financial ratios identified in the literature as useful in the prediction of the

4 THE PREDICTION OF BANKRUPTCY OF SMALL FIRMS IN THE UK USING LOGISTIC REGRESSION 35 default of a firm. Chen and Shimerda (1981) show that out of more than 100 financial ratios, almost 50 per cent were found useful in at least one empirical study. Previous empirical research has found that a firm is more likely to go bankrupt if it is unprofitable, highly leveraged, and suffers liquidity difficulties (Myers 1977). All our financial ratios have been employed in prior research, such as Altman (1968), Altman et al. (2010) and Ohlson (1980). In total, 14 financial ratios are considered in this paper 9. These ratios are categorized into five categories according to the financial aspects of the business that the variables measure: leverage, liquidity, profitability, activity, and size of the firm. Table 1 describes these ratios and how they are calculated 10. In previous studies, leverage ratios have appeared to be strong predictors related to bankruptcy. Therefore, the leverage ratios are a key measure of risk. Financially distressed firms would be expected to have larger liabilities relative to shareholder funds than healthier entities. In this study, four leverage ratios are employed: capital employed/total liabilities, short-term liabilities/total assets, total liabilities/current assets, and net worth/total assets. Liquidity is a common category in most credit decisions and represents the ability to convert an asset into cash quickly. Four ratios are considered in this paper, cash/total assets, current assets/current liabilities, quick assets/current assetsand cash/net worth. Cash/total assets expresses cash as a proportion of total assets, and, according to Chen et al. (2011) is the most important single variable relative to default in the unlisted dataset. Current assets/current liabilities is mainly used to give an idea of the firm s ability to pay back its short-term liabilities (debts and payables) with its short-term assets (cash, inventory, receivables). If a firm is in default, its current ratio should be low. However, just as the cash in one s wallet does not necessarily imply wealth, a high current ratio does not necessarily imply wealth. The ratio quick assets/current assetsdetermines the extent to which current assets consist of liquid assets. Cash/net worth measures net cash as a proportion of net worth. Financially distressed firms are more likely to have a declining and/or negative net worth. Many firms fail owing to a lack of liquid assets and thus financially distressed firms would be expected to have lower values for these ratios. A profitability ratio was considered in our analysis, retained profit/total assets. This is a measure of the cumulative profitability of the firm.firms that are unable to accumulate profit from sales have lower values of this variable. The ratios trade debtors/total assets, trade creditors/total liabilities and trade creditors/trade debtors depend on the sector and characteristics of each firm, and for this reason they are considered activity ratios. Activity ratios capture important bankruptcy information and are frequently used when performing fundamental analysis. Smaller companies often rely heavily on trade finance from suppliers when bank finance is not available. Moreover, microenterprises extend trade credit to customers as a means of gaining and retaining customers. The literature on trade credit suggests that smaller firms both extend more credit to customers and take extended credit from suppliers when facing decline and financial stress (Altman et al. 2010). Moreover, Hudson (1987) argues that trade credit forms a large proportion of a firm s liabilities, especially for small firms. He proposes that bankruptcy of a small firm is mainly influenced by trade creditors rather than bondholders. Therefore, we assume that all these activity ratios have a negative relationship with respect to bankruptcy. The variable total asset is almost indistinguishable as indicator of size risk. In accordance with the general trend in the literature, the neperian logarithm of the total assets and total assets without performing any transformation are considered in this study. With respect to the size of the firm, many previous studies found that large firms are less likely to encounter credit constraints thanks to the effect of a good reputation, and therefore their studies conclude that a firm s small size may lead to insolvency (Dietsch and Petey 2004; Saurina and Trucharte 2004).

