A Study on Risk Factors Impact on the Efficiency of Commercial Banks in China—Based on Fuzzy DEA Model

Abstract

Based on the study of risk factors impact on the efficiency of commercial Banks in China, is mainly to make use of the non-performing loan ratio, capital adequacy and overdue loans to risk adjustment of input and output, and then uses the fuzzy SBM and Super-SBM model to estimate the efficiency. Results show that: the influence of risk factors on the 13 commercial banks in 2011 has varying degrees. The less risk faced by banks, the smaller influence on the efficiency affected by the risk factors. Compared with the state-owned commercial banks, the risk factors have a higher influence on the joint-stock commercial banks.

1 Introduction

Banking industries in China have changed intensively after reform and opening-up. Many commercial banks transformed into joint-stock enterprises and appear on the market. The factors of external sources of finance and economic crisis all have influence on the development of banks. With the rapid development of the banking industries, the problem of poor efficiency is serious and it also brings about many questions—such as poor operation management level, lack of innovation about financial products, less competition, high bad loan ratio, nonstandard capital risk management and so on. All of these will influence the development of bank industries.

Indian commercial banks have been classified into two ownership groups (publicly owned and privately owned) by Sathye [1]. Then the measurement of efficiency is done using data envelopment analysis. The study shows that the efficiency of private sector commercial banks is lower than that of public sector banks. Hsiao et al. [2] used a fuzzy super-efficiency slack-based measure DEA to analyze the efficiency of 24 Taiwan’s commercial banks with vague characteristics. The study shows that the fuzzy super-efficiency slack-based measure of efficiency can not only characterize uncertain input variables and output variables, but also have a higher capability to evaluate bank efficiency than the conventional Fuzzy DEA approach. Eslami et al. [3] put fuzzy probabilistic constraints into classic DEA, and analyzed the data sources consist of monthly reports of 20 Iranian bank branches over a period of May 2008–February 2009. The study shows that the fuzzy super-efficiency slack-based measure of efficiency can not only characterize uncertain input variables and output variables, but also have a higher capability to evaluate bank efficiency than the conventional Fuzzy DEA approach. Compared with bank efficiency studies abroad, domestic studies started off not so early, which, moreover, lag behind in research methods together with philosophy to some degree. For example, Huang Xian et al. [8] gave an study on the technical efficiency of 13 commercial banks in China during the period 1998~2005 by applying the three-stage DEA model. When Tang Zhuangzhi [9] measured the efficiency of Chinese banks by applying DEA model, he used Factor Analysis Method to screen out suitable input-output variable, moreover put risk factor into it. Hu Han and Song Yuanliang [10] used SFA model to study Chinese commercial bank’s efficiency and risk factor’s influence on it. From the efficiency value, we will find that Chinese bank’s total efficiency level is low during sample period. From that, state-owned commercial banks’ efficiency is higher than joint-stock commercial banks’ efficiency. The study of risk factors shows that loan-deposit ratio and capital adequacy ratio have positive correlation relationship with bank’s efficiency, in addition, bad loan ratio has negative correlation relationship with bank’s efficiency.

Above-mentioned available literature about commercial bank’s efficiency all hypothesized that risk is neuter or that risk factors have so little influence on bank’s efficiency that we can ignore it. So the measurement of bank’s efficiency deviated from the true value. However, the risk itself is an uncertain message, and it also brings about capital’s fluctuation and income’s instability. It is suitable to show this uncertainty by applying vague number which is flexible structure. So we try to study risk factors influence on Chinese commercial banks’ efficiency by using fuzzy DEA model.

2 Fuzzy DEA Model

From the econometrics view, conventional DEA method is a pure linear programming process, moreover, compared with classical DEA, fuzzy DEA obviously increase calculate amount because fuzzy DEA fuzzify the index which is quantification by using L-R vague number and triangle vague number. In addition, the efficiency from conventional DEA is simple maximum value, but the efficiency from fuzzy DEA is a minimum one among the maximum value, and it is a precise maximum value.

2.1 Basic Definition of Fuzzy Number

Definition 1

Assuming X is an Universe of Discourse, $\tilde{A}$ a fuzzy set in X, is shown by a real-valued function which is subject to $X,\mu_{{\tilde{A}}} :X \to [0,1], x \to \mu_{{\tilde{A}}} (x)$, for $x \in X$, the function $\mu_{{\tilde{A}}}$ is a subjection function of $\tilde{A}$. The function value of $\mu_{{\tilde{A}}} (x)$ is called the membership of x for $\tilde{A}$, denoted as: $\tilde{A} = \left\{ {(x,\mu_{{\tilde{A}}} (x)) |x \in X} \right\}$.

Definition 2

Assuming R is real numbers set, and $\tilde{A}$ is fuzzy set, if for any real number $< y < z$, there are $\mu_{{\tilde{A}}} \ge min\left[ {\mu_{{\tilde{A}(x)}} ,\mu_{{\tilde{A}(z)}} } \right]$, so we call $\tilde{A}$ as convex fuzzy set on R.

Definition 3

Assuming the domain of discourse of X = R is real number set, if

• (1) $\tilde{A}$ is a convex fuzzy set on X;

• (2) $\exists \,a_{1} ,a_{2} \in X\, {{\rm and}}\, a_{1} \le a_{2}$, subject to $\forall x \in [a_{1} ,a_{2} ],\tilde{A}(x) = 1,\,\lim_{x \to + \infty } \tilde{A}(x) = 0 ,$ then $\tilde{A}$ is a fuzzy number on X.

Definition 4

If $\tilde{A} = [a_{1} , a_{M} , a_{2} ],\,0 \le a_{1} \le a_{M} \le a_{2}$, then call $\tilde{A} = [a_{1} , a_{M} , a_{2} ]$ the triangular fuzzy number. The membership function of $\tilde{A}$ can be represented (Van Laarhoven and Pedrycz [11]):

$$\alpha = \mu_{{\tilde{A}}} (x) = \left\{ {\begin{array}{*{20}c} {\frac{{x - a_{1} }}{{a_{M} - a_{1} }}, a_{1} \le x \le a_{M} } \\ {\frac{{x - a_{2} }}{{a_{M} - a_{2} }}, a_{M} \le x \le a_{2} } \\ {0,\quad others} \\ \end{array} } \right.$$

(1)

This article mainly use input-output variable to construct triangular fuzzy number, transforming triangular fuzzy number to interval number by applying α–cuts, then getting the result by using interval DEA model.

2.2 Constructing Fuzzy Number

By applying classical BCC model, we not only can measure the efficiency of every decision unite, but also can get the input differential value of every DMU. We use SFA model to separate the influence of risk factor and stochastic interference factor on input differential value, then have a risk-adjustment on input variable, at last, do a fuzzy DEA analysis on input variables’ fuzzy number.

