Introduction

In recent decades, financial and, in particular, banking crises have become more frequent, both in developing and industrialized countries, which translate into a severe shrinkage in production and heavy tax and financial losses (^{Morón, 2003}; ^{Redondo and Rodríguez, 2014}). In developing countries, an example of this is the crisis towards the end of 1994 that hit the Mexican banking sector, following its reprivatization four years before in 1991-1992, due to the high NPL ratios caused by high interest rates, the shrinking supply of loanable funds, the decrease in the level of financial intermediation and the economic slowdown, which brought the intervention of the State in the capitalization of said sector. For their part, industrialized countries have also suffered through such situations, as exemplified by the worst financial crises in recent memory, which began in August 2007 and was linked to problems in the real-estate sector of the United States and, particularly, with the subprime mortgages. In the current global economy, no country is exempt from contagion in the event of a crisis as deep as the American financial crisis, since it not only affects mortgage credit but rather the entire financial system as a whole (^{Saavedra, 2008}).

In order to avoid systemic banking crises, the Financial Stability Forum (FSB), substituted by the Financial Stability Board (FSB) in 2009, delegated the development of international financial supervisory guidelines applicable to weak banks to the BCBS in 2001, which were updated in 2015 in response to the 2007-2009 financial crisis. These guidelines stablish that, in order to identify weak banks, supervisors can employ methods based mainly on quantitative financial information that include an analysis of the financial statements and early warning systems (BCBS, 2001; ^{Redondo and Rodríguez, 2014}). The term “weak banks” refers to the report, issued by the Work Group in 2002, where it established that “a weak bank is that in which its liquidity or solvency is or will be affected, unless an improvement occurs in its financial resources, risk profile, strategic direction of the company, risk management capacity, or quality management”.

For its part, the Institute for the Protection of Banking Savings, based on the analysis of the financial information of the Mexican banking institutions, carried out activities at the national level allowing to preemptively detect whether a bank is in difficulties. Such is the case of the “Early Warning System” that sends a warning regarding a deteriorating situation that could lead to possible insolvency or bank failure (^{IPAB, Annual report 2002}).

In the last decade, the market instability that began in mid-2007 highlighted once more the importance of liquidity in the functioning of the financial markets and the banking sector. The change in the market conditions revealed the speed with which liquidity can be depleted, and exemplified that the lack thereof can be prolonged for a long time; thus, laying the foundation for its liquidity framework, the bcbs144 was published (see Table 5). Basel III (bcbs188, see Table 5) emerged in response to this international financial crisis, since Basel I and II focused on the reserve levels that banks must maintain for bank losses.

One of the reasons why the economic and financial crisis that erupted in 2007 was so severe is that the banking sectors of numerous countries had accumulated an excessive amount of leverage in and out of balance (bcbs189, see Table 5).This was in addition to the gradual erosion of the level and quality of its capital base. At the same time, numerous banks maintained insufficient levels of liquidity. Due to all of that, the banking system was not able to absorb the systemic losses suffered by the trading and credit portfolios. The crisis was aggravated by a procyclical displacement process and by the interconnexions between systemic institutions through complex operations. At the peak of the crisis, the market stopped trusting in the solvency and liquidity of numerous banking institutions. The deficiencies in the banking sector were quickly transmitted to the rest of the financial system and the real economy, causing a generalized shrinkage of the liquidity and credit available. Finally, the public sector had to intervene with unprecedented liquidity injections and with the provision of capital and guarantees (bcbs189).

Due to the aforementioned, the BCBS has currently established strong regulations at the international level to prevent the severity of the economic and financial crisis. The Early Warning indicators, among others, are considered measurement instruments within these regulations (bcbs144, see Table 5). Similarly, the Financial System Stability Council (CESF for its acronym in Spanish, Annual report, 2011) has been created in our country for the same purpose.

Thus, and given the fact that weak banks can drive or worsen financial crises, the need for early identification of said banks, through prediction methodologies, is evident (^{Redondo & Rodríguez, 2014}). This would allow to take corrective measures to restore their solidity and limit their potential loses. Therefore, this work proposes a model that allows identifying and predicting a weak banks, considering that even when most explanations of such difficulties are of the macroeconomic type, there are also works that relate them to the indicators of the bank itself, that is, explanations of the microeconomic type (^{Hernández and López, 2001}).