5 36 ANÁLISIS FINANCIERO In contrast, Berger (2006) concludes that lending to large corporations is riskier than lending to SMEs. Finally, Altman et al. (2010) found that the relationship between asset size and insolvency risk appears to be non-linear. In short, there is no consensus in the literature on how the size of a firm affects the probability of bankruptcy. 4. BANKRUPTCY MODEL 4.1. Selecting financial ratios It is important to obtain a parsimonious default prediction model. Therefore, only statistically significant financial ratios must be considered. There are two main methods for selecting significant ratios, forward and backward stepwise selection (for more details on these two procedures see Falkenstein et al. 2000). However, both selection techniques may be prone to problems. For this reason, in accordance with Altman and Sabato (2007), the procedure outlined below is followed here for the selection of the most important financial ratios. Once the potential candidate predictors have been defined and calculated, the accuracy ratio (AR) as defined by Keenan and Sobehart (1999) is observed for each financial variable 11. In order to avoid the problem of multicollinearity between the independent variables of the model, only one variable is selected from each ratio category. The variable selected is that which has the highest accuracy ratio from each group. A statistical forward stepwise selection procedure 12 is then applied in order to create the model using the five most significant variables of each category (celt, cashta, prta, tdta and ln_asset) 13. Table 2 shows all the financial ratios, the category which they pertain, and their AUC and AR values. In this study, the cut-off rate chosen for all the models is in terms of the level of sample default rate (0.50) 14. However, on observing the distributions of each of the selected variables for the two dependent variable groups, a large range of values is clearly visible. This high variability of the financial ratios for SMEs can be due to the variety of sectors in which these companies operate (financial data on real estate firms greatly differs from that of agricultural companies), or to the wide range of ages of the firms in the sample, or to their various levels of financial health. Therefore, certain researchers (see Altman and Sabato, 2007) use logarithmic transformations for selected financial variables in order to reduce the range of possible values, and increase the importance of the information given by each variable. They found that the accuracy of the ratio model which uses logarithmic transformations is higher than that which uses original variables. Therefore, four of the five original financial variables were transformed 15. We make hyperbolic tangent transformations, instead of logarithmic transformations, of selected original financial variables since our financial ratios can take negative values 16. Therefore, we run a new model (Model 2) which considers the transformation of the variables included in the previous Model 1 (that is, the variables celt,cashta, prta and tdta are substituted by four new variables tanh_celt, tanh_cashta, tanh_prta and tanh_tdta). The coefficients and significance level of all the variables finally considered in each model are collected in Table 3. As shown in this table, all the slopes (signs) follow our expectations. The relevance of these variables on the failure of firms can also be analyzed by the absolute values of coefficients of each variable. Cashta and ln_asset are the most relevant variables in both models.

6 THE PREDICTION OF BANKRUPTCY OF SMALL FIRMS IN THE UK USING LOGISTIC REGRESSION Results and Model Validation Considering the validation sub-sample 17, Model 2 has an average correct classification rate of 71.48% (see Table 4), which is higher than that of Model 1 (70.50%). Therefore, the hyperbolic tangent transformations clearly add value to the model; with an improvement of nearly 1% compared with the model which uses the original variables (see Table 4). Moreover, the incidence of Type I errors decreases considerably (nearly 4%), and increases Type II errors smoothly (approximately 1.5%). Similar results were obtained using the training sample. Therefore, we find, like Altman and Sabato (2007), that the transformations of financial variables improve the accuracy performance of bankruptcy models. This finding is produced across the high variability of the financial ratios for small firms due either to the different sectors in which these companies operate or to the different ages of the firms. Table 5 summarizes the results in terms of the AUC test of the two separate models tested on the training and validation sample 18. As the results revealed in this table, the AUC test of the model which includes hyperbolic tangent transformations of the original variables (Model 2) is, at 77.90%, higher than that of the model which contains the original financial variables with no transformation (77.00%) CONCLUSIONS AND FUTURE RESEARCH LINES In this study we improve upon existing models from the literature on bankruptcy modelling for small firms in two ways. First, we design a bankruptcy model specifically designed for small firms, and find that bankruptcy models can work well in the case of this kind of firm. This has major positive consequences: for lenders since (a) they would be able to reduce their capital requirements and, (b) they would increase credit availability for microenterprises, thereby augmenting their profitability thanks to the strong positive effect that smallbusiness lending has on bank profitability; and for microenterprises since they could obtain more financial resources.

