**3. Methodology**

#### **3.1 Data envelopment analysis (DEA)**

It is a nonparametric methodology of advanced linear programming, in which a double process of optimization is carried out, establishing the relative efficiency of Decision Making Unit—DMU, for [28] specify that this is done by generating an efficient frontier that locates the individual relative indices without having prior knowledge of the production function. More specifically, according to [29], DEA is used to evaluate the relative efficiency of a set of *n* DMUs, by posing linear programming problems for each unit according to the data of resource utilization or inputs and product or output generation. More broadly, according to [30], the

**109**

*The Colombian Banking Sector: Analysis from Relative Efficiency*

efficiency of each DMU is defined as the relationship between weighted outputs and weighted inputs, so that it is obtained by solving a double problem of linear programming in order to determine the optimum set of weights that maximizes. DEA has experienced a dynamic development, for [31] it gradually becoming a set of concepts and methodologies, which have materialized in a series of models. The first DEA model to be developed was the CRS (constant returns to scale) model that results in the categorical classification of each DMU [32]; after a few years, the VRS (variable returns to scale) model appears, through which not only constant returns can be worked [33], but also as [34] clarifies these returns can be incremental and decremental.

Consider a set composed of *n* DMU denoted as DMUj (j = 1, …, n), which uses resources xij (i = 1, …, m) and generate *s* outputs yrj (r = 1, …, s), part of that the multipliers *vi*, *ur* associated with *i* inputs and *r* outputs respectively are known. So, specifically, if the DMU0 is under study, this model is giving the solution to the problem of fractional programming for the measure of efficiency of that DMU0 as well [31]:

> *u*r y r*j* − ∑ *i*

where ε is a nonarquimidian value designated strictly positive.

 *u*r y ro / ∑ *i*

*v*i x i*j* ≤ 0, for all *j*

The theory of fractional programming expressed in [35] is applied and the fol-

The initial problem can be transformed into the following linear programming

*v*i x i0 = 1

*v*i x ij ≤ 0, for all *j*

*i*

 *μr* y *rj* − ∑ *i*

*v*i x io (1)

*μr* y *ro* (2)

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

<sup>ᧉ</sup>0 = max ∑*<sup>r</sup>*

*ur* , *vi* ≥ ε, for all *r*, *i*

μ *r* = *tur y*, *vi* = *tvi*

*t* = (∑*<sup>i</sup> v*i x i0)−1.

<sup>ᧉ</sup>0 = max ∑*<sup>r</sup>*

μ *r* , *vi* ≥ ε, for all *r*, *i*

∑

<sup>∑</sup>*<sup>r</sup>*

<sup>∑</sup>*<sup>r</sup>*

lowing changes of variables are made:

**3.2 CRS model**

Subject to:

where:

Subject to:

model:

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

efficiency of each DMU is defined as the relationship between weighted outputs and weighted inputs, so that it is obtained by solving a double problem of linear programming in order to determine the optimum set of weights that maximizes.

DEA has experienced a dynamic development, for [31] it gradually becoming a set of concepts and methodologies, which have materialized in a series of models. The first DEA model to be developed was the CRS (constant returns to scale) model that results in the categorical classification of each DMU [32]; after a few years, the VRS (variable returns to scale) model appears, through which not only constant returns can be worked [33], but also as [34] clarifies these returns can be incremental and decremental.

## **3.2 CRS model**

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

in 1812, when the City Bank of New York (today Citibank) was founded [22]. Citibank Colombia is a Citigroup franchise. The group is composed of the controlling company, Citibank Colombia S.A. and the subordinates [23]. According to [24], Citi in Colombia opened the first branch in 1916; since 1986, it has organized

according to the model of Consumer Banking and Corporate Banking.

consists of the Banco GNB Sudameris and 7 companies [25].

the two cooperatives owned), 10 foreign, and one public.

subsidiary companies.

immediately previous year.

captures places only 0.92 pesos.

**3.1 Data envelopment analysis (DEA)**

**3. Methodology**

Grupo Citibank: the history of Citibank in the world began in the United States

Grupo GNB Sudameris: in 1920, the bank was born as a Colombian mercantile company; after acquisitions and sales in 2004 Banco Sudameris acquires the majority shareholding of Banco Tequendama and Servibanca. At the beginning of 2014, the incorporation of the HSBC operation was formalized. The group currently

Grupo Corpbanca: Banco CorpBanca was created in 1997 from the merger of Banco Concepción Chileno and Banco Corp Group. Since then, it has positioned itself in the Chilean market as the fourth most important bank. For [26], in 2012 arrived at the Colombian market. Currently, Banco Corpbanca Colombia has four

Otherwise, at the December 2016 cutoff and using the SFC as an information source, it can be seen that within the Colombian financial system, there are 25 banks that represent 11% of the total of the entities in the sector, but in terms of participation in assets, banks represent 93%. For example, savings and housing cooperatives have a 78% stake in the entities, but only 2.1% participation in the sector's assets. Within the aforementioned 25 banks, there are 14 national (including