Chronological description of the financial situation of Mexican Banks and factors that have influenced it during the 1982-2009 period

The Mexican banking system was nationalized in 1982 and its operations were controlled until 1988; it was in that year that the financial liberation began, culminating with the reprivatization of the banking system and the introduction of foreign competition in 1994 (López and Snowden, 2000). Similarly, due to different factors-among which was the fact that the total non-performing portfolio of the banks exceeded their total equity-on December 1994 a devaluation of almost 100% occurred which, in addition to high interest rates at the beginning of 1995, generated an exchange rate and credit imbalance in bank operations and some of these went from technical bankruptcy to full bankruptcy (^{Correa, 1999}).

During the period of 1995-1998, actions were taken to face the Mexican banking crisis (^{Turrent, 2003}; ^{Murillo, 2005}). The first of these was to solve the liquidity problems, followed by solving the solvency problems. In the case of liquidity, the Banco de Mexico (BANXICO) established a dollar credit window to prevent banks from defaulting on the payment of foreign currency liabilities; and in the case of solvency, four programs were implemented:

The aforementioned banking crisis evidenced the protection mechanism of the resources of the depositors, since it was not enough and required the support of the Federal Government (Hazera et al., 2015) to prevent the collapse of the financial and payment systems. Thus, on January 19, 1999, the LPAB was published giving rise to the IPAB on May 21, 1999, which absorbed practically all of the liabilities of the FOBAPROA. This legal framework made possible a greater regulation of the protection of savings and the need to act in coordination with all other financial authorities in the execution of the necessary actions to procure the solidity of full-service banks. Additionally, the IPAB made possible the acquisition or merger of one bank with another with greater solvency, as well as the disappearance or capitalization of some banking institutions. It should be mentioned that with these measures-combined with the sale of non-strategic assets, the increase of the reserves, and the reorganization of the total non-performing portfolio -the generation of profits allowed to improve the bank capitalization indices in 1999, which, based on the Basel Accords (^{Hernández and López, 2001}), exceeded the 8% minimum required by the authorities by a wide margin. It is therefore reasonable to consider the year 1999 as ideal to represent the sample space and, according to ^{Mora, 1994} seems ideal to select a sample with data corresponding to the same period.

With the conclusion of the New Program in 2005, the IPAB exchanged instruments issued by the FOBAPROA for new securities and led to the replacement of the PCCC in which some of the banks analyzed in this work participated and survived the financial crisis.

In 2007, a segment of the international financial system fell into an intense crisis caused by an increase, in the United States, of the past due loans indices of the lower quality mortgage loans known as “subprime” (^{BANXICO, Annual Report, 2007}) which, in turn, led to the bankruptcy of Lehman Brothers in September 2008 and, consequently, the intensification of the international financial crisis and the propagation of its effects to emerging economies. Thus, the cost of interbank credit rose abruptly, and the liquidity of the financial markets dropped even more, complicating the situation as a result of the feedback links between the financial and the real sectors. Therefore, in addition to the liquidity crisis, the financial markets faced a solvency problem that led to the risk of collapse of the international financial system. In this context, the cash flows of emerging economies contracted significantly, affecting the exchange rates and equity and bond markets of many of these countries. These facts affected Mexican banking; consequently, in coordination with the Secretariat of Finance and Public Credit (SHCP for its acronym in Spanish), BANXICO implemented different actions directed to maintaining the good functioning of the domestic financial markets (^{BANXICO, Annual Report, 2008}). This helped Mexican banks continue generating profits and be well-capitalized, with sufficient credit reserves to face the rise of the past due loans indices in 2008 (^{BANXICO, Financial System Report, 2009}).

Classification of the nine Mexican Banks analyzed

To apply the discriminant analysis, it is necessary to classify the individuals into alternative groups, for this purpose Table 1 shows a few of the IPAB guidelines that helped classify weak banks and nonweak banks.

In relation to Table 1, it is indicated that according to the ^{IPAB (Annual Report, 2002)} and the regulating criteria (^{BANXICO, Financial System Report, 2006}), the solvency of a bank is detected through the “early warning” mechanism by calculating the Capitalization Index (ICAP). Nevertheless, the solvency of a bank also depends on its capacity to generate profits and the coverage index of its total non-performing portfolio with allowance for loan losses. Considering what is set out in Table 1 and the actions taken by the banks shown in Tables 2 and 3, the latter present the classification of the analyzed banks as weak banks and nonweak banks, respectively. It is necessary to note that ^{Altman et al. (2014)} consider a classification of weak banks and those nonweak banks very similar to that found in this work, and ^{Marín et al. (2011)} note those institutions that were intervened by the FOBAPROA as a business failure.

