Logistic Model of Credit Risk during the COVID-19 Pandemic

Bin Zhao1* and Jinming Cao2

Keywords: Data Analysis; Monte Carlo Model; OpenBUGS; Overdue Risk

Introduction

The Markov Chain Monte Carlo method (MCMC), originated in the early 1950s, is a Monte Carlo method that is simulated by computer under the framework of Bayesian theory. This method introduces Markov process into Monte Carlo simulation, and achieves dynamic simulation in which the sampling distribution changes as the simulation progresses, which makes up for the shortcoming that traditional Monte Carlo integral can only simulate statically. MCMC is a simple and effective computing method, which is widely used in many fields, such as statistics, Bayes problems, computer problems and so on. Credit business also known as credit assets or loan business, which is the most important asset business of commercial Banks. By lending money, the principal and interest are recovered, and profits are obtained after deducting costs. Therefore, credit is the main means of profit for commercial Banks during the COVID-19 pandemic.

By expanding the loan scale, the bank can bring more income, and inject more power into the social economy, so that the economy can develop faster and better. However, with the expansion of credit scale, it is often accompanied by risks such as overdue credit. Banks can reduce credit overdue risk from two aspects, one way is to increase the credit overdue penalties, such as lowering the personal credit, dragging into the blacklist, and so on. With the rapid development of Internet, personal credit registry has more and more influence on the individual. A bad credit report will bring much inconvenience for the individual, so, in order to avoid the adverse impact on your credit report, borrowers tend to repay the loan on time, but these means are all belong to afterwards, although reduced the frequency of overdue frequency, but still caused a certain loss to the bank.

Selectively lending to “quality customers” can reduce Banks’ credit costs even more if they anticipate the likelihood of delinquency in advance, before the customer takes out the loan.

How to identify whether the client is the “good customer” will need to collect overdue related information about the customer in advance, through the establishment of probability model between relevant variables and overdue, thus to rank the customer’s risk grade of overdue, if customers’s risk level is extremely high, the bank should choose to increase loan interest or refuse to reduce the risk of credit bank loans [1].

This paper includes the following three parts: model introduction and data description, data preprocessing and OpenBUGS simulation, and summary.

Overview of Logistic Model

If we want to use linear regression algorithm to solve the problem of a classification, (for classification, y value equal to0 or 1), but if you are using the linear regression, then assumes that the function of the output value may be greater than 1, or much less than zero, even if all the training sample label y is 0 or 1 but if algorithms get value is greater than 1 or far less than zero, will feel very strange.

So the algorithm we’re going to study in the next section is called the logistic regression algorithm, and this algorithm has the property that its output value is always between 0 and 1. So logistic regression is a classification algorithm whose output is always between 0 and 1.

First, let’s take a look at the LR of the dichotomy. The specific method is to map the regression value of each point to between 0 and 1 by using the SIGmoid function shown in Figure 1.

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As shown in the figure, let z = w × x + b, When z > 0, the greater z is, the closer the sigmoid returns to 1 (but never more than 1). On the contrary, when z < 0, the smaller z is, the closer the sigmoid return value is to 0 (but never less than 0).

This means that when you have a binary classification task (positive cases corresponding labeled 1, counterexample corresponding labels 0) and samples of each of the sample space for linear regression z = w × x + b, then the mapping using sigmoid function of g = sigmoid (z), and finally output the corresponding class label each sample (all value between 0 and the one greater than 0.5 is marked as positive example), then, two classification is completed. The final output can actually be regarded as the probability that the sample points belong to the positive example after the model calculation.

Thus, we can define the general model of the dichotomous LR as follows:

, {0,1}, , n n X R Y w R b R ∈ ∈ ∈ ∈

exp( . ) (Y 1| ) 1 exp( . ) 1 (Y 0 | ) 1 exp( . )

w x b p X w x b

$$ = 1 \mid X) = \frac {\exp (w . x + b)}{1 + \exp (w . x + b} $$

p X w x b $$ [ = 0 \mid X) = \frac {1}{1 + \exp (w . x + b} $$

For a given input x,

$$ p (Y = 1 \mid X) \mathrm {a n d} p (Y = 0 \mid X) $$

can be obtained, and the instance x will be classified into the

category with high probability value.

Odds of an event refer to the ratio between the probability of its occurrence and the probability of its non-occurrence. If the probability of its occurrence is P, the probability of the event is P/(1-P), and the log odds or logit function of the event is $$ \log i t (p) = \log \frac {p}{1 - p} $$ logistic regression can be obtained $$ \log \frac {p (Y = 1 \mid X)}{1 - p (Y = 1 \mid X)} = w. x $$ That is, the logarithmic probability of output Y=1 in the logistic regression model is a linear function of input X.