If there are n decision unites, every decision unite have m input methods, and there are p risk variables having influence on input differential value. The input differential regression function is:

$$S_{ij} = f^{i} \left( {z_{j} ;\beta^{i} } \right) + v_{ij} + u_{ij} , i = 1,2, \ldots m;j = 1,2, \ldots ,n$$

(2)

In function (2), $S_{ij}$ represent i th input’s difference of j th decision making unit; $z_{j} = (z_{1j} ,z_{2j} , \ldots , z_{pj} )$ represent p exogenous risk variable, $\beta^{j}$ is unestimated parameter of risk explanatory variable; $f^{i} (z_{j} ;\beta^{i} )$ shows the function relationship between risk variable and input difference. $v_{ij}$ is ith input’s random term of jth DMU, $u_{ij}$ is risk inefficiency’s nonnegative random variable of ith input of jth DMU. We assume $v_{ij}$ complying with normal distribution $(0,\sigma_{vi}^{2} )$, while $u_{ij}$ obeying truncated normal distribution $N^{ + } \left( {\mu^{j} , \sigma_{ui}^{2} } \right)$, in addition $v_{ij}$ and $u_{ij}$ are mutual independence.

By using Frontier Version software, we estimate the unknown parameter of function (2) by applying Maximum likelihood estimate. Meanwhile, we can estimate $\hat{E}[u_{ij} /v_{ij} + u_{ij} ]$ by using the method of Jondrow et al. [4]. Based on this, the value estimator of $v_{ij}$ is:

$$\hat{E}\left[ {\frac{{v_{ij} }}{{v_{ij} + u_{ij} }}} \right] = S_{ij} - z_{j} \hat{\beta }^{i} - \hat{E}[u_{ij} /(v_{ij} + u_{ij} )]$$

(3)

This paper uses estimated parameter to have a risk-adjustment on every DMU, the principle is to adjust all of the DMU into a same risk condition, at the same time, consider of the influence of stochastic disturbance term, so we can get the efficiency which reflect decision unite in risk condition. The adjustment expression is:

$$x_{ij}^{U} = x_{ij} + \left\{ {max_{j} \left[ {z_{j} \hat{\beta }^{i} } \right] - z_{j} \hat{\beta }^{i} } \right\} + \left\{ {max_{j} [\hat{v}_{ij} ] - \hat{v}_{ij} } \right\}$$

(4)

$$x_{ij}^{L} = x_{ij} - \left\{ {z_{j} \hat{\beta }^{i} - min_{j} [z_{j} \hat{\beta }^{i} ]} \right\} - \{ \hat{v}_{ij} - min_{j} [\hat{v}_{ij} ]\}$$

(5)

Among it, $x_{ij}$ is section i’s real value of section j of DMU, $\hat{\beta }^{i}$ is the estimator of risk explanatory variable, $\hat{v}_{ij}$ is the estimator of stochastic disturbance term.

In Eq. (4), the first bracket represent an adjustment of risk variable, and it makes all sample banks face the same great risk condition. Among Eq. (5), the first bracket also represent an adjustment of risk variable, but it makes all sample banks face the same low risk. The second bracket of the two equations depicts the adjustment of stochastic disturbance term, and all the sample banks are confronted with a same random error because of it. $x_{ij}^{U}$ represents input which has been adjust by enlarging risk, $x_{ij}^{L}$ represents input which has been adjust by decreasing risk, then we can build up triangular fuzzy number as $\left[ {x_{ij}^{L} , x_{ij} ,x_{ij}^{U} } \right]$ for every input variable.

2.3 SBM Model and It’s Solution in Fuzzy Condition

Fuzzy DEA model expands based on classical DEA models, so different classical DEA model has different fuzzy DEA model. This article chooses non-radial SBM model to study, so conventional CCR model and BCC model have incomplete measurement. Their model either is input-oriented or output-oriented, so we can only get input efficiency or output efficiency of $DMU_{s}$ while can not get the measurement of input and output efficiency at the same time, and their efficiency index miss the slack quantity of nonzero input-output, therefore, they can’t get the inefficiency of all $DMU_{s}$. While non-radial SBM model directly put the slack quantity input and output into objective function and constraint function, so on the one hand, it solves the measurement problem of slack quantity of input and output, on the other hand, it gives the efficiency based on input and output, so the estimate of efficiency about fuzzy $DMU_{s}$ is more perfect and more reasonable.

2.3.1 Non-radial SBM Model

SBM model, which is proposed by Tone [5], is used to measure non-radial efficiency of decision making unite. We assume there are n $DMU_{s}$, each DMU with m inputs, and s outputs. Based on this description, the efficiency evaluation model could be expressed:

$$\begin{array}{*{20}l} & {\text{Min}} \qquad \quad\,\, {\rho_{k} = \frac{{1 - \frac{1}{m}\mathop \sum \nolimits_{i = 1}^{m} \frac{{s_{i}^{ - } }}{{x_{ik} }}}}{{1 + \frac{1}{s}\mathop \sum \nolimits_{r = 1}^{s} \frac{{s_{r}^{ + } }}{{y_{rk} }}}}} \hfill \\ \hfill & \begin{aligned} {{\text{Subject}}\,{\text{to}}} \quad x_{ik} = \sum\nolimits_{j = 1}^{n} {x_{ij} \lambda_{j} + s_{i}^{ - } , i = 1,2, \ldots ,m} \hfill \\ y_{rk} = \sum\nolimits_{j = 1}^{n} {y_{rj} \lambda_{j} - s_{r}^{ + } , r = 1,2, \ldots ,s} \hfill \\ \end{aligned} \hfill \\ \end{array}$$

(6)

$$\begin{array}{*{20}l} {\sum\nolimits_{j = 1}^{n} {\lambda_{j} = 1} } \hfill \\ {\lambda_{j} \ge 0 , j = 1,2, \ldots ,n} \hfill \\ {s_{i}^{ - } \ge 0 , s_{r}^{ + } \ge 0} \hfill \\ \end{array}$$

where $(x_{ik} ,y_{rk} )$ represents the kth DMU, the ith input, and rth outputs. The $\lambda_{j}$ is represented by the kth DMU weighting for evaluating efficiency. The $s_{i}^{ - }$ is represented by input slack variable, while $s_{r}^{ + }$ is represented by output slack variable. Model (6) is difficult to calculate, so we can multiply non-negative scalar variable q to the denominator a numerator, and assume that the objective function denominator is equal to 1. As such, Model (6) forms a simple linear programming similar to Model (7):

$$\begin{array}{*{20}l} {\text{Min}} \hfill & {\tilde{\rho }_{k} = q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{S_{i}^{ - } }}{{x_{ik} }}} } \hfill \\ {\text{Subject to}} \hfill & {\begin{array}{*{20}l} {q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{S_{r}^{ + } }}{{y_{rk} }} = 1} } \hfill \\ {\begin{array}{*{20}l} {qx_{ik} = \sum\nolimits_{j = 1}^{n} {x_{ij} \lambda_{j}^{{\prime }} + S_{i}^{ - } , i = 1, 2, \ldots ,m} } \hfill \\ {qy_{rk} = \sum\nolimits_{j = 1}^{n} {y_{rj} \lambda_{j}^{{\prime }} - S_{r}^{ + } , r = 1, 2, \ldots ,s} } \hfill \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array}$$