Discriminant Analysis Model

It should be noted that, even when it was formulated by Altman in 1968, discriminant analysis is still used today to classify entities into alternative groups as can be observed in the works of ^{Crespo, 2011}, and of ^{Redondo and Rodríguez, 2014}. Discriminant analysis seems to have predominated in studies on the difficulties of financial entities from an empirical point of view and has not lost its prominence, perhaps because it is a well-studied technique and, above all, due to the good results it has produced (^{Redondo and Rodríguez, 2014}). This analysis is used to classify different individuals into alternative groups based on the values of a set of variables of those to be classified. Thus, each individual may belong to a single group and their belonging is introduced into the analysis by means of a categorical (dependent) variable that takes as many values as there are groups. For their part, the variables used for the classification of the individuals are called classifying (independent) variables and their information is summarized in functions called discriminant functions. The discriminant analysis can be implemented for explanatory and predictive purposes; in the former (see section V), the aim is to determine the contribution of each classifying variable to the correct classification of each individual, while in the latter (see section VI), the aim is to determine the group to which an individual belongs and for which the values of the classifying variables are known (^{Uriel and Aldás, 2005}; ^{Carvallo, 2007}). The statistical hypotheses of this analysis are the homoscedasticity and multivariate normality of the groups (^{Hair et al., 2005}), which are shown in sections V.1 and V.2, respectively.

Explanatory application of the Discriminant Analysis to the nine Mexican Banks

The discriminant analysis was applied to the nine Mexican Banks presented in Tables 2 and 3 for the period of 1999. Thus, the categorical (dependent) variable is defined as “Weak banks” (Group 1) or “Nonweak banks” (Group 2).

Regarding the classifying (independent) variables used in this work, it is indicated that they are thirteen of some of the financial indicators that have been used in Beaver’s univariate analysis (1966), in the Altman’s multiple discriminant analysis (1968), and in Ayala’s early warning models for financial crises (1999), with a context currently considered in the Basel Accords II and III of the BCBS. The selection of these indicators was made in accordance with what was proposed by different authors who have worked on the prediction of a business or banking crisis; their specific comments are:

(X ₁ ).- In ^{1968, Altman} used this financial indicator, mentioning that it is frequently found in studies of companies with financial problems and, of the three liquidity indicators that he evaluated, he considered it the most valuable. Similarly, in 1966, Beaver considered it to be the second best for prediction and ^{Alfaro, Gámez and García (2008)} used it to predict the business failure of 1180 companies (commercial and industrial). Likewise,^{Lifschutz (2010)}, ^{Alkhatib and Eqab (2011)}, and ^{Naresh and Sree (2014)} have published works that indicate the good prediction of bankruptcy by means of Altman’s model and, thus, the use of this indicator.

(X ₂ ) Return on assets.- In 1968, Beaver examined a large number of failed companies for which there had never been a prediction of failure. Through this examination he concluded that this indicator had good predictive power. A more recent study (^{Cools and van Toor, 2015}) used this indicator, among others, to make a comparison between weak and strong banks. Similarly, ^{Gamlath and Rathiranee (2014)} used this indicator to carry out a regression analysis of a sample of seven commercial banks in Sri Lanka over the period of 2007-2011 to determine the relation between working capital management and profitability. ^{El-Ansary and Hafez (2015)}, in their empirical study of Egyptian banks, related the Capital Adequacy Ratio as a dependent variable with the Return on Assets to analyze Profitability between other independent variables, concluding that they are significantly correlated during the 2003-2013 period. ^{Labatut et al. (2009)} consider this indicator to be very efficient in detecting company insolvency. It should be noted that the work of Labatut was done for small and micro enter-prises, but the author himself mentions that his study can be extrapolated to other contexts where similar situations occur, which is the case of banking. For their part, ^{Chávez and Córdova (2017)} considered this indicator in the study of the financial situation of Ecuadorian companies using discriminant analysis.

