*4.2. Value-Based DEA Results*

The value-based DEA was applied for the evaluation of the 94 banks, for the time interval 2011–2016, considering that the factors to be minimized (inputs) and factors to be maximized (outputs) are the same considered in the GMM, attending to the negative and positive coefficient signals (Table 4). The ROAA is on the side of factors to be maximized because it is the assumed measure of profitability. Therefore, it is considered to be the "more-the-better" type of performance measure.



Let *DMUj*, *j* = 1, ... ,94 be observed in *t* = 1, ... ,6 consecutive years. Then the sample used has 6 × 94 DMUs (*DMUtj*). The matrices of inputs and outputs of the 564 DMUs in evaluation are *X* = (*x*11, *x*12, ... , *x*194, *x*21, *x*22, ... , *x*294, ... , *x*61, *x*62, ... , *x*694) and *Y* = (*y*11, *y*12, ... , *y*194, *y*21, *y*22, ... , *y*280, ... , *y*61, *y*62, ... , *<sup>y</sup>*694), respectively.

Considering that the value *ptcj* is the performance of DMU *j* in factor *c*, for the year *t*, the factors performances are linearly converted into values following the procedure: Firstly, two limits, *MLc* and *MUc* , are defined for each factor, such that *MLc* < *minptcj* , *j* = 1, ... , 94;*t* = 1, ... , 6 and *MUc* > *maxptcj* , *j* = 1, ... , 94;*t* = 1, ... , 6, for each *c* = 1, ... , 5. Secondly, the values for each DMU are computed using:

$$w\_c^l(DMI\_i) = \begin{cases} \frac{p\_{cj}^l - M\_c^l}{M\_c^{l1} - M\_c^{l'}} \text{ if the factor c is to maximize} \\\frac{M\_c^{l1} - p\_{cj}^l}{M\_c^{l1} - M\_c^{l'}} \text{, if the factor c is to minimize} \end{cases}, \quad j = 1, \dots, 94; \ t = 1, \dots, 6; \ c = 1, \dots, 5 \quad \text{(4)}$$

The *MLc* and *MUc* values of the factors to minimize and the factor to maximize that were considered for all DMUs and for the interval 2011–2016 are displayed in Table 2.

The different DEA models have been widely used for performance evaluation in different practical applications, however, it is very common to find factors that have negative or zero values. For radial measures of efficiency, as the classical models (CCR and BCC), the presence of negative data is a problematical matter. The valued-based DEA overcomes this drawback by converting the performances on each factor into a value scale. Hence after being converted into value functions, all factors are to be maximized.

Value functions could also be obtained from the DMs' preferences and this may lead to piecewise and nonlinear value functions (see, for instance, Almeida and Dias 2012; Gouveia et al. 2015, 2016; and Gouveia and Clímaco 2018).

For this study, a unifying reference set for the whole period was considered, and then the optimal value difference *d*∗*k* was computed for each bank *k*, in each year, making it possible to compare all of them across years.

The statistic of the scores *d*∗ obtained with the evaluation of DMU's efficiency across the six years, using the value-based DEA method is depicted in Table 5. Attending to the results of the problem (2), the lower the value of *d*∗ is, the better, and if *d*∗ is negative then the DMU under analysis is efficient. The DMUs that have *d*∗ = 0 are weakly efficient and the ones that have *d*∗ > 0 are inefficient (Gouveia et al. 2013).


**Table 5.** Score statistics.

The years 2011 and 2014 are the ones that show more efficient banks, however they display the very different average of *d*<sup>∗</sup>. The year 2011 has the banks with the highest average score (more negative values of *d*∗) for the efficient banks, however, it also has the banks with the highest average of *d*∗ for the inefficient banks (more positive values). The overall average of the bank scores, considering the different years, are better for 2011, 2012 and 2016 (>0.8).

There are three e fficient banks for the remaining years, but the scores of the e fficient banks are on average better for 2012 and 2013.

Probably these results are reflective of the financial help that banks were getting, gradually, after the global financial crisis (Gulati and Kumar 2016), and that impact the di fferent Eurozone countries at di fferent times (Wild 2016). Faced with serious economic di fficulties in Greece, the European Union has adopted an aid plan, including loans and supervision of the European Central Bank. Our results are in line with Christopoulos et al. (2019) since they show that the PIIGS countries (Portugal, Ireland, Italy, Greece, and Spain) have a high degree of ine fficiency, which is aggravated after the sovereign debt crisis since these countries pursued a fragile economic policy for the macroeconomic characteristics of these countries

The largest number of e fficient banks in 2011 can be explained by the fact that these banks are German (3 banks) and French (2 banks), See Table 6. Data from the Statistical O ffice of the European Communities (Eurostat) show that in 2011, despite the severe sovereign debt crisis in some countries, Europe accelerates expansion through Germany and France. The two biggest heads of the European Union's economy announced quarterly and annualized growth data above all analysts' forecasts. Both countries had an increase in the Gross Domestic Product (GDP).


**Table 6.** The scores of the banks classified as e fficient at least once in 2011–2016.

Table 6 exhibits the banks that were classified as e fficient at least once in 2011–2016. The negative values of e fficient DMUs are highlighted in bold. We decide to designate DMUs by banks to make it easier to follow.