According to [27], the banking sector closed in 2016 with total assets of \$ 5 48 billion, an increase of 8.6% over the previous year and 126% in relation to the result for 2010, with an increasing trend in this period; the entities with the greatest assets at the end of 2016 are the Bancolombia, the Banco de Bogotá and Davivienda. With respect to the gross portfolio, 2016 closed with \$ 39 4 billion, representing an increase of 12% with respect to 2015 and 148% with respect to 2010, with an ever-increasing trend. By the end of 2016, the commercial portfolio of the banking sector participates with 58%, while the consumer portfolio with 27%, housing with 13%, an d microcredit with only 3%; these shares are very similar to those of the

Looking at the behavior of the liability, it is established that at the end of 2016, it is 475 billion with a growth of 9% for this last year, and between the years 2010 and 2016 with a growth of 125% with an ever-increasing trend. The ratio between the granted portfolio and the deposits of the public (savings, CDT, and current accounts) for the sector is 0.92, which represents that the sector for each peso that

It is a nonparametric methodology of advanced linear programming, in which a double process of optimization is carried out, establishing the relative efficiency of Decision Making Unit—DMU, for [28] specify that this is done by generating an efficient frontier that locates the individual relative indices without having prior knowledge of the production function. More specifically, according to [29], DEA is used to evaluate the relative efficiency of a set of *n* DMUs, by posing linear programming problems for each unit according to the data of resource utilization or inputs and product or output generation. More broadly, according to [30], the

**108**

Consider a set composed of *n* DMU denoted as DMUj (j = 1, …, n), which uses resources xij (i = 1, …, m) and generate *s* outputs yrj (r = 1, …, s), part of that the multipliers *vi*, *ur* associated with *i* inputs and *r* outputs respectively are known. So, specifically, if the DMU0 is under study, this model is giving the solution to the problem of fractional programming for the measure of efficiency of that DMU0 as well [31]:

$$\mathbf{e}\_{\diamond} = \max \sum\_{\mathsf{T}} u\_{\mathsf{r}} \mathbf{y}\_{\mathsf{ro}} / \sum\_{\mathsf{i}} v\_{\mathsf{i}} \mathbf{x}\_{\mathsf{io}} \tag{1}$$

Subject to:

$$\sum\_{\mathbf{r}} u\_{\mathbf{r}} \mathbf{y}\_{\mathbf{r}\mathbf{j}} - \sum\_{\mathbf{i}} v\_{\mathbf{i}} \mathbf{x}\_{\mathbf{i}\mathbf{j}} \preceq \mathbf{o} \text{, for all } j$$

$$u\_r, v\_i \ge e, \text{for all } r, i$$

where ε is a nonarquimidian value designated strictly positive.

The theory of fractional programming expressed in [35] is applied and the following changes of variables are made:

$$\mathfrak{u}\_r = t\mathfrak{u}\_r \mathfrak{y}\_r \mathfrak{v}\_i = \,\_r t\mathfrak{v}\_i \mathfrak{v}\_i$$

where:

$$t = \left(\sum\_{i} \nu\_{i} \mathbf{x}\_{i\odot}\right)^{-1}$$

The initial problem can be transformed into the following linear programming model:

$$\mathbf{e}\_{\diamond} = \mathbf{max} \sum\_{r} \mu\_{r} \mathbf{y}\_{rv} \tag{2}$$

Subject to:

$$\sum\_{i} \boldsymbol{\nu}\_{i} \mathbf{x}\_{i\boldsymbol{\alpha}} = \mathbf{1}$$

$$\sum\_{r} \mu\_r \mathbf{y}\_{rj} - \sum\_{i} \nu\_i \mathbf{x}\_{ij} \le \mathbf{o}, \text{for all } j$$

μ *r* , *vi* ≥ ε, for all *r*, *i*

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

### **3.3 VRS model**

According to [31] and all the above parameters, for this model we have a mathematical approach:

$$\mathbf{e}\_{\circ}{}^{\*} = \max \left[ \sum\_{r} u\_{r} \mathbf{y}\_{r\circ} - u\_{\circ} \right] / \sum\_{i} v\_{i} \mathbf{x}\_{i\circ} \tag{3}$$

Subject to:

$$\sum\_{\mathbf{r}} \boldsymbol{\mu}\_r \mathbf{y}\_{\mathbf{r}\mathbf{j}} - \boldsymbol{\mu}\_\mathbf{o} - \sum\_{\mathbf{i}} \boldsymbol{\nu}\_\mathbf{i} \mathbf{x}\_{\mathbf{i}\mathbf{j}} \preceq \odot \mathbf{j} \quad = \mathbf{1}, \ldots, m$$

$$u\_r \succeq e, \upsilon\_i \succeq e, \text{for all } i, r$$

## *uo* not restricted in sign

With its equivalent in linear programming:

$$\mathbf{e}\_o \stackrel{\*}{}=\max \sum\_{\mathbf{r}} \mu\_r \,\mathbf{y}\_{ro} - \mu\_o \tag{4}$$

Subject to:

 ∑ *i v*i x io = 1

$$
\sum\_{\tau} \mu\_{\tau} \mathbf{y}\_{\tau j} - \mu\_o - \sum\_{\mathbf{i}} \nu\_{\mathbf{i}} \mathbf{x}\_{\mathbf{i}\mathbf{j}} \preceq \odot, j = \mathbf{1}, \dots, m
$$

$$
\mu\_r \ge \varepsilon, \upsilon\_i \succeq \varepsilon, \text{ for all } i, r
$$

## μ *o* , unrestricted

The conventional measurement of DEA is based on the hypothesis that resources 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 multi-input and multi-output context.