(X ₃ ) Productivity (Altman).- In ^{1968, Altman} used this indicator to measure the real productivity of the company assets and, given that its definitive existence is based on the power of its asset earnings, it seems appropriate for studies related to business failures and outperforms other measures of profitability. In 1999, Carrasco indicated that, according to ^{Altman (1968)}, Productivity is fundamental, given that the existence of an entity is determined by the power of profitability of its assets. For their part, ^{Alfaro, Gámez and García (2008)} used this indicator, which was very important in their discriminant model, and mentioned that companies with greater economic profitability have more options to remain active, thus Productivity is lower for failed banks (^{Laffarga, Martín and Vázquez, 1987}). Similarly, ^{Lifschutz (2010)}, ^{Alkhatib and Eqab (2011)}, and ^{Naresh and Sree (2014)} have published works that indicate good bankruptcy prediction using Altman’s model. ^{Labatut et al. (2009)} also used this indicator.

(X ₄ ) Solvency.- In ^{1999, Carrasco} indicated that, according to ^{Altman (1968)}, the financial indicator market value of shares/book value of total debts is an effective predictor of bankruptcy. However, since it is difficult to obtain this type of information, the financial indicator Internal Leverage was used as predictor instead. It has also been used by ^{Millán de la Lastra et al. (2013)}, especially in the appearance of losses and particularly in the loan portfolio, which has a significant impact on the deterioration of its assets. Therefore, the accumulation of leverage of the banks must be avoided. Furthermore,^{Gutiérrez and Abad (2014)} show in the results that the equity structure of the entities, defined as financial autonomy (own funds/liabilities), is one of the two variables with the greatest predictive capacity. ^{Labatut et al. (2009)} also used this indicator. ^{Chávez and Córdova (2017)} applied the inverse of this indicator.

(X ₅ ) Liquidity (MACRO).- This indicator belongs to the Liquidity Risk category in the MACRO acronym (to the Liquidity category of CAMEL). Given that banks are financial intermediaries with liabilities that are short-term deposits and their assets are long-term loans, the imbalance between assets and liabilities is a natural consequence of the financial activity that, if not properly managed, can lead to solvency problems (^{Ayala, 1999}). For their part, ^{Laffarga, Martín and Vázquez (1991)} used it as one of the fifteen indicators that they selected for their bankruptcy prediction analysis of Spanish Banks, while in 2008, Alfaro, Gámez and García also used this financial indicator. In 2010, Gamlath and Rathiranee used this indicator in the regression analysis of seven commercial banks in Sri Lanka for the 2007-2011 period. ^{Labatut et al. (2009)} and ^{Chávez and Córdova (2017)} also applied this indicator.

(X ₆ ) Past due loans.- This indicator belongs to the Asset quality category within MACRO (Asset quality of CAMEL). ^{Hernández and López (2001)} suggest that this indicator is statistically significant to explain the banking crises and it reflects credit risk. The higher this relation is, the lower the probability of survival of the bank. Banks with this situation must contribute with a considerable portion of assets due to losses, which leads to a decrease in earnings and, consequently, of capital.

The increase of past due loans in the 2007-2012 period had two serious consequences in the Spanish financial system leading to the disappearance of Savings Banks, which represented 50% of the Spanish credit system (^{Climent, 2017}).

(X ₇ ) Provision coverage.- This indicator belongs to the Asset quality category within MACRO. ^{Carrasco (1999)} indicates that Whalen and Thomson (1988) used this indicator in models to predict the deterioration in the bank’s situation through changes in the CAMEL valuation. Similarly, in this sense, ^{Climent (2017)} did a study to predict the performance of the provisions for the impairment of Spanish credit institutions at different stages of the economic cycles they have gone through and thus obtain more real information on the needs of these provisions for loan impairment provisions.

(X ₈ ) Credit risk.- This indicator belongs to the Capital adequacy within MACRO (CAMEL Capital). ^{Carrasco (1999)} mentions that the results of the models built by Whalen and Thomson (1988), using only this indicator, are as good as those obtained by other more complex models and that this variable seems to be a substitute for the qualification of assets in the CAMEL valuation. For his part, ^{Ayala (1999)} indicates that the variations in the exchange rate can have significant effects on the financial situation of a bank; the risk of this exchange rate can become a credit risk if the profitability of the debtors does not correctly incorporate the expected devaluation.

(X ₉ ) Return on interest-earning assets.- This financial indicator belongs to the Operative Results category within MACRO (CAMEL Earnings), it is used to measure the risk inherent to the deadline transformation process and can degenerate into crisis when the return on assets falls systematically below the one paid for the liabilities (^{Ayala, 1999}; ^{Ballester et al., 2009}).