7 38 ANÁLISIS FINANCIERO And secondly, we suggest, in line with other authors, (for example, see Altman and Sabato, 2007), that when the original variables are transformed through the hyperbolic tangent function, the accuracy performance of bankruptcy models increases. In the existing literature, solutions to address financial distress for small firms have never been fully provided. For this reason, financial institutions should carefully consider the results of this study when setting their internal systems and procedures to manage credit risk for small firms. Finally, this study can be further improved in future research through the introduction of non-financial variables since previous literature (see Grunert et al, 2005) suggests that these kinds of variables significantly improve the prediction accuracies of bankruptcy models. REFERENCES Altman, E.I. (1968). Financial ratios, discriminant analysis and prediction of corporate bankruptcy. Journal of Finance, 23, Altman, E.I., Sabato, G. (2007). Modeling Credit Risk for SMEs: Evidence from US Market. A Journal of Accounting, Finance and Business Studies (ABACUS), 43(3), Altman, E.I., Sabato, G., Wilson, N. (2010). The value of non-financial information in small and medium-sized enterprise risk management. Journal of Credit Risk, 6(2), Ang, J. (1991). Small Business Uniqueness and the Theory of Financial Management. The Journal of Small Business Finance, 1(1), Basel Committee on Banking Supervision. (2001). The New Basel Capital Accord. Consultative Document. Beaver, W. (1966). Financial ratios as predictors of failure, empirical research in accounting: Selected studied. Journal of Accounting Research, 4, Becchetti, L., Sierra J. (2002). Bankruptcy risk and productive efficiency in manufacturing firms. Journal of Banking and Finance, 27, Berger, A., Udell, G. (1998). The Economics of Small Business Finance. : The Roles of Private Equity and Debt Markets in the Financial Growth Cycle. Journal of Banking and Finance, 22(6-8), Berger, A.N. (2006). Potential Competitive Effects of Basel II on Banks in SME Credit Markets in the United States. Journal of Financial Services Research, 29(1), Chen, S., Härdle, W., Moro, R. (2011). Modeling default risk with support vector machines. Quantitative Finance, 11 (1), Chen, K.H., Shimerda, T.A. (1981). An empirical analysis of useful financial ratios. Financial Management, 10 (1), Dietsch, M., Petey, J. (2004). Should SME Exposures be treated as Retail or as Corporate Exposures? A Comparative Analysis of Default Probabilities and Asset Correlation in French and German SMEs. Journal of Banking and Finance, Eisenbeis, R.A. (1978). Problems in Applying Discriminant Analysis in Credit Scoring Models. Journal of Banking and Finance, 2, Engelmann, B., Hayden, E., Tasche, D. (2003). Testing rating accuracy. Risk. Falkenstein, E., Boral, A., Carty, L. (2000). Riskcalc for private companies: Moody s default model. Report Number: 56402, Moody s Investors Service, Inc., New York. Fletcher, D., Goss, E. (1993). Forecasting with neural networks: An application using bankruptcy data. Information and Management, 24, Gentry, J.A., Newbold P., Whitford D.T. (1985). Classifying bankrupt firms with fund flow components. Journal of Accounting Research, 23, Grunert, J., Norden, L., Weber M. (2005). The Role of Non-Financial Factors in Internal Credit Ratings. Journal of Banking and Finance, 29, Headd, B. (2003). Redefining Business Success: Distinguishing Between Closure and Failure. Small Business Economics, 21, Hendry, D.F., Doornik, J.A. (1994). Modelling linear dynamic econometric systems. Scottish Journal of Political Economy, 41, Hudson, J. (1987). The Age, Regional and Industrial Structure of Company Liquidations. Journal of Business Finance and Accounting, 14(2), Karels, G., Prakash, A. (1987). Multivariate normality and forecasting of business bankruptcy. Journal of Business Finance Accounting, 14(4), Keenan, S.C., Sobehart J.R. (1999). Performance Measures for Credit Risk Models. Moody s Risk Management Services. Research Report.