(7)

$$\begin{array}{*{20}l} {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{\prime }} = q ,q > 0} } \hfill \\ {\lambda_{j}^{{\prime }} = q\lambda_{j} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {S_{i}^{ - } \ge 0 , S_{i}^{ - } = qs_{i}^{ - } } \hfill \\ {S_{r}^{ + } \ge 0 , S_{r}^{ + } = qs_{r}^{ + } } \hfill \\ \end{array}$$

When the optimum solution of Model (7) $\tilde{\rho }_{k}^{*} = 1, S_{i}^{ - *} = S_{r}^{ + *} = 0,$ the efficiency of decision making unite is 1, and there aren’t input excess and output shortfall, $DMU_{k}$ is SBM’s efficiency.

2.3.2 Fuzzy Expand and Solution of SBM Model

Among a $DMU_{s}$ set, in order to represent fuzzy input and output variables, we used $\mu_{{\tilde{x}_{ij} }} ,$ $\mu_{{\tilde{y}_{ij} }}$ as their membership functions, such that $\tilde{x}_{ij} ,\tilde{y}_{rj}$ are the fuzzy number of input and output, respectively. Usually, we assume all variables as fuzzy number, while if some variables are crisp data, then we can use degenerated member functions to describe them, such that this range will only have one value. Fuzzy SBM model is as follows:

$$\begin{array}{*{20}l} & {\text{Min}} \hfill {\tilde{\rho }_{k} = q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{S_{i}^{ - } }}{{\tilde{x}_{ik} }}} } \hfill \\ & {\begin{array}{*{20}l}{\text{Subject to}} \hfill {q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{S_{r}^{ + } }}{{\tilde{y}_{rk} }} = 1} } \hfill \\ \qquad \qquad\,\, {q\tilde{x}_{ik} = \sum\nolimits_{j = 1}^{n} {\tilde{x}_{ij} \lambda_{j}^{{\prime }} + S_{i}^{ - } , i = 1, 2, \ldots ,m} } \hfill \\ \end{array} } \hfill \\ \end{array}$$

(8)

$$\begin{array}{*{20}l} {q\tilde{y}_{rk} = \sum\nolimits_{j = 1}^{n} {\tilde{y}_{rj} \lambda_{j}^{{\prime }} - S_{r}^{ + } , r = 1, 2, \ldots ,s} } \hfill \\ {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{\prime }} = q ,q > 0} } \hfill \\ {\lambda_{j}^{{\prime }} = q\lambda_{j} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {S_{i}^{ - } \ge 0 , S_{i}^{ - } = qs_{i}^{ - } } \hfill \\ {S_{r}^{ + } \ge 0 , S_{r}^{ + } = qs_{r}^{ + } } \hfill \\ \end{array}$$

We assume $(\tilde{x}_{ij} ),S(\tilde{y}_{rj} )$ are $\tilde{x}_{ij} ,\tilde{y}_{rj}$’s support sets, the $\tilde{x}_{ij}$ and $\tilde{y}_{rj}$ y α–cut is therefore defined as:

$$(X_{ij} )_{\alpha } = \left\{ {x_{ij} \in S\left( {\tilde{x}_{ij} } \right) |\mu_{{\tilde{x}_{ij} }} (x_{ij} ) \ge \alpha } \right\} , \forall j, i$$

(9)

$$(Y_{rj} )_{\alpha } = \left\{ {y_{rj} \in S\left( {\tilde{y}_{rj} } \right) |\mu_{{\tilde{y}_{rj} }} (y_{rj} ) \ge \alpha } \right\} , \forall r, j$$

(10)

Thus, fuzzy inputs and outputs can be represented in intervals at different α–cut levels $\left\{ {(\tilde{x}_{ij} )_{\alpha } |0 < \alpha \le 1} \right\}$ and $\left\{ {\left( {\tilde{y}_{rj} } \right)_{\alpha } |0 < \alpha \le 1} \right\}$. Based on the above, we can express (Eqs. (9) and (10) as Model (11) and Model (12)):

$$\begin{aligned} (X_{ij} )_{\alpha } & = \left\{ {x_{ij} \in S\left( {\tilde{x}_{ij} } \right) |\mu_{{\tilde{x}_{ij} }} (x_{ij} ) \ge \alpha } \right\} = \left[ {(X_{ij} )_{\alpha }^{L} , (X_{ij} )_{\alpha }^{U} } \right] \\ & = \left[ {min_{{x_{ij} }} \left\{ {x_{ij} \in S(\tilde{x}_{ij} ) |\mu_{{\tilde{x}_{ij} }} (x_{ij} ) \ge \alpha } \right\} , max_{{x_{ij} }} \left\{ {x_{ij} \in S(\tilde{x}_{ij} ) |\mu_{{\tilde{x}_{ij} }} (x_{ij} ) \ge \alpha } \right\}} \right] \\ \end{aligned}$$

(11)

$$\begin{aligned} (Y_{rj} )_{\alpha } & = \left\{ {y_{rj} \in S(\tilde{y}_{rj} ) |\mu_{{\tilde{y}_{rj} }} (y_{rj} ) \ge \alpha } \right\} = [\left( {Y_{rj} )_{\alpha }^{L} , (Y_{rj} )_{\alpha }^{U} } \right] \\ & = \left[min_{{y_{rj} }} \left\{ {y_{rj} \in S(\tilde{y}_{rj} ) |\mu_{{\tilde{y}_{rj} }} (y_{rj} ) \ge \alpha } \right\} , max_{{y_{rj} }} \left\{ {y_{rj} \in S(\tilde{y}_{rj} ) |\mu_{{\tilde{y}_{rj} }} (y_{rj} ) \ge \alpha } \right\}\right] \\ \end{aligned}$$

(12)

This article assume input as triangular fuzzy number $X_{ij} = \left( {x_{ij}^{L} , x_{ij}^{M} , x_{ij}^{U} } \right)$, output as $Y_{rj} = \left( {y_{rj}^{L} , y_{rj}^{M} , y_{rj}^{U} } \right)$, its corresponding α–cut sets based on given α–level are:

$$(X_{ij} )_{\alpha } = \left[ {x_{ij}^{L} + \alpha \left( {x_{ij}^{M} - x_{ij}^{L} } \right), x_{ij}^{U} - \alpha \left( {x_{ij}^{U} - x_{ij}^{M} } \right)} \right]$$

(13)

$$(Y_{rj} )_{\alpha } = \left[ {y_{rj}^{L} + \alpha \left( {y_{rj}^{M} - y_{rj}^{L} } \right), y_{rj}^{U} - \alpha \left( {y_{rj}^{U} - y_{rj}^{M} } \right)} \right]$$