(X ₁₀ ) Implied liability rate.- This financial indicator belongs to the Liquidity Risk category within MACRO (CAMEL Liquidity). As with the Liquidity financial indicator, the Implicit Liability Rate measures the liquidity risk allowing the evaluation of the volatility risk of the liabilities that fund the assets (^{Ayala, 1999}). ^{Anastasi, Burdisso, Grubisic and Lencioni (1998)} mention that a greater rate reflects a greater difficulty in finding depositors in the market. Therefore, this indicator is associated with a greater implicit risk of the entity.

(X ₁₁ ) Efficiency.- This indicator belongs to the Handling (Management) category within MACRO (CAMEL Management). ^{Arango and Botero (2001)} indicate that, for Aristóbulo de Juan, an efficient management can help solve macroeconomic disturbances, whereas a deficient management leads banks to insolvency. The results of ^{Ríos and Gómez in 2015} indicate that banks with greater profit margins and with the lowest levels of technical efficiency are the ones that contribute more to the stability of the banking sector.

(X ₁₂ ) Productivity and (X ₁₃ ) Resource management.- These indicators belong to the Handling (Management) category of MACRO and were used to calculate the MACRO qualification index (^{Ayala, 1999}).

Table 4 shows the thirteen classifying variables (X₁-X₁₃) considered in this work.

Additionally, Table 5 shows how the context of these indicators is currently considered within the Basel Accords II and III, as a result of the instability of 2007.

With the values shown in Table 6 and using Equations 1(a)-1(c), the Observation, Group Means, and Total Means vectors were obtained, respectively. In said equations, x _i,g,k (see Table 6) is the observation of variable k of element i of group g; G is the number of groups; n _g is the number of elements of group g; K is the number of classifying variables; and n = n ₁ + n ₂ + ... + n _g is the size of the sample. In our case, G = 2, n ₁ = 4, n ₂ = 5, n = n ₁ + n ₂ = 9, and K = 13.

Table 7 shows the Group Means vectors and the Total Means vector.

Homoscedasticity Hypothesis

The null hypothesis is that the covariate matrices of the two groups are equal, whereas the alternative hypothesis is that said matrices are not equal. Thus, to verify the null hypothesis Van Valen’s test was used, which requires Student’s t-distribution and certain values dependent on the standardization of the data given by (^{Acuña, 2000}):

Where xs _i,g,k are the standardized observations, and M _g,k is the Median of the k-th standardized variable in the g-th Group (^{Acuña, 2000}). The results obtained are shown in Table 8, where it can be observed that the p-value (0.10) is greater than 0.05, indicating that the equality hypothesis of the covariance matrices is accepted.

Hypothesis of Multivariate Normality of the groups

The null hypothesis is H0:xg≈N(μg,σg), and the alternative is H1:xgN0≈N(μg,σg).

Due to the complexity of the normality analysis in the multivariate case, the variable to variable normality test was used to contrast the hypothesis of the two groups (^{Uriel and Aldás, 2005}). Thus, given that the samples are small, the Shapiro & Wilk test (^{González, 2006}) was used, which consists in comparing the statistic:

with a W _tab value. If W _cal is greater than W _tab for a particular level of significance, the data normality hypothesis is accepted. Based on this, Table 9 shows the results obtained for the thirteen independent variables and it can be observed that only four variables do not comply with this hypothesis. However, ^{Mora (1994)} mentions that different authors demonstrate that it is not absolutely necessary to fulfill the normality condition to obtain good results. Similarly, ^{Redondo and Rodríguez (2014)} indicate that the difficulty in practice to comply with the restrictive assumptions underlying the discriminatory analysis may not be as serious an inconvenience as it might seem at first glance, given its relative robustness in the face of non-compliance.

Decomposition of the Covariance Matrix

With the Observation, Group Means and Total Means vectors, the variation matrix between groups can be obtained through the sums of squares and cross products (SSCP) matrix. Thus,

For its part, the intra-group variation matrix, defined as the sums of squares and residual cross products (SSRCP), is:

The sum of the two matrices mentioned above provides the sums of squares and total cross products (SSTCP). Thus,

Discriminant Variable Selection

The selection of the variables that contributed the most to the discrimination of the groups was done using the step by step forward (^{Ferrán, 2001}) method and the Wilks’ Lambda criterion to measure the discriminant power gain/loss when introducing/removing a variable from the set. In each step, a classifying variable can enter if its corresponding statistic, F-to-enter, is greater than a prefixed minimum F value (F-min-entry) or it can leave if its corresponding statistic, F-to-leave, is lower than a prefixed maximum F value (F-max-leave). The F statistic is:

Where Aq=|Wq||Tq| is Wilks’ Lambda based on the first q variables selected.