When learning logistic regression models, for a given data set International Journal of Biochemistry & Physiology

$$T = \{(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)\},$$
$$x_i \in R^n, y_i \in \{0, 1\}$$

The maximum likelihood estimation method can be used to estimate the model parameters, and then the logistic regression model can be obtained. Set

$$p(Y = 1 \mid x) = \pi(x), p(Y = 0 \mid x) = 1 - \pi(x)$$

The likelihood function is

$$\prod_{i=0}^{N} [\pi(x_i)]^{y_i} [1 - \pi(x_i)]^{1-y_i}$$

The logarithmic likelihood function is

$$L(w) = \sum_{i=1}^{N} [y_i \log \pi(x_i) + (1-y_i) \log(1-\pi(x_i))]$$
$$= \sum_{i=1}^{N} [y_i \log \frac{\pi(x_i)}{1-\pi(x_i)} + \log(1-\pi(x_i))]$$
$$= \sum_{i=1}^{N} [y_i (w.x_i) - \log(1 + \exp(w.x_i))]$$

By gradient descent algorithm and newton method can get the maximum value in the L(w) and the estimates of w: $\bar{w}$, then the logistic regression model:

$$p(Y = 1 \mid X) = \frac{\exp(w.x + b)}{1 + \exp(w.x + b)}$$
$$p(Y = 0 \mid X) = \frac{1}{1 + \exp(w.x + b)}$$

MCMC

The formula of Markov Chain is as follows

$$p(X_{t+1} = x \mid X_{t-1}, \dots) = p(X_{t+1} = x \mid X_{t})$$

That is, the state transition probability value is only related to the current state. Let $P$ be the transition probability matrix, where $P_{ij}$ represents the probability of the transition from $i$ to $j$. So we can prove that

$$\lim_{n \to \infty} p_{ij}^n = \pi(j)$$

Where $\pi$ is the solution to $\pi p = \pi$. Since the probability of $x$ obeys $\pi(x)$ after each transfer, it is possible to sample from $\pi(x)$ by transferring the different bases to this probability matrix. Then, given $\pi(x)$, we can construct the transition probability matrix by the Gibbs algorithm.

Gibbs Algorithm:

Random initialization $\{x_i : i = 1, 2, \dots, n\}$

for $t = 0, 1, 2, \dots, \text{Cycle Sampling}$

$$x_1^{(t+1)} \square p(x_1 | x_2^{(t)}, x_3^{(t)}, \dots, x_n^{(t)})$$
$$x_2^{(t+1)} \square p(x_2 | x_2^{(t+1)}, x_3^{(t)}, \dots, x_n^{(t)})$$
$$\dots$$
$$x_j^{(t+1)} \square p(x_j | x_1^{(t+1)}, \dots, x_{j-1}^{(t+1)}, x_{j+1}^{(t+1)}, \dots, x_n^{(t+1)})$$
$$\dots$$
$$x_n^{(t+1)} \square p(x_n | x_1^{(t+1)}, x_2^{(t+1)}, \dots, x_{n-1}^{(t+1)})$$

Data Description and Preprocessing

The German credit card data set is adopted in this paper, which contains 20 variables, including 7 quantitative variables and 13 qualitative variables. The details are shown in Table 1.

The data set includes 20 variables, the influence of different variables on credit overdue is different, adopting too many variables will not only increase the cost of collecting data, and wast customer's, also increases the complexity of the model, reduce the accuracy of prediction, so before to fitting the model we need to screen all the indicators which have a significant effect. The following content will be introduced from the screening of quantitative indicators and the screening of qualitative indicators.

Model Training and Prediction

The significance level was set as 0.05. A total of 10 variables were screened to establish the model:

0 1 1 2 2 10 10 log (p ) [ ]

[ ] .....

$$ g i t \left(\mathrm {p} _ {i}\right) = \beta_ {0} + \beta_ {1} X _ {1} [ i ] + \beta_ {2} X _ {2} [ i ] + \dots \dots \beta_ {1 0} X _ {1 0} [ i ] $$ Where o

β is the constant term,

$$ \beta_ {i} (\mathrm {i} = 1, 2, \dots 1 0) \mathrm {i s t h e} $$

partial regression coefficient of independent variable;

Parameters of the model have been given independent “non-

informative” prior distribution, and OpenBUGS software is

used for modeling and sampling, as well as Doodle modeling

through OpenBUGS, to specify the distribution type and

logical relationship of various parameters, as shown in the

figure3:

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Each ovals represent a node IN the graph, rectangle with constant node, single arrow from the parent node to the random child nodes, hollow double arrows indicate the parent node to the logical type child nodes, the rectangular outside for tablet, the lower left corner “for (I IN 1: N)” said for loop, is used to calculate the likelihood function of all samples, and the overall likelihood function is obtained [3].

The posterior distribution statistics for each parameter were obtained using OpenBUGS software, as shown in Table 6.

Where, MC_error represents the error of monte Carlo simulation and is used to measure the variance of the mean value of parameters caused by simulation.Val2.5 PC and VAL97.5 PC represent the lower and upper limits of the 95% confidence interval of the median, respectively; Median is usually more stable than mean; Start represents the starting point of Gibbs sampling. In order to eliminate the influence of initial value on sampling, sampling is started after 1001 times. Sample represents the total number of samples extracted. A total of 10,000 samples were extracted in this paper [4].