(14)

Moreover, we can define the $DMU_{k}$ membership functions using Zadeh’s (1978) extension principle, as seen in Model (15):

$$\mu_{{\tilde{E}_{k} }} (z) = \sup\nolimits_{x,y} min\left\{ {\mu_{{\tilde{x}_{ij} }} (x_{ij} ), \mu_{{\tilde{y}_{rj} }} (y_{rj} ), \forall j, r, i |z = E_{k} (x, y)} \right\}$$

(15)

where $E_{k} (x, y)$ is represented by the kth DMU efficiency score at a set of $(x, y)$ by conventional SBM model and the $\mu_{{\tilde{E}_{k} }}$ is the minimum of $\left( {\mu_{{\tilde{x}_{ij} }} (x_{ij} ), \mu_{{\tilde{y}_{rj} }} (y_{rj} )} \right)$. From Model (15), we need $\mu_{{\tilde{x}_{ij} }} (x_{ij} ) \ge \alpha , \mu_{{\tilde{y}_{rj} }} (y_{rj} ) \ge \alpha$ and at least one $\mu_{{\tilde{x}_{ij} }} (x_{ij} )$ or $\mu_{{\tilde{y}_{rj} }} (y_{rj} )$ equal to α, such that $E_{k} \left( {x, y} \right)$ satisfies $\mu_{{\tilde{E}_{k} }} \left( z \right) = \alpha$. Based on given α level, all α–cut sets form a nested structure, when $0 < \alpha_{2} < \alpha_{1} < 1$, there are:

$\left[ {\left( {X_{ij} } \right)_{{\alpha_{1} }}^{L} , \left( {X_{ij} } \right)_{{\alpha_{1} }}^{U} } \right] \subseteq \left[ {\left( {X_{ij} } \right)_{{\alpha_{2} }}^{L} , \left( {X_{ij} } \right)_{{\alpha_{2} }}^{U} } \right],\left[ {\left( {Y_{rj} } \right)_{{\alpha_{1} }}^{L} ,\left( {Y_{rj} } \right)_{{\alpha_{1} }}^{U} } \right] \subseteq \left[ {\left( {Y_{rj} } \right)_{{\alpha_{2} }}^{L} ,\left( {Y_{rj} } \right)_{{\alpha_{2} }}^{U} } \right]$, so $\mu_{{x_{ij} }} \left( {x_{ij} } \right) \ge \alpha$ and $\mu_{{\tilde{x}_{ij} }} \left( {x_{ij} } \right) \ge \alpha$ have a same domain and the same with $\mu_{{\tilde{y}_{rj} }} \left( {y_{rj} } \right) \ge \alpha$ and $\mu_{{\tilde{y}_{rj} }} \left( {y_{rj} } \right) \ge \alpha$.

Based on the above formulas, we can find that $\mu_{{\tilde{E}_{k} }}$ has a lower and upper bound by α–cut, as seen in Model (16) and Model (17):

$$\left( {\rho_{k} } \right)_{\alpha }^{U} = { \hbox{max} }_{{\begin{array}{*{20}c} {\left( {X_{ij} } \right)_{\alpha }^{L} \le x_{ij} \le \left( {X_{ij} } \right)_{\alpha }^{U} } \\ {\left( {Y_{rj} } \right)_{\alpha }^{L} \le y_{rj} \le \left( {Y_{rj} } \right)_{\alpha }^{U} } \\ {\forall j,i,r} \\ \end{array} }} \left\{ {\begin{array}{*{20}l} {{ \hbox{min} }\quad q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{S_{i}^{ - } }}{{x_{ik} }}} } \hfill \\ {subject\,to\quad q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{S_{r}^{ + } }}{{y_{rk} }} = 1} } \hfill \\ {qx_{ik} = \sum\nolimits_{j = 1}^{n} {x_{ij} \lambda_{j}^{{\prime }} + S_{i}^{ - } , i = 1, 2, \ldots ,m} } \hfill \\ {qy_{rk} = \sum\nolimits_{j = 1}^{n} {y_{rj} \lambda_{j}^{{\prime }} - S_{r}^{ + } , r = 1, 2, \ldots ,s} } \hfill \\ {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{\prime }} = q ,q > 0} } \hfill \\ {\lambda_{j}^{{\prime }} = q\lambda_{j} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {S_{i}^{ - } \ge 0,S_{r}^{ + } \ge 0} \hfill \\ \end{array} } \right.$$

(16)

$$\left( {\rho_{k} } \right)_{\alpha }^{L} = { \hbox{min} }_{{\begin{array}{*{20}c} {\left( {X_{ij} } \right)_{\alpha }^{L} \le x_{ij} \le \left( {X_{ij} } \right)_{\alpha }^{U} } \\ {\left( {Y_{rj} } \right)_{\alpha }^{L} \le y_{rj} \le \left( {Y_{rj} } \right)_{\alpha }^{U} } \\ {\forall j,i,r} \\ \end{array} }} \left\{ {\begin{array}{*{20}l} {{ \hbox{min} }\quad q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{S_{i}^{ - } }}{{x_{ik} }}} } \hfill \\ {subject\,to \quad q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{S_{r}^{ + } }}{{y_{rk} }} = 1} } \hfill \\ {qx_{ik} = \sum\nolimits_{j = 1}^{n} {x_{ij} \lambda_{j}^{{\prime }} + S_{i}^{ - } , i = 1, 2, \ldots ,m} } \hfill \\ {qy_{rk} = \sum\nolimits_{j = 1}^{n} {y_{rj} \lambda_{j}^{{\prime }} - S_{r}^{ + } , r = 1, 2, \ldots ,s} } \hfill \\ {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{\prime }} = q ,q > 0} } \hfill \\ {\lambda_{j}^{{\prime }} = q\lambda_{j} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {S_{i}^{ - } \ge 0,S_{r}^{ + } \ge 0 } \hfill \\ \end{array} } \right.$$

(17)