For our case, a F-min-entry of 2.58 and a F-max-leave of 2.43 were specified. Table 10 shows the selected classifying variables, in order of appearance and following the step by step forward method and Wilks’ Lambda criterion. Even when the results are not shown, none of the selected variables were removed from the discriminant set.

Canonical Discriminant Function

Canonical discriminant functions are defined as (^{Pérez, 2005}):

Where i = min(G-1, q), with q being the number of classifying variables selected. In our case i = min(1,4) = 1, meaning there is only one discriminant function. Thus:

Considering matrices [F], [W], and [T] of the multiple variance analysis, vector {u}₁ is obtained maximizing λ ₁ , given by:

The maximization of λ ₁ leads to the following characteristic values problem:

Where, with the selected classifying variables, matrices [F] and [W] are:

[F]=0.0040.003-0.6790.0210.0030.003-0.6150.019-0.6790.021-0.6150.019120.194-3.763-3.7630.118      y       [W]=0.004-0.006-1.497-0.001-0.0060.0163.791-0.011-1.497-0.0013.791-0.011999.3540.4550.4550.268

Equation 11 was solved using Cholesky decomposition of matrix [W] and power iteration algorithm (^{Burden and Faires, 1985}). Thus, the result of the characteristic λ ₁ value was equal to 20.0173 and the characteristic vector was:

u1=28.028942.1010-0.12232.4827

Normalizing vector u1 so that with u1T[W]u1=1 we obtain vector v Thus,

v1=19.941529.9533-0.08701.7663

For its part, the standardized coefficients vector of the canonical discriminant function {u _e }₁ is obtained with:

Where matrix [S] is a diagonal matrix with elements that are the square roots of the elements of the main diagonal of matrix [W]. Thus,

ue1=1.18093.8127-2.75070.9138

The structure vector, which contains the combined intra-group correlations among the standardized canonical discriminant variables, is:

S-1[W]v1=0.23370.0984-0.07750.1483

For its part, the non-standardized coefficients vector of the canonical discriminant function is:

une1=(n-G)12v1=(9-2)1219.941529.9533-0.08701.7663=52.760379.2489-0.23024.6733

The constant c ₁ of the canonical discriminant function (Equation 8) with non-standardized coefficients is obtained through the following expression:

Where X¯T is the transposed vector of total means of the selected classifying variables. Thus, c ₁ = -12.0032 and the canonical discriminant function with non-standardized coefficients is:

For its part, the i-th canonical correlation has a value of

CRi=λi1+λi=20.01731+20.0173

To know whether a discriminant function is significant, we will contrast the null hypothesis below in the (k+1)-th step (the process begins with k = 0):

through the contrast statistic (Salvador, 2000):

Which is distributed as a χ ² with (q-k)(G-k-1) degrees of freedom if H₀ is true.

For its part, the T statistic corresponds to a Wilks’ Lambda given by equation 17.

The p-value associated to the contrast is given by:

Where T _obs is the observed value of T. Thus, substituting the corresponding values, the following is obtained:

Λ=0.048.Tobs.=9-1-4+22∑j=11In(1+λj)=(5)In(1+20.0173)=15.227.  y  PX(4-0)(2-0-1)2≥Tobs.=PX42≥15.227=0.004

With the canonical correlation (0.976) and the p-value (0.004), it is concluded that the obtained function has a high discriminant power and is significant. Additionally, the square of the canonical correlation (0.953) leads to the conclusion that 95.3% of the variance of the dependent variable can be explained by this model, which includes four independent variables (X ₉, X ₆, X ₁₁, and X ₅).

Linear Discriminant Analysis

This function can be obtained for each of the groups by using:

In equations 19 and 20, wi,j* are elements of matrix [W]-1, X¯i,j are the elements of vector X.j that correspond to the selected classifying variables; and π_j are the probabilities a priori. Thus, for this case study, we have:

Grupo 1X1,1X2,1X3,1X4,1=4.390×10-26.804×10-28.208×1005.683×10-1 y π1=0.5   Grupo 2X1,2X2,2X3,2X4,2=8.544×10-21.056×10-18.531×10-17.985×10-1 y π2=0.5

⟹b1,2b2,2b3,2b4,2=556.703613.3457-1.455745.1338 y a1=-40.6282

⟹b1,2b2,2b3,2b4,2=975.72291242.6354-3.283882.2430 y a2=-139.4446

Thus, Fisher’s Linear Discriminant Analysis for Groups 1 and 2, respectively, are:

F1=556.770X9+613.346X6-1.456X11+45.134X5-40.628   and   F2=975.723X9+1242.635X6-3.284X11+82.243X5-139.445

For both groups, the coefficients and constant are:

b1b2b3b4=b1,2b2,2b3,2b4,2-b1,2b2,2b3,2b4,2=418.953629.290-1.82837.109 y a=a2-a1=-98.816

Thus, for both groups, Fisher’s Linear Discriminant Classification Analysis is:

Using the function above, Table 11 shows the classification of the banks under study.