According to the parameters of Bayesian estimation, the error of model Colot simulation is generally relatively small, which indicates that the model has a good effect. With each parameter of the Gibbs sampling sample mean as a parameter to estimate, from the point of the results, the variable whether checking_account_status values for A13 (greater than 200 DM) and A14 (no checking account), variable credit_history whether values for A30 (not credit) and A31 (have to pay all the bank’s credits) have bigger influence on the overdue risk, relative variable savings for

1 2 3 4 5 6 7 8 9 10 log ( ) 1.732 0.4183

[ ] 1.578

[ ] 0.6884 [ ] 0.8637 [ ] 0.7879 [ ] 0.8786

[ ] 1.585

$$ g i t \left(p _ {i}\right) = - 1. 7 3 2 - 0. 4 1 8 3 X _ {1} [ i ] - 1. 5 7 8 X _ {2} [ i ] - 0. 6 8 8 4 X _ {3} [ i ] + 0. 8 6 3 7 X _ {4} [ i ] - 0. 7 8 7 9 X _ {5} [ i ] + 0. 8 7 8 6 X _ {6} [ i ] + 1. 5 8 5 X _ {7} [ i ] + 0. 6 7 5 5 X _ {8} [ i ] + 0. 5 6 0 3 X _ {9} [ i ] - 0. 3 7 5 5 X _ {1 0} [ i ] $$ When dividing the overdue risk level of customers, there may be two wrong divisions, that is, dividing “high-quality customers” into high-risk customers and high-risk customers into “high-quality customers”. Generally speaking, the economic costs of these two wrong divisions are different. For Banks, the cost matrix is shown in the table7 (0=Good, 1=Bad) [6].

A61 values (<100 dm) and A62 (100<x<500DM) has little impact on the overdue risk, indicating that the customer’s historical credit history and the current check status have a greater impact on the overdue risk, that is, the customer’s historical credit and current economic status have a greater impact on the overdue risk. Banks should focus on these two aspects when judging the customer’s credit risk level [5].

The logistic regression equation can be obtained

The rows represent the actual classification and the columns the predicted classification. It is worse to class a customer as good when they are bad (cost=5), than it is to class a customer as bad when they are good (cost=1). Define the loss function as

1000 $$ r = \sum_ {i = 1} ^ {1 0 0 0} f \left(p r e (i)\right) - r $$

1 ( ( ) ( ))

i Loss f pre i real i pre(i) and real(i) is The classification results of the i t h sample and The actual category, respectively, and f (x) is a piecewise function:

5 0 x $$ 0 = \left\{ \begin{array}{l l} 5 x < 0 \\ 0 x = 0 \\ 1 x > 0 \end{array} \right. $$ f(x) 0 0 x

1 0 x Each sample input the results of logistic regression model as a probability value, if the probability value is greater than a given probability value is the sample classification is 1, otherwise the classification of 0, due to the loss of the two types of error, according to the different probability of loss matrix can be calculated at a given value under the condition of overall losses, the results as shown. It can be seen that when the given probability value is 0.21, the overall loss is the smallest. The Confusion matrix are shown in the table 8. The precision of the model is 85%. The model identifies the vast majority of high-risk customers.

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Figure 5~7 shows the iteration history diagram, autocorrelation function diagram and kernel density diagram of all parameters.

Monte Carlo simulation starts from the initial value given for each parameter. Due to the randomness of extraction, the first part of extracted value is used as an independent sample obtained by annealing algorithm. Therefore, we must judge the convergence of the extracted Markov Chain. The convergence of Markov chains can be analyzed according to the results of parameter extraction [7].

Conclusion

This paper constructs a binomial logistic regression model based on the customer characteristic data of Banks.

Content mainly includes two parts, the first is the part of data pretreatment, the original data contains 20 variables, in order to make the model more concise, and improve the accuracy of classification model, reduce the cost of information collection and the time cost of customers, using “Boruta” method of screening of three quantitative indicators, and use the optimal segmentation method will be treated as continuous variable section.

Then, three qualitative variables were selected into the model by calculating the IV value of the variable, and the qualitative variables were treated with a unique heat type.

Two logIST-IC regression of SPSS software was used to screen out 10 variables with significance less than 0.05 into the model.

All the selected variables were brought into OpenBUGS software to obtain the parameter Bayesian estimation of the binomial logistic regression model. From the estimation results, it can be seen that the customer’s historical credit (Credit_history) and current economic status (Checking_ account_status) have the greatest impact on credit delinquency. Banks should pay more attention to these two aspects when evaluating the customer’s credit risk level [10].

Conflict of Interest

We have no conflict of interests to disclose and the manuscript has been read and approved by all named authors.

Acknowledgement

This work was supported by the Philosophical and Social Sciences Research Project of Hubei Education Department (19Y049), and the Staring Research Foundation for the Ph.D. of Hubei University of Technology (BSQD 2019054), Hubei Province, China.