Models (16) and (17) are two-stage mathematical programming model, so we should apply the formula of Model (16) and (17) to a classical one-stage mathematical programming model. If we attempt to ascertain the “minimum” efficiency of specific DMU, we should use specific DMU lower bound outputs and other DMUs with lower bound inputs, and specific DMU upper bound inputs and other DMUs with upper bound outputs. Alternatively, we can find the specific decision making units to have the “maximum” efficiency, we should take the specific decision making unit’s upper bound outputs and others’ DMUs upper bound inputs, and the specific DMU lower bound inputs and others’ DMU lower bound outputs. Then, the Model (16) and Model (17) can be transformed as follows:

$$\begin{array}{*{20}l}& {\text{Min}} \hfill {\left( {\rho_{k} } \right)_{\alpha }^{U} = q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{\left( {S_{i}^{ - } } \right)^{L} }}{{(x_{ik} )_{\alpha }^{L} }}} } \hfill \\ \hfill & {\begin{array}{*{20}l} {\text{Subject to}}\hfill {q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{\left( {S_{r}^{ + } } \right)^{U} }}{{\left( {y_{rk} } \right)_{\alpha }^{L} }} = 1} } \hfill \\ \quad\quad \quad \quad \quad \quad \quad \quad {\begin{array}{*{20}l} {q(x_{ik} )_{\alpha }^{L} = \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {(x_{ij} )_{\alpha }^{U} \lambda_{j}^{{\prime }} + (x_{ik} )_{\alpha }^{L} \lambda_{k}^{{\prime }} + \left( {S_{i}^{ - } } \right)^{L} , i = 1, 2, \ldots ,m} } \hfill \\ {q\left( {y_{rk} } \right)_{\alpha }^{L} = \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {y_{rj} } \right)_{\alpha }^{L} \lambda_{j}^{{\prime }} + \left( {y_{rk} } \right)_{\alpha }^{U} \lambda_{k}^{{\prime }} - \left( {S_{r}^{ + } } \right)^{U} , r = 1, 2, \ldots ,s} } \hfill \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array}$$

(18)

$$\begin{array}{*{20}l} {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{\prime }} = q ,q > 0} } \hfill \\ {\lambda_{j}^{{\prime }} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {\left( {S_{i}^{ - } } \right)^{L} \ge 0,\left( {S_{r}^{ + } } \right)^{U} \ge 0} \hfill \\ \end{array}$$

$$\begin{array}{*{20}l}& {\text{Min}} \hfill {\left( {\rho_{k} } \right)_{\alpha }^{L} = q - \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{\left( {S_{i}^{ - } } \right)^{U} }}{{\left( {x_{ik} } \right)_{\alpha }^{U} }}} } \hfill \\ \hfill & {\begin{array}{*{20}l} {\text{Subject to}}\hfill {q + \frac{1}{S}\sum\nolimits_{r = 1}^{S} {\frac{{\left( {S_{r}^{ + } } \right)^{L} }}{{\left( {y_{rk} } \right)_{\alpha }^{L} }} = 1} } \hfill \\ {\begin{array}{*{20}l} {q\left( {x_{ik} } \right)_{\alpha }^{U} = \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {(x_{ij} )_{\alpha }^{L} \lambda_{j}^{{\prime }} + (x_{ik} )_{\alpha }^{U} \lambda_{k}^{{\prime }} + \left( {S_{i}^{ - } } \right)^{U} , i = 1, 2, \ldots ,m} } \hfill \\ {q\left( {y_{rk} } \right)_{\alpha }^{L} = \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {y_{rj} } \right)_{\alpha }^{U} \lambda_{j}^{{\prime }} + \left( {y_{rk} } \right)_{\alpha }^{L} \lambda_{k}^{{\prime }} - \left( {S_{r}^{ + } } \right)^{L} , r = 1, 2, \ldots ,s} } \hfill \\ \end{array} } \hfill \\ \end{array} } \hfill \\ \end{array}$$

(19)

$$\begin{array}{*{20}l} {\sum\nolimits_{j = 1}^{n} {} \lambda_{j}^{{\prime }} = q ,q > 0} \hfill \\ {\lambda_{j}^{{\prime }} \ge 0 , j = 1, 2, \ldots , n} \hfill \\ {\left( {S_{i}^{ - } } \right)^{U} \ge 0,\left( {S_{r}^{ + } } \right)^{L} \ge 0} \hfill \\ \end{array}$$

The optimal value $\left( {\rho_{k} } \right)_{\alpha }^{L}$ and $\left( {\rho_{k} } \right)_{\alpha }^{U}$ of Model (18) and Model (19) formulate $DMU_{k}$’s interval efficiency $\left( {\rho_{k} } \right)_{\alpha } = \left[ {\left( {\rho_{k} } \right)_{\alpha }^{L} ,\left( {\rho_{k} } \right)_{\alpha }^{U} } \right]$ on given α–cut level.

According to the interval efficiency on given α–cut level, we can classify all the fuzzy decision making units $DMU_{k} \left( {k = 1, 2, \ldots , n} \right)$ into three classes:

• (1) $E_{\alpha }^{ + } = \left\{ {DMU_{k} |\left( {\rho_{k} } \right)_{\alpha }^{L} = 1, k = 1, 2, \ldots ,n} \right\}$, if $DMU_{k} \in E_{\alpha }^{ + }$, we say that it is fuzzy DEA efficiency on α–cut level;

• (2) $E_{\alpha } = \left\{ {DMU_{k} |\left( {\rho_{k} } \right)_{\alpha }^{L} < 1,\left( {\rho_{k} } \right)_{\alpha }^{U} = 1, k = 1, 2, \ldots , n } \right\}$, if $DMU_{k} \in E_{\alpha }$, we call that $DMU_{k}$ is fuzzy DEA efficiency on α–cut level;

• (3) $E_{\alpha }^{ - } = \left\{ {DMU_{k} |\left( {\rho_{k} } \right)_{\alpha }^{U} = 1, k = 1, 2, \ldots , n} \right\}$, if $DMU_{k} \in E_{\alpha }^{ - }$, we say that it is fuzzy DEA inefficiency on α–cut level.

2.4 Super-SBM Model and Its Solution in Fuzzy Condition

Data Envelopment Analysis (DEA) mainly use various input and output of decision making units to construct efficient productive frontier, so we can measure the same kind decision making units’ relative efficiency, and this is a nonparametric method. When using classical DEA models, we may find that many decision making units’ efficiency are 1, thus, we can’t have an efficiency order on them. In order to solve this problem, Andersen and Ptersen proposed Super-efficiency DEA in [12]. The difference between this model and conventional DEA model (CCR Model and BCC Model) is: classical DEA model, which put the linear combination of all decision making units as reference set of model evaluation, while Super-efficiency DEA model’s reference set removes unvalued decision making units, consisting by the linear combination of other decision making units. So, when there are two or more than two decision making units (the efficiency value is 1), the measurement of inefficient units’ efficiency is consistent between Super-efficiency DEA model and classical DEA model. While for the efficient decision making units, their efficiency can exceed 1, so the efficiency is different, thus we can have a complete order for all the decision making units.

