**4. Review of current literature**

The study of relative efficiency through DEA applied to banks is one of the most recurrent issues, so that some of the most recently published research is presented below. Ref. [37] measured the efficiency of the offices of a state bank in India by applying a DEA model based on slack. The diffuse DEA models are used in those

**111**

orientation to the outputs.

*The Colombian Banking Sector: Analysis from Relative Efficiency*

occasions in which it is considered that the accuracy of the data is not the best; for this reason, [38] used this type of model to analyze the banking sector in India. The CRS and VRS models were used by [39] in their research to measure the efficiency of commercial banks in Slovenia, Poland, Austria, Hungary, Slovenia, and Czech Republic. These two models were also applied by [40] to study the banks of Côte d'Ivoire in West Africa, for the period 2008–2010. The VRS model was also used by [41] to study 79 bank branches in Canada, only this time they used the

Islamic banks were studied by [42] comparing them with traditional banks using the Meta-Frontier Analysis (MFA) model. The Nash negotiation game model was combined with the centralized two-stage DEA model for [43] to study banks in China. Through an additive efficiency decomposition approach in DEA [44] evaluated the management and investment efficiencies of ICTs in Taiwan's banks for the 2007–2011 period. The Iranian banks were studied by [45] through the CRS model with output orientation. The research of [46] proposes a model with a multi-stage procedure that integrates robust methods, cluster analysis, and DEA to identify and

study the efficiency of management in the different branches of the banks. The main banks in Cambodia were studied by [47] through the DEA panel model for 13 years. [48] studied Taiwanese banks developing a new model based on DEA gaps to decompose their different components. Research from [49] focused on measuring the efficiency of marketing as a measure of performance after the merger, and this was investigated through DEA applied to 20 merger and acquisi-

[50] studied the relative efficiency of 23 Colombian commercial banks for a period of 10 years, with the CRS and VRS models oriented to inputs and outputs. [51] investigated the efficiency of Colombian banking from the year 2000 until 2012; they applied the VRS model with orientation to the outputs. On the other hand, [52] compared the relative efficiency of the real sector with that of the financial sector of the Colombian economy for 2014 using the VRS model oriented to the outputs.

Information source: Superintendencia Financiera de Colombia.

shows the number of banks that are part of the investigation for each year.

Determining the specific DEA model, as recommended by [55], since there is no evidence of constant returns to scale, we choose to use the DEA VRS model, and having a particular interest to evaluate how to obtain best results, we work with

Delimitation of inputs and outputs: given that DEA is a nonparametric boundary model, in which it is not necessary to previously establish the production function, and that the determinant variables of the model are the resources used (inputs) and

Delimitation of the DMU: given that the Colombian banking sector is studied, the DMUs are the banks that year after year, and from 2002 and until 2016, reported their financial statements to the SFC. Reaffirming the statement by [53], DEA is generally interpreted using the notion of production technology generated by the set of observed units. For the study period, 5 government banks reported to the SFC, because they were from the government; they withdrew from the database, accepting what was expressed by [54] as to which institutions of the Government alone are comparable to each other. Additionally, a private bank that was liquidated in 1999 also reported its financial statements, but because it was not comparable, because it was not fully operational, it was also removed from the database. **Table 1**

tion agreements within the US commercial banking.

**5. Specific methodological design**

*DOI: http://dx.doi.org/10.5772/intechopen.84585*

orientation to the inputs.

#### *The Colombian Banking Sector: Analysis from Relative Efficiency DOI: http://dx.doi.org/10.5772/intechopen.84585*

*Accounting and Finance - New Perspectives on Banking, Financial Statements and Reporting*

∑*r*

*i*

According to [31] and all the above parameters, for this model we have a math-

*ur* y r0 − *u*o]

 / ∑ *i*

*v*i x ij ≤ 0 j = 1, … , *n*

*v*i x i0 (3)

*μr* y *ro* − μ *<sup>o</sup>* (4)

**3.3 VRS model**

ematical approach:

Subject to:

Subject to:

ᧉ<sup>0</sup>

<sup>∑</sup>*<sup>r</sup>*

∗ = max [

*ur* ≥ ε, *vi* ≥ ε, for all *i*, *r*

*uo* not restricted in sign

With its equivalent in linear programming:

ᧉ<sup>0</sup>

<sup>∑</sup>*<sup>r</sup>*

∑

*ur* y r*j* − *u*o − ∑

∗

*μr* y *rj* − μ *o* − ∑

μ *r* ≥ ε, *vi* ≥ ε, for all *i*, *r*

μ *o* , unrestricted

multi-input and multi-output context.

**4. Review of current literature**

= max ∑*<sup>r</sup>*

*i*

*i*

*v*i x io = 1

The conventional measurement of DEA is based on the hypothesis that resources

The study of relative efficiency through DEA applied to banks is one of the most recurrent issues, so that some of the most recently published research is presented below. Ref. [37] measured the efficiency of the offices of a state bank in India by applying a DEA model based on slack. The diffuse DEA models are used in those

or inputs should be minimized, and products or outputs should be maximized according to [36]. Additionally, for each of these basic models, there is the orientation to the entrances and the orientation to the exits, depending on whether you want to prioritize the maximum decrease of the inputs keeping the outputs constant or the total maximization of the outputs with the constant inputs. One of the strengths of DEA is that a single efficiency result (%) is obtained for each unit, in a

*v*i x ij ≤ 0, *j* = 1, … , *n*

**110**

occasions in which it is considered that the accuracy of the data is not the best; for this reason, [38] used this type of model to analyze the banking sector in India.

The CRS and VRS models were used by [39] in their research to measure the efficiency of commercial banks in Slovenia, Poland, Austria, Hungary, Slovenia, and Czech Republic. These two models were also applied by [40] to study the banks of Côte d'Ivoire in West Africa, for the period 2008–2010. The VRS model was also used by [41] to study 79 bank branches in Canada, only this time they used the orientation to the inputs.

Islamic banks were studied by [42] comparing them with traditional banks using the Meta-Frontier Analysis (MFA) model. The Nash negotiation game model was combined with the centralized two-stage DEA model for [43] to study banks in China. Through an additive efficiency decomposition approach in DEA [44] evaluated the management and investment efficiencies of ICTs in Taiwan's banks for the 2007–2011 period. The Iranian banks were studied by [45] through the CRS model with output orientation. The research of [46] proposes a model with a multi-stage procedure that integrates robust methods, cluster analysis, and DEA to identify and study the efficiency of management in the different branches of the banks.

The main banks in Cambodia were studied by [47] through the DEA panel model for 13 years. [48] studied Taiwanese banks developing a new model based on DEA gaps to decompose their different components. Research from [49] focused on measuring the efficiency of marketing as a measure of performance after the merger, and this was investigated through DEA applied to 20 merger and acquisition agreements within the US commercial banking.

[50] studied the relative efficiency of 23 Colombian commercial banks for a period of 10 years, with the CRS and VRS models oriented to inputs and outputs. [51] investigated the efficiency of Colombian banking from the year 2000 until 2012; they applied the VRS model with orientation to the outputs. On the other hand, [52] compared the relative efficiency of the real sector with that of the financial sector of the Colombian economy for 2014 using the VRS model oriented to the outputs.