8 THE PREDICTION OF BANKRUPTCY OF SMALL FIRMS IN THE UK USING LOGISTIC REGRESSION 39 Leshno, M., Spector, Y. (1996). Neural network prediction analysis: The bankruptcy case. Neurocomputing, 10, Myers, S. (1977). Determinants of corporate borrowing. Journal Financial Economy, 5(2), Ohlson, J.A. (1980). Financial ratios and the probabilistic prediction of bankruptcy. Journal of Accounting Research, 18 (1), Pampel, F.C. (2000). Logistic regression: A primer. Sage Quantitative Applications in the Social Sciences Series, n.º 132. Thousand Oaks, CA: Sage Publications. Phillips, B., Kirchhoff, B. (1989). Formation, Growth and Survival; Small-Firm Dynamics in the U.S. Economy. Small Business Economics, 1, Piramuthu, S., Shaw, M. J., Gentry, J. A. (1994). A classification approach using multi-layered neural networks. Decision Support Systems, 11(5), Platt, H.D., Platt, M.B. (1990). A Note on the Use of Industry-relative Ratios in Bankruptcy Prediction. Journal of Banking and Finance, 15, Saurina, J., Trucharte, C. (2004). The Impact of Basel II on Lending to Small-and-Medium-Sized Firms: A Regulatory Policy Assessment Based on Spanish Credit Register Data. Journal of Finance Services Research, 26 (2), Sobehart, J., Keenan, S., Stein, R. (2001). Benchmarking Quantitative Default Risk Models: A Validation Methodology. Algorithmic Research Quarterly, 4 (1/2). West, D. (2000). Neural network credit scoring models. Computers and Operations Research, 27, Endnotes: 1.- This model is widely known as the Altman Z-score. 2.- Henceforth, AUC or AUC test. 3.- For a detailed analysis of problems on applying discriminant analysis in default prediction models, see Eisenbeis (1978). 4.- Other researchers also used a logistic model in order to examine the default firms (for example, see Becchetti and Sierra 2002; Gentry et al. 1985; Platt and Platt 1990). 5.- In line with other studies, we define corporate failure as entry into liquidation, administration or receivership between 1999 and 2008 since only one third of new businesses closed under circumstances that owners considered unsuccessful (Headd 2003). The accounts analyzed for failed companies are the last set of accounts filed in the year preceding insolvency. 6.- In our study, missing cases are those that have at least one instance of missing data for any independent variable. 7.- Moreover, since the failure rate of start-ups in their primary years is in the range of 50%-60% (Headd 2003; Phillips and Kirchhoff 1989), and approximately 85% of the firms in our sample are under ten years old, the fact that our sample contains 50% of failed firms is justified. 8.- For an introduction to the validation framework, see Sobehart et al Ratios are selected depending on the information available in our dataset Table 1 of Appendix 1 summarizes the descriptive statistics of all variables for both the insolvency and solvency sample According to Engelmann et al. (2003), the accuracy ratio (AR) is calculated as 2*(AUC - 0.5) This method considers that the variables enter the model if their significance level is less than 0.05 (5%) and the variables remain outside the model when this level is greater than 0.10 (10%) In some statistical studies, criticism of the forward stepwise selection procedure has been raised as it can yield theoretically implausible models and select irrelevant variables. For this reason, we make a twostep analysis, first by choosing the most relevant variables for our study through the accuracy ratio (AR) and then by applying the forward stepwise selection procedure (Hendry and Doornik 1994). Forward and backward stepwise procedures were also applied thereby directly obtaining the same results as the process The optimum cut-off value cannot be found without careful consideration of each peculiarity of each particular bank, such as tolerance of risk, profit-loss objectives, recovery process costs and efficiency, possible marketing strategies, etc The variable ln_asset is transformed. Therefore, it was not transformed again Thus, if we apply logarithmic transformations to a negative variable, the results will be missing a value The results obtained in the training sample can be seen in Table 2 of Appendix The validation sample is also known as the hold-out sample Both results are referred to the validation sample. However, the results in the training sample are similar.

9 40 ANÁLISIS FINANCIERO APPENDIX 1

10