2.4.1 Super-SBM Model

Assuming a production possibility set ${\text{P}}\left( {x_{0} , y_{0} } \right)$ which covers $\left( {X,Y} \right)$ but excluding $\left( {x_{0} , y_{0} } \right)$ and $n$ $DMU_{s}$, each with m inputs and s outputs, for the kth decision unit, its efficiency rating model is:

$$\begin{array}{*{20}l}& {\text{Min}} \hfill {\tau_{k} = \frac{{\frac{1}{m}\mathop \sum \nolimits_{i = 1}^{m} \frac{{\bar{x}_{i} }}{{x_{ik} }}}}{{\frac{1}{s}\mathop \sum \nolimits_{r = 1}^{s} \frac{{\bar{y}_{r} }}{{y_{rk} }}}}} \hfill \\ \hfill & {\begin{array}{*{20}l}{\text{Subject to}}\hfill {\bar{x}_{i} { \ge }\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {x_{ij} \lambda_{j} , i = 1, 2, \ldots m} } \hfill \\ \qquad \qquad \,\, {\bar{y}_{r} \le \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {y_{rj} \lambda_{j} , r = 1, 2, \ldots , s} } \hfill \\ \end{array} } \hfill \\ \end{array}$$

(20)

$$\begin{array}{*{20}l} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {} \lambda_{j} = 1, j = 1, 2, \ldots , n} \hfill \\ {\bar{x}_{i} > x_{ij} ,0 \le \bar{y}_{r} \le y_{rk} ,\lambda_{j} \ge 0} \hfill \\ \end{array}$$

where $\left( {\bar{x},\bar{y}} \right) \in {\text{P}}\backslash \left( {x_{k} ,y_{k} } \right) \cap \left\{ {\bar{x} \ge x_{k} , \bar{y} \le y_{k} } \right\}$. Since Model (20) offers fractional linear programming, it may produce an infinite number of solutions. Tone [6] solved the above problem through multiplying non-negative scalar variable q for the denominator and numerator, assuming that the objective function of the denominator is equal to 1, so Model (20) transforms a simple linear programming similar to Model (21):

$$\begin{array}{*{20}l}& {\text{Min}} \hfill {\tilde{\tau }_{k} = \frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{\bar{x}_{i}^{{\prime }} }}{{\tilde{x}_{ik} }}} } \hfill \\ & {\begin{array}{*{20}l} {\begin{array}{*{20}l}{\text{Subject to}} \hfill {\frac{1}{s}\sum\nolimits_{r = 1}^{s} {\frac{{\bar{y}_{r}^{{\prime }} }}{{\tilde{y}_{rk} }} = 1} } \hfill \\ {\bar{x}_{i}^{{\prime }} \ge \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\tilde{x}_{ij} \lambda_{j}^{{\prime }} , i = 1, 2, \ldots , m} } \hfill \\ \end{array} } \hfill \\ {\bar{y}_{r}^{{\prime }} \le \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\tilde{y}_{rj} \lambda_{j}^{'} , r = 1, 2, \ldots , } s} \hfill \\ \end{array} } \hfill \\ \end{array}$$

(21)

$$\begin{array}{*{20}l} {\sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\lambda_{j}^{{\prime }} = q, j = 1, 2, \ldots , n} } \hfill \\ {\bar{x}_{i}^{{\prime }} > \tilde{x}_{ij} ,0 \le \bar{y}_{r}^{{\prime }} \le \tilde{y}_{rj} ,\lambda_{j}^{{\prime }} \ge 0} \hfill \\ \end{array}$$

2.4.2 Super-SBM Model’s Fuzzy Expand and Solution

In a $DMU_{s}$ set, we assume input $\tilde{x}_{ij}$ and output $\tilde{y}_{rj}$ are approximate available, also, $\tilde{x}_{ij} , \tilde{y}_{rj}$ of the membership functions are defined as $\mu_{{\tilde{x}_{ij} }} ,\mu_{{\tilde{y}_{rj} }}$. According to the process of fuzzy expanding and solution of SBM model, we can get the optimal efficiency interval of $DMU_{k}$ based on α–cut level:

$$\begin{aligned} \left( {\tau_{k} } \right)_{\alpha }^{U} & = min\frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{\left( {\bar{x}_{i}^{{\prime }} } \right)^{L} }}{{\left( {x_{ik} } \right)_{\alpha }^{L} }}} \\ {\text{Subject to}}\quad & \frac{1}{s}\sum\nolimits_{r = 1}^{s} {\frac{{\left( {\bar{y}_{r}^{{\prime }} } \right)^{U} }}{{\left( {y_{rk} } \right)_{\alpha }^{U} }} = 1} \\ \left( {\bar{x}_{i}^{{\prime }} } \right)^{L} & \ge \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {x_{ik} } \right)_{\alpha }^{L} \lambda_{j}^{{\prime }} , i = 1, 2, \ldots , m} \\ \left( {\bar{y}_{r}^{{\prime }} } \right)^{U} & \le \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {y_{rk} } \right)_{\alpha }^{U} \lambda_{j}^{{\prime }} , r = 1, 2, \ldots , s} \\ & \quad \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\lambda_{j}^{{\prime }} = q > 0, j = 1, 2, \ldots , n} \\ \left( {\bar{x}_{i}^{{\prime }} } \right)^{L} & \ge q\left( {x_{ik} } \right)_{\alpha }^{L} , 0 \le \left( {\bar{y}_{r}^{{\prime }} } \right)^{U} \le q\left( {y_{rk} } \right)_{\alpha }^{U} , \lambda_{j}^{{\prime }} \ge 0 \\ \end{aligned}$$

(22)

$$\begin{aligned} \left( {\tau_{k} } \right)_{\alpha }^{L} & = min\frac{1}{m}\sum\nolimits_{i = 1}^{m} {\frac{{\left( {\bar{x}_{i}^{{\prime }} } \right)^{U} }}{{\left( {x_{ik} } \right)_{\alpha }^{U} }}} \\ {\text{Subject to}} & \quad \frac{1}{s}\sum\nolimits_{r = 1}^{s} {\frac{{\left( {\bar{y}_{r}^{{\prime }} } \right)^{L} }}{{\left( {y_{rk} } \right)_{\alpha }^{L} }} = 1} \\ \left( {\bar{x}_{i}^{{\prime }} } \right)^{U} & \ge \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {x_{ik} } \right)_{\alpha }^{U} \lambda_{j}^{{\prime }} , i = 1, 2, \ldots , m} \\ \left( {\bar{y}_{r}^{{\prime }} } \right)^{L} & \le \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\left( {y_{rk} } \right)_{\alpha }^{L} \lambda_{j}^{{\prime }} , r = 1, 2, \ldots , s} \\ & \quad \sum\nolimits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} {\lambda_{j}^{{\prime }} = q > 0, j = 1, 2, \ldots , n} \\ \left( {\bar{x}_{i}^{{\prime }} } \right)^{U} & \ge q\left( {x_{ik} } \right)_{\alpha }^{U} , 0 \le \left( {\bar{y}_{r}^{{\prime }} } \right)^{L} \le q\left( {y_{rk} } \right)_{\alpha }^{L} , \lambda_{j}^{{\prime }} \ge 0 \\ \end{aligned}$$

(23)

