2.1 Collateral and moral hazard

Collateral is the bank's right over the borrower's assets in case of bankruptcy. It is an alternative resource for paying the debt beyond the cash flows (^{Tirole, 2005}). One of the main reasons why banks use guarantees is that they reduce the risk of credit loss in default situations (^{Godlewski a Weill, 2011}). It also prevents companies from accessing financing while others do not (credit rationing) (^{Besanko & Thakor, 1987}).

Credit rationing occurs because the lender is im-perfectly informed of the characteristics and behavior of the borrower when accessing credit, known as asymmetric information. Under a scenario of asymmetric information, the collateral, prior to the signing of the credit agreement, serves as a differentiator (screening) between good and bad borrowers, mitigating adverse selection problems and, once the credit is granted, can act as a driver of the effort needed in activities that require financing, that is, to reduce moral hazard problems. (^{Agarwal et al., 2015}).

Moral hazard refers to conflicts of interest that arise in the development of a contractual relationship where one of the parties (Agent) may engage in conduct that goes against the benefit of the other (Principal). An example of this can be found in the insurance field, where someone ensures an asset and, therefore, has less incentive to maintain it properly. The insurance company usually cannot monitor or verify the behavior of the insured (^{Lee & Shin, 2008}), which means that moral hazard has retrospective effects (^{Mirrlees, 1999}).

Regarding the issue of moral hazard in credit, there is a body of theoretical work, as in ^{Csóka et al. (2015)} and empirical works, such as ^{Uesugi et al. (2010)} and ^{Castillo et al. (2018)} , among others. These last authors incorporate variables to measure moral hazard within the framework of credit risk. These variables are defined with a positive sign and within four categories: i) asset substitution, aimed at seeking investments with higher risk and better returns; ii) low effort, characterized by high costs and expenses; iii) underinvestment, characterized by the use of financial resources for non-operating expenses; and iv) different use of the invested capital, when the financing resource is used for activities other than those established in the credit agreement. Similarly, ^{Castillo et al. (2021)} established an index to measure and monitor the influence of moral hazard on short-term credit performance for SMEs. However, the moral hazard variable within the calculation of the probability of default of a credit portfolio has not been used yet. This is the situation in which the National Guarantee Fund of Colombia is immersed in its role of SME credit guarantor when it needs to calculate the probability of default of its portfolio, given that the institution works as a guarantee provided by a third party, and this type of guarantee does not solve moral hazard problems (^{Menkhoff et al., 2011}). Thus, the calculation of the probability of default of said portfolio corresponds to the objective of this work.

On the other hand, when analyzing the credit risk of a portfolio, the individual effect of each element of the portfolio overall must be considered, and this is where the issue of copulas becomes relevant.

2.2 Copulas

A copula is a multivariate distribution such that its marginal distributions are uniform and describes how the distributions are jointly related to determine the multivariate distribution (^{Bluhm et al., 2010}).

There is a large body of literature on copulas, including ^{Fredheim (2008)} and ^{Rank (2006)} . An enlightening literature review on copula applications to financial and economic time series can be found in ^{Patton (2012)} and, more recently, in ^{Dewick and Liu (2022)} . In particular, for the credit risk case, the remarkable work of ^{Frey and McNeil (2003)} , the authors present recommendations on credit risk model choice. Moreover, the financial literature presents applications of Gaussian (^{Bourgey et al., 2020}) and non-Gaussian copula models on portfolio credit risk, such as the t-copula (^{Chan and Kroese, 2010}), composite Berstein copula (^{Guo et al., 2017}), and Archimedean copulas (^{Fenech et al., 2015}; ^{Cui et al., 2022}).

A related work to our paper is ^{Liu et al. (2019)} , where the authors employ an Archimedean copula to examine the credit risk of Chinese SMEs loans. To understand the usefulness of a copula, the Vasicek model is considered, which is based on the Gaussian, and it is still used for credit risk modeling (e.g., ^{García-Céspedes and Moreno, 2022}). In order to calculate the default probability of a portfolio, the probability of default for each credit is initially considered in the model. Then, the joint probability is calculated, considering the individual impact on the whole credit portfolio (^{Hull, 2015}).

3.1 Data

The data for calculating the default probabilities come from crossing two sources of information: the FNG and the Colombian Superintendency of Corporations. Based on this, the financial ratios are calculated for traditional and moral hazard ratios of the companies that have obtained loans under the FNG's guarantee.

The data is described below, which is based on the article by ^{Castillo et al. (2021)} , where the descriptive analysis of the variables used in the modeling can be observed.

From the FNG source, companies short term credit information (working apital loan) is taken for a period less than or equal to 12 months. This is because the database does not allow identifying the dates on which a customer may have defaulted on loan payments, such default occurs when the collateral is paid to the financial intermediary by the FNG. Therefore, with data for the period indicated above, the company's financial information for the year in which the collateral became effective and the completion of the credit term will not be significantly different, unless the company entered bankruptcy, which was verified. This organizes the work into annual cohorts, which provides an instant and complete picture of the SMEs' credit performance (guaranteed by the FNG) for each year. Balance sheets and income statements for these same companies with FNG credit guarantees were obtained from the Colombian Superintendency of Corporations.

The number of records obtained (filtered by company size)¹ regarding the number of SMEs in Colombia was 2,234, 2,451, 3,469, 3,498, 3,896, 3,760, 3,615, and 3228 for the years 2007-2014, respectively. These records consist of two groups. The first corresponds to companies for which the collateral was paid to the financial intermediary by the FNG; these companies defaulted. There were: 8, 5, 10, 6, 9, 17, 17 and 20 companies that defaulted in each of the years examined, respectively. The second group consists of the remaining companies for which the collateral was terminated without a claim; that is, the borrower paid back the entire loan.

In the modeling, the focus is on the behavior of the company at the end of the collateral period because either the company defaulted or it did not. Additionally, given that a company can have multiple loans, the company's response regarding the entire credit package was considered; that is, whether it complied or not. A single loan default was enough to consider the entire credit package to be in breach of contract.

In the two logit models used, the dependent variable, in all cases, corresponded to the state of compliance with the terms of the credit package (failure or not). In the first model, the independent variables were the traditional financial ratios and the number of loans; a one (1) was assigned when there was a single loan, otherwise a zero (0) was assigned for all other cases. In the second model, the moral hazard ratios were added. (p. 314)

Figure 1 shows the variables for the logit models, corresponding to the traditional financial ratios, the moral hazard ratios, and the dummy variable (A-NUMBER-C) associated with the number of SME loans

Source: own elaboration.

Figure 1 Logit models variables. 

3.2 Method

Based on the probabilities of default obtained from the logit models shown in Figure 1, the probabilities of default in the portfolio from 2007 to 2014 are calculated using copulas, using the Vasicek model, as described in ^{Hull (2015)} :

(1)

where:

g(DR) = probability density function, DR = probability of individual default, PD = probability of joint (portfolio) default, ρ = joint (portfolio) dependence coefficient, and N^-1 (.) is the quantile function of a standard normal distribution.