The best-ranked bank, in terms of annual performance, was Bank 7, a bank of Malta. This bank has the best performance value for the return on average assets and equity to assets factors in 2011, when compared to all others in this and other years. This is likely to be related to good risk managemen<sup>t</sup> practices necessarily implemented after the crisis. However, it also has a good performance value for the number of employees, which guarantees it to be classified as e fficient in the following years. This is likely to be related to the good risk managemen<sup>t</sup> practices necessarily implemented after the crisis (Bezzina et al. 2014).

It should be noted that, besides the Bank 7, banks that are e fficient more than once, are German banks. However, Bank 4, an Italian bank, appears to be e fficient as often as Spain's Bank 5.

In Table 7 the results of value-based DEA Formulation (2) are presented. Each DMU chooses its best feasible weights for factors to be classified as well as possible factors relative to the set of all DMUs (banks). That is, the e fficiency scores were obtained by allowing DMUs to ignore some factors from the assessment since the DMU under evaluation is free to choose the weights associated with factors (value functions) that minimize the di fference of value to the "best" DMU (bank), according to the "min-max regret" rule.


**Table 7.** Results of value-based data envelopment analysis (DEA) (efficiency score and optimal weights) and number of times as benchmarks, for efficient banks.

Considering the banks that were classified as efficient in 2011, it could be observed that most of them disregard *y*ROAA and *y*ETA (*w*<sup>∗</sup>ROAA<sup>=</sup> 0 and *<sup>w</sup>*<sup>∗</sup>*ETA* = 0). In the context of the economic and financial crisis, the profitability of banks suffered a significant reduction and, in some banks, fell to negative values. This may justify the fact that banks do not consider the return on average assets to be a factor in their evaluation; they are not "good" enough in it. Most of the efficient banks chose the *y*NLTA as a relevant factor. This factor is the one that is more often chosen for the efficiency status and only four banks disregarded it from the evaluation (Bank 7 (2011), Bank 9 (2011), Bank 2 (2013) and Bank 8 (2015)). The German banks are placed at the efficiency frontier because they are the ones with the best performances associated with the risk factor and elect it as the most prevalent.

In order to find the cases where a DMU emphasizes the pure self-evaluation, in detriment to being evaluated as an organizational unit with a balanced set of factors, it is common to use a measure which consists of recording the frequency with which this DMU appears in the peer group of other DMUs (see the last column of Table 7). The higher the number of times that a DMU belongs to the linear combination that generates the projected points of other DMUs, the more likely it will be a good performance model (Charnes et al. 1984). Thus, in the set of inefficient banks, the bank that appears most often (289 times) in the linear combination that comprises the projected point (the target) is the Bank 7 (2011). This bank is followed by Bank 8 (2015), which is the second most chosen by inefficient banks. The inefficient banks choose as peers those who form the efficient frontier, the ones that have the best practices, and those who are similar to them in the way that they want to make the smallest effort on the factors towards improvement.

The solution obtained from Formulation (3) of the value-based DEA method is a proposal of an efficiency target (projection) for each inefficient bank. To attain an efficiency status, these inefficient banks must change their value in each factor by the amount indicated by *<sup>s</sup>\**. Table 8 shows the results of Phase 2 only for the first 12 inefficient banks. It is interesting to observe that in the first 12 banks classified as inefficient, 11 were already classified as efficient in other years.


**Table 8.** Results of value-based DEA (Phase 2) for the first 12 inefficient banks.

All the banks in Table 8, being close to the efficiency frontier, need to make a small effort on the factors towards improvement. However, all need to increase the *y*ROAA. In fact, across the sample, 457 banks need to improve on this factor, considering that the same bank that has 6 years of evaluation. The positive slacks with higher average values are the ones associated with the factor *x*CIR, which may indicate that most important sources of inefficiency are the return on average assets and cost-to-income ratio.

Considering all the inefficient banks, the factor that most often appears with null slacks is the *x*SIZE (186 times). However, in 564 banks, 2/3 need to improve (reduce) also in this factor to be efficient. This result is noteworthy insofar as the banks listed in the sample are also considered the largest banks in each country and throughout this article, it is possible to verify that the size of the banks is a determinant of the profitability and consequent efficiency of the banks.

In short, the post-crisis period brought the Basel III agreement, which increased regulatory costs (Anagnostopoulos and Kabeega 2019). Thus, the cost-to-income ratio contributes to the banks' inefficiency, corresponding to the adjustment that the bank had to make after the crisis, accommodating the new regulatory costs.

The bank size and the composition of their assets appear as a promoter of efficiency and profitability and this is also in line with the post-crisis period. Small banks had more financial difficulties, as a result of capital inadequacy, and a lack of financial security margin (Anagnostopoulos and Kabeega 2019). The composition of the assets also shows that in the post-crisis period, banks with fewer impairments become more efficient.

Moreover, this study shows that the number of efficient banks remains constant in the period of the sovereign debt crisis (2011–2014) and the following two years (2015–2016). This result sugges<sup>t</sup> that banks restricted their funding to the economy (Kevork et al. 2018), making bank assets important for efficiency and profitability. Therefore, our results sugges<sup>t</sup> that the sovereign debt crisis will have consequences in this sector until 2016 and that this will naturally condition the economy.