2.5 Ranking of Decision Making Units in Fuzzy Condition

It is difficult to determine the ranking, because every decision making units’ interval efficiency with α–cut level are measured in fuzzy condition. So we use the area measurement method proposed by Chen and Klien [7] to rank the fuzzy number at unknown membership functions:

$${\text{I}}\left( {\tilde{E}_{k} , \tilde{R}} \right) = \lim\nolimits_{m \to \infty } \left( {\frac{{\sum\nolimits_{i = 0}^{m} {\left[ {\left( {E_{k} } \right)_{{\alpha_{i} }}^{U} - c} \right]} }}{{\sum\nolimits_{i = 0}^{m} {\left[ {\left( {E_{k} } \right)_{{\alpha_{i} }}^{U} - c} \right]} - \sum\nolimits_{i = 0}^{m} {\left[ {\left( {E_{k} } \right)_{{\alpha_{i} }}^{L} - d} \right]} }}} \right)$$

(24)

where $c = min_{i, k} \left\{ {\left( {E_{k} } \right)_{{\alpha_{i} }} } \right\}$, and $= max_{i, k} \left\{ {\left( {E_{k} } \right)_{{\alpha_{i} }} } \right\},i = 0 , 1 , \ldots , m$. If ${\text{I}}\left( {\tilde{E}_{k} , \tilde{R}} \right)$ is higher than the represented the kth DMU ranking is higher.

3 Sample Data Introduction

3.1 The Source of Banks’ Input-Output Data

As a financing institution which provide various services, banking industry’s input-output is far different from general industry. According to existing literatures, [13] indicated that there are many methods to define banks’ input-output, such as “Intermediary Approach”, “Asset Approach”, “Value Added Approach”, “Production Approach”, and “User Cost Approach”, and the first three methods are usually been applied.

This article chooses the input-output index by applying Intermediary Approach and Asset Approach. And from abroad literatures about index selecting, we define input variable as deposit, net value of fixed assets and operating expenses; and define output variable as net credit and pretax profit.

In order to maintain consistency of statistical caliber and availability of data, this paper selects 13 listed commercial banks as sample. Including 5 state-owned banks: Agricultural Bank of China, Industrial and Commercial Bank of China, China Construction Bank, Bank of China and Bank of Communications; and 8 joint-stock commercial banks: China Citic Bank, China Minsheng Bank, Hua Xia Bank, China Everbright Bank, China Merchants Bank, Soci é t é G é n é rale, Shanghai Pudong Development Bank, and Ping An Bank. Ping An Bank came out after Shenzhen Development Bank Co., Ltd. Merged with Ping An Bank Co, SZPA at 2012. This paper uses the name of Ping An Bank to replace Shenzhen Development Bank during 2008–2011. The sample period is from the year 2008–2012, and sample data come mainly from every bank’s annual report during the year 2008–2012.

3.2 How to Select Risk Factor Variable

Credit risks are the main risk during banks’ operating activities. Commercial banks always use the bad loan ratio as one of the most important index measuring the credit risk. And it reflects the risk safety degree of banks’ credit capital. The non-performing loan of commercial banks mainly includes substandard loan, doubtful loan and loss loan. The non-performing loan ratio = year-end non-performing loan of commercial bank/total loan of commercial bank. The higher bad loan ratio, the bigger risk of credit capital comes to commercial banks, and the less chance that banks can call in this loan, relatively, the greater loss may bring to the bank, and the bank’s operating cost become bigger, so it will influence profitability and efficiency of operation.

Capital adequacy ratio = the capitalization of the bank/the weighted total assets. Capital adequacy ratio reflects how much loss the bank can undertake only depending on its own assets and shows the bank’s ability to resist the risk when the depositors’ and creditors’ assets are impaired.

Past due loans are used to depict that repayment of bank loans are not on time, and reflect the service condition about this loans. Though past due loans can’t transform to bad loan, we couldn’t make sure if it will be get back, so it will cause uncertain loss to banks. And that continuous increase of past due loans would make the market have a negative attitude to the quality of banks’ credit assets. Therefore, managing the past due loans not only can solve banks’ past due loans quickly and reduce the uncertain loss, but also can promote the market’s confidence to banks’ credit security.

So, non-performing loan ratio, capital adequacy ratio and past due loans ratio are chosen as risk factor index to adjust Chinese commercial banks’ input-output variables, then we get the efficiency by applying fuzzy DEA model.

This paper selects the research of the efficiency which has been adjusted by risk at 2011, so, we add the data of commercial banks’ non-performing loan ratio, capital adequacy ratio and past due loans ratio which collected in 2011 to it based on former sample data base.

3.3 Adjust the Risk of Input-Output Variable

This article can not only measure 13 commercial banks’ various efficiency in 2011 by applying BCC model and DEAP software, but also can get the difference about the input data. At the same time, we substitute the data which gained from every bank’s non-performing loan ratio and capital adequacy ratio in 2011 into SFA model. Then, we can get the conclude by using frontier software. The results list in Table 1.

Table 1 Parameter estimation of commercial banks’ various efficiency in 2011

From Table 1, we can know that the non-performing loan ratio has a positive influence on the three input difference, and they are all significant at 0.01 level. It indicates that the rising of non-performing loan ratio will aggravate the input difference, and the increase of the difference means the expansion of banks’ input redundancy. Because of that, banks will be at a disadvantage operating condition and its efficiency will turn down. Nevertheless, capital adequacy ratio has a negative influence on the three input difference, and they are all significant at 0.01 level. The rising of capital adequacy ratio will decrease the input difference and banks’ input redundancy. Because of that, banks’ operating condition has improved and its efficiency also enhances. All of these are in line with theory and reality. Then, we can use Model (3), Model (4) and Model (5) to calculate triangle vague number based on risk adjustment. Moreover, the method to adjust output risk is to fuzzify net credit, and the vague number expression is $\tilde{L} = \left[ {L - L_{0} ,L + L_{0} } \right]$, among it, L is net credit, while L ₀ is past due loan.

4 Empirical Analysis

4.1 Empirical Result Analysis After Risk Adjustment

First, put the variable without risk adjustment into SBM model and Super-SBM model to calculate the efficiency of 13 commercial banks. Then, put the input-output variable which has been adjust by risk into SBM model and Super-SBM model, after that, we can get the 13 commercial banks’ technical efficiency based on different α–cut set level within risk adjustment by applying DEA-Solver. On Table 2.