The probabilities of belonging to the groups were obtained with:

Where π _g and F _g are the probability a priori and Fisher’s Linear Discriminant Analysis, respectively, of the g-th group.

Table 12 shows the probabilities of belonging to Groups 1 and 2 according to Fisher’s Linear Discriminant Analyses and the probabilities a priori.

As can be observed in Tables 11 and 12, the proposed model perfectly identifies the classification of the nine Mexican banks that were part of the Mexican banking crisis.

Predictive Application of the Financial Situation of Four Mexican Banks and One American Bank

Using Fisher’s Linear Discriminant Classification Analysis (ec. 21) the situation of the Mexican banks 1, 5, 6 and 7 was predicted, shown in Table 13. The empirical results of the prediction are generally consistent with the analyses reported by the IPAB 1999-2005 (see Table 1) and BANXICO in its annual reports for the 2007-2008 period. The recovery of the banks in 2008, which is consistent with what is set forth in section II, should be noted. The same prediction was done for an American bank, the results of which are shown in Table 14. As can be observed, the predicted result is not consistent with the annual reports provided by the presidents of said bank for the 2000-2006 period but are consistent for the 2007-2010 period. For this last period, it should be mentioned that the proposed model has a recovery for 2009 and 2010, which is reflected by the tendency for positive values of the aforementioned function. This coincides with the fact that this bank adhered to the Trouble Asset Relief Program (TARP), which was approved by the U.S. Congress in October 2008 for the restoration of price formation mechanisms in the markets through the purchase of assets with valuation problems by the country’s Treasury Department (^{Calvo and Martín de Vidales, 2014}). It should be noted that Cools and van Toor (2010) considered Bank of America as a week bank during the 2002-2006 period.

Based on the aforementioned, it is possible to indicate the goodness of the proposed model with regard to prediction. Additionally, even when the IPAB determined that the institutions analyzed here-in relation to capitalization levels-comply with the requirements in the New Program of the PCCC in the 2000-2003 period and, according to the CNBV, were classified in category I from 2004 to 2008, it should be noted that the international financial crisis raised various questions regarding capitalization rules in line with the recommendations of the BCBS. Specifically, the goodness of the capitalization index as a banking solvency indicator has come into doubt (^{BANXICO, Financial System Report, 2009}). On the other hand, the total provisions-calculated as a proportion of the total non-performing portfolio of the Banorte, Banamex, Santander, BBVA Bancomer, and HSBC banks-remained stable in 2006. However, in 2007 these banks reduced their profits due to lower intermediation revenue and higher provisioning expenses. Subsequently, in 2008, provisions were affected by a substantial increase in full-service bank credit ratings below “A” due to an increase in delinquency (^{BANXICO; Financial System Report, 2009}). In this manner, it can be observed that, although capital insufficiency is prevented by considering the capitalization index, the insufficiency of preventive reserves is determined using the level of banks loans.

Even though financial crises are not the only source of economic slowdown, quantified by the Gross Domestic Product (GDP), the turbulence of the international financial markets significantly affected the performance of the world economy during and after 2008 (^{BANXICO, Annual Report, 2008}). In particular, given that the financial situation of four Mexican and one American banks are analyzed in a predictive manner in this work, Table 15 shows the annual variation of the GDP (as a percentage) of Mexico and the U.S. for the 2006-2010 period. As can be observed in the aforementioned table, the growth of the economy was very low in said period, which coincides with the predictions presented in Tables 13 and 14.