Table 2 13 commercial banks’ technical efficiency within risk adjustment in 2011

The second row in Table 2 gives out the 13 commercial banks’ technical efficiency by applying SBM model in 2011. There are 5 commercial banks’ efficiency less than 1, including Agricultural Bank of China, China Minsheng Bank, Hua Xia Bank, China Everbright Bank and China Merchants Bank, and they all are inefficiency unite of SBM. The other 8 commercial banks are efficiency unites of SBM, and they are all at efficiency frontier, meanwhile, there are neither input redundancy nor output deficiency. In order to rank them better, the third row lists out the 8 commercial banks’ super efficiency by using Super-SBM model. From fifth row to ninth row, it depicts 5 commercial banks’ technical efficiency by using Fuzzy SBM model and 8 commercial banks’ technical efficiency by using Fuzzy Super-SBM model based on different α–cut set level. There are 5 levels about α–cut set, and it’s a = 0 $\upalpha = 0, 0.3, 0.5, 0.7, 1$. We can use α–cut set level as different degree of the risk adjustment about input-output, and the lower α–cut set level, the greater the degree of risk adjustment is. We also can get that the efficiency change interval is wider from the table. This means that the risk bank faced is bigger, and the variation of its efficiency is more intensive. For example, when $\upalpha = 0$, it means that all of the banks need get their input-output risk changed in a greatest degree, and the degree of the corresponding efficiency is maximum. On the contrary, when $\upalpha = 1$, it means that we don’t do any risk adjustment to input-output, while the variables stay the same, at the same time, the upper limit value and lower limiting value of efficiency are equal to the efficiency calculated by SBM model and Super-SBM model.

From interval efficiency score in Table 2, we can get that risk factor has different influence on 13 commercial banks, and as the α–cut set level increasing, from $\upalpha = 0$ to $\upalpha = 1$, the amount of variation of commercial banks’ efficiency is smaller. It means that the degree of banks’ operating efficiency affected by the risk is lower as the risk which banks encounter become smaller, and the change is stable.

Among the state owned commercial banks, only Agricultural Bank of China is relative inefficiency, and the efficiency value is 0.6234. Meanwhile, the degree of its efficiency affected by the risk is greatest, because the efficiency interval is [0.4772, 1.000] when $\upalpha = 0$. Amplitude of variation is bigger than other four state owned commercial banks at the same α–cut set level. When $\upalpha = 0.5$, The upper limit efficiency value of Agricultural Bank of China has declined to 0.8736, so it is the weakest ability to resist risks among the state-owned commercial banks, and its risk-adjusted efficiency is lowest. From the Super-SBM model in Table 2, we can see that the efficiency of Industrial and Commercial Bank of China is highest, then it’s Bank of Communications, following is Bank of China and China Construction Bank. From the degree affected by risk, Bank of China is minimally affected by risk, and the fluctuation range of efficiency is 0.023(=1.0415−1.0392) when $\upalpha = 0$. The amount efficiency of variation of Industrial and Commercial Bank of China, China Construction Bank and Bank of Communications respectively are 0.064, 0.0772 and 0.1688. Above all, we know that 5 state-owned commercial banks’ efficiency affected by risk ranking from weakest to greatest is Bank of China, Industrial and Commercial Bank of China, China Construction Bank, Bank of Communications and Agricultural Bank of China, while it also it risk adjustment order from highest to lowest of the 5 state-owned commercial banks.

Among joint-stock owned commercial banks, there are 4 banks which are SBM inefficiency, according to the efficiency order from highest to lowest, they orderly are China Merchants Bank, China Minsheng Bank, China Everbright Bank and Hua Xia Bank. There are 4 banks having efficiency, from Super-SBM model in Table 2, we can get that the efficiency of Ping An bank is highest, following is Soci é t é G é n é rale, Shanghai Pudong Development Bank and China Citic Bank. From the degree of risk impact, because joint-stock owned commercial banks start late, their risk management system is incomplete, and the management system is imperfect, the management level is not high, so the risk factor has a bigger influence on joint-stock owned commercial banks than state-owned commercial banks. Besides, the inefficient joint-stock owned commercial banks are greater affected by risk factor than state-owned commercial banks from Table 2. In the 4 inefficient commercial banks, Hua Xia Bank’s efficiency amplitude fluctuation reach to 0.9252(=1.0000−0.0748) at $\upalpha = 0$, and it is greatest affected by risk factor, following is China Everbright Bank, China Minsheng Bank and China Merchants Bank, the efficiency amplitude fluctuation respectively are 0.8734, 0.8150 and 0.7342. Among the 4 efficient banks, Shanghai Pudong Development Bank’s efficiency amplitude fluctuation reach to 0.0168(1.0278−1.0110) at $\upalpha = 0$, and it is weakest affected by risk factor, while Ping An Bank’s fluctuation range of efficiency reach to 0.4784, and it is greatest affected by risk factor, the rest are Soci é t é G é n é rale (0.1503) and China Citic Bank (0.0447). Above all, we can rank the risk degree affecting joint-stock owned commercial banks’ efficiency, from small to large, successively are Shanghai Pudong Development Bank, China Citic Bank, Soci é t é G é n é rale, Ping An Bank, China Merchants Bank, China Minsheng Bank, China Everbright Bank and Hua Xia Bank, so, Shanghai Pudong Development Bank’s risk management level is highest, while Hua Xia Bank’s risk management level is lowest.

4.2 Commercial Banks’ Efficiency Ordering After Risk Adjustment

From Table 2, we can see that every commercial banks’ efficiency based on α–cut set in the fuzzy condition. It is difficult to have a total ordering of all banks’ efficiency affected by risk factor, so, using the interval evaluation method proposed by Chen and Klien [7] to rank the 13 commercial banks’ fuzzy index, the result showing in Table 3.

Table 3 Commercial banks’ efficiency ordering with risk adjustment in 2011

Table 3 list the 13 commercial banks ordering according SBM efficiency and Fuzzy SBM interval efficiency, among that, interval efficiency is ordered by fuzzy index (k). According to the result, Ping An Bank’s efficiency is the greatest among the 13 commercial banks, while the efficiency of Industrial and Commercial Bank of China is the highest among state-owned commercial banks, the gap between them is small. The efficiency of Agricultural Bank of China, Hua Xia Bank and China Everbright Bank have a different order after risk adjustment, because that SBM efficiency ranking doesn’t count on risk factor, while interval efficiency ranking count in the risk factor. Agricultural Bank of China, as the state-owned bank, its risk management, risk system and risk control is better than Hua Xia Bank and China Everbright Bank, so we get this order.

5 Conclusion

This paper firstly uses input-output to have risk adjustment, then using Fuzzy SBM model and Super-SBM model to get Chinese 13 commercial banks’ efficiency under risk condition based α–cut set interval efficiency in the year 2011. The results are as follows, first, the influence of risk factors on the 13 commercial banks in 2011 has varying degrees, and the less risk faced by banks, the smaller influence on the efficiency affected by the risk factors. Second, by comparing 13 commercial banks’ efficiency fluctuation range of risk factors and find that among the state-owned commercial banks, Bank of China is the strongest ability to resist risks, Agricultural Bank of China is the weakest one; rather than the joint-stock commercial banks generally affected by a high degree of risk, among the joint-stock owned commercial banks, the risk management level of Shanghai Pudong Development Bank is highest, and Hua Xia Bank is the weakest one. Third, on 13 commercial banks sorting of risk-adjustment efficiency, we find that Soci é t é G é n é rale is the highest efficiency, Hua Xia Bank is the least efficiency, and the average ranking of the state-owned commercial bank is higher than the joint-stock commercial banks.