Conclusions

We have presented a discriminant analysis model to identify and predict the financial situation of full-service banking institutions through a combination of financial indicators belonging to different methodologies that show an eminently predictive approach. As far as identification is concerned, it is observed that this analysis has a verified explicative application as demonstrated by the discriminant function obtained, which is significant having a high discriminant power since the canonical correlation value was of 0.976 and, for its part, the probabilities of belonging to one or another group (Table 11) were 1. Regarding prediction, it was also shown that the predictive application to the four Mexican banks analyzed in the 2000-2008 period, and to the one American bank analyzed in the 2000-2010 period, is reliable since it generally coincides with the analysis issued by the IPAB, BANXICO, and the presidents of the Bank of America. Both applications were done based on the classifying variables resulting from the step by step method such as Return on interest-earning assets, Past due loans, Efficiency and Liquidity (MACRO), which are closely related to the national and world financial crisis. In the case of Return on interest-earning assets , which contemplates operations with derivative financial instruments, it can be said that, even when it was selected with data from 1999 through the proposed model, it still plays an important role in financial crises as evidenced by the Dodd-Frank Wall Street Reform in the United States (^{González and Marqués, 2010}) and the European Market Infrastructure Regulation (EMIR) in Europe (Regulation of the European Parliament and of the Council, 2012), where it is stated that the common denominator of these initiatives is that they mainly focus on the derivatives, which contributed to the destabilization of the markets (^{Asobancaria, 2013}). On the other hand, the cited financial products, in addition to the structuring and proliferation of other low-quality loans and, in general, the incidence of economic policy within the real economy (2004-2011 period) and on their performance, can be cited for the case of Spain, where their immediate consequence was the deterioration of liquidity and financing leading it to a recession toward the end of 2008 (^{Marín et al., 2015}). In relation to the gross financial margin, it is influenced by variations in the exchange rate that directly affect the cash and revenue streams of banking entities (^{Ballester et al., 2009}).

Regarding delinquency, Ana Fernández-Sainz and Felipe Llaugel, emphasized in 2011 that some variables relative to the banking sector helped explain the possibility of problems, although not the time of the crisis. Among the specific variables considered were: the risk-ad-justed capital ratio, the total non-performing portfolio as a proportion of the total loan portfolio, and the credit concentration as a proportion of the total loan portfolio.

Previous authors also state that numerous studies have demonstrated that the individual analysis of the indicators can be tricky, especially when the aim is to determine a threshold within which each indicator must be located and that, when it exceeds said threshold, an alert signal is generated. Thus, this entails the development of a methodology that does not individually analyze the indicators, but rather the pattern of the set of relevant indicators. Additionally, it can be said that diverse investigations strengthen the hypothesis that it is not the individual indicators that must be analyzed but rather the behavioral pattern that they present. Furthermore, said authors mention that some works (Todorov et al., 1999) indicate that the classification function (statistical technique of multivariate analysis) can be created with a relatively reduced number of indicators due to the great correlation that is commonly observed between them.

Regarding the financial indicators used in this research work, it should be mentioned that ^{Altman et al. (2014)}, in their article “Anatomy of Banks in Crisis”, contemplate financial indicators belonging to the CAMEL system that are similar or equal to those analyzed here, originating from the Bankscope database. There is no general theory that guides the selection process of said indicators (^{Labatut et al., 2009}). However, the context of each of the thirteen indicators considered in this work has been or will be contemplated-with lesser or greater relevance-in Basel Accords II and III of the BCBS as of 2008 as shown in Table 5, which highlights the relevance of the selection and effective employment of each of the indicators utilized here.

Without a doubt, the current situation and possible future economic development of a country or world area normally reflects, first of all, on the financial economy and it first shows itself in the financial information of the Banks. For this reason, it is of great interest to have methodologies, verified in the field of praxis that contribute with real and timely information regarding this possible development and that additionally control key variables in banking such as solvency through own resources, as well as adequate records of possible delinquents via credit provisions. Following the bibliographic review carried out, we can confirm that the proposed methodology is in line with the latest trends in ex ante analysis of bank solvency prediction, which are analyzed and supported by the BIS, the Basel Committee, the European Banking Stress Test and the national analysis, among others. This analysis and information would allow developing a national and/or supranational economic and financial policy in due course that will mitigate the effects on the real economy and its incidence in all the key variables for economic development.

Based on the above, it is concluded that the proposed model can be used, in addition to the institutional early warning models, to predict whether a bank is in distress. In that regard, it should be mentioned that said model, even though it does not take the ICAP into account as early warning models do, does consider the provision for credit losses. Additionally, this model can be applied to any bank of any country and for any year after those analyzed in this work, considering that, for its application, the necessary and important updated information in the banking sector (^{Gras et al., 2015}) should be available to calculate the financial indicators proposed in this work.